Finite Element Modeling Using COMSOL Multiphysics
Alexandria 25-27 September, 2012
PROGRAM Elnady Engineering and Agencies 2 Elgabal Elakhdar Buildings, Nasr City, 11471 Cairo, Egypt. Tel: +20 2 23425763 Fax: +20 2 23421793 www.elnadycompany.com
Finite Element Modeling Using COMSOL Multiphysics Day1 Agenda 09:00 – 11:00
Introduction to COMSOL Multiphysics
11:00 – 12:30
COMSOL Demonstraion Installation on attendees’ laptop Coffee Break
12:30 – 1:00
01:00 – 03:00
Tutorial session: Getting started with COMSOL - Parallel plate capacitor - Thermal cube - Stresses on a wrench - Electrical heating and thermal stresses in a busbar
Day2 Agenda -
09:00 – 12:00
-
Background on Finite Element Analysis Advanced Meshing Techniques
Coffee Break
12:00– 12:30 12:30 – 02:00
Solver Sequence and settings Model of Day: Heat Sink
Day3 Agenda 09:00 – 10:30
Results and Post Processing
10:30 – 12:00
Advanced Structural Mechanics examples - Elbow Bracket, different Force Analysis - Elbow Bracket, Elasto-plastic Analysis - Tube Connector - Fluid Structure Interaction
Welcome Letter & Trial License Details Dear Colleague, It is our pleasure to welcome you to the Multiphysics High Performance Computing Workshop. During your registration, you should receive the following items: • •
The Program book including step-by-step tutorials for the minicourse DVD with the latest COMSOL Multiphysics version 4.3. For installation, see the information about the trial license below
Your Trial License • • • • • •
You will find an installation DVD of COMSOL Multiphysics 4.3. The installation Code is 9FFFBFFF0-TYSW-121009-17073896-324157832 This is a full functional trial license with all modules enabled. Expiry date: 2012-10-09 You are entitled for technical Support. Send an email to
[email protected] For pricing and ordering information, please contact
[email protected]
We wish you an informative and great Workshop! Elnady Engineering & Agencies
Introductory Models
Tutorials • Parallel plate capacitor • Thermal cube • Stresses in a wrench
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Tutorial 1: Parallel Plate Capacitor What you will learn • More experience with the interface • Sketching geometries in COMSOL • How to have a domain “not active” in the analysis
Tutorial 2: Thermal cube • More Practice with the User Interface • Adding a 2nd Physics • Adding Thermal Expansion
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Tutorial 3: Stresses in a Wrench What you will learn • Hands on with the interface • The basic steps in all COMSOL models • How to show deformed plots
Tutorial 4: Electrical Heating and Thermal Stresses in a Busbar
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Tutorial 1: Parallel Plate Capacitor What you will learn • More experience with the interface • Sketching geometries in COMSOL • How to have a domain “not active” in the analysis
• • • •
Select “2D” AC / DC > Solid Mechanics Stationary Checkered flag to Finish
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Create Geometry • • • •
Open Model 1 Pick “Geometry 1” Change Length Unit to “cm” Draw Three Rectangles
Set Materials • Right Click on “Materials” • Choose “Open Model Library” • Pick “Built-In” Material Library • Select “Air” • Right Click on it, “Add to Model” (This will assign it to all subdomains but we will only use it in the air domain: Electrodes will not be included in the analysis)
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Electrostatics – Active only in Air Domain • • • •
Pick “Electrostatics” in Model Builder Clear all Selections (hit “ X “ ) Only Select Air Domain Right-click to confirm
Set Boundary Condition – Ground • • • • •
Right-Click “Electrostatics” Choose “Ground” Choose “Box Selection” Select Bottom Electrode Confirm
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Set Boundary Cond – Electric Potential • • • • •
Right-Click “Electrostatics” Choose “Electric Potential” Set to 100 Volts Choose “Box Selection” Select Top Electrode
Mesh – Free Triangular (Extra Fine)
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Solve • Pick “Study 1” • Hit “ = “
Tutorial 2: Thermal cube • More Practice with the User Interface • Adding a 2nd Physics • Adding Thermal Expansion
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Choose Physics and Study • • • •
Choose File > New Select “3D” Select “Heat Transfer in Solids” in “Physics” Choose “Stationary” under “Study”
Draw Geometry • Right click on “Geometry” • Choose “Block” • Leave as 1 x 1 x 1 [m]
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Set Materials • Right click on “Materials” • Choose “Open Material Library” • Go to “Built-in” Section • Right Click on “Aluminum 6063” • Choose “Add to Model” • Assign it to the block
Set Boundary Conditions • • • •
Right Click on “Heat Transfer” Select “Convective Cooling” Choose “All Boundaries” h = 1000 Text = 500
• • • • •
Right Click on “Heat Transfer” Select “Temperature” Choose bottom surface Right click to confirm Set Temperature as “300”
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Mesh and Solve • • • •
Right Click on “Mesh” Select “Free Tetrahedral” Select “Size” Choose “Coarser” mesh size
• Right Click on “Study” • Select “Compute”
Add Structural Equations • • • • •
Right Click on “model 1” node Choose “Add Physics” Open “Structural Mechanics” Pick “Solid Mechanics” Choose the finish (checkered) flag
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Add Thermal Expansion • Open “Structural Mechanics” in Model Tree • Right Click on “Linear Elastic Material Model” • Choose “Thermal Expansion” • Change “Temperature” from “user defined” to “Temperature (ht/solid1)” • Leave Ref Temperature as 293 K
Constrain Bottom Surface + Solve • Right Click on “Solid Mechanics” • Choose “Fixed Constraint” • Select & Confirm bottom surface • Right Click on “Study” • Pick “Compute” • Open “3D Plot group 1” • Right Click on “Surface” • Add “Deformation”
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Tutorial 3: Stresses in a Wrench What you will learn • Hands on with the interface • The basic steps in all COMSOL models • How to show deformed plots
1) Set Dimensions, Physics & Study-Type • • • •
Select “3D” Structural Mechanics > Solid Mechanics Stationary Then … the checkered flag to Finish
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2) Import Geometry • • • • •
Right click on “Geometry” Choose “Import” Then “Browse” C:\COMSOL41\models\COMSOL_Multiphysics\Structural_Mechanics Choose “Wrench.mphbin” “Import”
3) Set Materials In Model Builder: • Right Click on “Materials” • “Open Model Library” • Pick “Built-In” • Select “High Strength Alloy Steel” • Right Click, “Add to Model” • Select “Steel AISI 4340” • Right Click, “Add to Model” • Pick Bolt, right-click to add
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4 a) Constrain Bolt In Model Builder: • Right Click on Solid Mechanics • Choose “Fixed Constraint” In Graphics Window: • Pick the Bolt Cut End • Right Click to confirm
4 b) Add Load In Model Builder: • Right click on Solid Mechanics • Choose “Face Load” • Pick end of the wrench • Enter ”F” for Fz
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5) Mesh In Model Tree: • Right Click on “Mesh 1” • Choose “Add Tetrahedral” • Pick “Size • Change to “Coarser” • Right click “Mesh 1” • Pick “Build All”
6) Solve In Model Builder: • Pick “Study 1” • Choose “Compute”
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Display Stress on Deformed Geometry • Open “Results” > “3D Plot Group 1” • Pick “Surface 1” > Replace Expression – Plot Von Mises Stress
• Right click on “Surface 1” • Add “Deformation”
Tutorial 7: Electrical Heating and Thermal Stresses in a Busbar
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Your first model
Exercises: Electrical Heating in a Busbar • Bus bar made of copper. • Bolts made of titanium. • Direct current 10 mV from a transformer to an electric device. • Heat is produced due to the resistive losses--joule heating. • Calculate the thermal expansion. • Work though exercises, but not the appendix
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Your first training model: A Copper Busbar • A busbar is an electric current conductor – typically found in highpower applications. • You will investigate: – Joule Heating – Thermal Expansion – Flow with Cooling for a certain busbar geometry
Boundary Conditions: Electric Currents
Electric Potential 10mV
Ground
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Boundary Conditions: Heat Transfer
Cooling with a heat transfer coefficient of 5 W/m^2/K and surrounding air temperature at 293.15 K
Now you’re ready for your own simulations! The only difference between the hands-on and your problem is… • Perhaps a few more physics • A bit more geometry • Some more loads & b.c.’s
• Go to it (!!!)
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Capture the Concept
TM
Questions & Answers
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An Example: Electrical Heating in a Busbar
An Example: Electrical Heating in a Busbar In order to get acquainted with COMSOL Multiphysics, it is best to work through a basic example step by step. This section will cover the model building procedure, highlighting several features and demonstrating common simulation tasks. The example shows how to: • Set up a multiphysics simulation • Run a geometric parametric sweep • Expand a model, taking more phenomena into account, after successfully
running a first simulation By following the directions in this section, you will create a model that analyzes a busbar designed to conduct a direct current from a transformer to an electric device (see below). The current conducted in the busbar, from bolt 1 to bolts 2a and 2b, produces heat due to the resistive losses, a phenomenon referred to as Joule heating. Titanium Bolt 2a
Titanium Bolt 2b Titanium Bolt 1
This presents a classic example of multiphysics, since the electric conductivity of the conductor is temperature dependent while the heat source in the heat transfer problem depends on the electric current density. In fact, the phenomena are strongly coupled, meaning one affects the other and vice versa. 6
An Example: Electrical Heating in a Busbar
Once you have captured the basic multiphysics phenomena, you will also have the chance to investigate thermal expansion yielding structural stresses and strains in the busbar and the effects of cooling by an air stream. The Joule heating effect is described by conservation laws for electric current and energy. Once solved for, the two conservation laws give the temperature and electric field, respectively. All surfaces, except the bolt contact surfaces, are cooled by natural convection in the air surrounding the busbar. You can assume that the electric potential at the upper-right vertical bolt surface is 10 mV and that the potential at the two lower horizontal surfaces of the lower bolts is 0 V. The Model Wizard Start COMSOL by double-clicking its icon on the desktop. When the Model Wizard opens, select a space dimension; the default is 3D. Click the Next button to continue to the physics page.
In the Add Physics step, click the Heat Transfer folder, right-click Joule Heating and Add Selected. Click the Next button .
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An Example: Electrical Heating in a Busbar
The last Model Wizard step is to select the Study Type. Preset Studies contains studies that have solver and equation settings adapted to the Selected physics, in this example, Joule Heating. Select the Stationary study type and click the Finish button . Any selection from the Custom Studies list needs manual fine-tuning. Global Definitions The Global Definitions in the Model Builder stores Parameters, Variables, and Functions with a global scope. You can use these operations in several Models. In this case, there is only one Model 1 node where the parameters are used. Since you will run a geometric parameter study later in the example, define the geometry using parameters from the start. Do this by entering parameters for the length for the lower part of the busbar, L, the radius of the titanium bolts, rad_1, the thickness of the busbar, tbb, and the width of the device, wbb.
2xrad_1
wbb
L tbb
Also add the parameters that will control the mesh, mh, a heat transfer coefficient for cooling by natural convection, htc, and a value for the voltage across the busbar, Vtot.
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An Example: Electrical Heating in a Busbar
Right-click Global Definitions and select Parameters to create a list of parameters containing L, rad_1, tbb, wbb, mh, htc, and Vtot. In the Parameters table, click the first row under Name and enter L. Click the first row under Expression and enter the value of L, 9[cm]. Note that you can enter any unit inside the square brackets. Continue adding the other parameters according to the Parameters list. It is a good idea to type in descriptions for variables, in case you forget or share the model with others. Geometry Here you can take a shortcut by loading the model geometry from the Model Library, which you open by choosing Open Model Library from the File menu. In the Model Library, select COMSOL Multiphysics>Multiphysics>busbar_geom and then click Open. Alternatively, you can construct the busbar geometry using the parameters entered above with the COMSOL geometry tools. If you would like to see this now, have a look at “Appendix: Creating Geometry Sequences” on page 42. Once you have created or imported the geometry, experimenting with different dimensions is easy: update the values of L, tbb, or wbb, and rerun the geometry sequence. Under Global Definitions click Parameters. In the Settings window, select the wbb parameter’s Expression column and enter 10[cm] to change the value of the width wbb. In the Model Builder, click the Form Union node and then
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An Example: Electrical Heating in a Busbar
the Build All button wider busbar.
wbb=5cm
to rerun the geometry sequence. You should get a
wbb=10cm
Return to the Parameters table and change the value of wbb back to 5[cm]. Rerun the geometry sequence. Experienced users of other CAD programs are already familiar with this approach since all major CAD platforms include parameterized geometries. To support this class of users and avoid redundancy, COMSOL offers the LiveLink family of products. These products connect COMSOL Multiphysics directly with a separate CAD program, so that all parameters specified in CAD can be interactively linked with your simulation geometry. The current product line includes LiveLink™ for SolidWorks®, LiveLink™ for Inventor®, and LiveLink™ for Pro/ENGINEER®. It is also worth noting that LiveLink™ for Matlab® is available for those who want to incorporate a COMSOL Multiphysics model into an extended programming environment. Having completed the geometry for your model, it is time to define the materials.
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An Example: Electrical Heating in a Busbar
Materials The Materials node administrates the material properties for all physics and all domains in a Model node. The busbar is made of copper, and the bolts are made of titanium. Both these materials are available from the Built-In material database. In the Model Builder, right-click Materials and select Open Material Browser. In the Material Browser, expand the Built-In materials folder, right-click Copper and select Add Material to Model. Click the Material Browser window tab and repeat the above procedure for Titanium beta-21S. In the Model Builder, collapse the Geometry 1 node to get a better overview of the model. Under the Materials node, click Copper. In the Settings window, locate the Material Contents section. The Material Contents section provides useful feedback on the model’s material property usage. Properties that are both required by the physics and available from the material are marked with a green check mark . Properties required by the physics but missing in the material will result in an error and are therefore marked with a stop sign . A property that is not used in the model is unmarked. You may note that the Thermal expansion coefficient at the bottom of the list is not used, but you will need it later for expanding your model with heat induced stresses and strains.
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An Example: Electrical Heating in a Busbar
In the Settings window, click the Clear Selection button selection list.
to clear the
Click in the Graphics window to highlight the copper domain. Right-click anywhere in the Graphics window to add the highlighted domain to the copper Selection list in the Settings window. Domain number 1 should now display in the list.
In the Model Builder, click the Titanium beta-21S node. Click the upper titanium bolt to highlight it and then right-click to add it to the titanium Selection list. Repeat this procedure for the two remaining bolts. These should correspond to the domains 2, 3, and 4. Be sure to investigate the Material section in the Settings window. All the properties used by the physics interfaces should have a green check mark . Close the Material Browser.
Contents
Physics The domain settings for the Joule Heating physics interface are complete now that you have set the material properties for the different domains. Next you will set the proper boundary conditions for the heat transfer problem and the conduction of electric current.
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An Example: Electrical Heating in a Busbar
In the Model Builder, expand the Joule Heating node to examine the default physics interface nodes. Joule Heating Model 1 contains the settings for heat conduction and current conduction. The heating effect for Joule heating is set in Electromagnetic Heat Source 1. Thermal Insulation 1 contains the default boundary condition for the heat transfer problem and Electric Insulation 1 corresponds to the conservation of electric current. Initial Values 1 contains initial guesses for the nonlinear solver for stationary problems and initial conditions for time-dependent problems. Right-click the Joule Heating node. In the second section of the context menu—the boundary section—select Heat Transfer >Heat Flux.
Boundary Section
Domain Section
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An Example: Electrical Heating in a Busbar
In the Settings window, select All boundaries in the Selection list. Rotate the busbar to inspect the back. Click one of the titanium bolts to highlight it in green. Right-click anywhere in the Graphics window to remove this boundary from the Selection list. Repeat this for the other two bolts. This removes boundaries 8, 12, and 28 from the Selection list. Inspect the list to make sure only these boundaries are removed. In the Settings window, click the Inward heat flux radio button. Enter htc in the Heat transfer coefficient field, h. Continue by setting the boundary conditions for the electric current.
In the Model Builder, right-click the Joule Heating node. In the second section of the context menu—the boundary section— select Electric Currents>Electric Potential. Click the circular face of the upper titanium bolt to highlight it and right-click anywhere to add it to the Selection list. The boundary face is numbered 28.
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An Example: Electrical Heating in a Busbar
In the Settings window, enter Vtot in the Voltage field. The last step in the physics settings is to set the two remaining bolt surfaces to ground. In the Model Builder, right-click the Joule Heating node. In the boundary section of the context menu, select Electric Currents>Ground. In the Graphics window, click one of the remaining bolts to highlight it. Right-click anywhere to add it to the Selection list. Repeat this procedure for the last bolt. Click the Go to Default 3D View button
on the Graphics toolbar.
Mesh The simplest way to mesh is to create an unstructured tetrahedral mesh. This will do nicely for the busbar. In the Model Builder, right-click the Mesh 1 node and select Free Tetrahedral. Click the Size node . In the Settings window, select the Custom button under Element Size. Enter mh in the Maximum element size field to set the maximum element size to 6 mm— the value entered earlier as a global parameter. Click the Build All button
to create the mesh.
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An Example: Electrical Heating in a Busbar
Later, in Studying Mesh Dependence and Convergence on page 29, you will create meshes for running mesh convergence studies and to investigate the magnitude of the change in a solution parameter after refining the mesh. Study To run a simulation, in the Model Builder, right-click Study 1 and select Compute . The Study node automatically defines a solution sequence for the simulation based on the selected physics and the study type. The simulation only takes a few seconds to solve.
Results The default plot displays the temperature in the busbar. The temperature difference in the device is quite small, less than 1 K, due to the high thermal conductivity of copper and titanium. The temperature is substantially higher than the ambient temperature 293 K. With this solution in hand, you can create an image that will be displayed by COMSOL when browsing for model files. Go to the File menu and select Save Model Image. There are two other ways to create images from this plot. One way is to use the Image Snapshot button in the Graphics toolbar for directly creating an image, and another is to add an Image node to the Report
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An Example: Electrical Heating in a Busbar
by right-clicking the Plot Group of interest. The second option lets you reuse the Image Settings if you update the model. The temperature distribution is symmetric with a vertical mirror plane running between the two lower titanium bolts and running across the middle of the upper bolt. In this case, the model does not require much computing power and you can model the whole geometry. For more complex models, you should consider using symmetries in order to reduce the size of the model. Another plot of interest shows the current density in the device. In the Model Builder, right-click Results and select 3D Plot Group. Right-click 3D Plot Group 3 and select Surface. In the Settings window, in the Expression section, . Select Electric Currents> click the Replace Expression button Current density norm. This is the variable for the magnitude, or absolute value, of the current density vector.
Click the Plot button . The resulting plot is almost uniform due to the high current density at the contact edges with the bolts. You can change the Range of the color table to visualize the current density distribution in the Settings window. Under Range, select the Manual color range check box. Enter 0.6e7 in the Maximum field. Click the Plot button . Save the model as busbar.mph. This plot shows how the current takes the shortest path in the 90-degree bend in the busbar. Moreover, the edges of the busbar outside of the bolts are hardly utilized for current conduction.
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An Example: Electrical Heating in a Busbar
Rotate the device to show the back of the busbar where you will see the high current density around the contact surfaces of the bolts. Continue the exercise by adding your own plots and investigating ways of generating cross section plots and cross section line plots. Now you have completed a basic multiphysics simulation. The following sections are designed to increase your understanding of the steps you implemented up to this point as well as to extend your simulation to include other relevant effects, like thermal expansion and fluid flow.
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Importing CAD Files When importing a design into COMSOL Multiphysics from a CAD file, the CAD Import Module translates it into a format understood by COMSOL Multiphysics. After importing a CAD file you continue the modeling process with meshing and analysis in the usual fashion.
CAD Geometry for Finite Element Analysis (FEA) A geometry created in a 3D CAD software might not automatically be suitable for FEA. One reason is that the geometry description requirements are rather stringent. For example, a 3D CAD geometry might contain very small gaps, edges, or surfaces that are difficult for the user of the CAD software to control. These can make meshing of the imported geometry impossible. A common type of problem in CAD designs is the presence of sliver faces, which are very long and narrow faces. While the CAD Import Module provides tools to detect and remove such faces, in some cases the topology of the geometry is such that the sliver face can not be removed by such tools. Such an example is illustrated in the figure below.
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CHAPTER 2: CAD IMPORT
After meshing this geometry, a very dense mesh is unfortunately created to resolve the narrow strip of the face.
Alternatively the narrow strip can be easily avoided during the design of the geometry in the CAD software, through making sure that the neighboring fillets have large enough radii. As the example above illustrates, meshing can be useful for diagnosing sliver faces and other small features in your geometry, and you can use it as a complement to the tools provided by the CAD Import Module. In COMSOL Multiphysics you can quickly create a surface or volume mesh of the part, then examine it for areas of dense mesh, which usually indicate small features. Careful design in the CAD software is also important during the design of assemblies. Touching parts in a CAD assembly might not be accurately positioned or might contain slight mismatches between dimensions. Such unintentional errors also lead to the presence of small features and sliver faces in the decomposed geometry. For this reason it is recommended that parts of the CAD assembly are designed in the context of the assembly. Even geometry features on a larger scale can make meshing of imported geometries difficult. Such features could be the subdivision of faces into smaller ones, or smaller holes, fillets and chamfers. You can usually control these features in the CAD software. Most modern CAD packages allow you to save different configurations of a part in the same file. It is often beneficial to save a configuration specially for FEA, where you
IMPORTING CAD FILES
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suppress features not important for the analysis. This way you can reduce the number of mesh elements and thus the amount of memory needed to solve the problem. 3D CAD GEOMETRY OF AN ENGINE PISTON
In this example you import and mesh two versions of a 3D geometry of a diesel-engine piston. The first version of the geometry is intended for FEA and has many features suppressed. With the use of two symmetry planes you only need to analyze a quarter of the original geometry. The second version is the complete 3D geometry. 1 Start COMSOL by double-clicking its icon on the desktop. 2 In the Model Wizard (
) the default space dimension is 3D. Since you do not need any physics modeling, you can click the Finish ( )button.
)node is added to the Model Builder. The Geometry 1 node The Geometry 1 ( contains the geometry sequence, where the geometry operations are stored. In the Settings window to the right of the Model Builder you are able to change various geometry settings. 3 To add an import feature to the geometry sequence right-click Geometry 1 and select Import (
).
4 In the Settings window click the Browse button. 5 Locate the course CD on the hard disk and select the file piston_for_fem.x_b,
then click Open.
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CHAPTER 2: CAD IMPORT
6 Click Import in the Settings window.
As soon as the import is finished the geometry appears in the Graphics window.
Continue by creating a simple unstructured tetrahedral mesh. 7 Right-click the Mesh 1 node and select Free tetrahedral.
The Mesh 1 node contains the operations that make up the mesh sequence.
IMPORTING CAD FILES
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8 To build the mesh click the Build All (
) button.
The Messages window indicates that there are roughly 34,000 tetrahedral elements in this geometry. For the sake of comparison, import and mesh the complete 3D CAD geometry. 1 In the Model Builder select the Import 1 node. 2 In the Settings window click the Browse button. 3 Locate the course CD and select the file piston_design.x_b, then click Open. 4 Click the Mesh 1 node.
Since you have changed a setting, the Import 1 node is automatically rebuilt as you leave the geometry sequence. This means that the newly specified file is imported instead.
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CHAPTER 2: CAD IMPORT
5 Click the Build All button to mesh the geometry.
According to the Messages window, this part consists of approximately 201,000 tetrahedral elements. Observe the differences between the two geometries and how this effects the mesh. The regions around small features such as fillets contain many small elements, which contribute significantly to the total number of elements.
IMPORTING CAD FILES
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I mport of I GES F i les The Initial Graphics Exchange Specification, or IGES, data format was first published in 1980, and supports data exchange for a wide range of formats including 3D solid modeling representations. Although it is not as suitable for CAD data transfer as modern file formats, the neutral IGES data format is still wide-spread for exchange of geometry data. In the IGES format, 3D data is handled by a trimmed surface representation, which, in its basic form, does not include information about how the surfaces are connected. Thus, the faces that make up a solid geometry have to be stitched together during the import of the geometry, making the IGES format very sensitive to tolerances. The latest update of the IGES format, version 5.3, addresses this issue by including support for objects called Manifold Solid B-rep Objects (entity type 186), which can contain surface connectivity information. Provided it is supported by your CAD software, it is recommended that you use this option for IGES file export.
Knitting Faces after Import In the majority of cases, IGES files do not contain any connectivity information and the process of knitting the faces together into a solid object is sensitive to tolerances. In this example, you can test how the knit tool included with the CAD Import Module can help you with creating a solid object, even though there are gaps between faces due to either design flaws or tolerance issues. 1 Click the New (
) button on the main toolbar.
2 In the Model Wizard make sure that the space dimension is 3D, then click the Finish
button. 3 To add an import feature to the geometry sequence right-click Geometry 1 and select Import. 4 In the Settings window click the Browse button. 5 Locate the course CD on the hard disk and select the file sphere.igs, then click Open.
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CHAPTER 2: CAD IMPORT
6 Click Import in the Settings window.
The geometry looks like a solid sphere, whose outer surface is split into two halves.
Continue by creating a simple unstructured tetrahedral mesh. 7 Right-click the Mesh 1 node and select Free tetrahedral. 8 To build the mesh click the Build All button.
The mesh appears in the Graphics window, but a warning is displayed in the Settings window that tetrahedral elements were not created since the geometry contains no domains. Add a Measurements node to the geometry sequence to examine the imported geometry. 9 Right-click Geometry 1 then select Measurements.
IMPORT OF IGES FILES
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10 Click the sphere object in the Graphics window to highlight it. Right-click anywhere
to add it to the Selection list. The following information is displayed in the Measurements window:
The information indicates that the geometry is not a solid object since it does not contain any domains. By default, during import the CAD Import Module tries to knit faces and form solids. If a solid is not created, it usually means that there were gaps, larger than the default import tolerance of 10-5 in the geometry units, between geometry faces. In these next steps you will create a surface mesh of the sphere to visualize the gap between the two halves. 11 Right-click the Free Tetrahedral 1 node and select Size. 12 From the Geometric entity level list select Boundary. 13 In the Graphics window click one of the faces of the sphere to highlight it.
Right-click anywhere to add it to the Selection list. 14 From the Predefined list select Fine.
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CHAPTER 2: CAD IMPORT
15 Click the Build All button.
Examine the meshed surfaces, note that the mesh only connects on a portion of the common edge. On the portion where the mesh does not connect there is a narrow gap between the faces.
Even when the geometry includes such a gap you can create a solid by using the knit tool. 16 Right-click the Geometry 1 node and select CAD Repair, then Knit to Solid.
The Knit to Solid 1 node is inserted into the geometry sequence after the Import 1 node. 17 Select the sphere in the Graphics window, then enter 1e-4 in the Absolute repair tolerance edit field. 18 To build the feature click the Build All button.
To check if a solid was indeed created you can use the Measurements tool. 19 Right-click the Geometry 1 node and select Measurements. 20 Add the sphere to the selection list, by selecting it in the Graphics window.
The displayed information still indicates that there are no domains in the sphere. This means that the gap was larger than 10-4 m and the tool could not create a solid.
IMPORT OF IGES FILES
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21 Click the Knit to Solid 1 node, then enter 1e-2 in the Absolute repair tolerance edit
field. 22 Click the Build all button to build the feature. 23 Right-click the Geometry 1 node and select Measurements, then select the sphere.
This time the object contains a domain, which indicates that a solid object was created by the knit to solid tool. 24 Mesh the geometry by clicking the Mesh 1 node, then the Build all button.
This time a properly connected volume mesh for the solid object was created.
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CHAPTER 2: CAD IMPORT
Modeling with Assemblies A CAD file which contains an assembly or a multi-body part is a collection of solid bodies. When importing such a design, each body becomes a geometry object in COMSOL Multiphysics. In addition you can have additional geometry objects created by various other features of the geometry sequence. The collection of geometry objects resulting from the operations in the geometry sequence needs to be prepared for physics modeling. This is done by the last feature node of the geometry sequence, called the Finalize ( ) feature. The Finalize feature cannot be deleted from the sequence, and its label in the Model Builder changes according to the finalization method. There are two methods for finalization of geometry: • forming a union of all objects, which becomes the modeling domain - The modeling domain, created by the union operation, consists of domains separated by interior boundaries. COMSOL ensures continuity in the field variables across interior boundaries. • forming an assembly object for modeling - The geometry objects become separate domains for modeling. Identity pairs (or contact pairs) need to be defined to ensure continuity. - Imprints can optionally be created on touching boundaries. Each of these methods influences the type of mesh you can create. Forming a union imposes the largest constraints on the meshing procedure, while an assembly object without imprints leaves you with maximum freedom for mesh generation. In the following exercise, you import a 3D CAD assembly and test each of the above methods for handling several geometry objects. You will also be able to test how to prepare the same assembly for contact modeling by setting up contact pairs. The last exercise of this section will demonstrate how gaps and dimensional mismatches can influence the final mesh of an imported geometry.
Form a union 1 Click the New (
) button on the main toolbar.
2 In the Model Wizard make sure that the space dimension is 3D, then click the Finish
button.
MODELING WITH ASSEMBLIES
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3 To add an import feature to the geometry sequence right-click Geometry 1 and select Import. 4 In the Settings window click the Browse button. 5 Locate the course CD on the hard disk and select the file piston_assembly.x_b,
then click Open. 6 Click Import in the Settings window.
When the import is completed, examine the geometry. Notice that there are three geometry objects: the piston, the pin, and the rod. The pin is connected to the rod and the piston.
pin
rod
piston
7 Click the Form Union button in the Model Builder, then click the Build Selected (
)
button. The default finalization method is Form a union. After the operation is completed, examine the resulting geometry. Note that COMSOL Multiphysics has combined the three geometry objects into just one with three subdomains corresponding to the piston, pin, and rod, respectively. These subdomains are delimited by inner surfaces. 8 Right-click the Geometry 1 node and select Measurements. 9 Select all three geometry objects in the Graphics window.
Note that each of the 3 objects contains one domain.
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10 From the Type of geometry list select Finalized geometry. 11 From the Geometric entity level list select Entire geometry.
Note that the finalized geometry is a single object containing three domains. The total number of faces, edges, and points does not correspond to the number of entities before union. Mesh two of the subdomains with different sized meshes. 12 Right-click the Mesh 1 node and select Free Tetrahedral. 13 From the Geometric entity level list select Domain. 14 Add the pin (domain 3) and the rod (domain 2) to the selection list by selecting
them in the Graphics window. 15 Right-click the Free Tetrahedral 1 node and select Size. 16 From the selection list select domain 2, then click the Remove from Selection button. 17 From the Predefined list select Extremely fine. 18 Right-click the Free Tetrahedral 1 node and select Size. 19 From the selection list select domain 3, then click the Remove from Selection button. 20 From the Predefined list select Extremely coarse. 21 Click Build All button.
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17
Examine the mesh. The Wireframe Rendering ( ) or Transparency ( ) buttons are useful when details of a geometry are hidden behind other parts. Notice that the elements are connected at the surface between the subdomains.
Form an Assembly and Create Imprints Now change the finalization method to form an assembly and create imprints. 1 Click the Form Union node in the Model Builder. 2 From the Finalization method list select Form an assembly. 3 Select the Create imprints check box. 4 Click the Build Selected button.
The operation creates imprints on the pin surface that is in contact with the piston and the rod. IDENTITY PAIRS
To ensure continuity between the domains you can create identity pairs between the boundaries of the domains. 1 Right-click the Definitions node and select Identity Pair (
). This adds an identity
pair feature to the Model Builder. To make the selection of surfaces easier first hide the rod and the piston. 2 Click the Select Domain (
) button.
3 In the Graphics window click the rod, then click the Hide Selected ( 4 In the Graphics window click the piston, then click the Hide Selected ( 5 Click the Select Boundaries (
) button. ) button.
) button to switch back to boundary selection.
6 Add the boundary of the pin in contact with the rod (boundary 105) to the list of Source Boundaries.
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7 To add a second identity pair, between the pin and the piston, right-click the Definitions node and select Identity Pair. 8 Add the boundary of the pin in contact with the piston (boundary 104) to the list
of Source Boundaries.
Before selecting the destination boundaries, you can make the rod and the piston visible again and hide the pin instead. 9 Click first the Reset Hiding (
) button, then click the Select Domain (
10 In the Graphics window click the pin, then click the Hide Selected ( 11 In the Settings window of Identity Pair 2 click Activate Selection (
) button.
) button. ) next to the
Destination Boundaries section. 12 Add the boundary of the piston in contact with the pin (boundary 42) to the list of Destination Boundaries.
13 Click the Identity Pair 1 node. 14 Click Activate Selection (
) next to the Destination Boundaries section.
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19
15 Add the boundary of the rod in contact with the pin (boundary 53) to the list of Destination Boundaries.
16 To show all domains again click the Reset Hiding (
) button.
MESHING
The three domains of the assembly object all have their own boundary surfaces. This option gives you greater freedom in meshing the different parts than the union method. 17 Right-click the Mesh 1 node and select Build All.
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As you can see, you can now freely mesh the two objects with different sized meshes. Note that this mesh is for illustrative purposes and is not appropriate for FEA, due to the quality of the mesh.
Form an Assembly Now test the third, and last, method of finalizing the geometry by creating an assembly object without any modification of the geometry. 1 Click the Form Assembly node in the Model Builder. 2 De-select the Create imprints check box. 3 Click the Build Selected button.
The operation forms an assembly object for modeling, but the geometry objects are not modified. MESHING
This procedure gives you even greater freedom in meshing the objects. Because there are no imprints on the boundary surfaces, you can create a swept mesh of the pin. 1 Click the Free Tetrahedral 1 node. 2 In the Selection list select domain 3 and click the Remove from Selection (
) button.
3 Right-click the Size 1 node and select Delete. 4 Right-click Mesh 1 and select Swept. 5 From the Geometric entity level list select Domain. 6 Select the pin (domain 3). 7 Right-click the Swept 1 node and select Size. 8 From the Predefined list select Extremely Fine.
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21
9 Click the Build All (
) button.
In this example you can see that after importing CAD assemblies, the method that gives you the greatest freedom in meshing is to form an assembly object without creating imprints. This method is also suitable for complex CAD geometries where a geometry analysis by the other two methods fails.
Gaps in Assemblies The CAD model in this exercise contains some gaps on the assembly level, which can prevent the creation of a good mesh. One of the parts also contains holes we would like to remove. You can start by importing the CAD file. 1 Click the New (
) button on the main toolbar.
2 In the Model Wizard make sure that the space dimension is 3D, then click the Finish
button. 3 To add an import feature to the geometry sequence right-click Geometry 1 and select Import. 4 In the Settings window click the Browse button.
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5 Locate the course CD on the hard disk and select the file assembly_gaps.x_b,
then click Open. 1 Click Import in the Settings window.
plate 2
plate 1
The assembly consists of two parts, plate 1 and plate 2. Plate 2 has four screw holes, and two feet that fit into corresponding holes on plate 1. Besides the imported parts, the surrounding air volume is also important for the analysis. Therefore, you can continue by creating a block around the objects. 2 Right-click the Geometry 1 node and select Block. 3 Use data from the following table to draw a block: POSITION: BASE: CORNER
SIZE
x
-0.01
Width
0.06
y
-0.03
Depth
0.06
z
-0.06
Height
0.12
4 Click the Build All (
) button.
5 Right-click the Mesh 1 node and select Free Tetrahedral.
The finalize operation is performed automatically as you click the Mesh 1 node. During the finalize operation, COMSOL Multiphysics determines the geometric
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23
difference of the objects and creates one object with three subdomains delimited by single surfaces. 6 Click the Build All (
) button.
After the meshing is completed, note that there are about to 130,000 tetrahedral elements in this seemingly simple geometry. 7 Click the Transparency (
) button to visualize the inner domains.
Dense mesh regions
You can see that there is a very fine mesh in several boundary regions between the two parts of the assembly. Usually this occurs if small, dimensional discrepancies exist between parts. They might be present in the design to provide important clearance for assembling the final product. In this case the clearance between the feet
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of plate 1 and the corresponding slots on plate 2 causes a very thin air gap. However, such fine details do not affect the overall physics phenomena. narrow gaps
You cannot change the dimension of imported objects with the CAD Import Module. Yet, you can, easily remove both the feet and the slots. To this you need to use the defeaturing tools. 8 Right-click the Geometry 1 node, then select CAD Defeaturing>Delete Faces.
In the Tools window that appears you can select faces of the geometry to be removed. When a delete operation is completed a feature node will be added to the geometry sequence. To make the selection of faces easier first hide the block and plate 1. 9 Click the Select Objects (
) button.
10 In the Graphics window click the block, then click the Hide Selected ( 11 In the Graphics window click plate 1, then click the Hide Selected ( 12 Click the Select Boundaries (
) button. ) button.
) button to switch back to boundary selection.
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25
13 Add the faces highlighted in the figure below, boundaries 1, 2, 5, 6, 7, 8 on object
imp1(1), to the list of Faces to delete.
The selected faces delimit the features to be removed. 14 Click the Delete Selected button to delete the feet.
The default heal method, Patch, heals the wound that results from removing the faces, by shrinking and growing neighboring faces to cover the hole. When the operation completes a Delete Faces 1 feature node is added to the Model Builder, right after the current feature in the geometry sequence, which was the Block 1 feature. With the feet removed you can continue with removing the corresponding slots on plate 1. Hide some objects again to make face selection easier, this time the block and plate 2. 15 Click the Select Objects (
) button.
16 In the Graphics window click the block, then click the Hide Selected ( 17 In the Graphics window click plate 2, then click the Hide Selected ( 18 Click the Select Boundaries (
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) button. ) button.
) button to switch back to boundary selection.
19 Add the faces highlighted in the figure below, boundaries 5-8, 9-12 on object
imp1(2), to the list of Faces to delete.
20 Select the Heal as through hole check box.
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27
21 Click the Delete Selected button to delete the holes.
Before continuing with meshing, you can also delete the four mounting holes, shown in the figure below, since they are assumed to not influence the analysis.
You can easily select the faces delimiting the holes by using the Select Box tool. 22 Click the Select Box (
) button, then in the Graphics window draw a rectangle around the faces delimiting one of the holes. Confirm the selection by right-clicking.
23 Repeat the previous step for the other three holes. 24 The Faces to delete list should now contain faces 7-14 and 17-24 of object dfa1. 25 Click the Delete Selected button to delete the holes.
The last step is to create a new mesh of the geometry.
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26 Right-click the Mesh 1 node in the Model Builder, then select Build All.
Dense mesh regions
The mesh now consists of about 31,000 tetrahedral elements. As you can see in the figure, there are still two regions of high mesh density left. The reason for this is a slight difference in width between the two objects.
Objects are not of the same width
If this dimensional mismatch is unintentional and assumed to not be important for the analysis, it can easily fixed in a CAD software. With the CAD Import Module, you can export defeatured parts to a Parasolid file, which you can open and edit with a CAD program.
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29
27 Right-click the Geometry 1 node and select Export to File. 28 The Input list shows all three objects selected (dfa2, blk1, and dfa3). To accept and
export the objects click Export. 29 From the Save as type list select either Parasolid binary file or Parasolid text file. 30 Select a directory, enter a file name, then click Save.
The resulting Parasolid file can be imported into most CAD software. You can also use the CAD Import Module to fix the dimensional discrepancy, by for example using the repair functionality of the Finalize feature. 31 Click the Form Union feature node. 32 In the Relative repair tolerance edit field enter 1e-3. 33 Right-click the Mesh 1 node in the Model Builder, then select Build All.
As you can see the resulting mesh contains only about 5000 elements.
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The Meshing Sequence COMSOL Multiphysics provides an interactive meshing environment where, with a few mouse clicks, you can easily mesh individual faces or domains. Each meshing operation is added to the meshing sequence. The final mesh is the result of building all the operations in the meshing sequence. This example demonstrates how to use the meshing sequence. On a realistic geometry you create a mesh consisting of different element types. You learn how to mesh certain parts of a geometry, modify this mesh, change the parameters to your liking, and rebuild the mesh.
Meshing with Default Settings To start, import a 3D CAD geometry and mesh it with the default settings using the free mesher to generate a mesh with tetrahedral elements. 1 Click the New (
) button on the main toolbar.
2 In the Model Wizard make sure that the space dimension is 3D, then click the Finish
button. 3 To add an import feature to the geometry sequence right-click Geometry 1 and select Import. 4 In the Settings window click the Browse button. 5 Locate the course CD on the hard disk and select the file solder_joints.x_b,
then click Open. 6 Click Import in the Settings window.
Note that for many of the imported objects a warning is displayed that the default import tolerance, 10-5 m, might be too large for the geometry, which includes parts on the order of a tenth of a millimeter in size. Features smaller than the import tolerance are automatically removed during import. Therefore, if you know that your geometry contains small features you can change the import tolerance in the
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Settings window of the Import feature. There you also have the option of turning off the repair feature during import. In this particular case, this is not necessary.
The geometry represents a small part of a circuit board with an electronic component mounted by means of several solder ball joints. Electronic component
Solder joints Circuit board
Continue by creating a simple unstructured tetrahedral mesh. 7 Click the Mesh 1 node.
As you click the Mesh 1 node the Finalize feature is automatically built, which means that the various objects are combined into the one object with inner boundaries and subdomains. 8 Click the small plus sign (or triangle) in front of the Mesh 1 node to expand the
meshing sequence. Initially the meshing sequence contains only the Size feature node. This feature is called a global attribute feature, since it influences all subsequent operation features in the meshing sequence. This first Size feature node cannot be deleted from the meshing sequence. You can also add attribute features under an operation feature node, in which case it is called a local attribute feature. 9 Right-click the Mesh 1 node and select Free tetrahedral.
The Free Tetrahedral 1 feature node is added after the Size feature node. COMSOL Multiphysics always inserts new nodes in the meshing sequence after the current feature node. To indicate the current feature node, it appears with a quadratic frame around its icon. As soon as it is inserted, the Free Tetrahedral 1 node becomes the current feature.
THE MESHING SEQUENCE
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33
10 To build the mesh click the Build All button.
According to information displayed in the Messages window this mesh consists of approximately 35,000 elements. While the geometry is resolved quite well by this mesh, you may want to reduce the number of elements to reduce the memory required for solving the problem.
Adding Local Attribute Features Assume that you are investigating the solder joints and would therefore like to keep the detailed mesh in the spherical subdomains, but create a mesh with fewer elements in the remaining objects. 1 Right-click the Free Tetrahedral 1 feature node and select Size. 2 From the Geometric entity level list select Domain.
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3 Use the Graphics window to select the domains 1-3, which represent the circuit
board and the electronic component.
4 From the Predefined list select Coarser. 5 Click the Build All button to build the new mesh.
According to the Messages window the mesh now consists of approximately 24,000 elements.
The Swept Mesher For even fewer elements in the circuit board and electronic component, you can create a swept mesh. This technique sweeps a boundary mesh through the domains to create a structured mesh in the sweep direction. The swept mesher operates on a 3D subdomain by first meshing a source face, and then sweeping the resulting face mesh through the subdomain to an opposite target face. For straight and circular sweep paths, you can use several connected faces as source
THE MESHING SEQUENCE
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35
faces. All faces that encompass a domain are classified as either source faces, a target face, or boundary faces. The boundary faces are the faces connecting the source and target faces. MESHING A SET OF BOUNDARIES
First you can modify the free tetrahedral mesh operation to operate only on the solder joints. 1 Click the Free Tetrahedral 1 feature node. 2 From the Geometric entity level list select Domain. 3 From the Selection list select domains 1,2, and 3, then click the Remove from Selection (
) button.
You can also remove the Size 1 feature node. 4 Right-click the Size 1 node, then select Delete.
You can now create a coarse mesh on the source boundaries for the sweep mesh operation. Build the mesh before continuing. 5 Click the Build All (
) button.
6 Right-click the Mesh 1 node then select More Operations then Free Triangular. 7 From the Graphics window select the two faces (boundaries 2 and 14) highlighted
in the figure below.
To make selection easier click the Wireframe Rendering ( to turn of wireframe rendering. 8 Right-click the Free Triangular 1 node and select Size. 9 From the Predefined list select Coarser.
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) button. Click it again
10 Click the Build All (
) button to build the mesh.
The mesh is built, but a Problems node appears under the Free Triangular 1 feature node. 11 Expand the Problems 1 node, then click the Warning 1 node to read the message.
COMSOL Multiphysics generates the warning because the Coarser predefined mesh size you have applied to the two surfaces has been overridden by the relatively finer Normal mesh settings specified in the first Size feature node of the meshing sequence. The reason is to avoid bad quality meshes, that may result if the boundary of a domain is meshed with a coarser mesh size then the volume. To avoid this situation a good practice is to set the first global Size setting to the coarsest mesh that you plan to have in the geometry, then specify local size feature nodes for the mesh operations that need finer mesh. 12 Click the Size node, then select Coarser from the Predefined list. 13 Right-click the Free Tetrahedral 1 node, and select Size. 14 From the Predefined list select Normal.
Since the global setting is Coarser, the size attribute to the Free Triangular 1 node is no longer necessary. 15 Right-click the Size 1 node under the Free Triangular 1 node, then select Delete.
THE MESHING SEQUENCE
|
37
16 Right-click the Mesh 1 node then select Build All (
) to build the mesh.
Notice that the triangular mesh is coarser this time. CREATING A SWEPT MESH
Now that the source faces are meshed, you can sweep this mesh through the subdomains. 1 Right-click the Mesh 1 node and select Swept.
The Geometric entity level list is set to Remaining by default for new mesh operations. In this case the remaining domains correspond to the domains we would like to sweep mesh.
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2 Click the Build All (
) button to build the mesh.
The mesh now consists of approximately 15,000 elements. The swept mesher used the Coarser predefined mesh size to determine the number of elements along the sweep direction. By specifying a distribution for the swept mesher you have the possibility to manually control the number and distribution of elements along the sweep direction. 3 Right-click the Swept 1 feature node and select Distribution. 4 In the Number of elements edit field enter 2. 5 Click the Build All button to build the mesh.
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39
This latest mesh consists of approximately 18,000 elements, keeping the higher resolution for the subdomains which are important for the analysis, while providing a less dense structured mesh for the remaining subdomains.
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Free Mesh Parameters The free meshing algorithm using tetrahedral elements is the most general meshing technique, and does not pose any constraints on the structure of the geometry. Hence it can be used to mesh any arbitrary object. This exercise shows how to use the various parameters of the free mesh algorithm. There are a number of predefined parameter sets, and each parameter can be tuned individually.
Predefined Parameter Settings There are nine predefined parameter sets ranging from “extremely fine” to “extremely coarse.” These settings result in a good mesh for most geometries and simulation problems. Follow the next steps to import a 3D CAD model and to test two of these settings. 1 Click the New (
) button on the main toolbar.
2 In the Model Wizard make sure that the space dimension is 3D, then click the Finish
button. 3 To add an import feature to the geometry sequence right-click Geometry 1 and select Import. 4 In the Settings window click the Browse button. 5 Locate the course CD on the hard disk and select the file piston_for_fem.x_b,
then click Open. 6 Right-click the Mesh 1 node and select Free tetrahedral. 7 To build the mesh click the Build All (
) button.
The Messages window indicates that there are roughly 34,000 tetrahedral elements in this geometry. Use the mouse to rotate the object and look at the geometry more closely. You can see that it contains some small details such as fillets and chamfers. Assume that the existing mesh does not resolve these details sufficiently for your simulation needs and so a finer parameter setting is required. 8 Right-click the Free Tetrahedral 1 node, then select Size.
FREE MESH PARAMETERS
|
41
9 From the Predefined list select Fine.
This changes the predefined parameter set from “Normal” to “Fine”. 10 To build the mesh click the Build All button. 11 In the Messages window, note that this mesh consists of approximately 91,000
elements. If you examine the mesh you can see that, even with the Fine settings, some narrow areas are not resolved sufficiently. You could decrease the mesh size once more but, to achieve a fine mesh only on certain types of boundaries, you also have the possibility to tune individual mesh parameters. This is demonstrated in the following sections.
Mesh Curvature Assume that you want a better resolution on regions with a high curvature, such as fillets. To find the right setting you can test it first on just one of the fillets in the geometry. 1 Click the Size 1 node. 2 From the Predefined list select Normal. 3 From the Geometric entity level list select Boundary. 4 From the Graphics window select the boundary highlighted (boundary 12) in the
figure below.
5 In the Element Size section click the Custom radio button. 6 Select the Minimum element size check box. 7 Change the value of Minimum element size to 5e-4.
With the Minimum element size parameter, you can prevent the generation of a fine mesh around small curved regions of the geometry. When the radius of curvature is smaller than this value, the mesh generator considers the radius of curvature as being this value.
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8 To build the mesh click the Build All (
) button.
9 Select the Resolution of curvature check box. 10 In the Resolution of curvature enter 0.5.
The Resolution of curvature parameter determines the size of boundary elements compared to the curvature of the geometric boundary. The curvature radius multiplied by the resolution of curvature gives the maximum allowed element size along the boundary. A lower value gives a finer mesh along curved boundaries. 11 To build the mesh click the Build All (
) button.
The selected fillet is resolved even finer with this mesh.
FREE MESH PARAMETERS
|
43
Resolution of Narrow Regions Assume that you also want better resolution on narrow regions with no curvature such as chamfers. 12 From the Graphics window add the boundary highlighted (boundary 44) in the
figure below to the Selection list of the Size 1 feature node.
13 Select the Resolution of narrow regions check box. 14 In the Resolution of narrow regions edit field enter 2.1.
The mesh Resolution of narrow regions mesh parameter approximately controls the number of layers of elements that are created in narrow regions (approximately). If the value of this parameter is less than one, the mesh generator might create elements that are anisotropic in size in narrow regions. 15 To build the mesh click the Build All (
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) button.
Element Growth Rate Assume that you are happy with the new parameter settings for curved and narrow regions. Now use these to mesh the entire geometry. 1 Click the Zoom Extents (
) button to see the entire geometry.
2 From the geometric entity level list in the Size 1 feature node select Entire geometry. 3 To build the mesh click the Build All (
) button.
The fine details of the geometry are resolved satisfactorily. However, the mesh consists of approximately 308,000 elements, which is too many. Assume that the inner part of the volume does not need such a high resolution. You can change it by specifying the rate of growth from the small elements on the surface towards the inner part of the geometry. 4 Select the Maximum element growth rate check box. 5 In the Maximum element growth rate edit field enter 2.3.
The Maximum element growth rate determines the maximum rate at which the element size can grow from a region with small elements to a region with larger elements. 6 To build the mesh click the Build All (
) button.
7 In the Messages window you can see that this mesh consists of approximately
112,000 elements.
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|
45
Mesh Quality Changing various mesh parameters can influence the quality of the mesh. In general, the default settings result in a mesh with rather good quality. However, this quality can decrease drastically when changing individual parameters. First take a look at the mesh statistics for the current mesh on the piston. 1 Right-click the Mesh 1 node and select Statistics.
The Statistics window appears, which contains details about the mesh, including the number and type of elements, and a histogram of element quality.
The element quality is a number between 0 and 1, where 1 describes a perfectly isotropic element, and 0 describes a flat element. For 3D meshes in general a minimum quality of about 0.1 means a satisfactory mesh. However, this depends also on the type of geometry and physics application. Note that the quality number is calculated based on the linear elements.
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In the following, you learn how to visualize the elements with the worst mesh quality. Assume that you want to achieve the opposite goal as described in the previous example—you want much to resolve the narrow regions with less mesh. 1 Click the Size 1 feature node. 2 Select the Predefined radio button, then select the Custom radio button.
This way the custom mesh parameters are reset to values corresponding to the “Normal” predefined mesh size. 3 Select the Resolution of narrow regions check box. 4 Change the value of Resolution of narrow regions to 0.1. 5 To build the mesh click the Build All (
) button.
6 Right-click the Mesh 1 node and select Statistics.
Now your mesh consists of only 28,000 elements. However, in the area of the narrow regions the quality becomes worse. The minimum element quality has decreased, and according to the histogram there is a relatively small number of elements that have the lowest quality. Visualize this situation with the following steps. 7 Right-click the Mesh 1 node and select Plot.
The Mesh 1 plot is added to the 3D Plot Group 1 under the Results section of the Model Builder. The mesh plot that appears in the Graphics window contains the volume elements colored according to quality. 8 Expand the Element Filter section, and select Enable filter. 9 From the Criterion list select Worst quality. 10 In the Fraction edit field enter 0.005.
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|
47
11 Click the Plot button.
You can now see those 0.5% of elements with the worst quality, where almost all of them are located around the narrow faces. As anticipated, decreasing the mesh resolution in narrow regions detrimentally influences the quality of the mesh.
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Mapped and Swept Meshing COMSOL Multiphysics provides the ability to create structured meshes, specifically, mapped meshes on surfaces and swept meshes in subdomains. Each of these techniques can be used interactively and has its own set of parameters. This exercise demonstrates how to use the mapped and swept meshing techniques. It shows how to create these meshes interactively so as to achieve a desired distribution of elements.
Swept Mesh with Default Settings First create a swept mesh with the default settings. 1 Click the New (
) button on the main toolbar.
2 In the Model Wizard make sure that the space dimension is 3D, then click the Finish
button. 3 To add an import feature to the geometry sequence right-click Geometry 1 and select Import. 4 In the Settings window click the Browse button. 5 Locate the course CD on the hard disk and select the file thin_layer.mphbin,
then click Open. 6 Right-click the Mesh 1 node and select Swept. 7 To build the mesh click the Build All (
) button.
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49
This procedure creates a very coarse mesh consisting of hexahedral elements. Assume that you want dense layers at the bottom because you expect high gradients in the simulation. In addition, you want a better resolution in the curved part of the geometry. The following section shows how to achieve these goals.
Mapped Mesh Parameters In order to force a certain element distribution in a cross-section of the geometry, you must create a mesh on a source face. This mesh is later swept through the subdomains of the geometry. 1 Click the Size node in the Model Builder. 2 Click the Build Selected (
) button.
By building the Size feature, it becomes the current feature node in the mesh sequence after which the mapped mesh operation can be inserted. 3 Right-click the Mesh 1 node and select More Operations>Mapped. 4 Add boundary 1, which is highlighted in the figure below, to the Selection list.
5 Right-click the Mapped 1 node and select Distribution. 6 Add Edges 1 and 6 to the Selection list of the Distribution 1 feature node.
7 From the Distribution properties list select Predefined distribution type. 8 Type 8 in the Number of elements edit field.
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9 In the Element ratio edit field type 5.
This parameter controls the ratio in size between the last element and the first element along the edge. 10 In the Distribution method list select Geometric sequence. 11 Right-click the Mapped 1 node and select Distribution. 12 Add Edges 2 and 4 to the Selection list of the Distribution 2 feature node.
13 Type 2 in the Number of elements field. 14 To build the mesh click the Build All (
) button.
15 Use the zoom and rotation features to magnify the modified mesh.
You have achieved the desired element distribution in the cross section. The mesh looks to be adequate on the straight parts. However, the curved parts are not resolved satisfactorily.
Swept Mesh Parameters Change the swept mesh parameters in order to achieve the desired element distribution along the subdomains.
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51
1 Right-click on the Swept 1 node and select Distribution. 2 In the Selection list select domains 2, 3 and 4, then click the Remove from Selection
(
) button.
3 From the Distribution properties list select Predefined distribution type. 4 Type 5 in the Number of elements edit field. 5 In the Element ratio edit field type 3. 6 From the Distribution method list select Geometric sequence. 7 Select the Reverse direction check box.
This defines the distribution in the opposite direction for the selected subdomain. 1 Right-click on the Swept 1 node and select Distribution. 2 In the Selection list select domains 1, 3 and 4, then click the Remove from Selection
(
) button.
3 From the Distribution properties list select Predefined distribution type. 4 Type 5 in the Number of elements edit field. 5 In the Element ratio edit field type 3. 6 From the Distribution method list select Geometric sequence.
The last step is to define the distribution for the curved parts. 7 Right-click on the Swept 1 node and select Distribution. 8 In the Selection list select domains 1 and 2, then click the Remove from Selection
(
) button.
9 Type 10 in the Number of elements edit field. 10 To build the mesh click the Build All (
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) button.
Solved with COMSOL Multiphysics 4.1
Heat Sink Introduction This model is intended as a first introduction to simulations of fluid flow and conjugate heat transfer. It is also a very useful model for exploring the modeling procedure in COMSOL Multiphysics. More specifically, it demonstrates the use of three important concepts: • The concept of sequencing. Each operation that you can use to set up a simulation is displayed as a node in the Model Tree. As you proceed with your model, the Model Tree forms a sequence of operations. You can make changes to any of the nodes in the tree and then re-run the sequence of operations to update your model according to the changes in the node settings. • The selection of materials in the Materials node. In this node, you can specify and investigate materials and material properties for all the domains and for all physics interfaces in your model. • The use of selections. In the Selections node, you can define selections of domains, boundaries, edges, and points, which you can later use in other steps in the modeling procedure.
Model Definition The modeled system consists of an aluminum heat sink for cooling of components in electronic circuits mounted inside a channel of rectangular cross section; see Figure 1. Such a set-up is used in order to measure the cooling capacity of heat sinks. Air enters the channel at the inlet and exits the channel at the outlet. The base surface of the heat sink is kept at a constant temperature through an external heat source. All other external faces are thermally insulated.
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inlet
outlet
base surface
Figure 1: The model set-up including channel and heat sink. The cooling capacity of the heat sink can be determined by measuring the power required to keep the base surface of the heat sink at a constant temperature. The model solves a thermal balance for the heat sink and the air flowing in the rectangular channel. Thermal energy is transported through conduction in the aluminum heat sink and through conduction and convection in the cooling air. The temperature field is continuous across the internal surfaces between the heat sink and the air in the channel. The temperature is set at the inlet of the channel and at the base of the heat sink. Alternatively, you can also simulate the presence of a layer of adhesive material between the heat sink and the heating device used to keep a constant temperature. In such case, you have to define a heat transfer coefficient for the adhesive layer and then set the temperature at the heater side of the layer. The transport of thermal energy at the outlet is dominated by convection. The flow field is obtained by solving one momentum balance for each space coordinate (x, y, and z) and a mass balance. The inlet velocity is defined by a parabolic velocity profile for fully developed laminar flow. At the outlet, a constant pressure is combined the assumption that there are no viscous stresses in the direction perpendicular to the outlet. At all solid surfaces, the velocity is set to zero in all three spatial directions.
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The thermal conductivity of aluminum, the thermal conductivity of air, the heat capacity of air, and the air density are all temperature-dependent material properties. You can find all of the settings mentioned above in the physics interface for Conjugate Heat Transfer in COMSOL Multiphysics. You also find the material properties, including their temperature dependence, in the Material Browser
Results In Figure 2, the hot wake behind the heat sink visible in the plot is a sign of the convective cooling effects.
Figure 2: The surface plot visualizes the temperature field on the channel walls and the heat sink surface, while the arrow plot shows the flow velocity field around the heat sink.
Model Library path: Heat_Transfer_Module/Tutorial_Models/heat_sink
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Modeling Instructions MODEL WIZARD
1 Go to the Model Wizard window. 2 Click Next. 3 In the Add Physics tree, select Heat Transfer>Conjugate Heat Transfer>Laminar Flow (nitf). 4 Click Next. 5 In the Studies tree, select Preset Studies>Stationary. 6 Click Finish. GLOBAL DEFINITIONS
Parameters 1 In the Model Builder window, right-click Global Definitions and choose Parameters.
Define some parameters that you can use when specifying the channel dimensions. 2 Go to the Settings window for Parameters. 3 Locate the Parameters section. In the Parameters table, enter the following settings: NAME
EXPRESSION
DESCRIPTION
L_channel
7[cm]
Channel length
W_channel
3[cm]
Channel width
H_channel
1.5[cm]
Channel height
U0
5[cm/s]
Mean inlet velocity
GEOMETRY 1
Import 1 1 In the Model Builder window, right-click Model 1>Geometry 1 and choose Import. 2 Go to the Settings window for Import. 3 Locate the Import section. Click the Browse button. 4 Browse to the model’s Model Library folder and double-click the file heat_sink_n19.mphbin.
5 Click the Build Selected button. 6 In the Model Builder window, right-click Geometry 1 and choose Work Plane.
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Rectangle 1 1 In the Model Builder window, right-click Geometry and choose Rectangle. 2 Go to the Settings window for Rectangle. 3 Locate the Size section. In the Width edit field, type L_channel. 4 In the Height edit field, type W_channel. 5 Locate the Position section. In the x edit field, type -4.5e-2. 6 In the y edit field, type -W_channel/2. 7 Click the Build Selected button.
Extrude 1 1 In the Model Builder window, right-click Work Plane 1 and choose Extrude. 2 Click the Go to Default 3D View button on the Graphics toolbar. 3 Go to the Settings window for Extrude. 4 Locate the Distances from Work Plane section. In the table, enter the following
settings: DISTANCES (M)
H_channel
5 Click the Build Selected button. MATERIALS
1 In the Model Builder window, right-click Model 1>Materials and choose Open Material Browser. 2 Go to the Material Browser window. 3 Locate the Materials section. In the Materials tree, select Built-In>Air. 4 Right-click and choose Add Material to Model from the menu.
Air By default, the first material you add applies to all domains. Typically, you can leave this setting and add other materials that override the default material where applicable. In this example, specify aluminum for Domain 2: 1 In the Model Builder window, right-click Materials and choose Open Material Browser. 2 Go to the Material Browser window. 3 Locate the Materials section. In the Materials tree, select Built-In>Aluminum 3003-H18.
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4 Right-click and choose Add Material to Model from the menu.
Aluminum 3003-H18 1 In the Model Builder window, click Aluminum 3003-H18. 2 Select Domain 2 only. C O N J U G A T E H E A T TR A N S F E R
Fluid 1 1 In the Model Builder window, right-click Model 1>Conjugate Heat Transfer and choose Fluid. 2 Select Domain 1 only.
Inlet 1 1 In the Model Builder window, right-click Conjugate Heat Transfer and choose the
boundary condition Laminar Flow>Inlet. 2 Select Boundary 115 only. 3 Go to the Settings window for Inlet. 4 Locate the Boundary Condition section. From the Boundary condition list, select Laminar inflow. 5 In the Uav edit field, type U0.
Outlet 1 1 In the Model Builder window, right-click Conjugate Heat Transfer and choose the
boundary condition Laminar Flow>Outlet. 2 Select Boundary 1 only.
Temperature 1 1 In the Model Builder window, right-click Conjugate Heat Transfer and choose the
boundary condition Heat Transfer>Temperature. 2 Select Boundary 115 only.
The default temperature, corresponding to 20 degrees Celsius or 68 degrees Fahrenheit, applies at the inlet.
Temperature 2 1 In the Model Builder window, right-click Conjugate Heat Transfer and choose the
boundary condition Heat Transfer>Temperature. 2 Select Boundary 8 only. 3 Go to the Settings window for Temperature.
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4 Locate the Temperature section. In the T0 edit field, type 393.15[K].
This value corresponds to 120 degrees Celsius or 248 degrees Fahrenheit.
Outflow 1 1 In the Model Builder window, right-click Conjugate Heat Transfer and choose the
boundary condition Heat Transfer>Outflow. 2 Select Boundary 1 only. MESH 1
Free Tetrahedral 1 In the Model Builder window, right-click Model 1>Mesh 1 and choose Free Tetrahedral.
Size 1 1 In the Model Builder window, right-click Free Tetrahedral 1 and choose Size. 2 Go to the Settings window for Size. 3 Locate the Geometric Scope section. From the Geometric entity level list, select Domain. 4 Select Domain 1 only. 5 Locate the Element Size section. From the Predefined list, select Finer. 6 Click the Build All button. DEFINITIONS
In the Model Builder window, right-click Study 1 and choose Compute.
Selection 1 1 In the Model Builder window, right-click Model 1>Definitions and choose Selection. 2 Go to the Settings window for Selection. 3 Locate the Geometric Scope section. From the Geometric entity level list, select Boundary. 4 Right-click Selection 1 and choose Rename. 5 Go to the Rename Selection dialog box and type walls in the New name edit field. 6 Click OK. 7 Right-click Selection 1 and choose Select Box. 8 Select Boundaries 3 and 5–114 only.
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RESULTS
Data Sets 1 In the Model Builder window, right-click Results>Data Sets>Solution 1 and choose Add Selection. 2 Go to the Settings window for Selection. 3 Locate the Geometric Scope section. From the Geometric entity level list, select Boundary. 4 From the Selection list, select walls.
3D Plot Group 1 1 In the Model Builder window, click Surface 1. 2 Go to the Settings window for Surface. 3 Locate the Coloring and Style section. From the Color table list, select Thermal. 4 In the Model Builder window, right-click 3D Plot Group 1 and choose Arrow Volume. 5 Go to the Settings window for Arrow Volume. 6 In the upper-right corner of the Expression section, click Replace Expression. 7 From the menu, choose Conjugate Heat Transfer (Laminar Flow)>Velocity field (u, v, w). 8 Locate the Arrow Positioning section. Find the x grid points subsection. In the Points
edit field, type 40. 9 Find the y grid points subsection. In the Points edit field, type 20. 10 Find the z grid points subsection. From the Entry method list, select Coordinates. 11 In the Coordinates edit field, type 5e-3. 12 Right-click Arrow Volume 1 and choose Color Expression. 13 Go to the Settings window for Color Expression. 14 In the upper-right corner of the Expression section, click Replace Expression. 15 From the menu, choose Conjugate Heat Transfer (Laminar Flow)>Velocity magnitude (nitf.U). 16 Click the Plot button.
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Various Analyses of an Elbow Bracket Introduction The component depicted in Figure 1 is part of a support mechanism and is subjected to both mechanical loads and thermal loads. This tutorial model takes you through the steps to carry out a detailed analysis of the part using the Structural Mechanics Module.
Figure 1: Geometry of the elbow bracket. In the various parts of the model you are introduced to using the available basic analysis types, together with numerous postprocessing possibilities. These analysis types are: • Static analysis • Eigenfrequency analysis • Damped eigenfrequency analysis • Transient analysis • Modal based transient analysis • Frequency response analysis • Modal based frequency response analysis • Parametric analysis • Linear buckling analysis This tutorial model consists of a single model, with nine studies, corresponding to these analysis types, which are described in the section Available Study Types in the Structural Mechanics Module User’s Guide. The chapter Structural Mechanics Modeling in the same manual provides further assistance.
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Note: If you have already built the short model version described in Static and Eigenfrequency Analyses of an Elbow Bracket, you can proceed directly to the section Time-Dependent Analysis.
Model Definition The geometry for this part, see Figure 1, has been created with a CAD software, and it is available for you to import into COMSOL Multiphysics.
Material Structural steel, as taken from the material library, with Young’s modulus of 200 GPa, Poisson’s ratio of 0.33, and thermal expansion coefficient 12.3·10-6 K−1.
Damping The Structural Mechanics Module supports Rayleigh damping and loss factor damping. You can also use no damping, which is the default option. In some of the analyses here you will use Rayleigh damping, where you specify damping parameters that are proportional to the mass (αdM) and stiffness (βdK) in the following way: C = α dM M + β dK K where C is the damping matrix, M is the mass matrix, and K is the stiffness matrix. The damping is specified locally in each domain; this means that you can specify different damping parameters in different parts of the model. To find the values for the Rayleigh damping, you can use the relations between the critical damping ratio and the Rayleigh damping parameters. It is often easier to interpret the critical damping ratios, which are given by dM α --------- ω i + β dK ⋅ ω i ξ i = -------------------------------------------2
where ξi is the critical damping ratio at a specific angular frequency ωi. Knowing two pairs of corresponding ξi and ωi results in a system of equations
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1 - ω -----------------------1 ( 2 ⋅ ω 1 ) 2 α dM ω β dK 1 ------------------- ------2 ( 2 ⋅ ω2 ) 2
=
ξ1 ξ2
with the damping parameters as the unknown variables. Assume that the structure has a constant damping ratio of 0.1. Select two frequencies near the excitation frequency, 400 Hz and 600 Hz, to calculate the damping parameters. You can do this in MATLAB with the following commands: b = [0.1;0.1]; A = [1/(2*400*2*pi) 2*pi*400/2; 1/(2*600*2*pi) 2*pi*600/2]; % A*damp = b damp = A\b; alphadM = damp(1) betadK = damp(2)
The result is αdM = 300 and βdK = 3.2·10−5. For more information see the section Modeling Damping and Losses in the Structural Mechanics Module User’s Guide. If modal based dynamic response studies are performed it is usually easier to give the critical damping ratios directly. This will also give more detailed control over the damping properties over a large frequency range.
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Loads and Constraints The displacement are fixed in all directions on the face shown in Figure 2. The load is described under each study, but in all cases it is distributed over the face shown in this figure.
Fixed face
Loaded face
Figure 2: Constraint and loading of the bracket. The Model Library note immediately below appears in the discussion of every model. The path indicates the location of the model file in the Model Library root directory. The most convenient way to open it is from the Model Library window in the COMSOL Desktop, which you can open from the File menu.
Model Library path: Structural_Mechanics_Module/Tutorial_Models/ elbow_bracket
Modeling Instructions MODEL WIZARD
1 Go to the Model Wizard window. 2 Click Next. 3 In the Add physics tree, select Structural Mechanics>Solid Mechanics (solid). 4 Click Next. 5 Find the Studies subsection. In the tree, select Preset Studies>Stationary.
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6 Click Finish. GEOMETRY 1
Import 1 1 In the Model Builder window, right-click Model 1>Geometry 1 and choose Import. 2 Go to the Settings window for Import. 3 Locate the Import section. From the Geometry import list, choose COMSOL Multiphysics file. 4 Click the Browse button. 5 Browse to the model’s Model Library folder and double-click the file elbow_bracket.mphbin.
6 Click the Import button. 7 Click the Wireframe Rendering button on the Graphics toolbar.
The view in the Graphics window should look like that in the image below.
8 Click the Wireframe Rendering button on the Graphics toolbar to return to the
default surface rendering.
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MESH 1
Size 1 In the Model Builder window, right-click Model 1>Mesh 1 and choose Free Tetrahedral.
The default free mesher has nine predefined combinations of mesh parameter settings. They range from Extremely fine to Extremely coarse, with Normal as the default setting. Unless any other mesh parameters are set, this is the setting that is used if you use Build All or Build Selected to generate the mesh. 2 Go to the Settings window for Size. 3 Locate the Element Size section. From the Predefined list, choose Coarse. 4 Click the Build All button. MATERIALS
Next, specify the material properties. You can do this either by explicitly typing them in or by selecting a library material in the Material Browser. For this model, use a library material. 1 In the Model Builder window, right-click Model 1>Materials and choose Open Material Browser. 2 Go to the Material Browser window. 3 Locate the Materials section. In the Materials tree, select Built-In>Structural steel. 4 Right-click and choose Add Material to Model from the menu.
Static Analysis A static analysis has no explicit or implicit time dependencies. This situation corresponds to the steady state with constant (in time) boundary conditions and material properties. The purpose of such analysis can be to find the maximum stress level and compare it with the material’s yield strength, as well as to check that the deformation of the component is within the limits of the design criteria.
Results and Discussion The analysis shows that the von Mises effective stress has a maximum value of 187 MPa, which, compared with the material’s yield strength of 350 MPa, results in a utilization factor of 53%.
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The analysis also gives the maximum static displacements as 1.14 mm Three different representations of the stress state are shown in Figure 3 through Figure 5.
Figure 3: Effective stresses on the boundary of the volume.
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Figure 4: Isosurface plot of the effective stress.
Figure 5: Arrow plot of the principal stresses.
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Modeling Instructions SOLID MECHANICS
Fixed Constraint 1 1 In the Model Builder window, right-click Model 1>Solid Mechanics and choose Fixed Constraint. 2 Select Boundary 1 only.
Boundary Load 1 1 In the Model Builder window, right-click Solid Mechanics and choose Boundary Load. 2 Select Boundary 29 only. 3 Go to the Settings window for Boundary Load. 4 Locate the Force section. Specify the FA vector as 3[MPa]
X
0
Y
3[MPa]
Z
STUDY 1
In this model, where there are many different studies, it is a good idea to assign manual names to some features. 1 In the Model Builder window, right-click Study 1 and choose Rename. 2 Go to the Rename Study dialog box and type Study 1 (Static) in the New name
edit field. 3 Click OK.
The default settings in the generated solver sequence are OK for this model. STUDY 1 (STATIC)
1 In the Model Builder window, right-click Study 1 (Static) and choose Compute.
Before moving on to analyzing the solution, rename the solver sequence. 2 In the Model Builder window, expand the Study 1 (Static) node.
Solver 1 1 In the Model Builder window, expand the Study 1 (Static)>Solver Configurations node. 2 Right-click Study 1 (Static)>Solver Configurations>Solver 1 and choose Rename.
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3 Go to the Rename Solver dialog box and type Static Sequence in the New name
edit field. 4 Click OK. RESULTS
Similarly, rename the solution data set.
Data Sets 1 In the Model Builder window, right-click Results>Data Sets>Solution 1 and choose Rename. 2 Go to the Rename Solution dialog box and type Static Solution in the New name
edit field. 3 Click OK.
In the Results branch, you can create various plot types, evaluate expressions, or animate the results. The result features can visualize any expression containing, for example, the solution variables, their derivatives, and the space coordinates. Many frequently used expressions are predefined as postprocessing variables, and they are directly available in the Expression section menus for the various plot types. When the solver finishes, two default plots appear. One shows total displacement as a surface plot and the other shows the same quantity as a slice plot. Re-use these plot groups, beginning by creating a volume plot of the von Mises stress with markers for maximum and minimum values together with the deformed shape of the component. For this purpose, modify the first plot group.
Stress (solid) 1 In the Model Builder window, right-click Results>Stress (solid) and choose Rename. 2 Go to the Rename 3D Plot Group dialog box and type Static Stress Contour in
the New name edit field. 3 Click OK. 4 Click the Zoom Extents button on the Graphics toolbar.
To evaluate the maximum displacement, use a Maximum coupling operator. DEFINITIONS
Maximum 1 1 In the Model Builder window, right-click Model 1>Definitions and choose Maximum. 2 Go to the Settings window for Maximum.
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3 Locate the Source Selection section. From the Geometric entity level list, choose Boundary. 4 From the Selection list, choose All boundaries.
Variables 1a 1 In the Model Builder window, right-click Definitions and choose Variables. 2 Go to the Settings window for Variables. 3 Locate the Variables section. In the Variables table, enter the following settings: NAME
EXPRESSION
DESCRIPTION
U_max
maxop1(solid.disp)
Maximum deflection
STUDY 1 (STATIC)
Static Sequence 1 In the Model Builder window, right-click Study 1 (Static)>Solver Configurations>Static Sequence and choose Solution>Update.
This step is necessary in order to access variables that were created after the solution was performed. RESULTS
Derived Values 1 In the Model Builder window, right-click Results>Derived Values and choose Global Evaluation. 2 Go to the Settings window for Global Evaluation. 3 In the upper-right corner of the Expression section, click Replace Expression. 4 From the menu, choose Definitions>Maximum deflection (U_max). 5 Locate the Expression section. From the Unit list, choose mm. 6 Click the Evaluate button.
The result, approximately 1.1 mm appears in the Results window. Next, replace the second default plot group by an isosurface plot. The resulting plot should resemble that in Figure 4.
3D Plot Group 2 1 In the Model Builder window, right-click Results and choose 3D Plot Group. 2 Right-click Results>3D Plot Group 2 and choose Isosurface.
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3 Go to the Settings window for Isosurface. 4 In the upper-right corner of the Expression section, click Replace Expression. 5 From the menu, choose Solid Mechanics>Stress>von Mises stress (solid.mises). 6 Right-click Isosurface 1 and choose Deformation. 7 Right-click 3D Plot Group 2 and choose Rename. 8 Go to the Rename 3D Plot Group dialog box and type Static Stress Isosurface
in the New name edit field. 9 Click OK. 10 Click the Go to Default 3D View button on the Graphics toolbar.
With the following steps you can reproduce the principal stress arrow plot shown in Figure 5:
3D Plot Group 3 1 Right-click Results and choose 3D Plot Group. 2 In the Model Builder window, right-click Results>3D Plot Group 3 and choose More Plots>Principal Stress Volume. 3 Go to the Settings window for Principal Stress Volume. 4 Locate the Positioning section. Find the X grid points subsection. In the Points edit
field, type 10. 5 Find the Y grid points subsection. In the Points edit field, type 15. 6 Find the Z grid points subsection. In the Points edit field, type 10. 7 Click the Plot button. 8 In the Model Builder window, right-click 3D Plot Group 3 and choose Rename. 9 Go to the Rename 3D Plot Group dialog box and type Static Principal Stress Arrow Plot in the New name edit field.
10 Click OK. 11 Click the Go to Default 3D View button on the Graphics toolbar.
Eigenfrequency Analysis An eigenfrequency analysis finds the eigenfrequencies and modes of deformation of a component. The eigenfrequencies f in the structural mechanics field are related to the eigenvalues λ returned by the solvers through
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–λ f = − --------2πi
(1)
In COMSOL Multiphysics you can choose between working with eigenfrequencies and working with eigenvalues according to your preferences. Eigenfrequencies are the default option for all physics interfaces in the Structural Mechanics Module. If no damping is included in the material, the undamped natural frequencies are computed. The purpose of the following eigenfrequency analysis is to find the six lowest eigenfrequencies and corresponding mode shapes.
Results and Discussion The first six eigenfrequencies are: EIGENFREQUENCY
FREQUENCY
f1
417 Hz
f2
571 Hz
f3
1930 Hz
f4
2450 Hz
f5
3110 Hz
f6
3930 Hz
The mode shapes corresponding to the two lowest eigenfrequencies are shown in Figure 6. The deformed plot indicates an oscillation in the xy-plane for the lowest eigenfrequency, while the second lowest eigenmode shows an oscillation in the yz-plane.
Figure 6: Eigenmodes of the two lowest eigenfrequencies.
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Notes About the COMSOL Implementation Any loads present on the model, such as the load from the static load case above, are ignored in the default eigenfrequency analysis. It is also possible to include effects from prestress. You can find an example of such an analysis in the model Vibrating String.
Modeling Instructions Add a new study to your model. MODEL WIZARD
1 In the Model Builder window, right-click the root node and choose Add Study. 2 Go to the Model Wizard window. 3 Find the Studies subsection. In the tree, select Preset Studies>Eigenfrequency. 4 Click Finish. STUDY 2
1 In the Model Builder window, right-click Study 2 and choose Rename. 2 Go to the Rename Study dialog box and type Study 2 (Eigenfrequency) in the New name edit field. 3 Click OK. STUDY 2 (EIGENFREQUENCY)
Solver 2 1 In the Model Builder window, right-click Study 2 (Eigenfrequency) and choose Compute. 2 Expand the Study 2 (Eigenfrequency)>Solver Configurations node. 3 Right-click Study 2 (Eigenfrequency)>Solver Configurations>Solver 2 and choose Rename. 4 Go to the Rename Solver dialog box and type Eigenfrequency Sequence in the New name edit field. 5 Click OK.
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RESULTS
Data Sets 1 In the Model Builder window, right-click Results>Data Sets>Solution 2 and choose Rename. 2 Go to the Rename Solution dialog box and type Eigenfrequency Solution in the New name edit field. 3 Click OK.
Mode Shape (solid) As a default, the first eigenmode is shown. Follow these steps to reproduce the plot in the left panel of Figure 6: 1 In the Model Builder window, expand the Mode Shape (solid) node, then click Surface 1. 2 Go to the Settings window for Surface. 3 Locate the Coloring and Style section. Clear the Color legend check box.
The displacement values do not have any real significance for an eigenmode plot such as this one. Take a look at the second mode as well. 4 In the Model Builder window, click Mode Shape (solid). 5 Go to the Settings window for 3D Plot Group. 6 Locate the Data section. From the Eigenfrequency list, choose 570.790707. 7 Click the Plot button.
Compare the resulting plot to that to the right in Figure 6. You can give the plot a more descriptive name: 8 Right-click Mode Shape (solid) and choose Rename. 9 Go to the Rename 3D Plot Group dialog box and type Mode Shapes in the New name
edit field. 10 Click OK. 11 Right-click Mode Shape (solid) and choose Player to create a movie showing how the
elbow bracket would deform if subjected to a harmonic load with a frequency near the selected eigenfrequency, in this case 570.6 Hz. To play the movie again, click the Play button on the Graphics toolbar.
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Export The default sweep type is dynamic data extension. If you set the Sweep type to Stored solutions and then click the Generate Frame button, you get an animation where each frame corresponds to an eigenmode in the Eigenfrequency list. By using the Frame number slider in the Frames section you can browse the eigenmodes. 1 In the Model Builder window, right-click Results>Export>Player 1 and choose Rename. 2 Go to the Rename Player dialog box and type Mode Shapes in the New name edit
field. 3 Click OK.
Damped Eigenfrequency Analysis If the material has damping, the eigenvalue solver will automatically switch to computation of the damped eigenfrequencies. The damped eigenfrequencies and eigenmodes are complex. The real part of the eigenfrequency corresponds to the frequency and the imaginary part is the damping.
Results and Discussion The first six eigenfrequencies are given below, and can be compared with the results from the undamped model.: EIGENFREQUENCY
FREQUENCY
UNDAMPED FREQUENCY
f1
415+41.3i Hz
417 Hz
f2
568+56.6i Hz
571 Hz
f3
1885+397i Hz
1930 Hz
f4
2372+629i Hz
2450 Hz
f5
2946+996i Hz
3110 Hz
f6
3600+1577i Hz
3930 Hz
The relative damping of a certain mode is the ratio between the imaginary and the real part. It can be seen that the relative damping increases rapidly as the natural frequency increases. This is an effect of the Rayleigh damping model.
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Notes About the COMSOL Implementation As the eigenvalues exist as complex conjugate pairs, the first six damped eigenfrequencies correspond to the first three undamped eigenfrequencies. For this reason, twelve computed frequencies are requested.
Modeling Instructions MODEL WIZARD
1 In the Model Builder window, right-click the root node and choose Add Study. 2 Go to the Model Wizard window. 3 Find the Studies subsection. In the tree, select Preset Studies>Eigenfrequency. 4 Click Finish. SOLID MECHANICS
Add damping and specify the mass and stiffness parameters.
Damping 1 1 In the Model Builder window, right-click Linear Elastic Material Model 1 and choose Damping. 2 Go to the Settings window for Damping. 3 Locate the Damping Settings section. In the αdM edit field, type 300. 4 In the βdK edit field, type 3.2e-5. STUDY 3
1 In the Model Builder window, right-click Study 3 and choose Rename. 2 Go to the Rename Study dialog box and type Study 3 (Damped Eigenfrequency)
in the New name edit field. 3 Click OK. STUDY 3 (DAMPED EIGENFREQUENCY)
Solver 3 1 In the Model Builder window, right-click Study 3 (Damped Eigenfrequency) and choose Compute. 2 Expand the Study 3 (Damped Eigenfrequency)>Solver Configurations node.
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3 Right-click Study 3 (Damped Eigenfrequency)>Solver Configurations>Solver 3 and
choose Rename. 4 Go to the Rename Solver dialog box and type Damped Eigenfrequency Sequence
in the New name edit field. 5 Click OK. RESULTS
Data Sets 1 In the Model Builder window, right-click Results>Data Sets>Solution 3 and choose Rename. 2 Go to the Rename Solution dialog box and type Damped Eigenfrequency Solution in the New name edit field.
3 Click OK.
Mode Shape (solid) 1 In the Model Builder window, right-click Results>Mode Shape (solid) and choose Rename. 2 Go to the Rename 3D Plot Group dialog box and type Damped Mode Shapes in the New name edit field. 3 Click OK.
The mode shape is very similar to the one obtained when solving the undamped problem.
Damped Mode Shapes 1 In the Model Builder window, expand the Results>Damped Mode Shapes node, then
click Surface 1. 2 Go to the Settings window for Surface.
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3 Locate the Coloring and Style section. Clear the Color legend check box.
Time-Dependent Analysis This analysis solves for the transient solution of the displacements and velocities as functions of time. The material properties, forces, and boundary conditions can vary in time. The purpose of this analysis is to find the transient response from a harmonic load during the first five periods. The excitation frequency is 500 Hz, which is between the first and second eigenfrequencies found in the eigenfrequency analysis. This load is applied on the face indicated in Figure 2. The expression for the load can be written as F x = 1.5 ⋅ [ 1 + sin ( 2π ⋅ 500 ⋅ t – π ⁄ 2 ) ] MPa
(2)
where t denotes the time in seconds.
Results and Discussion Because the loading is harmonic, the expected solution will consist of an initial transient, and after long time the response will be a stationary harmonic solution with its amplitude controlled by the damping of the system.
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The following plot shows the x-displacement at a point on the loaded face:
Figure 7: x-displacement at a point on the loaded face. The figure below shows the von Mises stress in the bracket at 0.0036 s. The maximum value is 252 MPa.
Figure 8: von Mises stress at t = 3.6 ms.
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Notes About the COMSOL Implementation In order to get a good accuracy, you should set the absolute tolerance value to a reasonable value smaller than the actual expected displacements. The solution is always performed in the selected base unit system, so tolerances should in this case be expressed in m. When a harmonic load is used, the time step can sometimes oscillate in an inefficient manner, causing longer solution times. This can be avoided by using the more restrictive time stepping obtained by selecting the check box for Time step increase delay. For more information on the settings for the time-dependent solver, see Time Dependent in the COMSOL Multiphysics Reference Guide.
Modeling Instructions If you are working from the beginning of this example, ignore the next four instructions. If you are starting from the short model version described in Static and Eigenfrequency Analyses of an Elbow Bracket, load that model as described here: 1 From the View menu, choose Model Library. 2 Go to the Model Library window. 3 In the Model Library tree, select Structural Mechanics Module>Tutorial Models>elbow bracket brief. 4 Click Open. MODEL WIZARD
1 In the Model Builder window, right-click elbow_bracket_brief.mph and choose Add Study. 2 Go to the Model Wizard window. 3 Find the Studies subsection. In the tree, select Preset Studies>Time Dependent. 4 Click Finish. STUDY 4
Solving for five periods with an excitation frequency of 500 Hz means solving for 10 ms. Save the solution every 0.2 ms.
Step 1: Time Dependent 1 In the Model Builder window, click Study 4>Step 1: Time Dependent.
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2 Go to the Settings window for Time Dependent. 3 Locate the Study Settings section. In the Times edit field, type range(0,2e-4,10e-3).
4 In the Model Builder window, right-click Study 4 and choose Rename. 5 Go to the Rename Study dialog box and type Study 4 (Time-Dependent) in the New name edit field. 6 Click OK. 7 Right-click Study 4 and choose Show Default Solver. STUDY 4 (TIME-DEPENDENT)
Solver 4 1 In the Model Builder window, expand the Study 4 (Time-Dependent)>Solver Configurations node. 2 Right-click Study 4 (Time-Dependent)>Solver Configurations>Solver 4 and choose Rename. 3 Go to the Rename Solver dialog box and type Time-Dependent Sequence in the New name edit field. 4 Click OK.
Time-Dependent Sequence 1 In the Model Builder window, expand the Study 4 (Time-Dependent)>Solver Configurations>Time-Dependent Sequence node, then click Time-Dependent Solver 1. 2 Go to the Settings window for Time-Dependent Solver. 3 Click to expand the Absolute Tolerance section. 4 In the Tolerance edit field, type 1e-5. 5 Click to expand the Time Stepping section. 6 Select the Time step increase delay check box. Keep the default value of 15.
This setting instructs the solver not to increase the time step until 15 consecutive steps have been successful. 7 In the Amplification for high frequency edit field, type 0.95.
By raising this value from its default value of 0.75 you reduce the damping of high frequencies. You can reduce the file size significantly by not storing time derivatives when not needed.
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8 Click to expand the Output section. 9 Clear the Store time-derivatives check box. RESULTS
Before computing the solution, prepare a plot for displaying the results during the solution process.
Data Sets 1 In the Model Builder window, right-click Results>Data Sets>Solution 4 and choose Rename. 2 Go to the Rename Solution dialog box and type Time-Dependent Solution in the New name edit field. 3 Click OK.
1D Plot Group 6 1 Right-click Results and choose 1D Plot Group. 2 Go to the Settings window for 1D Plot Group. 3 Locate the Data section. From the Data set list, choose Time-Dependent Solution. 4 Right-click Results>1D Plot Group 6 and choose Point Graph. 5 Select Point 30 only. 6 Go to the Settings window for Point Graph. 7 In the upper-right corner of the y-Axis Data section, click Replace Expression. 8 From the menu, choose Solid Mechanics>Displacement>Displacement field (Material)>Displacement field, X component (u). 9 Locate the y-Axis Data section. From the Unit list, choose mm. 10 Select the Description check box. 11 In the associated edit field, type X displacement on loaded face. 12 In the Model Builder window, right-click 1D Plot Group 6 and choose Rename. 13 Go to the Rename 1D Plot Group dialog box and type Time-Dependent Displacement Graphs in the New name edit field.
14 Click OK. STUDY 4 (TIME-DEPENDENT)
Step 1: Time Dependent 1 In the Model Builder window, click Study 4 (Time-Dependent)>Step 1: Time Dependent.
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2 Go to the Settings window for Time Dependent. 3 Click to expand the Results While Solving section. 4 Select the Plot check box. 5 From the Plot group list, choose Time-Dependent Displacement Graphs. SOLID MECHANICS
A new load is needed for this study. You could just change the existing one, but when you have multiple studies it is better to have individual load features and disable the ones not currently used.
Boundary Load 1 1 In the Model Builder window, right-click Boundary Load 1 and choose Rename. 2 Go to the Rename Boundary Load dialog box and type Static Load in the New name
edit field. 3 Click OK. 4 Right-click Boundary Load 1 and choose Disable.
Boundary Load 2 1 Right-click Solid Mechanics and choose Boundary Load. 2 In the Model Builder window, right-click Boundary Load 2 and choose Rename. 3 Go to the Rename Boundary Load dialog box and type Time-Dependent Load in the New name edit field. 4 Click OK. 5 Select Boundary 29 only. 6 Go to the Settings window for Boundary Load. 7 Locate the Force section. Specify the FA vector as 1.5[MPa]*(1+sin(2*pi*500[Hz]*t-pi/2))
X
0
Y
0
Z
STUDY 4 (TIME-DEPENDENT)
In the Model Builder window, right-click Study 4 (Time-Dependent) and choose Compute.
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RESULTS
Time-Dependent Displacement Graphs Compare the result for the x-displacement with the graph shown in Figure 7.
3D Plot Group 7 1 In the Model Builder window, right-click Results and choose 3D Plot Group. 2 Right-click Results>3D Plot Group 7 and choose Surface. 3 Go to the Settings window for Surface. 4 In the upper-right corner of the Expression section, click Replace Expression. 5 From the menu, choose Solid Mechanics>Stress>von Mises stress (solid.mises). 6 Right-click Surface 1 and choose Deformation. 7 In the Model Builder window, click 3D Plot Group 7. 8 Go to the Settings window for 3D Plot Group. 9 Locate the Data section. From the Data set list, choose Time-Dependent Solution. 10 From the Time list, choose 0.0036. 11 Click the Plot button.
Before generating a movie of the solution, you need to fix the scale factor for the deformation; otherwise this factor is automatically adjusted for each movie frame in such a way that the elbow does not move in the Graphics window. 12 In the Model Builder window, click Surface 1>Deformation 1. 13 Go to the Settings window for Deformation. 14 Locate the Scale section. Select the Scale factor check box. 15 In the Model Builder window, right-click 3D Plot Group 7 and choose Rename. 16 Go to the Rename 3D Plot Group dialog box and type Time-Dependent Stress Contour in the New name edit field.
17 Click OK. 18 Right-click 3D Plot Group 7 and choose Player.
Time-Dependent Modal Analysis In a modal-based analysis, the problem is reduced by representing the dynamics of the structure as a combination of small number of its most significant eigenmodes. This is very efficient when the frequency content of the loads applied to the structure is limited, so that only a small number of modes will be excited.
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You solve the same problem as in the previous section. Just as for the direct solution, it is important that you set the absolute tolerance small enough.
Results and Discussion The plot in Figure 9 below shows the same x-displacement as in the previous section but with results from both the full and the modal based time-dependent analysis. The correspondence between the solutions is good even though only the six first eigenmodes are used.
Figure 9: x-displacement at a point on the loaded surface for full and modal analyses. Figure 10 below shows the von Mises stress in the bracket at 0.0036 s. The maximum value is 237 MPa, which can be compared with the 252 MPa computed using the direct solution above. In general, more modes than what is needed to compute accurate displacements are required to obtain good stress solutions.
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Figure 10: von Mises stress for modal solution.
Notes About the COMSOL Implementation When you create a new study, it is possible to directly select “Time-Dependent Modal.” Such a study, however, generates a complete solver sequence including the eigenvalue computation step. Because the eigenvalues are already available, you create an “Empty Study” and add the study steps manually. The undamped eigenmodes are used as the base, and the damping is provided by the material. In the modal procedure, all loads must have the same variation in time, specified in the study step. This means that you should not enter any time-dependent loads (that is, loads with an explicit dependency on the time using the time variable t).
Modeling Instructions MODEL WIZARD
1 In the Model Builder window, right-click the root node and choose Add Study. 2 Go to the Model Wizard window.
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3 Find the Studies subsection. In the tree, select Custom Studies>Empty Study. 4 Click Finish. SOLID MECHANICS
Time-Dependent Load In the Model Builder window, right-click Model 1>Solid Mechanics>Time-Dependent Load and choose Disable.
Boundary Load 3 1 Right-click Solid Mechanics and choose Boundary Load. 2 In the Model Builder window, right-click Boundary Load 3 and choose Rename. 3 Go to the Rename Boundary Load dialog box and type Modal Time-Dependent Load in the New name edit field.
4 Click OK. 5 Select Boundary 29 only. 6 Go to the Settings window for Boundary Load. 7 Locate the Force section. Specify the FA vector as 1.5[MPa]
X
0
Y
0
Z
STUDY 5
1 In the Model Builder window, right-click Study 5 and choose Rename. 2 Go to the Rename Study dialog box and type Study 5 (Modal Time-Dependent)
in the New name edit field. 3 Click OK. 4 Right-click Study 5 and choose Time-Dependent Modal. STUDY 5 (MODAL TIME-DEPENDENT)
Step 1: Time-Dependent Modal 1 Go to the Settings window for Time-Dependent Modal. 2 Locate the Study Settings section. In the Times edit field, type range(0,2e-4,10e-3).
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Solver 5 1 In the Model Builder window, right-click Study 5 (Modal Time-Dependent) and choose Show Default Solver. 2 Expand the Study 5 (Modal Time-Dependent)>Solver Configurations node. 3 Right-click Study 5 (Modal Time-Dependent)>Solver Configurations>Solver 5 and
choose Rename. 4 Go to the Rename Solver dialog box and type Modal Time-Dependent Sequence
in the New name edit field. 5 Click OK.
Modal Time-Dependent Sequence 1 In the Model Builder window, expand the Study 5 (Modal Time-Dependent)>Solver Configurations>Modal Time-Dependent Sequence node, then click Modal Solver 1. 2 Go to the Settings window for Modal Solver. 3 Locate the General section. Find the Tolerance subsection. In the Absolute global tolerance edit field, type 1e-5. 4 Locate the Eigenpairs section. From the Solution list, choose Eigenfrequency Sequence. 5 Click to expand the Advanced section. 6 In the Load factor edit field, type 1+sin(2*pi*500*t-pi/2). 7 In the Model Builder window, right-click Study 5 (Modal Time-Dependent) and choose Compute. RESULTS
Data Sets 1 In the Model Builder window, right-click Results>Data Sets>Solution 5 and choose Rename. 2 Go to the Rename Solution dialog box and type Modal Time-Dependent Solution
in the New name edit field. 3 Click OK.
Stress (solid) Reproduce the plot in Figure 10 by following these instructions: 1 In the Model Builder window, click Stress (solid). 2 Go to the Settings window for 3D Plot Group. 3 Locate the Data section. From the Time list, choose 0.0036.
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4 Click the Plot button. 5 In the Model Builder window, right-click Stress (solid) and choose Rename. 6 Go to the Rename 3D Plot Group dialog box and type Modal Time-Dependent Stress Contour in the New name edit field.
7 Click OK.
Add the results from this study to the graph of the direct time dependent results, so that the methods can be compared.
Time-Dependent Displacement Graphs 1 In the Model Builder window, right-click Results>Time-Dependent Displacement Graphs and choose Point Graph. 2 Select Point 30 only. 3 Go to the Settings window for Point Graph. 4 In the upper-right corner of the y-Axis Data section, click Replace Expression. 5 From the menu, choose Displacement field, X component (u). 6 Locate the Data section. From the Data set list, choose Modal Time-Dependent Solution. 7 Locate the y-Axis Data section. From the Unit list, choose mm. 8 Click to expand the Coloring and Style section. 9 Find the Line style subsection. From the Color list, choose Green. 10 Click the Plot button.
Frequency Response Analysis A frequency response analysis solves for the linearized steady-state response from harmonic excitation loads. The loads can have amplitudes and phase shifts that may depend on the excitation frequency, f: π F freq = F ( f ) ⋅ cos 2πf ⋅ t + F Ph ( f ) ⋅ ---------- 180
(3)
where F(f) is the amplitude and FPh(f) is the phase shift of the load. The result of a frequency response analysis is a complex time-dependent displacement field, which can be interpreted as an amplitude, uamp, and a phase angle, uphase. The actual displacement at any point in time is the real part of the solution
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u = u amp cos ( 2πf ⋅ t + u phase )
(4)
The software can visualize the amplitudes and phases as well as the solution at a specific iφ angle (time). When plotting, COMSOL Multiphysics multiplies the solution by e , where φ is the angle in radians that corresponds to the angle specified in degrees in the Solution at angle (phase) edit field in the solution feature under the Data sets. The plot then shows the real part of the evaluated expression u = u amp cos ( φ + u phase )
(5)
The angle φ is available as the variable phase (radians) and can be used in plot expressions. The purpose of this analysis is to find the response from a harmonic load with an excitation frequency in the range 350–650 Hz, which includes the first two eigenfrequencies found in the eigenfrequency analysis. The loads are F x = 3 ⋅ 10
6
F z = 3 ⋅ 10
6
N/m N/m
2 2
and there is no time shift between the load components.
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Results and Discussion The amplitudes of the x-, y-, and z-displacements as functions of excitation frequency, at a point on the face where the load is applied, appear in the following figure:
Figure 11: Displacement amplitudes vs. excitation frequency. The peaks in the displacement amplitude curves are associated with the two lowest eigenfrequencies of the bracket. Note that the lowest eigenfrequency, around 410 Hz, corresponds to the peak on the x-displacement amplitude curve, while the next eigenfrequency, around 570 Hz, corresponds to the peak on the z-displacement amplitude curve. This is just as indicated by the eigenmode shapes obtained from the eigenfrequency analysis.
Notes About the COMSOL Implementation The loads are given without explicit time dependencies, since the harmonic variation is an underlying assumption in this analysis type. The loads can have different phase angles. This can either obtained by adding a Phase feature to the load, or by writing the load in complex form. Usually when performing a frequency response analysis you want to sweep over a frequency range. This can be done using the parametric solver, using the frequency as parameter.
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Modeling Instructions MODEL WIZARD
1 In the Model Builder window, right-click the root node and choose Add Study. 2 Go to the Model Wizard window. 3 Find the Studies subsection. In the tree, select Preset Studies>Frequency Domain. 4 Click Finish.
The load is the same as in the stationary study, so you can reuse the first load feature. SOLID MECHANICS
Static Load 1 In the Model Builder window, right-click Model 1>Solid Mechanics>Modal Time-Dependent Load and choose Disable. 2 Right-click Static Load and choose Enable. STUDY 6
1 In the Model Builder window, right-click Study 6 and choose Rename. 2 Go to the Rename Study dialog box and type Study 6 (Frequency Domain) in the New name edit field. 3 Click OK. STUDY 6 (FREQUENCY DOMAIN)
Step 1: Frequency Domain 1 In the Model Builder window, click Study 6 (Frequency Domain)>Step 1: Frequency Domain. 2 Go to the Settings window for Frequency Domain. 3 Locate the Study Settings section. In the Frequencies edit field, type range(350,10,650).
Solver 6 1 In the Model Builder window, right-click Study 6 (Frequency Domain) and choose Compute. 2 Expand the Study 6 (Frequency Domain)>Solver Configurations node. 3 Right-click Study 6 (Frequency Domain)>Solver Configurations>Solver 6 and choose Rename.
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4 Go to the Rename Solver dialog box and type Frequency Domain Sequence in the New name edit field. 5 Click OK. RESULTS
Data Sets 1 In the Model Builder window, right-click Results>Data Sets>Solution 6 and choose Rename. 2 Go to the Rename Solution dialog box and type Frequency Domain Solution in
the New name edit field. 3 Click OK.
Stress (solid) 1 In the Model Builder window, expand the Stress (solid) node, then click Surface 1. 2 Go to the Settings window for Surface. 3 Locate the Expression section. From the Unit list, choose MPa. 4 Click the Plot button.
5 In the Model Builder window, right-click Stress (solid) and choose Rename. 6 Go to the Rename 3D Plot Group dialog box and type Frequency-Response Stress Contour in the New name edit field.
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7 Click OK.
Add a 1D plot group and reproduce the displacement amplitude graphs in Figure 11.
1D Plot Group 10 1 Right-click Results and choose 1D Plot Group. 2 Go to the Settings window for 1D Plot Group. 3 Locate the Data section. From the Data set list, choose Frequency Domain Solution. 4 Locate the Title section. From the Title type list, choose Manual. 5 In the Title text area, type Displacement amplitudes. 6 Right-click Results>1D Plot Group 10 and choose Rename. 7 Go to the Rename 1D Plot Group dialog box and type Frequency Response Displacement Graphs in the New name edit field.
8 Click OK.
Frequency Response Displacement Graphs 1 In the Model Builder window, right-click Results>Frequency Response Displacement Graphs and choose Point Graph. 2 Select Point 30 only. 3 Go to the Settings window for Point Graph. 4 In the upper-right corner of the y-Axis Data section, click Replace Expression. 5 From the menu, choose Solid Mechanics>Displacement>Displacement amplitude (Material)>Displacement amplitude, X component (solid.uAmpX). 6 Click to expand the Legends section. 7 Select the Show legends check box. 8 From the Legends list, choose Manual. 9 In the table, enter the following settings: LEGENDS
X
10 In the Model Builder window, right-click Frequency Response Displacement Graphs and
choose Point Graph. 11 Select Point 30 only. 12 Go to the Settings window for Point Graph. 13 In the upper-right corner of the y-Axis Data section, click Replace Expression.
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14 From the menu, choose Solid Mechanics>Displacement>Displacement amplitude (Material)>Displacement amplitude, Y component (solid.uAmpY). 15 Locate the Coloring and Style section. Find the Line style subsection. From the Color
list, choose Green. 16 Locate the Legends section. Select the Show legends check box. 17 From the Legends list, choose Manual. 18 In the table, enter the following settings: LEGENDS
Y
19 In the Model Builder window, right-click Frequency Response Displacement Graphs and
choose Point Graph. 20 Select Point 30 only. 21 Go to the Settings window for Point Graph. 22 In the upper-right corner of the y-Axis Data section, click Replace Expression. 23 From the menu, choose Solid Mechanics>Displacement>Displacement amplitude (Material)>Displacement amplitude, Z component (solid.uAmpZ). 24 Locate the Coloring and Style section. Find the Line style subsection. From the Color
list, choose Red. 25 Locate the Legends section. Select the Show legends check box. 26 From the Legends list, choose Manual. 27 In the table, enter the following settings: LEGENDS
Z
Click the Plot button.Compare the resulting plot with that in Figure 11.
Frequency Response Modal Analysis You can also solve the same frequency response problem using the modal method. The same remarks as for the Time-dependent Modal analysis above are relevant.
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Results and Discussion In Figure 12 below, the results from the modal frequency response analysis are overlaid on the results from the previous direct frequency response analysis (see Figure 11). The curves are almost indistinguishable. The response is to a large degree controlled by the lowest eigenmodes, which are used in the modal analysis. The modal method is however more effective in terms of computer resources.
Figure 12: Displacement amplitudes vs. excitation frequency for direct and modal frequency-response analyses.
Modeling Instructions SOLID MECHANICS
Static Load In the Model Builder window, right-click Model 1>Solid Mechanics>Static Load and choose Disable.
Boundary Load 4 1 Right-click Solid Mechanics and choose Boundary Load. 2 In the Model Builder window, right-click Boundary Load 4 and choose Rename. 3 Go to the Rename Boundary Load dialog box and type Modal Frequency Load in
the New name edit field.
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4 Click OK. 5 Select Boundary 29 only. 6 Go to the Settings window for Boundary Load. 7 Locate the Force section. Specify the FA vector as linper(3)[MPa]
X
0
Y
linper(3)[MPa]
Z
MODEL WIZARD
1 In the Model Builder window, right-click the root node and choose Add Study. 2 Go to the Model Wizard window. 3 In the Studies tree, select Custom Studies>Empty Study. 4 Click Finish. STUDY 7
1 In the Model Builder window, right-click Study 7 and choose Rename. 2 Go to the Rename Study dialog box and type Study 7 (Modal Frequency Response) in the New name edit field.
3 Click OK. STUDY 7 (MODAL FREQUENCY RESPONSE)
Step 1: Frequency-Domain Modal 1 In the Model Builder window, right-click Study 7 (Modal Frequency Response) and
choose Study Steps>Frequency-Domain Modal. 2 Go to the Settings window for Frequency-Domain Modal. 3 Locate the Study Settings section. In the Frequencies edit field, type range(350,10,650).
Solver 7 1 In the Model Builder window, right-click Study 7 (Modal Frequency Response) and
choose Show Default Solver. 2 Expand the Study 7 (Modal Frequency Response)>Solver Configurations node. 3 Right-click Study 7 (Modal Frequency Response)>Solver Configurations>Solver 7 and
choose Rename.
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4 Go to the Rename Solver dialog box and type Modal Frequency-Domain Sequence
in the New name edit field. 5 Click OK.
Modal Frequency-Domain Sequence 1 In the Model Builder window, expand the Study 7 (Modal Frequency Response)>Solver Configurations>Modal Frequency-Domain Sequence node, then click Modal Solver 1. 2 Go to the Settings window for Modal Solver. 3 Locate the Eigenpairs section. From the Solution list, select Eigenfrequency Sequence. 4 In the Model Builder window, right-click Study 7 (Modal Frequency Response) and
choose Compute. RESULTS
Data Sets 1 In the Model Builder window, right-click Results>Data Sets>Solution 7 and choose Rename. 2 Go to the Rename Solution dialog box and type Modal Frequency-Domain Solution in the New name edit field.
3 Click OK.
Stress (solid) 1 In the Model Builder window, expand the Stress (solid) node, then click Surface 1. 2 Go to the Settings window for Surface. 3 Locate the Expression section. From the Unit list, select MPa.
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4 Click the Plot button.
5 In the Model Builder window, right-click Stress (solid) and choose Rename. 6 Go to the Rename 3D Plot Group dialog box and type Modal Frequency-Response Stress Contour in the New name edit field.
7 Click OK.
Frequency Response Displacement Graphs Reproduce the plot in Figure 12 as follows: 1 In the Model Builder window, right-click Results>Frequency Response Displacement Graphs and choose Point Graph. 2 Go to the Settings window for Point Graph. 3 Locate the Data section. From the Data set list, choose Modal Frequency-Domain Solution. 4 Select Point 30 only. 5 In the upper-right corner of the y-Axis Data section, click Replace Expression. 6 From the menu, choose Solid Mechanics>Displacement>Displacement amplitude (Material)>Displacement amplitude, X component (solid.uAmpX). 7 Locate the Coloring and Style section. Find the Line style subsection. From the Color
list, choose Blue. 8 From the Line list, choose Dashed.
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9 Find the Line markers subsection. From the Marker list, choose Asterisk. 10 Locate the Legends section. Select the Show legends check box. 11 From the Legends list, choose Manual. 12 In the table, enter the following settings: LEGENDS
X (modal)
13 In the Model Builder window, right-click Frequency Response Displacement Graphs and
choose Point Graph. 14 Go to the Settings window for Point Graph. 15 Locate the Data section. From the Data set list, choose Modal Frequency-Domain Solution. 16 Select Point 30 only. 17 In the upper-right corner of the y-Axis Data section, click Replace Expression. 18 From the menu, choose Solid Mechanics>Displacement>Displacement amplitude (Material)>Displacement amplitude, Y component (solid.uAmpY). 19 Locate the Coloring and Style section. Find the Line style subsection. From the Color
list, choose Green. 20 From the Line list, choose Dashed. 21 Find the Line markers subsection. From the Marker list, choose Asterisk. 22 Locate the Legends section. Select the Show legends check box. 23 From the Legends list, choose Manual. 24 In the table, enter the following settings: LEGENDS
Y(modal)
25 In the Model Builder window, right-click Frequency Response Displacement Graphs and
choose Point Graph. 26 Go to the Settings window for Point Graph. 27 Locate the Data section. From the Data set list, choose Modal Frequency-Domain Solution. 28 Select Point 30 only. 29 In the upper-right corner of the y-Axis Data section, click Replace Expression.
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30 From the menu, choose Solid Mechanics>Displacement>Displacement amplitude (Material)>Displacement amplitude, Z component (solid.uAmpZ). 31 Locate the Coloring and Style section. Find the Line style subsection. From the Color
list, choose Red. 32 From the Line list, choose Dashed. 33 Find the Line markers subsection. From the Marker list, choose Asterisk. 34 Locate the Legends section. Select the Show legends check box. 35 From the Legends list, choose Manual. 36 In the table, enter the following settings: LEGENDS
Z(modal)
37 Click the Plot button.
Parametric Analysis A parametric analysis solves for the response as a function of a parameter. You can freely define the parameter name and what it affects; it can be a material property, a load parameter, or some other expression. The purpose of this example is to find the response to static loading of the bracket as a function of the direction of the load parameterized by the angle α. Apply the load on the face shown in Figure 2. To control the direction of the load, introduce a parameter in the load expressions: F x = 3 ⋅ cos ( α ⋅ π ⁄ 180 ) MPa F z = 3 ⋅ sin ( α ⋅ π ⁄ 180 ) MPa
(6)
where α is the angle of the load direction in the xz-plane. Let −45°≤ α≤ 45°.
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VAR IOUS ANALYSES O F AN ELBOW BR ACKET
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2a
Results and Discussion The following plot shows the x-, y-, and z-displacements as functions of the direction of the load, α, at a point on the surface where the load is applied:
Figure 13: Displacement amplitudes versus load direction.
Notes About the COMSOL Implementation When you perform a parametric study, you will select the underlying study type (here: Stationary) first. The you add a Parametric feature, and define the parameter values.
Modeling Instructions MODEL WIZARD
1 In the Model Builder window, right-click the root node and choose Add Study. 2 Go to the Model Wizard window. 3 Find the Studies subsection. In the tree, select Preset Studies>Stationary. 4 Click Finish. STUDY 8
1 In the Model Builder window, right-click Study 8 and choose Rename.
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2 Go to the Rename Study dialog box and type Study 8 (Parametric Static) in
the New name edit field. 3 Click OK. 4 Right-click Study 8 and choose Parametric Sweep.
It is necessary to define the parameters to be used. The values given here do not influence the parametric solver, but can be used for tests outside it. GLOBAL DEFINITIONS
Parameters 1 In the Model Builder window, right-click Global Definitions and choose Parameters. 2 Go to the Settings window for Parameters. 3 Locate the Parameters section. In the Parameters table, enter the following settings: NAME
EXPRESSION
DESCRIPTION
alpha
-45
Solver parameter
Add a load with a parameter dependency. SOLID MECHANICS
Modal Frequency Load In the Model Builder window, right-click Modal Frequency Load and choose Disable.
Boundary Load 5 1 Right-click Solid Mechanics and choose Boundary Load. 2 In the Model Builder window, right-click Boundary Load 5 and choose Rename. 3 Go to the Rename Boundary Load dialog box and type Parametric Load in the New name edit field. 4 Click OK. 5 Select Boundary 29 only. 6 Go to the Settings window for Boundary Load. 7 Locate the Force section. Specify the FA vector as
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3[MPa]*cos(alpha*pi/180)
X
0
Y
3[MPa]*sin(alpha*pi/180)
Z
VAR IOUS ANALYSES O F AN ELBOW BR ACKET
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2a
STUDY 8 (PARAMETRIC STATIC)
Parametric Sweep 1 In the Model Builder window, click Study 8 (Parametric Static)>Parametric Sweep. 2 Go to the Settings window for Parametric Sweep. 3 Locate the Study Settings section. Under Parameter names, click Add. 4 Go to the Add dialog box. 5 In the Parameter names list, select alpha (Solver parameter). 6 Click the OK button. 7 Go to the Settings window for Parametric Sweep. 8 Locate the Study Settings section. In the Parameter values edit field, type range(-45,5,45).
Solver 8 1 In the Model Builder window, right-click Study 8 (Parametric Static) and choose Compute. 2 Expand the Study 8 (Parametric Static)>Solver Configurations node. 3 Right-click Study 8 (Parametric Static)>Solver Configurations>Solver 8 and choose Rename. 4 Go to the Rename Solver dialog box and type Parametric Static Sequence in the New name edit field. 5 Click OK. RESULTS
Data Sets 1 In the Model Builder window, right-click Results>Data Sets>Solution 8 and choose Rename. 2 Go to the Rename Solution dialog box and type Parametric Static Solution in
the New name edit field. 3 Click OK.
Stress (solid) In the Model Builder window, right-click Results>Stress (solid) and choose Delete.
1D Plot Group 12 1 Right-click Results and choose 1D Plot Group.
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2 Go to the Settings window for 1D Plot Group. 3 Locate the Data section. From the Data set list, choose Parametric Static Solution. 4 Click to expand the Title section. 5 From the Title type list, choose None. 6 Locate the Plot Settings section. Select the x-axis label check box. 7 In the associated edit field, type Force direction (degrees). 8 Select the y-axis label check box. 9 In the associated edit field, type Displacement (mm). 10 Right-click Results>1D Plot Group 12 and choose Point Graph. 11 Select Point 30 only. 12 Go to the Settings window for Point Graph. 13 Locate the y-Axis Data section. In the Expression edit field, type u. 14 Locate the Legends section. Select the Show legends check box. 15 From the Legends list, choose Manual. 16 In the table, enter the following settings: LEGENDS
X
17 Right-click Point Graph 1 and choose Duplicate. 18 Go to the Settings window for Point Graph. 19 Locate the y-Axis Data section. In the Expression edit field, type v. 20 Locate the Coloring and Style section. Find the Line style subsection. From the Color
list, choose Green. 21 Locate the Legends section. In the table, enter the following settings: LEGENDS
Y
22 Right-click Point Graph 2 and choose Duplicate. 23 Go to the Settings window for Point Graph. 24 Locate the y-Axis Data section. In the Expression edit field, type w. 25 Locate the Coloring and Style section. Find the Line style subsection. From the Color
list, choose Red.
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VAR IOUS ANALYSES O F AN ELBOW BR ACKET
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2a
26 Locate the Legends section. In the table, enter the following settings: LEGENDS
Z
27 Click the Plot button. 28 In the Model Builder window, right-click 1D Plot Group 12 and choose Rename. 29 Go to the Rename 1D Plot Group dialog box and type Parametric Response Graphs in the New name edit field.
30 Click OK.
Linear Buckling Analysis A structure under compression can sometimes become unstable due to buckling. The critical buckling load can be estimated using a Linear Buckling Analysis. To perform this analysis, you first run a stress analysis with an arbitrary load level. In a second study step, the buckling load is computed as a scale factor with respect to the load used in the first analysis.
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Results and Discussion The computed eigenvalue is 103.1. Because the load applied in the stationary step was 1 kN, the estimated buckling load is 103.1 kN. The shape of the buckling mode is shown below.
Figure 14: Buckling mode shape. It can be noted that the stresses caused by the pre-load are so large that for in this structure a plastic collapse would occur long before the buckling load was reached.
Notes About the COMSOL Implementation When a Linear Buckling study is selected, both study steps are automatically prepared. It is only necessary to define the reference load.
Modeling Instructions MODEL WIZARD
1 In the Model Builder window, right-click the root node and choose Add Study. 2 Go to the Model Wizard window. 3 Find the Studies subsection. In the tree, select Preset Studies>Linear Buckling. 4 Click Finish.
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VAR IOUS ANALYSES O F AN ELBOW BR ACKET
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Solved with COMSOL Multiphysics 4.2a
SOLID MECHANICS
Next, set up the buckling pre-load, beginning by disabling the parametric load.
Parametric Load In the Model Builder window, right-click Parametric Load and choose Disable.
Boundary Load 6 1 Right-click Solid Mechanics and choose Boundary Load. 2 In the Model Builder window, right-click Boundary Load 6 and choose Rename. 3 Go to the Rename Boundary Load dialog box and type Buckling Pre-Load in the New name edit field. 4 Click OK. 5 Select Boundary 29 only. 6 Go to the Settings window for Boundary Load. 7 Locate the Force section. From the Load type list, choose Total force. 8 Specify the Ftot vector as 0
X
1[kN]
Y
0
Z
STUDY 9
1 In the Model Builder window, right-click Study 9 and choose Rename. 2 Go to the Rename Study dialog box and type Study 9 (Linear Buckling) in the New name edit field. 3 Click OK. STUDY 9 (LINEAR BUCKLING)
Solver 9 1 In the Model Builder window, right-click Study 9 (Linear Buckling) and choose Compute. 2 Expand the Study 9 (Linear Buckling)>Solver Configurations node. 3 Right-click Study 9 (Linear Buckling)>Solver Configurations>Solver 9 and choose Rename. 4 Go to the Rename Solver dialog box and type Linear Buckling Sequence in the New name edit field.
©2011 COMSOL
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5 Click OK. RESULTS
Data Sets 1 In the Model Builder window, right-click Results>Data Sets>Solution 9 and choose Rename. 2 Go to the Rename Solution dialog box and type Linear Buckling Solution in the New name edit field. 3 Click OK. 4 Right-click Solution 10 and choose Rename. 5 Go to the Rename Solution dialog box and type Linear Buckling Pre-Load Solution in the New name edit field.
6 Click OK.
Mode Shape (solid) With the following steps you can reproduce the plot in Figure 14: 1 In the Model Builder window, expand the Mode Shape (solid) node, then click Surface 1. 2 Go to the Settings window for Surface. 3 Locate the Coloring and Style section. Clear the Color legend check box. 4 Click the Plot button. 5 In the Model Builder window, right-click Mode Shape (solid) and choose Rename. 6 Go to the Rename 3D Plot Group dialog box and type Buckling Shape Plot in the New name edit field. 7 Click OK.
Export 1 In the Model Builder window, right-click Results>Export>Player 2 and choose Rename. 2 Go to the Rename Player dialog box and type Time-Dependent Stress Contour
in the New name edit field. 3 Click OK.
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VAR IOUS ANALYSES O F AN ELBOW BR ACKET
©2011 COMSOL
Solved with COMSOL Multiphysics 4.0a. © COPYRIGHT 2010 COMSOL AB.
Tube Connection Introduction A tube connection consisting of a flange with four prestressed bolts is subjected to tensile forces. A sketch of the connection is shown below.
Model Definition The tube is made of steel and has an outer diameter of 220 millimeters and an inner diameter of 200 millimeters. The flange has a diameter of 360 millimeters and it is 30 millimeters thick. The connection consists of four prestressed M24 bolts. The bolts are prestressed to 80% of the yield strength. The tensile force in the tube varies from 0 to 500 kN. To compute the influence of the tensile force on the stress level in the bolt, the model includes a parametric analysis. Because of symmetry in both load and geometry, you only need to analyze one eighth of one of the flanges. The geometry has been created in the CAD software SolidWorks and is available as an IGES file. Two contact regions are modeled. One contact pair acts between the bottom surface of the flange and the top surface of an additional fixed solid which supplies the symmetry condition with respect to contact. The other contact pair acts between the washer under the bolt head and the flange.
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Note: This model requires both the Structural Mechanics Module and the CAD Import Module.
Results and Discussion After the pretension step there is a tensile stress in the bolt, and compressive stress in the flange under the bolt. This is illustrated in Figure 1.
Figure 1: The axial stress after the pretension step. The general stress state at maximum external load is shown in Figure 2. In addition to the stress that has developed in the fillet between tube and flange, there are some interesting features. The stress in the bolt has increased significantly and is no longer
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constant over the cross section. Furthermore a stress of the order 300 MPa has developed at the outer edge of the flange.
Figure 2: Effective stress at maximum external load. The applied external load in this example severely overloads the bolted joint. Up to about half the full load it works fairly well, but then the contact between the two mating flanges start to open up, so that the external force is transmitted directly by the bolts. This is displayed in Figure 3. Actually, the conditions are even worse than the average force indicates. The bolt is subjected to bending with a non-uniform stress distribution over the cross section. The maximum stress is well above the yield limit, and is approaching the ultimate stress. The development of the axial stress in two points on opposite sides of the bolt is displayed in Figure 4. The points are located in the x-z symmetry plane. One point is as close to the tube centerline as possible and the other as far out as possible.
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Figure 3: The bolt force as a function of the tensile force.
Figure 4: The development of the bolt stress at two different positions in the cross section.
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The plots of the contact pressure between the mating flanges shown in Figure 5 and Figure 6 are informative. In the first plot the contact pressure after bolt pretensioning is shown. It can be seen that there are too few bolts, since almost no contact pressure is established over a large part of the flange. When the full external load is applied, the flanges are separating even directly under the bolts and the contact region has moved outwards. This explains the high stress seen at the outer edge of the flange.
Figure 5: Contact pressure between flanges after pretensioning the bolts.
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Figure 6: Contact pressure between the flanges at full external load.
Notes About the COMSOL Implementation The analysis is performed in two steps, represented by two studies. In the first step, the effects of pretensioning the bolt are computed, and in the second step the external load on the tube is applied as a parametric sweep. The pretensioning of the bolt is modelled using an auxiliary variable, representing a prestrain on a piece of the bolt. The level of this prestrain is determined by an extra equation, stating that the prescribed pretension force must equal the integral of the axial stress over a section through the bolt. In the following step when the external load is applied, the prestrain is kept constant, while the axial force is free to adapt to the changes in external load.
Model Library path: Structural_Mechanics_Module/Contact_and_Friction/ tube_connection
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Modeling Instructions MODEL WIZARD
1 Go to the Model Wizard window. 2 Click Next. 3 In the Add Physics tree, select Structural Mechanics>Solid Mechanics (solid). 4 Click Add Selected. 5 In the Add Physics tree, select Mathematics>ODEs and DAEs (ge). 6 Click Add Selected. 7 Click Next. 8 In the Studies tree, select Preset Studies for Selected Physics>Stationary. 9 Click Finish. GLOBAL DEFINITIONS
Parameters 1 In the Model Builder window, right-click Global Definitions and choose Parameters. 2 Go to the Settings window for Parameters. 3 Locate the Parameters section. In the Parameters table, enter the following settings: NAME
EXPRESSION
DESCRIPTION
A_tube
pi/ 4*(0.22^2-0.2^2)[m^ 2]
Tube area
S_pre
800[MPa]*0.8*0.8
Prestress in the bolt (class 8.8)
Di_wash
25[mm]
Washer inner diameter
Do_wash
45[mm]
Washer outer diameter
T_wash
4[mm]
Washer thickness
Do_bolthead
38[mm]
Bolthead diameter
H_bolthead
15[mm]
Bolthead height
As_bolt
353[mm^2]
Stress area of bolt
Ds_bolt
sqrt(4/pi*As_bolt)
Effective bolt diameter
F_pre
S_pre*As_bolt
Pretension force
R_bc
150[mm]
Bolt circle radius
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NAME
EXPRESSION
F_tube
500[kN]
DESCRIPTION
Axial force in tube
loadpar
0
Tube load multiplier
GEOMETRY 1
Import 1 1 In the Model Builder window, right-click Model 1>Geometry 1 and choose Import. 2 Go to the Settings window for Import. 3 Locate the Import section. Click the Browse button. 4 Browse to the model’s Model Library folder and double-click the file tube_connection.igs.
5 Click the Import button. 6 Click the Build All button. 7 Click the Zoom Extents button on the Graphics toolbar.
Cut out 1/8 of the structure to make use of the repetitive symmetry.
Work Plane 1 1 In the Model Builder window, right-click Geometry 1 and choose Work Plane. 2 Click the Zoom Extents button on the Graphics toolbar.
Rectangle 1 1 In the Model Builder window, right-click Work Plane 1>Geometry and choose Rectangle. 2 Go to the Settings window for Rectangle. 3 Locate the Position section. In the y edit field, type -0.1. 4 Locate the Size section. In the Width edit field, type 0.2. 5 In the Height edit field, type 0.3. 6 Click the Build All button.
Revolve 1 1 In the Model Builder window, right-click Work Plane 1 and choose Revolve. 2 Go to the Settings window for Revolve. 3 Locate the Revolution Angles section. In the End angle edit field, type 45. 4 Click the Build All button.
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Intersection 1 1 In the Model Builder window, right-click Geometry 1 and choose Boolean Operations>Intersection. 2 Select the objects imp1 and rev1 only. 3 Click the Build All button. 4 Click the Zoom Extents button on the Graphics toolbar.
Next, create the bolt.
Work Plane 2 1 In the Model Builder window, right-click Geometry 1 and choose Work Plane. 2 Click the Zoom Extents button on the Graphics toolbar.
Bézier Polygon 1 1 In the Model Builder window, right-click Work Plane 1>Geometry and choose Bézier Polygon. 2 Go to the Settings window for Bézier Polygon. 3 Locate the Polygon Segments section. Click the Add Linear button. 4 Find the Control points subsection. In row 1, set x to R_bc. 5 In row 1, set y to -0.03. 6 In row 2, set x to R_bc+Ds_bolt/2 and y to -0.03. 7 Click the Add Linear button. 8 In row 2, set y to T_wash. 9 Click the Add Linear button. 10 In row 2, set x to R_bc+Di_wash/2. 11 Click the Add Linear button. 12 In row 2, set y to 0. 13 Click the Add Linear button. 14 In row 2, set x to R_bc+Do_wash/2. 15 Click the Add Linear button. 16 In row 2, set y to T_wash. 17 Click the Add Linear button. 18 In row 2, set x to R_bc+Do_bolthead/2. 19 Click the Add Linear button. 20 In row 2, set y to T_wash+H_bolthead.
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21 Click the Add Linear button. 22 In row 2, set x to R_bc. 23 Click the Add Linear button. 24 Click the Close Curve button. 25 Click the Build All button.
Add a region for prestrain application. The size is not important, but it is reasonable to make it of same order as the bolt diameter, in order not to impose any unnecessary constraints on the mesh size.
Rectangle 1 1 In the Model Builder window, right-click Work Plane 2>Geometry and choose Rectangle. 2 Go to the Settings window for Rectangle. 3 Locate the Position section. In the x edit field, type R_bc. 4 In the y edit field, type -0.03. 5 Locate the Size section. In the Width edit field, type Ds_bolt/2. 6 In the Height edit field, type 10[mm]. 7 Click the Build All button.
Copy 1 1 In the Model Builder window, right-click Work Plane 2>Geometry and choose Transforms>Copy. 2 Select the object r1 only. 3 Go to the Settings window for Copy. 4 Locate the Displacement section. In the y edit field, type 0.01. 5 Click the Build All button.
Union 1 1 In the Model Builder window, right-click Work Plane 2>Geometry and choose Boolean Operations>Union. 2 Select the objects r1, copy1, and b1 only. 3 Click the Build All button.
Revolve 2 1 In the Model Builder window, right-click 1>Work Plane 2 and choose Revolve. 2 Go to the Settings window for Revolve.
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3 Locate the Revolution Angles section. In the End angle edit field, type 180. 4 Locate the Point on the Revolution Axis section. In the x edit field, type R_bc. 5 Click the Build All button.
Form Union 1 In the Model Builder window, click Form Union. 2 Go to the Settings window for Finalize. 3 Locate the Finalize section. From the Finalization method list, select Form an assembly. 4 Click the Build All button.
Block 1 1 In the Model Builder window, right-click Geometry 1 and choose Block. 2 Go to the Settings window for Block. 3 Locate the Size and Shape section. In the Width edit field, type 0.15. 4 In the Depth edit field, type 0.01. 5 In the Height edit field, type 0.15. 6 Locate the Position section. In the x edit field, type 0.05. 7 In the y edit field, type -0.04. 8 In the z edit field, type -0.14. 9 Click the Build All button. 10 Click in the Graphics window, Press Ctrl+A, and then right-click to select all objects. 11 Click the Zoom Selected button on the Graphics toolbar.
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This completes the geometry modeling stage. The geometry should now look like that in the figure below.
DEFINITIONS
1 In the Model Builder window, right-click Model 1>Definitions and choose Contact Pair. 2 Select Boundary 5 only. 3 Go to the Settings window for Contact Pair. 4 In the upper-right corner of the Destination Boundaries section, click Activate Selection. 5 Select Boundary 9 only. 6 In the Model Builder window, right-click Definitions and choose Contact Pair. 7 Select Boundary 13 only. 8 Go to the Settings window for Contact Pair. 9 In the upper-right corner of the Destination Boundaries section, click Activate Selection. 10 Select Boundaries 20 and 43 only.
Set up an integration through the bolt, so that the axial force can be computed from the stress.
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Integration 1 1 In the Model Builder window, right-click Definitions and choose Model Couplings>Integration. 2 Go to the Settings window for Integration. 3 Locate the Operator Name section. In the Operator name edit field, type IntSec. 4 Locate the Source Selection section. From the Geometric entity level list, select Boundary. 5 Select Boundaries 33 and 40 only.
Create a variable for the current bolt force, taking the symmetry into account.
Variables 1 1 In the Model Builder window, right-click Definitions and choose Variables. 2 Go to the Settings window for Variables. 3 Locate the Variables section. In the Variables table, enter the following settings: NAME
EXPRESSION
DESCRIPTION
F_bolt
2*IntSec(solid.sy)
Bolt force
MATERIALS
1 In the Model Builder window, right-click Model 1>Materials and choose Open Material Browser. 2 Go to the Material Browser window. 3 Locate the Materials section. In the Materials tree, select Built-In>Structural steel. 4 Right-click and choose Add Material to Model from the menu. SOLID MECHANICS
Fixed Constraint 1 1 In the Model Builder window, right-click Model 1>Solid Mechanics and choose More>Fixed Constraint. 2 Select Domain 1 only.
Symmetry 1 1 In the Model Builder window, right-click Solid Mechanics and choose Symmetry.
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2 Select Boundaries 8, 14, 17, 18, 27, 28, 30, 34, 36, 39, and 42 only.
You can do this by first copying the text “8, 14, 17, 18, 27, 28, 30, 34, 36, 39, and 42” and then clicking the Paste Selection button next to the Selection box or clicking in the box and pressing Ctrl+V.
Contact 1 1 In the Model Builder window, right-click Solid Mechanics and choose Pairs>Contact. 2 Go to the Settings window for Contact. 3 Locate the Pair Selection section. In the Pairs list, select Contact Pair 1. 4 Locate the Normal Contact section. In the pn edit field, type min(1e-3*5^segiter,1)*solid.E/h*50.
5 Locate the Initial Values section. In the Tn edit field, type S_pre/30.
Contact 2 1 In the Model Builder window, right-click Solid Mechanics and choose Pairs>Contact. 2 Go to the Settings window for Contact. 3 Locate the Pair Selection section. In the Pairs list, select Contact Pair 2. 4 Locate the Normal Contact section. In the pn edit field, type min(1e-3*5^segiter,1)*solid.E/h*50.
5 Locate the Initial Values section. In the Tn edit field, type S_pre/3.
Boundary Load 1 1 In the Model Builder window, right-click Solid Mechanics and choose Boundary Load. 2 Select Boundary 10 only. 3 Go to the Settings window for Boundary Load. 4 Locate the Force section. Specify the F vector as 0
X
loadpar*F_tube/A_tube
Y
0
Z
ODES AND DAES
1 In the Model Builder window, click Model 1>ODEs and DAEs. 2 Go to the Settings window for ODEs and DAEs. 3 Locate the Interface Identifier section. In the Identifier edit field, type prestrain.
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Global Equations 1 1 In the Model Builder window, expand the ODEs and DAEs node, then click Global Equations 1. 2 Go to the Settings window for Global Equations. 3 Locate the Global Equations section. In the table, enter the following settings: NAME
EQUATION F(U,UT,UTT,T)
INITIAL VALUE (U)
eps_p
F_pre-F_bolt
-S_pre/2E11[Pa]*0.75*7
A reasonable initial guess for the prestrain is based on that 75% of the deformation occurs in the bolt, and that the region you are using for prestrain here is roughly 1/ 7 of the total bolt length. SOLID MECHANICS
Initial Stress and Strain 1 1 In the Model Builder window, right-click Linear Elastic Material Model 1 and choose Initial Stress and Strain. 2 Go to the Settings window for Initial Stress and Strain. 3 Locate the Domains section. Click Clear Selection. 4 Select Domain 5 only. 5 Locate the Initial Stress and Strain section. In the ε0 table, enter the following
settings: 0
0
0
0
eps_p
0
0
0
0
MESH 1
Mapped 1 1 In the Model Builder window, right-click Model 1>Mesh 1 and choose More Operations>Mapped. 2 Select Boundary 5 only.
Size 1 1 Right-click Mapped 1 and choose Size. 2 Go to the Settings window for Size.
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3 Locate the Element Size section. From the Predefined list, select Coarse. 4 Click the Build All button.
Swept 1 1 In the Model Builder window, right-click Mesh 1 and choose Swept. 2 Go to the Settings window for Swept. 3 Locate the Domains section. From the Geometric entity level list, select Domain. 4 Select Domain 1 only.
Distribution 1 1 Right-click Swept 1 and choose Distribution. 2 Go to the Settings window for Distribution. 3 Locate the Distribution section. In the Number of elements edit field, type 1. 4 Click the Build All button.
Free Tetrahedral 1 1 In the Model Builder window, right-click Mesh 1 and choose Free Tetrahedral. 2 Go to the Settings window for Free Tetrahedral. 3 Locate the Domains section. From the Geometric entity level list, select Domain. 4 Select Domain 2 only. 5 Click the Build All button.
Edge 1 1 In the Model Builder window, right-click Mesh 1 and choose More Operations>Edge. 2 Select Edge 98 only.
Distribution 1 1 Right-click Edge 1 and choose Distribution. 2 Select Edge 98 only. 3 Go to the Settings window for Distribution. 4 Locate the Distribution section. In the Number of elements edit field, type 3.
Swept 2 1 In the Model Builder window, right-click Mesh 1 and choose Swept. 2 Go to the Settings window for Swept. 3 In the upper-right corner of the Source Faces section, click Activate Selection. 4 Click to expand the Source Faces section.
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5 Select Boundaries 36, 39, and 42 only. 6 Click to expand the Destination Faces section. 7 In the upper-right corner of the Destination Faces section, click Activate Selection. 8 Select Boundaries 18, 27, and 30 only.
Size 1 1 Right-click Swept 2 and choose Size. 2 Go to the Settings window for Size. 3 Locate the Element Size section. From the Predefined list, select Extremely fine.
Distribution 1 1 In the Model Builder window, right-click Swept 2 and choose Distribution. 2 Go to the Settings window for Distribution. 3 Locate the Distribution section. In the Number of elements edit field, type 8. 4 Click the Build All button. STUDY 1
1 In the Model Builder window, right-click Study 1 and choose Show Default Solver. 2 Expand the Study 1>Solver Configurations node.
Solver 1 1 In the Model Builder window, expand the Solver Configurations>Solver 1 node, then
click Dependent Variables 1. 2 Go to the Settings window for Dependent Variables. 3 Locate the Scaling section. From the Method list, select Manual. 4 In the Scale edit field, type 1e8. 5 In the Model Builder window, expand the Dependent Variables 1 node, then click mod1_u. 6 Go to the Settings window for Field. 7 Locate the Scaling section. From the Method list, select Manual. 8 In the Scale edit field, type 1e-5. 9 In the Model Builder window, click Dependent Variables 1>mod1_ODE1. 10 Go to the Settings window for State. 11 Locate the Scaling section. From the Method list, select Manual.
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12 In the Scale edit field, type 1e-2.
The default solver sequence segregates the ODE variable from the displacement variables, which is not what you want to do. Modify the segregated groups as follows: 13 In the Model Builder window, expand the Stationary Solver 1>Segregated 1 node. 14 Right-click Stationary Solver 1>Segregated 1>Segregated Step 1 and choose Delete. 15 In the Confirm Delete dialog box, click Yes. 16 Go to the Settings window for Segregated Step. 17 Locate the General section. Under Variables, click Add. 18 Go to the Add dialog box. 19 In the Variables list, select mod1_ODE1. 20 Click the OK button. 21 In the Model Builder window, right-click Stationary Solver 1>Segregated 1>Lumped Step 1 and choose Delete. 22 In the Confirm Delete dialog box, click Yes. 23 In the Model Builder window, right-click Stationary Solver 1>Segregated 1 and choose Segregated Step. 24 Go to the Settings window for Segregated Step. 25 Locate the General section. Under Variables, click Add. 26 Go to the Add dialog box. 27 In the Variables list, select mod1_solid_Tn_p1 and mod1_solid_Tn_p2. 28 Click the OK button. 29 In the Model Builder window, right-click Study 1 and choose Compute. RESULTS
Reproduce the plot in Figure 1 with the following steps:
3D Plot Group 1 1 In the Model Builder window, expand the Results>3D Plot Group 1 node, then click Surface 1. 2 Go to the Settings window for Surface. 3 In the upper-right corner of the Expression section, click Replace Expression. 4 From the menu, choose Solid Mechanics>Stress tensor>Stress tensor y (solid.sy). 5 Click to expand the Range section.
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6 Select the Manual color range check box. 7 In the Minimum edit field, type -3e8. 8 In the Maximum edit field, type 5e8. 9 Click the Plot button. MODEL WIZARD
1 In the Model Builder window, right-click the root node and choose Add Study. 2 Go to the Model Wizard window. 3 In the Studies tree, select Preset Studies for Selected Physics>Stationary. 4 Click Finish. STUDY 2
In the Model Builder window, right-click Study 2 and choose Show Default Solver.
Solver 2 1 In the Model Builder window, expand the Study 2>Solver Configurations>Solver 2
node, then click Dependent Variables 1. 2 Go to the Settings window for Dependent Variables. 3 Locate the Initial Values section. From the Method list, select Solution. 4 From the Solution list, select Solver 1. 5 Locate the Scaling section. From the Method list, select Manual. 6 In the Scale edit field, type 1e8. 7 Locate the Variables Not Solved For section. From the Method list, select Solution. 8 From the Solution list, select Solver 1. 9 In the Model Builder window, expand the Dependent Variables 1 node, then click mod1_u. 10 Go to the Settings window for Field. 11 Locate the Scaling section. From the Method list, select Manual. 12 In the Scale edit field, type 1e-5.
The pre-tension strain is to be kept constant during the application of the external load, so it should not be solved for. 13 In the Model Builder window, click Dependent Variables 1>mod1_ODE1. 14 Go to the Settings window for State. 15 Locate the General section. Clear the Solve for this state check box.
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16 In the Model Builder window, expand the Stationary Solver 1>Segregated 1 node. 17 Right-click Stationary Solver 1>Segregated 1>Segregated Step 1 and choose Disable. 18 Right-click Stationary Solver 1>Segregated 1>Lumped Step 1 and choose Delete. 19 In the Confirm Delete dialog box, click Yes. 20 In the Model Builder window, right-click Stationary Solver 1>Segregated 1 and choose Segregated Step. 21 Go to the Settings window for Segregated Step. 22 Locate the General section. Under Variables, click Add. 23 Go to the Add dialog box. 24 In the Variables list, select mod1_solid_Tn_p1 and mod1_solid_Tn_p2. 25 Click the OK button. 26 In the Model Builder window, right-click Stationary Solver 1 and choose Parametric. 27 Go to the Settings window for Parametric. 28 Locate the General section. In the Parameter names edit field, type loadpar. 29 In the Parameter values edit field, type range(0,0.1,1).
Because the contact is well established, the contact iterations do not need the smooth ramping of the penalty factor. SOLID MECHANICS
Contact 1 1 In the Model Builder window, click Contact 1. 2 Go to the Settings window for Contact. 3 Locate the Normal Contact section. In the pn edit field, type solid.E/ solid.hmin_dst*50.
Contact 2 1 In the Model Builder window, click Contact 2. 2 Go to the Settings window for Contact. 3 Locate the Normal Contact section. In the pn edit field, type solid.E/ solid.hmin_dst*50. STUDY 2
In the Model Builder window, right-click Study 2 and choose Compute.
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RESULTS
The following steps reproduce the plot in Figure 2:
3D Plot Group 3 1 In the Model Builder window, expand the Results>3D Plot Group 3 node, then click Surface 1. 2 Go to the Settings window for Surface. 3 In the upper-right corner of the Expression section, click Replace Expression. 4 From the menu, choose Solid Mechanics>von Mises stress (solid.mises). 5 Go to the Settings window for Surface. 6 Locate the Range section. Select the Manual color range check box. 7 In the Maximum edit field, type 5e8. 8 Click the Plot button.
Proceed to plot the bolt force as a function of the tensile force as in Figure 3.
1D Plot Group 5 1 In the Model Builder window, right-click Results and choose 1D Plot Group. 2 Go to the Settings window for 1D Plot Group. 3 Locate the Data section. From the Data set list, select Solution 2. 4 Right-click Results>1D Plot Group 5 and choose Global. 5 Go to the Settings window for Global. 6 Locate the Expressions section. In the Expressions table, enter the following settings: EXPRESSION
F_bolt/1000
7 Click the Plot button. 8 Click to expand the Legends section. 9 Clear the Show legends check box. 10 Click to expand the Coloring and Style section. 11 Find the Line style subsection. In the Width edit field, type 2. 12 Locate the X-Axis Data section. From the Parameter list, select Expression. 13 In the Expression edit field, type 500*loadpar. 14 In the Model Builder window, click 1D Plot Group 5. 15 Go to the Settings window for 1D Plot Group.
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16 Locate the Plot Settings section. In the x-axis label edit field, type Axial force on tube (kN).
17 In the y-axis label edit field, type Axial force in bolt (kN). 18 Click the Plot button.
The following steps create the plot in Figure 4:
1D Plot Group 6 1 In the Model Builder window, right-click Results and choose 1D Plot Group. 2 Go to the Settings window for 1D Plot Group. 3 Locate the Data section. From the Data set list, select Solution 2. 4 Right-click Results>1D Plot Group 6 and choose Point Graph. 5 Select Vertices 33 and 51 only. 6 Go to the Settings window for Point Graph. 7 In the upper-right corner of the Expression section, click Replace Expression. 8 From the menu, choose Solid Mechanics>Stress tensor>Stress tensor y (solid.sy). 9 Locate the Expression section. From the Unit list, select MPa. 10 Locate the X-Axis Data section. From the Parameter list, select Expression. 11 In the Expression edit field, type 500*loadpar. 12 Click to expand the Coloring and Style section. 13 Find the Line style subsection. From the Line list, select Cycle. 14 In the Width edit field, type 2. 15 Click to expand the Legends section. 16 Select the Show legends check box. 17 From the Legends list, select Manual. 18 In the table, enter the following settings: LEGENDS
Outside Inside
19 In the Model Builder window, click 1D Plot Group 6. 20 Go to the Settings window for 1D Plot Group. 21 Locate the Plot Settings section. In the x-axis label edit field, type Axial force on tube (kN).
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22 In the y-axis label edit field, type Bolt stress (MPa). 23 Click the Plot button.
Finally, reproduce the contact pressure plots shown in Figure 5 and Figure 6:
3D Plot Group 7 1 In the Model Builder window, right-click Results and choose 3D Plot Group. 2 Right-click Results>3D Plot Group 7 and choose Surface. 3 Go to the Settings window for Surface. 4 In the upper-right corner of the Expression section, click Contact pressure, contact pair p1 (solid.Tn_p1). 5 Locate the Range section. Select the Manual color range check box. 6 In the Maximum edit field, type 1e8. 7 Click the Plot button. 8 Locate the Data section. From the Data set list, select Solution 2. 9 Click the Plot button.
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Tutorial: 6
A L E Flui d- S t r uc t u re In t erac t i on Introduction The following example demonstrates techniques for modeling fluid-structure interactions in COMSOL Multiphysics. It illustrates how fluid flow can deform structures and how to solve for the flow in a continuously deforming geometry using the arbitrary Lagrangian-Eulerian (ALE) technique. The model geometry consists of a horizontal flow channel in the middle of which is an obstacle, a narrow vertical structure (Figure 1). The fluid flows from left to right, except where the obstacle forces it into a narrow path in the upper part of the channel, and it imposes a force on the structure’s walls resulting from the viscous drag and fluid pressure. The structure, being made of a deformable material, bends under the applied load. Consequently, the fluid flow also follows a new path, so solving the flow in the original geometry would generate incorrect results. The ALE method handles the dynamics of the deforming geometry and the moving boundaries with a moving grid. COMSOL Multiphysics computes new mesh coordinates on the channel area based on the movement of the structure’s boundaries and mesh smoothing. The Navier-Stokes equations that solve the flow are formulated for these moving coordinates. The structural mechanics portion of the model does not require the ALE method, and COMSOL Multiphysics solves it in a fixed coordinate system as usual. However, the strains the model computes in this way are the only source for computing the deformed coordinates with ALE.
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Figure 1: Fluid flows into this horizontal flow channel from the left, and it enters with a parabolic velocity profile. A narrow vertical structure in the channel (the straight vertical structure) forces the flow into a narrow path. Due to fluid pressure and viscous drag, the originally vertical structure bends. This simulation models the fluid flow in a deformed, moving mesh that follows the movement of the bending structure.
Model Definition In this example the flow channel is 100 μm high and 300 μm long. The vertical structure—5 μm wide, 50 μm high, and with a semicircular top—sits 100 μm away from the channel’s left boundary. Assume that the structure is long in the direction perpendicular to the image. The fluid is a water-like substance with a density ρ = 1000 kg/m3 and dynamic viscosity η = 0.001 Pa·s. To demonstrate the desired techniques, assume the structure consists of a flexible material with a density ρ = 7850 kg/m3 and Young’s modulus E = 200 kPa.
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FLUID FLOW
The fluid flow in the channel is described by the incompressible Navier-Stokes equations for the velocity field, u = (u, v), and the pressure, p, in the spatial (deformed) moving coordinate system: ρ
∂u T – ∇ ⋅ [ – p I + η ( ∇u + ( ∇u ) ) ] + ρ ( ( u – um ) ⋅ ∇ )u = F ∂t
(1)
–∇ ⋅ u = 0 In these equations, I denotes the unit diagonal matrix and F is the volume force affecting the fluid. Assume that no gravitation or other volume forces affect the fluid, so that F = 0. The coordinate system velocity is um = (um, vm). At the channel entrance on the left, the flow has fully developed laminar characteristics with a parabolic velocity profile but its amplitude changes with time. At first flow increases rapidly, reaching its peak value at 0.215 s; thereafter it gradually decreases to a steady-state value of 5 cm/s. The centerline velocity in the x direction, uin (see Figure 4), with the steady-state amplitude U comes from the equation 2
U⋅t u in = ----------------------------------------------------------2 2 2 ( 0.04 – t ) + ( 0.1t )
(2)
where t must be expressed in seconds. At the outflow (right-hand boundary), the condition is p = 0. On the solid (nondeforming) walls, no-slip conditions are imposed, u = 0, v = 0, while on the deforming interface the velocities equal the deformation rate, u0 = ut and v0 = vt (the default condition; note that u and v on the right-hand sides refer to the displacement components). STRUCTURAL MECHANICS
The structural deformations are solved for using an elastic formulation and a nonlinear geometry formulation to allow large deformations. The obstacle is fixed to the bottom of the fluid channel. All other object boundaries experience a load from the fluid, given by T
F T = – n ⋅ ( – p I + η ( ∇u + ( ∇u ) ) )
(3)
where n is the normal vector to the boundary. This load represents a sum of pressure and viscous forces.
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MOVING MESH
The Navier-Stokes equations are solved on a freely moving deformed mesh, which constitutes the fluid domain. The deformation of this mesh relative to the initial shape of the domain is computed using Winslow smoothing. This is the default smoothing when using the Fluid-Structure Interaction interface. For more information, please refer to the chapter “The Fluid-Structure Interaction Interface” in the MEMS Module User’s Guide. Inside the obstacle, the moving mesh follows the deformations of the obstacle. At the exterior boundaries of the flow domain, the deformation is set to zero in all directions.
Results and Discussion Figure 2 shows the geometry deformation and flow at t = 4 s when the system is close to its steady state. Due to the channel’s small dimensions, the Reynolds number of the flow is small (Re Fluid-Structure Interaction (fsi). 5 Click Add Selected. 6 Click Next.
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7 In the Studies tree, select Preset Studies>Time Dependent. 8 Click Finish. GLOBAL DEFINITIONS
Parameters 1 In the Model Builder window, right-click Global Definitions and choose Parameters. 2 Go to the Settings window for Parameters. 3 Locate the Parameters section. In the Parameters table, enter the following settings: NAME
EXPRESSION
DESCRIPTION
U
3.33[cm/s]
Inlet mean velocity at steady state
Variables 1 1 In the Model Builder window, right-click Global Definitions and choose Variables. 2 Go to the Settings window for Variables. 3 Locate the Variables section. In the Variables table, enter the following settings: NAME
EXPRESSION
DESCRIPTION
u_in
U*t^2/ sqrt(t^4-0.07[s^2]*t^2+ 0.0016[s^4])
Inlet mean velocity
GEOMETRY 1
1 In the Model Builder window, click Model 1>Geometry 1. 2 Go to the Settings window for Geometry. 3 Locate the Geometry Settings section. Find the Units subsection. From the Length unit list, select µm.
Rectangle 1 1 Right-click Model 1>Geometry 1 and choose Rectangle. 2 Go to the Settings window for Rectangle. 3 Locate the Size section. In the Width edit field, type 300. 4 In the Height edit field, type 100. 5 Click the Build All button.
Rectangle 2 1 In the Model Builder window, right-click Geometry 1 and choose Rectangle.
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2 Go to the Settings window for Rectangle. 3 Locate the Size section. In the Width edit field, type 5. 4 In the Height edit field, type 47.5. 5 Locate the Position section. In the x edit field, type 100. 6 Click the Build All button.
Fillet 1 1 In the Model Builder window, right-click Geometry 1 and choose Fillet. 2 On the object r2, select Vertices 3 and 4 only. 3 Go to the Settings window for Fillet. 4 Locate the Radius section. In the Radius edit field, type 2.5. 5 Click the Build All button. The geometry should look like that in the figure below.
FLUID-STRUCTURE INTERACTION
By default the Fluid-Structure Interaction interface treats all domains as fluid. Add the appropriate domain selection to the default node for the solid domain and proceed to the material specification.
Linear Elastic Material Model 1 1 In the Model Builder window, expand the Model 1>Fluid-Structure Interaction node,
then click Linear Elastic Material Model 1.
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2 Select Domain 2 only. MATERIALS
Material 1 1 In the Model Builder window, right-click Model 1>Materials and choose Material. 2 Select Domain 1 only. 3 Go to the Settings window for Material. 4 Locate the Material Contents section. In the Material Contents table, enter the
following settings: PROPERTY
NAME
VALUE
Density
rho
1e3
Dynamic viscosity
mu
1e-3
Material 2 1 In the Model Builder window, right-click Materials and choose Material. 2 Select Domain 2 only. 3 Go to the Settings window for Material. 4 Locate the Material Contents section. In the Material Contents table, enter the
following settings: PROPERTY
NAME
VALUE
Young's modulus
E
2e5
Poisson's ratio
nu
0.33
Density
rho
7850
FLUID-STRUCTURE INTERACTION
1 In the Model Builder window, click Model 1>Fluid-Structure Interaction. 2 Go to the Settings window for Fluid-Structure Interaction. 3 Locate the Physical Model section. From the Compressibility list, select Incompressible flow.
Inlet 1 1 In the Model Builder window, right-click Fluid-Structure Interaction and choose the
boundary condition Laminar Flow>Inlet. 2 Select Boundary 1 only.
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3 Go to the Settings window for Inlet. 4 Locate the Boundary Condition section. From the Boundary condition list, select Laminar inflow. 5 In the Uav edit field, type u_in.
Outlet 1 1 In the Model Builder window, right-click Fluid-Structure Interaction and choose the
boundary condition Laminar Flow>Outlet. 2 Select Boundary 8 only.
Fixed Constraint 1 1 In the Model Builder window, right-click Fluid-Structure Interaction and choose the
boundary condition Solid Mechanics>Fixed Constraint. 2 Select Boundary 5 only. MESH 1
Free Triangular 1 1 In the Model Builder window, right-click Model 1>Mesh 1 and choose Free Triangular. 2 Go to the Settings window for Free Triangular. 3 Locate the Domains section. From the Geometric entity level list, select Entire geometry.
Size 1 In the Model Builder window, click Size. 2 Go to the Settings window for Size. 3 Locate the Element Size section. From the Calibrate for list, select Fluid dynamics. 4 Click the Build All button. STUDY 1
Step 1: Time Dependent 1 In the Model Builder window, click Study 1>Step 1: Time Dependent. 2 Go to the Settings window for Time Dependent. 3 Locate the Study Settings section. In the Times edit field, type range(0,0.005,1.5) range(1.7,0.02,4).
4 Select the Relative tolerance check box. 5 In the associated edit field, type 0.001.
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6 In the Model Builder window, right-click Study 1 and choose Show Default Solver.
Solver 1
You can improve the displacement scaling by changing the scale to 10−5. 1 In the Model Builder window, expand the Study 1>Solver Configurations>Solver 1>Dependent Variables 1 node, then click Dependent Variables 1>mod1_u_solid. 2 Go to the Settings window for Field. 3 Locate the Scaling section. From the Method list, select Manual. 4 In the Scale edit field, type 1.0e-5. 5 In the Model Builder window, right-click Study 1 and choose Compute. You can
ignore the non-ideal constraints related warning shown in the log. RESULTS
2D Plot Group 1 The default plot shows the pressure distribution in the fluid domain. To reproduce Figure 1, plot the velocity field. 1 Go to the Settings window for 2D Plot Group. 2 Locate the Data section. From the Time list, select 2. 3 In the Model Builder window, expand the 2D Plot Group 1 node, then click Surface 1. 4 Go to the Settings window for Surface. 5 In the upper-right corner of the Expression section, click Replace Expression. 6 From the menu, choose Fluid-Structure Interaction (Laminar Flow)>Velocity magnitude (fsi.U). 7 Locate the Coloring and Style section. Select the Wireframe check box. 8 Clear the Color legend check box. 9 Click the Plot button. 10 In the Model Builder window, right-click 2D Plot Group 1 and choose Arrow Surface. 11 Go to the Settings window for Arrow Surface. 12 In the upper-right corner of the Expression section, click Replace Expression. 13 From the menu, choose Fluid-Structure Interaction (Laminar Flow)>Velocity field (u_fluid, v_fluid). 14 Locate the Coloring and Style section. In the Scale factor edit field, type 1.5.
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2D Plot Group 2 Proceed to reproduce Figure 2, which shows flow velocity and streamlines. 1 In the Model Builder window, right-click Results and choose 2D Plot Group. 2 Go to the Settings window for 2D Plot Group. 3 Locate the Plot Settings section. From the Frame list, select Spatial (x, y, z). 4 Right-click Results>2D Plot Group 2 and choose Surface. 5 Go to the Settings window for Surface. 6 In the upper-right corner of the Expression section, click Replace Expression. 7 From the menu, choose Fluid-Structure Interaction (Laminar Flow)>Velocity magnitude (fsi.U). 8 Click the Plot button. 9 In the Model Builder window, right-click 2D Plot Group 2 and choose Streamline. 10 Go to the Settings window for Streamline. 11 In the upper-right corner of the Expression section, click Replace Expression. 12 From the menu, choose Fluid-Structure Interaction (Laminar Flow)>Velocity field (u_fluid, v_fluid). 13 Locate the Streamline Positioning section. From the Entry method list, select Coordinates. 14 In the x edit field, type 0^(range(1,15)) 125*1^(range(1,2)). 15 In the y edit field, type range(0,100/14,100) 20 5. 16 Locate the Coloring and Style section. From the Color list, select Red. 17 Click the Plot button.
Report To animate flow around the structure, do the following: 1 In the Model Builder window, right-click Results>Report and choose Player. 2 Go to the Settings window for Player. 3 Locate the Scene section. From the Subject list, select 2D Plot Group 2. 4 Locate the Parameter Sweep section. From the Select via list, select Interpolated times. 5 In the Times edit field, type range(0.025,0.025,0.5). 6 Right-click Player 1 and choose Play.
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To inspect the deformed geometry and deformed mesh near the top of the structure (Figure 5), proceed with the following steps.
Data Sets 1 In the Model Builder window, right-click Results>Data Sets and choose Solution. 2 Right-click Solution 2 and choose Add Selection. 3 Go to the Settings window for Selection. 4 Locate the Geometric Scope section. From the Geometric entity level list, select Domain. 5 Select Domain 2 only.
2D Plot Group 3 1 In the Model Builder window, right-click Results and choose 2D Plot Group. 2 Go to the Settings window for 2D Plot Group. 3 Locate the Plot Settings section. From the Frame list, select Spatial (x, y, z). 4 Locate the Data section. From the Time list, select 0. 5 Right-click Results>2D Plot Group 3 and choose Surface. 6 Go to the Settings window for Surface. 7 Locate the Expression section. In the Expression edit field, type 1. 8 Locate the Coloring and Style section. From the Coloring list, select Uniform. 9 From the Color list, select Blue. 10 Select the Wireframe check box. 11 Click to expand the Quality section. 12 Click the Plot button. 13 In the Model Builder window, right-click 2D Plot Group 3 and choose Surface. 14 Go to the Settings window for Surface. 15 Locate the Data section. From the Data set list, select Solution 2. 16 From the Time list, select 0. 17 Locate the Expression section. In the Expression edit field, type 1. 18 Locate the Coloring and Style section. From the Coloring list, select Uniform. 19 Click the Zoom Box button on the Graphics toolbar and then use the mouse to zoom
in on the obstacle. 20 In the Model Builder window, click 2D Plot Group 3. 21 Go to the Settings window for 2D Plot Group.
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22 Locate the Data section. From the Time list, select 2. 23 In the Model Builder window, click Surface 2. 24 Go to the Settings window for Surface. 25 Locate the Data section. From the Time list, select 2. 26 Click the Plot button.
Add the arrow plot, to reproduce Figure 3. 1 In the Model Builder window, click 2D Plot Group 3. 2 Go to the Settings window for 2D Plot Group. 3 Locate the Data section. From the Time list, select 0.15. 4 In the Model Builder window, click Surface 2. 5 Go to the Settings window for Surface. 6 Locate the Data section. From the Time list, select 0.15. 7 In the Model Builder window, right-click 2D Plot Group 3 and choose Arrow Surface. 8 Go to the Settings window for Arrow Surface. 9 Locate the Expression section. In the x component edit field, type xt. 10 In the y component edit field, type yt. 11 Locate the Coloring and Style section. In the Scale factor edit field, type 1.5. 12 Click the Plot button. RESULTS
Finally, plot the horizontal mesh velocity, the mesh deformation at the point beside the top of the structure, and inflow velocity (Figure 4).
1D Plot Group 4 1 In the Model Builder window, right-click Results and choose 1D Plot Group. 2 Right-click Results>1D Plot Group 4 and choose Global. 3 Go to the Settings window for Global. 4 In the upper-right corner of the Expressions section, click Replace Expression. 5 From the menu, choose Definitions>Inlet mean velocity (u_in). 6 Click the Plot button.
Data Sets 1 In the Model Builder window, right-click Results>Data Sets and choose Cut Point 2D. 2 Go to the Settings window for Cut Point 2D.
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3 Locate the Point Data section. In the x edit field, type 105. 4 In the y edit field, type 50.
1D Plot Group 4 1 In the Model Builder window, right-click Results>1D Plot Group 4 and choose Point Graph. 2 Go to the Settings window for Point Graph. 3 Locate the Data section. From the Data set list, select Cut Point 2D 1. 4 Locate the Expression section. In the Expression edit field, type xt. 5 From the Unit list, select mm/s. 6 Locate the Legends section. Select the Show legends check box. 7 From the Legends list, select Manual. 8 In the table, enter the following settings: LEGENDS
mesh velocity in the x direction (mm/s)
9 Right-click Point Graph 1 and choose Duplicate. 10 Go to the Settings window for Point Graph. 11 Locate the Expression section. In the Expression edit field, type x-X. 12 From the Unit list, select mm. 13 Locate the Legends section. In the table, enter the following settings: LEGENDS
mesh displacement in the x direction (mm)
14 Click the Plot button.
