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Nonlinear Response Topology Optimization using Equivalent Static Loads - Case Studies Abstract Purpose - Structural optimization considering the nonlinearity is fairly expensive and sensitivity analysis is difficult to calculate. The large number of nonlinear analysis is required due to the large number of design variables involved in the topology optimization. In the element density based topology optimization, many elements have the value close to 0, these are called low density elements. These low density elements create a mesh distortion that have a severe effect on the optimization results in the next cycles. The updating of finite element material with these low density elements make the problem more complex. In order to overcome these difficulties, nonlinear response optimization using equivalent loads (NROEL) primarily used for size and shape optimization is used for topology optimization. Design/Methodology/Approach - The nonlinear response optimization problem is converted to linear response optimization problem by equivalent static loads. Equivalent loads are defined as the loads for linear analysis, which generate the same response field as that of nonlinear analysis. The nonlinear analysis is performed with the given loading conditions in analysis domain and equivalent loads are calculated, then these equivalent loads are used to perform linear response topology optimization in optimization domain. The design is updated and process is repeated until the convergence criterion is satisfied. Findings - In this paper authors have presented results of five case studies with material, geometric and contact nonlinearities using a proposed modified method for topology optimization using equivalent static loads. The optimum results of the two case studies have been compared with numerical results. The results of the other case studies are compared with conventional linear response topology optimization. The optimum results are observed in good agreement providing justification of the proposed method. Originality/value - The modified equivalent loads method is an effective tool for nonlinear response topology optimization. The difficulties in the previous nonlinear topology optimization problem; mesh distortion due to the low density finite elements, the updating of the finite element material, and large number of nonlinear analysis required has been solved in this proposed method by introducing the transformation variable and new update method. Keywords Topology optimization, Equivalent loads, Nonlinear analysis, Nonlinear response optimization, Finite element method. Paper type Research Paper

1. Introduction Topology optimization is one of the most important types of structural optimization. The purpose of topology optimization is to find the optimal layout of a structure so that given applied loads are transferred to desired constraints in a specified domain using a given amount of material while equilibrium and design constraints are satisfied. Two kinds of methods are used in topology optimization; homogenization method and density method (Bendsoe et al., 2002). Bendsoe et al.

(1988) proposed the homogenization method. The density method was suggested by Bendsoe (1989). The density of the finite element is used as a design variable in the density method. The modulus of each finite element is calculated based on the density value. If the density is close to zero, the corresponding finite element is eliminated as it is considered empty. The density method is based on isotropic material (Park, 2007). The commercial software NASTRAN is based on the density method. The commercial software NASTRAN is used for topology optimization (MD Nastran, 2008). Gea (1996) proposed a design domain method which maintains the advantages from both the homogenization method and density function approach. Sigmund (1997) proposed the design of a compliant mechanism using topology optimization. Luo et al. (1998, 1998) studied the optimal stiffer design of 3D shell/plate structures. Most of the work mentioned above is based on linear finite element analysis to find the structural responses. It means that this work is based on the assumption of small deformation during the optimization process. In the cases of compliant mechanism and energy absorbing structure applications, the assumption of small deformation does not stand any more. In order to obtain a more reasonable design, appropriate nonlinearities must be considered. Yuge et al. (1999) modified the homogenization method for nonlinear deformation. Buhl et al. (2000) studied topology optimization with geometrically nonlinearity and compared the results of linear analysis and nonlinear analysis. Mayer et al. (1996) studied the application of topological optimization techniques to a crashworthiness problem using the elastoplastic material model. Gea et al. (2001) studied the stiffness optimization problems with geometrical nonlinearities. Bruns et al. (2001) studied the topology optimization of structures by considering the material and geometrical nonlinearities. Yoon et al. (2005, 2007) suggested the element connectivity parameterization method for topology optimization of material nonlinear continuum structures and geometrically nonlinear structures. Jung et al. (2004) studied the topology optimization of geometrically and materially nonlinear structures. Linear response structural optimization is very easy and convenient because the sensitivity information can be easily calculated. However, structural optimization using nonlinear analysis is comparatively expensive. It takes a long time to evaluate the design sensitivity of the objective function or constraints. Therefore the equivalent loads method has been developed to solve the difficulty of calculating the sensitivity information. Kang et al. (2001) proposed the equivalent static loads method for dynamic response optimization originally. Shin et al. (2007) modified the equivalent loads method for nonlinear response size and shape optimization. The main idea is that a nonlinear response optimization problem is converted to a linear response optimization with equivalent loads. Equivalent loads are defined as the loads for linear analysis, which generate the same response fields as those of nonlinear analysis. The method was originally proposed for size and shape optimization. A preliminary study was done for topology optimization with equivalent static loads to check the proposed method (Lee et al., 2010). The overview of the NROEL was published (Park, 2011). The concept of NROEL for nonlinear static response topology optimization was published (Lee et al., 2012). In this research, we solved few more case studies to verify the nonlinear static response topology optimization using equivalent static loads. The results strongly support the proposed method.

2. Background Theory 2.1.

Nonlinear Static Analysis

The response is directly proportional to the load in the linear analysis. Linearity may represent the real response of the structure or may be the expected response based on assumptions made for analysis purposes. In linear analysis, we consider that the displacements and rotations are small, stress is directly proportional to the strain, and also the loads maintain their unique direction when the structure is deformed. Many practical problems can be solved by the above assumptions made for linear analysis. But, sometimes the above assumptions are not valid. Sometimes the elastic material becomes plastic or the material does not have the linear stress-strain relationship, and it is also possible that due to the material failure and buckling, the structure may lose its stiffness. Displacement and rotations may be so large that it becomes necessary to write the equilibrium equations for the deformed structure rather than the original structure. Some parts are in contact with each other. The contact area is changed when the load is changed. In all the above cases, the structure shows nonlinear response. Therefore, nonlinearity should be considered in order to obtain accurate results. In structural mechanics, three kinds of nonlinearity are considered In material nonlinearity, the material properties are the functions of the state of stress or strain (Cook, 2002). The nonlinear material response may be nonlinear elastic or plastic. Material response that is nonlinear and maintains a permanent strain or returns slowly to an unstrained state on complete unloading is called as nonlinear elastic. Inelastic behavior of materials that maintains a permanent set on complete unloading is called plasticity. In geometric nonlinearity, the deformation is large enough that it is required to write the equilibrium equations of the deformed structural geometry. The loads also change direction, when deformation increases (Cook, 2002). Contact nonlinearity, a contact problem is a type of a geometrically nonlinear problem that takes place when different structures, or different parts of the same structure, either come into contact, separate, or slide into each other. All kinds of nonlinearity are considered for solving the case studies in this research.

2.2.

Topology Optimization

The concept of optimization is applied to the structural design in structural optimization. Structural optimization has been employed in various fields, especially in aerospace engineering to reduce the weight of airplane structures. Optimization is aggressively applied to various structural designs due to many reasons. First, structural design process and optimization process are almost similar. The reduction of weight in structural design is similar to minimization of the objective function in optimization. The structural design conditions can be transformed easily into optimization constraints. Second, the finite element method (FEM) can be converted in structural optimization. The objective function and constraints can be estimated accurately by defining it into definite forms by FEM. FEM is one of the most developed computational methods for the analysis of system.

Topology optimization is the distribution of materials in the structure. Topology optimization is applied to continuum structures and also to discrete structures with truss or beam elements. Two kinds of methods are used in topology optimization; homogenization method and density method. The structure is consisted of infinitely many holes in homogenization method and these holes are considered to be evenly distributed. The element is considered as an empty, if the size of the hole increase than certain size. The method is well established mathematically and full reliable. The density of the finite element is the design variable in the density method. The density is utilized to calculate the modulus of each element. If the density value is close to zero, then corresponding element is considered empty. The density method is weaker than the homogenization method mathematically to some extent; however, it is more popular due to the simple theory and easiness of implementing it numerically (Park, 2007). The purpose of the topology optimization is to maximize the stiffness of a structure. Maximization of the stiffness is equivalent to the minimization of the compliance. Therefore, the objective of topology optimization is generally to minimize the compliance. The linear response topology optimization theory is used in the nonlinear response topology optimization process using equivalent loads. The topology optimization formulation is as follows: Find to minimize subject to

b mean compliance K(b) z - f = 0 ∑ veb ≤ V

(1)

0.0 < bi min ≤ bi ≤ 1.0 Where the design variable vector b is the density of the elements, ve is the volume of each element, and V is the total volume of the structure.

3. Nonlinear Response Topology Optimization using Equivalent Static Loads Equivalent loads are defined as the loads for linear analysis, which generate the same response fields as those of nonlinear analysis. The optimization process using the equivalent loads consists of two kinds of domains. These are the analysis domain and design domain as shown in Figure 1. Nonlinear analysis under the given loading condition is carried out in the analysis domain. Equivalent loads are calculated based on the results of the analysis domain. Linear response optimization using the equivalent loads is performed in the design domain. Then the design is updated. The optimization process iterates between the domains until the convergence criterion is satisfied.

3.1.

Calculation of Equivalent Static Loads

The equivalent loads are calculated by the following below steps. The equilibrium equation of the structural nonlinear behavior can be represented as

Κ b, z N z N f

(2)

where K is the stiffness matrix, b is the design variable vector, zN is the nonlinear nodal displacement vector (responses) and f is the external load vector. The subscript N represents the nonlinear analysis. Equivalent loads are obtained by multiplying the linear stiffness matrix (KL) and nonlinear nodal displacement vector (zN) from Eq. (2).

f eq K L z N

(3)

where feq is the equivalent load vector. The subscript L represents the linear analysis. Eq. (4) is the equation of linear analysis with the equivalent loads.

Κ L b z L f eq

(4)

where zL is the linear displacement vector. When the linear analysis is performed with the equivalent loads from Eq. (3), the linear nodal displacement vector (zL) has same values as those of nonlinear nodal displacement vector (zN) as shown in Eq. (3) and Eq. (4). 3.2.

Topology Optimization process using Equivalent Loads

The topology optimization process using equivalent loads method is illustrated in flowchart as shown in Figure 2. The steps of the topology optimization process are as follows: Step 1: Set the initial cycle value k=0, and design variable b(k) = b(0) Step 2: Perform the nonlinear response analysis in Eq. (2) with b(k) Step 3: Calculate the equivalent loads in Eq. (3) Step 4: Perform the linear static topology optimization in Eq. (1) with equivalent loads. Step 5: The design variables values are transposed into transformation variables by using Eq. (5)

0 i 1

when

bi 1

when

bi 1

(5)

where, αi is a transformation variable of the ith design variable and ε1 is the separation parameter. The transformation variables are temporary variables and the total number of transformation variables is equal to the total number of design variables. The design variables in linear response

topology optimization do not have exact values, but have continuous values between bmin and bmax as given in Eq. (1). But few elements have value very close to the bmin. Therefore these low density finite elements create mesh distortion that results in non-reliable optimization results. Therefore, the separation parameter is needed in order to decide whether this finite element should be included or eliminated for optimization in the next cycle. Step 6: Update the design. The elements having transformation variables of value 0 are eliminated in the finite element model and elements corresponding to the transformation variables having value 1 remain in the FE model. Therefore, the number of elements in the present cycle is less than the previous cycle. Step 7: If k=0, then go to the next step. Otherwise check the convergence criterion.

countif b i

k

bi

k 1

2 n 3

(6)

Where ε2 is a small value close to 0, ε3 is the percentage and n is the total number of design variables. The number is counted if the difference of the design variables of the present cycle (k) and previous cycle (k-1) is more than the specified value (ε2), and the counted number is smaller than the percentage of the total design variables. Then the optimization process is terminated because the convergence criterion is satisfied. Otherwise go to the next cycle.

4. Case Studies Five case studies are solved by using the equivalent loads method and linear response optimization method. First two case studies; beam is solved by linear response optimization and equivalent loads method considering geometric nonlinearity. The optimum results are compared with the numerical results. Third and fourth case studies; Michell structure and spacer grid are solved by linear optimization method and equivalent loads method by considering the material and geometrical nonlinearities. The equivalent loads optimization results are compared with the linear response optimization result. Fifth case study, a flange with rectangular hole in the mid and a rectangular beam passing through it is solved with a contact condition. Different combinations of the nonlinearities are considered for it. The elimination method is used as the cycle update method. NASTRAN is used for nonlinear response analysis and optimization. Following terminologies are used in the paper: MNL: material nonlinearity GNL: Geometric nonlinearity CNL: contact nonlinearity

4.1.

Case Study 1

A long beam 800 mm long, 200 mm width, and 10 mm thickness is fixed at the midpoint of the both ends as shown in Figure 3. A 200 N concentrated force is applied at the center of the top edge. The number of design variables is 1600, minimizing the strain energy is the objective function, and the mass constraint is 20 % of the total mass. The Young’s modulus is 1 GPa and the Poisson’s ratio is 0.3. The nonlinear response topology optimization using the equivalent load is formulated as follows:

Find to minimize subject to

bi i = 1,……., 1600 strain energy KL zL – feq = 0 mass ≤ V mass total x 20 %

(7)

0.001 < bi ≤ 1.0 First linear topology optimization is performed and optimization results are obtained. The nonlinear response topology optimization using equivalent loads considering the geometrical nonlinearity are performed and optimization results are obtained. This example has been solved numerically and optimization results are available already (Gea et al., 2001). The linear response topology optimization results and nonlinear response topology optimization results using equivalent loads are compared with the numerical results as shown in Figure 4. The optimum results are very close to the numerical results. This shows that equivalent loads method is very efficient for nonlinear response topology optimization.

4.2.

Case Study 2

In order to check the consistency in the optimization method and results, the same design problem as in case study 1 is considered with different loading condition, boundary condition and material properties as shown in Figure 5. The formulation of the problem is same as given in Eq. (7). The beam is fixed on both ends and three concentrated forces 200N, 400N and 200N are applied on the bottom edge of the beam in the downward direction. The material used has Young’s modulus 100 MPa and the Poisson’s ratio is 0.3. Linear response topology optimization and nonlinear response topology optimization using equivalent loads are performed and optimization results are compared with the numerical results (Gea et al., 2001) as shown in Figure 6. The optimum results are very close to the numerical results in this case study too. This shows the consistency in the optimization method and results and justifies the equivalent loads method. 4.3.

Case Study 3

A cantilever plate 550 mm long and 400 mm high is fixed at one end as shown in Figure 7. A 2000 N force is applied at the center point of the free edge. The optimization problem is solved according

to the Eq. (8). The number of design variables is 8800 and objective function is the minimizing the strain energy. The mass constraint for the topology optimization is 50 % of the total mass. The material has bilinear elastoplastic properties. The Young’s modulus is 210 GPa, the hardening modulus is 105 GPa and Poisson’s ratio is 0.3. Find to minimize subject to

bi i = 1,……., 8800 strain energy KL zL – feq = 0 mass ≤ V mass total x 50 %

(8)

0.001 < bi ≤ 1.0 Linear response topology optimization and nonlinear response topology optimization using equivalent loads are performed. The optimization results are obtained as shown in Figure 8. The optimization results seem to be different from each other. The material and geometric nonlinearity is considered in this case. In order to get the nonlinear strain energy, the nonlinear analysis is performed on the linear response optimum result and nonlinear response optimum result with the same force and boundary conditions as applied on the initial model. The total strain energy of the equivalent loads optimization results is less than the linear response optimization result as given in Table 1. Since the main purpose of the topology optimization is to maximize the stiffness that is equal to minimize the strain energy. It shows that equivalent loads optimization results are better compared to the linear response optimization result. 4.4.

Case Study 4

The spacer grid set is one of the main components of a nuclear fuel assembly. There are four spacer grid in a unit spacer grid set. The fuel rod is inserted into the unit spacer grid set as shown in Figure 9. The function of the spacer grid set is; to support the fuel rod, to provide the path to a cooling fluent in order to increase the heat transfer from the hot fuel rod to the coolant. The spring and dimples are the actual supporting parts in the unit spacer grid (Shin et al., 2008). The optimization of the spring part is preferred. When the fuel rod is inserted into the unit spacer grid, the spring is deformed by about 3 mm. The fuel rod is in contact with the spring at the central line. So, the force is applied at the central line of the spring. Nonlinear analysis is performed using the condition that the spring is moved by 0.3 mm after the contact of the spring and the fuel rod. The central part of the spacer grid is used as the design domain in topology optimization as shown in Figure 10. The formulation to perform topology optimization on the spacer grid is given in Eq. (9). The number of design variables is 7016 and objective function is the minimizing the strain energy. The mass constraint for the topology optimization is 70 % of the total mass. The separation parameter used is 0.001.

Find to minimize subject to

bi i = 1,……., 7016 strain energy KL zL – feq = 0 mass ≤ V mass total x 70 %

(9)

0.001 < bi ≤ 1.0 The material is zircaloy-4; which is an alloy of zirconium and tartar. The Young’s modulus is 113.7 GPa, the yield strength is 379.5 MPa, the Poisson’s ratio is 0.296, and the density is 6550.0 kg/m3. The stress strain curve of the material is shown in Figure 11. We performed the linear response topology optimization and nonlinear response topology optimization using equivalent loads and optimization results are obtained as shown in Figure 12. The material and geometric nonlinearity are considered for solving this problem. The linear response topology optimization results are compared with nonlinear response optimum result by calculating the nonlinear strain energy. The nonlinear analysis is performed on the linear response optimum result and nonlinear response optimum result with the same force and boundary conditions as applied on the initial model. The optimum results obtained by nonlinear response optimization using equivalent loads are better compared to the linear response optimization result due to the lesser values of the strain energy values as given in Table 2, because stiffness is inversely proportional to strain energy. 4.5.

Case Study 5

The piece of block consist of rectangular hole at the mid of the block. A rectangular beam is passed through this hole. This block supports the beam and acts like a flange. The dimensions of the block, hole and beam are: (100x100x100), (30x30x30) and (300x30x30) respectfully. The finite element model of the block with rectangular beam is shown in Figure 13 (a) and this model is used for analysis. The finite element model of the block as shown in Figure 13 (b) is used for the optimization. 100 N of distributed force is applied on each side of the rectangular beam. The material properties; Young’s modulus is 210 GPa, the hardening modulus is 105 GPa and Poisson’s ratio is 0.3. The formulation of the topology optimization as given in Eq. (10) is used to perform the topology optimization. The number of design variables is 7280 and objective function is the minimizing the strain energy. The mass constraint for the topology optimization is 40 % of the total mass. Find to minimize subject to

bi i = 1,……., 7280 strain energy KL zL – feq = 0 mass ≤ V mass total x 40 %

(10)

0.001 < bi ≤ 1.0 The nonlinear response topology optimization using equivalent loads is performed by considering the different combinations of nonlinearities and optimization results are obtained as shown in

Figure 14. The linear response topology optimization cannot be performed because there is contact nonlinearity.

5. Conclusions The equivalent loads method is a valuable tool applied to the nonlinear response topology optimization. The problem of mesh distortion due to the low density finite elements has been solved by introducing the transformation variable and new update method. A separation parameter is used to define the transformation variables. This separation parameter defines the density value to remove the finite elements having density less than this value. The transformation variables of the finite elements with a value of 0 are removed from the finite element model for the next cycle. The updating of the finite element material becomes easy after the removal of low density finite elements. The equivalent loads method require only few nonlinear analysis and problem is converged to the optimum. Five different kinds of case studies are solved by linear response topology optimization and nonlinear response topology optimization using equivalent loads considering different nonlinearities. The first two case studies; beam has been solved by considering the geometric nonlinearity and results have been compared to the numerical results. The optimum results are very close to the numerical results that validate the proposed method. The consistency in the proposed method has been verified. The third and fourth case studies; Michell structure and spacer grid have been solved by considering the material and geometric nonlinearities. The optimum results are obtained. The nonlinear analysis is performed on the linear response optimum result and nonlinear response optimum result with the same force and boundary conditions as applied on the initial model to calculate the strain energy values. The objective of topology optimization is to maximize the stiffness. Lower the value of the strain energy higher will be the stiffness. The strain energy values of the equivalent loads optimization results are less than the strain energy values of the linear response optimization result. This shows that the equivalent loads optimization result is better compared to the linear topology optimization result. The fifth case study a flange with contact condition is also solved and optimum results are obtained. The linear response topology optimization has not been solved due to the contact condition. Nonlinearity should be considered in the design process, if necessary. Therefore, the equivalent loads method is an excellent tool to get the optimization results.

References Bendsoe, M.P. (1989), “Optimal shape design as a material distribution problem”, Structural Optimization, Vol. 1, No. 4, pp. 193-202.

Bendsoe, M.P. and Kikuchi, N. (1988), “Generating optimal topologies in structural design using a homogenization method”, Computer Method in Applied Mechanics and Engineering, Vol. 71, Issue 2, pp. 197-224. Bendsoe, M.P. and Sigmund, O. (2002), Topology Optimization: Theory, Methods and Application, Springer, Germany. Bruns, T.E. and Tortorelli, D.A. (2001), “Topology optimization of non-linear elastic structures and compliant mechanism”, Computer Methods in Applied Mechanics and Engineering, Vol. 190, Issues 26-27, pp. 3443-3459. Buhl, T., Pedersen, C.B.W. and Sigmund, O. (2000), “Stiffness design of geometrically nonlinear structures using topology optimization”, Structural and Multidisciplinary Optimization, Vol. 19, No. 2, pp. 93-104. Cook, R. D., Malkus, D. S., Plesha, M. E. and Witt, R. J. (2002), Concepts and Applications of Finite Element Analysis, 4th edition, pp. 595-638. Gea, H.C. (1996), “Topology optimization: A new microstructure based design domain method”, Computers and Structures, Vol. 61, Issue 5, pp. 781-788. Gea, H.C. and Luo, J.H. (2001), “Topology optimization of structures with geometrical nonlinearities”, Computers and Structures, Vol. 79, No. 20, pp. 1977-1985. Jung, D.Y. and Gea, H.C. (2004), “Topology optimization of nonlinear structures”, Finite Elements in Analysis and Design, Vol. 40, Issue 11, pp. 1417-1427. Kang, B.S., Choi, W.S. and Park, G.J. (2001), “Structural optimization under equivalent static loads transformed from dynamic loads based on displacement”, Computers and Structures, Vol. 79, Issue 2, pp. 145-154. Lee, H. A., Ahmad, Z. and Park, G. J. (2010), “Preliminary study on nonlinear static response topology optimization using equivalent load”, Transactions of the Korean society of Mechanical Engineers- A, Volume 34, Issue 12, pp.1811-1820 (In Korean) Lee, H.A. and Park, G.J. (2012), “Topology optimization for structures with nonlinear behavior using the equivalent static loads method”, Journal of Mechanical Design, Vol. 134, pp. 031004-14. Luo, J.H. and Gea, H.C. (1998), “Optimal based orientation of 3D shell/plate structures”, Finite Elements in Analysis and Design, Vol. 31, Issue 1, pp. 55-71. Luo, J.H. and Gea, H.C. (1998), “A systematic topology optimization approach for optimal stiffer design”, Structural Optimization, Vol. 16, No. 4, pp. 280-288.

Mayer, R.R., Kikuchi, N. and Scott, R.A. (1996) “Application of topological optimization techniques to structural crashworthiness”, International Journal for Numerical Methods in Engineering, Vol. 39, Issue 8, pp. 1383-1403. MD Nastran (2008), R3 Quick Reference Guide, MSC. Software Corporation. Park, G.J. (2007), Analytic Methods for Design Practice, Springer, Germany, pp. 237-243. Park, G. J., (2011), “Technical overview of the equivalent static loads method for non-linear static response structural optimization”, Structural and Multidisciplinary Optimization, Vol. 43, Issue 3, pp. 319-337. Shin, M.K., Park, K.J. and Park, G.J. (2007), “Optimization of structures with nonlinear behavior using equivalent loads”, Computers Methods in Applied Mechanics and Engineering, Vol. 196, Issue 4-6, pp. 1154-1167. Shin, M.K., Lee, H.A., Lee, J.J., Song, K.N. and Park, G.J. (2008), “Optimization of a nuclear fuel spacer grid spring using homology constraints”, Nuclear Engineering and Design, Vol. 238, Issue 10, pp. 624-2634. Sigmund, O. (1997), “On the design of compliant mechanism using topology optimization”, Mechanics of Structures and Machines, Vol. 25, Issue 4, pp. 493-524. Yoon, G.H. and Kim, Y.Y. (2005), “Element connectivity parameterization for topology optimization of geometrically nonlinear structures”, International Journal of Solids and Structures, Vol. 42, Issue. 7, pp. 1983-2009. Yoon, G.H. and Kim, Y.Y. (2007), “Topology optimization of material nonlinear continuum structures by the element connectivity parameterization”, International Journal for Numerical Methods in Engineering, Vol. 69, Issue 10, pp. 2196-2218. Yuge, K., Iwai, N. and Kikuchi, N. (1999), “Optimization of 2-D structures subjected to nonlinear deformations using the homogenization method”, Structural Optimization, Vol. 17, No. 4, pp. 286299.

Figure 1 Optimization process using equivalent loads

Analysis domain

Design domain

Update design

Nonlinear static analysis

Linear static topology optimization

Equivalent static loads

Figure 2 Topology optimization process using equivalent loads

Start

k=0

End

k=k+1 No

Yes

Nonlinear analysis

Convergence Calculate equivalent loads

Update design

Linear response topology optimization

Transform into transformation variables

Figure 3 Finite element model of the beam 1 200 N

200 mm

800 mm

Figure 4 Optimum results of the beam 1

Equivalent loads method

Numerical method

(a) Linear

(b) GNL

Figure 5 Finite element model of the beam 2

200 N

200 N 400 N

Figure 6 Optimum results of the beam 2

Equivalent loads method

Numerical method

(a) Linear

(b) GNL

Figure 7 Finite element model of the Michell structure

550 mm

400 mm

2000 N

Figure 8 Optimum results of the Michell Structure

(a) Linear

(b) MNL

(c) GNL

(d) MNL and GNL

Figure 9 Unit spacer grid

Dimple

Nuclear fuel rod Spring

Figure 10 Finite element model of the spacer grid Design domain (spring part)

x

y z z x (a) Design domain for the topology optimization

16.0 mm

12.85 mm z x (b) Boundary conditions (top view)

y

x

0.98 mm 0.91 mm

(c) Loading conditions (front view)

Figure 11 Stress-strain curve for spacer grid material

Figure 12 Optimum results of the spacer grid

(a) Linear

(b) MNL

(c) GNL

(d) MNL and GNL

Figure 13 Finite element model of the block

(a)

(b)

Figure 14 Optimum results of the flange with different nonlinearities Isometric view

Side view

(c) CNL

(d) MNL and CNL

(e) GNL and CNL

(f) MNL, GNL and CNL

Table 1 Strain energy values (Nm) of the Michell structure optimum results

MNL

GNL

MNL and GNL

Strain energy with linear optimum result

171.82

75.35

172.00

Strain energy with nonlinear optimum result

163.02

74.80

160.67

Table 2 Strain energy values (Nm) of the spacer grid optimum results

MNL

GNL

MNL and GNL

Strain energy with linear optimum result

10.64

15.45

13.62

Strain energy with nonlinear optimum result

9.82

14.02

11.89

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1. Introduction Topology optimization is one of the most important types of structural optimization. The purpose of topology optimization is to find the optimal layout of a structure so that given applied loads are transferred to desired constraints in a specified domain using a given amount of material while equilibrium and design constraints are satisfied. Two kinds of methods are used in topology optimization; homogenization method and density method (Bendsoe et al., 2002). Bendsoe et al.

(1988) proposed the homogenization method. The density method was suggested by Bendsoe (1989). The density of the finite element is used as a design variable in the density method. The modulus of each finite element is calculated based on the density value. If the density is close to zero, the corresponding finite element is eliminated as it is considered empty. The density method is based on isotropic material (Park, 2007). The commercial software NASTRAN is based on the density method. The commercial software NASTRAN is used for topology optimization (MD Nastran, 2008). Gea (1996) proposed a design domain method which maintains the advantages from both the homogenization method and density function approach. Sigmund (1997) proposed the design of a compliant mechanism using topology optimization. Luo et al. (1998, 1998) studied the optimal stiffer design of 3D shell/plate structures. Most of the work mentioned above is based on linear finite element analysis to find the structural responses. It means that this work is based on the assumption of small deformation during the optimization process. In the cases of compliant mechanism and energy absorbing structure applications, the assumption of small deformation does not stand any more. In order to obtain a more reasonable design, appropriate nonlinearities must be considered. Yuge et al. (1999) modified the homogenization method for nonlinear deformation. Buhl et al. (2000) studied topology optimization with geometrically nonlinearity and compared the results of linear analysis and nonlinear analysis. Mayer et al. (1996) studied the application of topological optimization techniques to a crashworthiness problem using the elastoplastic material model. Gea et al. (2001) studied the stiffness optimization problems with geometrical nonlinearities. Bruns et al. (2001) studied the topology optimization of structures by considering the material and geometrical nonlinearities. Yoon et al. (2005, 2007) suggested the element connectivity parameterization method for topology optimization of material nonlinear continuum structures and geometrically nonlinear structures. Jung et al. (2004) studied the topology optimization of geometrically and materially nonlinear structures. Linear response structural optimization is very easy and convenient because the sensitivity information can be easily calculated. However, structural optimization using nonlinear analysis is comparatively expensive. It takes a long time to evaluate the design sensitivity of the objective function or constraints. Therefore the equivalent loads method has been developed to solve the difficulty of calculating the sensitivity information. Kang et al. (2001) proposed the equivalent static loads method for dynamic response optimization originally. Shin et al. (2007) modified the equivalent loads method for nonlinear response size and shape optimization. The main idea is that a nonlinear response optimization problem is converted to a linear response optimization with equivalent loads. Equivalent loads are defined as the loads for linear analysis, which generate the same response fields as those of nonlinear analysis. The method was originally proposed for size and shape optimization. A preliminary study was done for topology optimization with equivalent static loads to check the proposed method (Lee et al., 2010). The overview of the NROEL was published (Park, 2011). The concept of NROEL for nonlinear static response topology optimization was published (Lee et al., 2012). In this research, we solved few more case studies to verify the nonlinear static response topology optimization using equivalent static loads. The results strongly support the proposed method.

2. Background Theory 2.1.

Nonlinear Static Analysis

The response is directly proportional to the load in the linear analysis. Linearity may represent the real response of the structure or may be the expected response based on assumptions made for analysis purposes. In linear analysis, we consider that the displacements and rotations are small, stress is directly proportional to the strain, and also the loads maintain their unique direction when the structure is deformed. Many practical problems can be solved by the above assumptions made for linear analysis. But, sometimes the above assumptions are not valid. Sometimes the elastic material becomes plastic or the material does not have the linear stress-strain relationship, and it is also possible that due to the material failure and buckling, the structure may lose its stiffness. Displacement and rotations may be so large that it becomes necessary to write the equilibrium equations for the deformed structure rather than the original structure. Some parts are in contact with each other. The contact area is changed when the load is changed. In all the above cases, the structure shows nonlinear response. Therefore, nonlinearity should be considered in order to obtain accurate results. In structural mechanics, three kinds of nonlinearity are considered In material nonlinearity, the material properties are the functions of the state of stress or strain (Cook, 2002). The nonlinear material response may be nonlinear elastic or plastic. Material response that is nonlinear and maintains a permanent strain or returns slowly to an unstrained state on complete unloading is called as nonlinear elastic. Inelastic behavior of materials that maintains a permanent set on complete unloading is called plasticity. In geometric nonlinearity, the deformation is large enough that it is required to write the equilibrium equations of the deformed structural geometry. The loads also change direction, when deformation increases (Cook, 2002). Contact nonlinearity, a contact problem is a type of a geometrically nonlinear problem that takes place when different structures, or different parts of the same structure, either come into contact, separate, or slide into each other. All kinds of nonlinearity are considered for solving the case studies in this research.

2.2.

Topology Optimization

The concept of optimization is applied to the structural design in structural optimization. Structural optimization has been employed in various fields, especially in aerospace engineering to reduce the weight of airplane structures. Optimization is aggressively applied to various structural designs due to many reasons. First, structural design process and optimization process are almost similar. The reduction of weight in structural design is similar to minimization of the objective function in optimization. The structural design conditions can be transformed easily into optimization constraints. Second, the finite element method (FEM) can be converted in structural optimization. The objective function and constraints can be estimated accurately by defining it into definite forms by FEM. FEM is one of the most developed computational methods for the analysis of system.

Topology optimization is the distribution of materials in the structure. Topology optimization is applied to continuum structures and also to discrete structures with truss or beam elements. Two kinds of methods are used in topology optimization; homogenization method and density method. The structure is consisted of infinitely many holes in homogenization method and these holes are considered to be evenly distributed. The element is considered as an empty, if the size of the hole increase than certain size. The method is well established mathematically and full reliable. The density of the finite element is the design variable in the density method. The density is utilized to calculate the modulus of each element. If the density value is close to zero, then corresponding element is considered empty. The density method is weaker than the homogenization method mathematically to some extent; however, it is more popular due to the simple theory and easiness of implementing it numerically (Park, 2007). The purpose of the topology optimization is to maximize the stiffness of a structure. Maximization of the stiffness is equivalent to the minimization of the compliance. Therefore, the objective of topology optimization is generally to minimize the compliance. The linear response topology optimization theory is used in the nonlinear response topology optimization process using equivalent loads. The topology optimization formulation is as follows: Find to minimize subject to

b mean compliance K(b) z - f = 0 ∑ veb ≤ V

(1)

0.0 < bi min ≤ bi ≤ 1.0 Where the design variable vector b is the density of the elements, ve is the volume of each element, and V is the total volume of the structure.

3. Nonlinear Response Topology Optimization using Equivalent Static Loads Equivalent loads are defined as the loads for linear analysis, which generate the same response fields as those of nonlinear analysis. The optimization process using the equivalent loads consists of two kinds of domains. These are the analysis domain and design domain as shown in Figure 1. Nonlinear analysis under the given loading condition is carried out in the analysis domain. Equivalent loads are calculated based on the results of the analysis domain. Linear response optimization using the equivalent loads is performed in the design domain. Then the design is updated. The optimization process iterates between the domains until the convergence criterion is satisfied.

3.1.

Calculation of Equivalent Static Loads

The equivalent loads are calculated by the following below steps. The equilibrium equation of the structural nonlinear behavior can be represented as

Κ b, z N z N f

(2)

where K is the stiffness matrix, b is the design variable vector, zN is the nonlinear nodal displacement vector (responses) and f is the external load vector. The subscript N represents the nonlinear analysis. Equivalent loads are obtained by multiplying the linear stiffness matrix (KL) and nonlinear nodal displacement vector (zN) from Eq. (2).

f eq K L z N

(3)

where feq is the equivalent load vector. The subscript L represents the linear analysis. Eq. (4) is the equation of linear analysis with the equivalent loads.

Κ L b z L f eq

(4)

where zL is the linear displacement vector. When the linear analysis is performed with the equivalent loads from Eq. (3), the linear nodal displacement vector (zL) has same values as those of nonlinear nodal displacement vector (zN) as shown in Eq. (3) and Eq. (4). 3.2.

Topology Optimization process using Equivalent Loads

The topology optimization process using equivalent loads method is illustrated in flowchart as shown in Figure 2. The steps of the topology optimization process are as follows: Step 1: Set the initial cycle value k=0, and design variable b(k) = b(0) Step 2: Perform the nonlinear response analysis in Eq. (2) with b(k) Step 3: Calculate the equivalent loads in Eq. (3) Step 4: Perform the linear static topology optimization in Eq. (1) with equivalent loads. Step 5: The design variables values are transposed into transformation variables by using Eq. (5)

0 i 1

when

bi 1

when

bi 1

(5)

where, αi is a transformation variable of the ith design variable and ε1 is the separation parameter. The transformation variables are temporary variables and the total number of transformation variables is equal to the total number of design variables. The design variables in linear response

topology optimization do not have exact values, but have continuous values between bmin and bmax as given in Eq. (1). But few elements have value very close to the bmin. Therefore these low density finite elements create mesh distortion that results in non-reliable optimization results. Therefore, the separation parameter is needed in order to decide whether this finite element should be included or eliminated for optimization in the next cycle. Step 6: Update the design. The elements having transformation variables of value 0 are eliminated in the finite element model and elements corresponding to the transformation variables having value 1 remain in the FE model. Therefore, the number of elements in the present cycle is less than the previous cycle. Step 7: If k=0, then go to the next step. Otherwise check the convergence criterion.

countif b i

k

bi

k 1

2 n 3

(6)

Where ε2 is a small value close to 0, ε3 is the percentage and n is the total number of design variables. The number is counted if the difference of the design variables of the present cycle (k) and previous cycle (k-1) is more than the specified value (ε2), and the counted number is smaller than the percentage of the total design variables. Then the optimization process is terminated because the convergence criterion is satisfied. Otherwise go to the next cycle.

4. Case Studies Five case studies are solved by using the equivalent loads method and linear response optimization method. First two case studies; beam is solved by linear response optimization and equivalent loads method considering geometric nonlinearity. The optimum results are compared with the numerical results. Third and fourth case studies; Michell structure and spacer grid are solved by linear optimization method and equivalent loads method by considering the material and geometrical nonlinearities. The equivalent loads optimization results are compared with the linear response optimization result. Fifth case study, a flange with rectangular hole in the mid and a rectangular beam passing through it is solved with a contact condition. Different combinations of the nonlinearities are considered for it. The elimination method is used as the cycle update method. NASTRAN is used for nonlinear response analysis and optimization. Following terminologies are used in the paper: MNL: material nonlinearity GNL: Geometric nonlinearity CNL: contact nonlinearity

4.1.

Case Study 1

A long beam 800 mm long, 200 mm width, and 10 mm thickness is fixed at the midpoint of the both ends as shown in Figure 3. A 200 N concentrated force is applied at the center of the top edge. The number of design variables is 1600, minimizing the strain energy is the objective function, and the mass constraint is 20 % of the total mass. The Young’s modulus is 1 GPa and the Poisson’s ratio is 0.3. The nonlinear response topology optimization using the equivalent load is formulated as follows:

Find to minimize subject to

bi i = 1,……., 1600 strain energy KL zL – feq = 0 mass ≤ V mass total x 20 %

(7)

0.001 < bi ≤ 1.0 First linear topology optimization is performed and optimization results are obtained. The nonlinear response topology optimization using equivalent loads considering the geometrical nonlinearity are performed and optimization results are obtained. This example has been solved numerically and optimization results are available already (Gea et al., 2001). The linear response topology optimization results and nonlinear response topology optimization results using equivalent loads are compared with the numerical results as shown in Figure 4. The optimum results are very close to the numerical results. This shows that equivalent loads method is very efficient for nonlinear response topology optimization.

4.2.

Case Study 2

In order to check the consistency in the optimization method and results, the same design problem as in case study 1 is considered with different loading condition, boundary condition and material properties as shown in Figure 5. The formulation of the problem is same as given in Eq. (7). The beam is fixed on both ends and three concentrated forces 200N, 400N and 200N are applied on the bottom edge of the beam in the downward direction. The material used has Young’s modulus 100 MPa and the Poisson’s ratio is 0.3. Linear response topology optimization and nonlinear response topology optimization using equivalent loads are performed and optimization results are compared with the numerical results (Gea et al., 2001) as shown in Figure 6. The optimum results are very close to the numerical results in this case study too. This shows the consistency in the optimization method and results and justifies the equivalent loads method. 4.3.

Case Study 3

A cantilever plate 550 mm long and 400 mm high is fixed at one end as shown in Figure 7. A 2000 N force is applied at the center point of the free edge. The optimization problem is solved according

to the Eq. (8). The number of design variables is 8800 and objective function is the minimizing the strain energy. The mass constraint for the topology optimization is 50 % of the total mass. The material has bilinear elastoplastic properties. The Young’s modulus is 210 GPa, the hardening modulus is 105 GPa and Poisson’s ratio is 0.3. Find to minimize subject to

bi i = 1,……., 8800 strain energy KL zL – feq = 0 mass ≤ V mass total x 50 %

(8)

0.001 < bi ≤ 1.0 Linear response topology optimization and nonlinear response topology optimization using equivalent loads are performed. The optimization results are obtained as shown in Figure 8. The optimization results seem to be different from each other. The material and geometric nonlinearity is considered in this case. In order to get the nonlinear strain energy, the nonlinear analysis is performed on the linear response optimum result and nonlinear response optimum result with the same force and boundary conditions as applied on the initial model. The total strain energy of the equivalent loads optimization results is less than the linear response optimization result as given in Table 1. Since the main purpose of the topology optimization is to maximize the stiffness that is equal to minimize the strain energy. It shows that equivalent loads optimization results are better compared to the linear response optimization result. 4.4.

Case Study 4

The spacer grid set is one of the main components of a nuclear fuel assembly. There are four spacer grid in a unit spacer grid set. The fuel rod is inserted into the unit spacer grid set as shown in Figure 9. The function of the spacer grid set is; to support the fuel rod, to provide the path to a cooling fluent in order to increase the heat transfer from the hot fuel rod to the coolant. The spring and dimples are the actual supporting parts in the unit spacer grid (Shin et al., 2008). The optimization of the spring part is preferred. When the fuel rod is inserted into the unit spacer grid, the spring is deformed by about 3 mm. The fuel rod is in contact with the spring at the central line. So, the force is applied at the central line of the spring. Nonlinear analysis is performed using the condition that the spring is moved by 0.3 mm after the contact of the spring and the fuel rod. The central part of the spacer grid is used as the design domain in topology optimization as shown in Figure 10. The formulation to perform topology optimization on the spacer grid is given in Eq. (9). The number of design variables is 7016 and objective function is the minimizing the strain energy. The mass constraint for the topology optimization is 70 % of the total mass. The separation parameter used is 0.001.

Find to minimize subject to

bi i = 1,……., 7016 strain energy KL zL – feq = 0 mass ≤ V mass total x 70 %

(9)

0.001 < bi ≤ 1.0 The material is zircaloy-4; which is an alloy of zirconium and tartar. The Young’s modulus is 113.7 GPa, the yield strength is 379.5 MPa, the Poisson’s ratio is 0.296, and the density is 6550.0 kg/m3. The stress strain curve of the material is shown in Figure 11. We performed the linear response topology optimization and nonlinear response topology optimization using equivalent loads and optimization results are obtained as shown in Figure 12. The material and geometric nonlinearity are considered for solving this problem. The linear response topology optimization results are compared with nonlinear response optimum result by calculating the nonlinear strain energy. The nonlinear analysis is performed on the linear response optimum result and nonlinear response optimum result with the same force and boundary conditions as applied on the initial model. The optimum results obtained by nonlinear response optimization using equivalent loads are better compared to the linear response optimization result due to the lesser values of the strain energy values as given in Table 2, because stiffness is inversely proportional to strain energy. 4.5.

Case Study 5

The piece of block consist of rectangular hole at the mid of the block. A rectangular beam is passed through this hole. This block supports the beam and acts like a flange. The dimensions of the block, hole and beam are: (100x100x100), (30x30x30) and (300x30x30) respectfully. The finite element model of the block with rectangular beam is shown in Figure 13 (a) and this model is used for analysis. The finite element model of the block as shown in Figure 13 (b) is used for the optimization. 100 N of distributed force is applied on each side of the rectangular beam. The material properties; Young’s modulus is 210 GPa, the hardening modulus is 105 GPa and Poisson’s ratio is 0.3. The formulation of the topology optimization as given in Eq. (10) is used to perform the topology optimization. The number of design variables is 7280 and objective function is the minimizing the strain energy. The mass constraint for the topology optimization is 40 % of the total mass. Find to minimize subject to

bi i = 1,……., 7280 strain energy KL zL – feq = 0 mass ≤ V mass total x 40 %

(10)

0.001 < bi ≤ 1.0 The nonlinear response topology optimization using equivalent loads is performed by considering the different combinations of nonlinearities and optimization results are obtained as shown in

Figure 14. The linear response topology optimization cannot be performed because there is contact nonlinearity.

5. Conclusions The equivalent loads method is a valuable tool applied to the nonlinear response topology optimization. The problem of mesh distortion due to the low density finite elements has been solved by introducing the transformation variable and new update method. A separation parameter is used to define the transformation variables. This separation parameter defines the density value to remove the finite elements having density less than this value. The transformation variables of the finite elements with a value of 0 are removed from the finite element model for the next cycle. The updating of the finite element material becomes easy after the removal of low density finite elements. The equivalent loads method require only few nonlinear analysis and problem is converged to the optimum. Five different kinds of case studies are solved by linear response topology optimization and nonlinear response topology optimization using equivalent loads considering different nonlinearities. The first two case studies; beam has been solved by considering the geometric nonlinearity and results have been compared to the numerical results. The optimum results are very close to the numerical results that validate the proposed method. The consistency in the proposed method has been verified. The third and fourth case studies; Michell structure and spacer grid have been solved by considering the material and geometric nonlinearities. The optimum results are obtained. The nonlinear analysis is performed on the linear response optimum result and nonlinear response optimum result with the same force and boundary conditions as applied on the initial model to calculate the strain energy values. The objective of topology optimization is to maximize the stiffness. Lower the value of the strain energy higher will be the stiffness. The strain energy values of the equivalent loads optimization results are less than the strain energy values of the linear response optimization result. This shows that the equivalent loads optimization result is better compared to the linear topology optimization result. The fifth case study a flange with contact condition is also solved and optimum results are obtained. The linear response topology optimization has not been solved due to the contact condition. Nonlinearity should be considered in the design process, if necessary. Therefore, the equivalent loads method is an excellent tool to get the optimization results.

References Bendsoe, M.P. (1989), “Optimal shape design as a material distribution problem”, Structural Optimization, Vol. 1, No. 4, pp. 193-202.

Bendsoe, M.P. and Kikuchi, N. (1988), “Generating optimal topologies in structural design using a homogenization method”, Computer Method in Applied Mechanics and Engineering, Vol. 71, Issue 2, pp. 197-224. Bendsoe, M.P. and Sigmund, O. (2002), Topology Optimization: Theory, Methods and Application, Springer, Germany. Bruns, T.E. and Tortorelli, D.A. (2001), “Topology optimization of non-linear elastic structures and compliant mechanism”, Computer Methods in Applied Mechanics and Engineering, Vol. 190, Issues 26-27, pp. 3443-3459. Buhl, T., Pedersen, C.B.W. and Sigmund, O. (2000), “Stiffness design of geometrically nonlinear structures using topology optimization”, Structural and Multidisciplinary Optimization, Vol. 19, No. 2, pp. 93-104. Cook, R. D., Malkus, D. S., Plesha, M. E. and Witt, R. J. (2002), Concepts and Applications of Finite Element Analysis, 4th edition, pp. 595-638. Gea, H.C. (1996), “Topology optimization: A new microstructure based design domain method”, Computers and Structures, Vol. 61, Issue 5, pp. 781-788. Gea, H.C. and Luo, J.H. (2001), “Topology optimization of structures with geometrical nonlinearities”, Computers and Structures, Vol. 79, No. 20, pp. 1977-1985. Jung, D.Y. and Gea, H.C. (2004), “Topology optimization of nonlinear structures”, Finite Elements in Analysis and Design, Vol. 40, Issue 11, pp. 1417-1427. Kang, B.S., Choi, W.S. and Park, G.J. (2001), “Structural optimization under equivalent static loads transformed from dynamic loads based on displacement”, Computers and Structures, Vol. 79, Issue 2, pp. 145-154. Lee, H. A., Ahmad, Z. and Park, G. J. (2010), “Preliminary study on nonlinear static response topology optimization using equivalent load”, Transactions of the Korean society of Mechanical Engineers- A, Volume 34, Issue 12, pp.1811-1820 (In Korean) Lee, H.A. and Park, G.J. (2012), “Topology optimization for structures with nonlinear behavior using the equivalent static loads method”, Journal of Mechanical Design, Vol. 134, pp. 031004-14. Luo, J.H. and Gea, H.C. (1998), “Optimal based orientation of 3D shell/plate structures”, Finite Elements in Analysis and Design, Vol. 31, Issue 1, pp. 55-71. Luo, J.H. and Gea, H.C. (1998), “A systematic topology optimization approach for optimal stiffer design”, Structural Optimization, Vol. 16, No. 4, pp. 280-288.

Mayer, R.R., Kikuchi, N. and Scott, R.A. (1996) “Application of topological optimization techniques to structural crashworthiness”, International Journal for Numerical Methods in Engineering, Vol. 39, Issue 8, pp. 1383-1403. MD Nastran (2008), R3 Quick Reference Guide, MSC. Software Corporation. Park, G.J. (2007), Analytic Methods for Design Practice, Springer, Germany, pp. 237-243. Park, G. J., (2011), “Technical overview of the equivalent static loads method for non-linear static response structural optimization”, Structural and Multidisciplinary Optimization, Vol. 43, Issue 3, pp. 319-337. Shin, M.K., Park, K.J. and Park, G.J. (2007), “Optimization of structures with nonlinear behavior using equivalent loads”, Computers Methods in Applied Mechanics and Engineering, Vol. 196, Issue 4-6, pp. 1154-1167. Shin, M.K., Lee, H.A., Lee, J.J., Song, K.N. and Park, G.J. (2008), “Optimization of a nuclear fuel spacer grid spring using homology constraints”, Nuclear Engineering and Design, Vol. 238, Issue 10, pp. 624-2634. Sigmund, O. (1997), “On the design of compliant mechanism using topology optimization”, Mechanics of Structures and Machines, Vol. 25, Issue 4, pp. 493-524. Yoon, G.H. and Kim, Y.Y. (2005), “Element connectivity parameterization for topology optimization of geometrically nonlinear structures”, International Journal of Solids and Structures, Vol. 42, Issue. 7, pp. 1983-2009. Yoon, G.H. and Kim, Y.Y. (2007), “Topology optimization of material nonlinear continuum structures by the element connectivity parameterization”, International Journal for Numerical Methods in Engineering, Vol. 69, Issue 10, pp. 2196-2218. Yuge, K., Iwai, N. and Kikuchi, N. (1999), “Optimization of 2-D structures subjected to nonlinear deformations using the homogenization method”, Structural Optimization, Vol. 17, No. 4, pp. 286299.

Figure 1 Optimization process using equivalent loads

Analysis domain

Design domain

Update design

Nonlinear static analysis

Linear static topology optimization

Equivalent static loads

Figure 2 Topology optimization process using equivalent loads

Start

k=0

End

k=k+1 No

Yes

Nonlinear analysis

Convergence Calculate equivalent loads

Update design

Linear response topology optimization

Transform into transformation variables

Figure 3 Finite element model of the beam 1 200 N

200 mm

800 mm

Figure 4 Optimum results of the beam 1

Equivalent loads method

Numerical method

(a) Linear

(b) GNL

Figure 5 Finite element model of the beam 2

200 N

200 N 400 N

Figure 6 Optimum results of the beam 2

Equivalent loads method

Numerical method

(a) Linear

(b) GNL

Figure 7 Finite element model of the Michell structure

550 mm

400 mm

2000 N

Figure 8 Optimum results of the Michell Structure

(a) Linear

(b) MNL

(c) GNL

(d) MNL and GNL

Figure 9 Unit spacer grid

Dimple

Nuclear fuel rod Spring

Figure 10 Finite element model of the spacer grid Design domain (spring part)

x

y z z x (a) Design domain for the topology optimization

16.0 mm

12.85 mm z x (b) Boundary conditions (top view)

y

x

0.98 mm 0.91 mm

(c) Loading conditions (front view)

Figure 11 Stress-strain curve for spacer grid material

Figure 12 Optimum results of the spacer grid

(a) Linear

(b) MNL

(c) GNL

(d) MNL and GNL

Figure 13 Finite element model of the block

(a)

(b)

Figure 14 Optimum results of the flange with different nonlinearities Isometric view

Side view

(c) CNL

(d) MNL and CNL

(e) GNL and CNL

(f) MNL, GNL and CNL

Table 1 Strain energy values (Nm) of the Michell structure optimum results

MNL

GNL

MNL and GNL

Strain energy with linear optimum result

171.82

75.35

172.00

Strain energy with nonlinear optimum result

163.02

74.80

160.67

Table 2 Strain energy values (Nm) of the spacer grid optimum results

MNL

GNL

MNL and GNL

Strain energy with linear optimum result

10.64

15.45

13.62

Strain energy with nonlinear optimum result

9.82

14.02

11.89

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