Topology Optimization and 3D Printing of the MBB Structure Using the 3D SIMP Method

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1 Topology Optimization and 3D Printing of the MBB Structure Using the 3D SIMP Method Yedige Tlegenov Department of Mechanical Engineering National University of Singapore 9 Engineering Drive 1, Singapore, , Singapore yedige@nus.edu.sg Abstract - The topology optimization of the 3D beam was performed under different filtering and solver techniques in the current study using 3D SIMP method. Five different filtering techniques were implemented and compared. The finite element analysis with three types of mesh sizes was conducted via direct solver and iterative solver. The optimized structure was prototyped using 3D printer. I. INTRODUCTION Recently with the development of additive manufacturing processes there are more opportunities on fabricating almost any geometry of the structure. Considering this, there is a need in developing computational methods that will allow shape optimization for reducing the weight of the structure. Topology optimization is a computational technique that allows the structures to withstand maximum loading with minimum material consumption. Thus, topology optimization is carried out by finite element analysis of the structure and then evaluation of the density distribution in order to remove material with least amount of stresses. The most common tool that is being used widely for finite element analysis and topology optimization is Matlab. For example, Sigmund (2001) presented a Matlab program for 2D topology optimization, which computes the stiffness matrix and performs filtering by nested loops. Another example can be a 3D topology optimization program (Liu and Tovar, 2014) developed for Matlab. The advantages of the 3D topology optimization program are not only in terms of extra dimension, but also in terms of relatively less solving time and more number of filtering techniques. This study presents the topology optimization of 3D beam using the Matlab program developed by Liu and Tovar (2014) using different filtering and solver techniques as well as prototyping the optimized structure using 3D printer. A Messerchimitt-Bolkow-Blohm (MBB) beam is considered with one load at the middle point in its top surface, as shown in Table 4, middle row in the paper (Liu and Tovar, 2014). The load can be considered as a unit load. The material has a modulus of elasticity of 100 MPa (or the 1

2 unity) and the Poisson s ratio of 0.3. Set L equal to 1.0 cm. The aim of current study is to perform two following parts: Part 1. Conduct the topology optimization of the MBB structure for the following cases of filters: 1. Density filter only; 2. Sensitivity filter only; 3. Grey scale filter only; 4. Density filter and grey scale filter; 5. Sensitivity filter and grey scale filter. Among the five cases, for this MBB structure, indicate which case provides the most blackand-white solution. Part 2. Conduct the topology optimization of the MBB structure, using the grey scale filter only, for the following cases of mesh size: a. 30 x 5 x 5 b. 60 x 10 x 10 c. 120 x 20 x 20 In addition, compare two methods, namely, the direct solver and the iterative solver and evaluate the results. Part 3. Prototype the resulting structure using 3D printer. II. METHODOLOGY Let us consider the given structure, which is illustrated in Figure 1. The applied central load is 1 N, and the structure is supported horizontally in all four bottom corners. The topology optimization of the MBB structure using the 3D SIMP method for the minimization of the strain energy function was conducted. The Matlab code was downloaded from reference paper (Liu and Tovar, 2014). The code was modified as per given problem with applied load and boundary conditions. Firstly, the loading condition was modified in line 12 as 12 [il,jl,kl] = meshgrid(nelx/2, nely, nelz/2); which is a point load acting centre of the top surface of the structure. Next, the supports were modified in line 16 as 16 iif = [0 0 nelx nelx]; jf = [ ]; kf = [0 nelz 0 nelz]; 2

3 which are four supports placed on the corners of the MBB structure. Figure 1. Structure with given boundary conditions and applied loads Part 1. Density filter and Sensitivity filter were performed by replacing the real sensitivities with the filtered sensitivities. The line 2 was changed by adding one input ft (ft = 1 for density filter, ft = 2 for sensitivity filter). Next, the sensitivity filter was added to the program, by changing lines as (Liu and Tovar, 2014): if ft == 1 dc(:) = H*(dc(:)./Hs); dv(:) = H*(dv(:)./Hs); elseif ft == 2 dc(:) = H*(x(:).*dc(:))./Hs.max(1e-3,x(:)); end Furthermore, the design variable update strategy (line 86) in the optimal search procedure was changed as (Liu and Tovar, 2014): 86 if ft == 1 xphys(:) = (H*xnew(:))./Hs; elseif ft == 2 xphys = xnew; end Gray scale filter was performed by changing the optimality criteria update, by adding input q (q=2) to the program and modifying line 85 as (Liu and Tovar, 2014): 85 xnew = max(0,max(x-move,min(1,min(x+move,(x.*sqrt(- dc./dv/lmid)).^q)))); Part 2. Iterative solver appears to be very useful when the mesh size is comparatively large and the direct solver in line 72 takes long time to compute the finite element analysis. Thus, line 72 3

4 was replaced by a built-in preconditioned conjugate gradients method function, which was called as pcg (Liu and Tovar, 2014): 72 tolit = 1e-8; maxit = 8000; M = diag(diag(k(freedofs, freedofs))); U(freedofs,:) = pcg (K(freedofs, freedofs), F(freedofs,:), tolit, maxit, M); After the above mentioned changes were made to the Matlab program (Liu and Tovar, 2014) we were able to perform topology optimization under different input parameters which will be discussed below. Part 3. The resulting optimized structure was prototyped using Makerbot Replicator 3D printer. The printing material is PLA plastic with no supports; all the sizes are in mm. III. RESULTS The results are elaborated as follows. The topology optimization of the MBB structure using the 3D SIMP method was performed under different filtering techniques and the iterative solver. Part 1. Figure 2 illustrates the optimized MBB structures after performing five different filtering techniques. It can be seen that density filter, sensitivity filter, and gray scale filter minimize the checkerboard patterns if there are any. From all five filtering methods, the sensitivity filter combined with the gray scale filter results in the most black-and-white pattern. Part 2. According to the Table 1, the comparison between two finite element analyses solvers is presented (namely, direct solver and iterative solver). As can be noted, the iterative solver takes about 20 times less time to solve the large mesh size structure. In contrast, the direct solver performs faster when the mesh size is relatively small. It can be concluded that the iterative solver is more efficient for large mesh size structure, and vice versa. Part 3. Figure 3 presents the 3D printed prototype of optimized structure. The material consumption for the given part was relatively minimal as compare to the solid rectangular beam, which resulted in 25 grams of PLA plastic usage for optimized structure. 4

5 Density Filter Sensitivity Filter Gray Scale Filter Density Filter and Grey Scale Filter Sensitivity Filter and Grey Scale Filter Figure 2. Topology optimized design using five different filtering techniques Table 1. Time usage of finite element analysis time for different solvers Mesh size Direct Solver Iterative Solver 30 x 5 x sec sec 60 x 10 x sec sec 120 x 20 x sec sec 5

front right top isometric Figure 3. 3D printed topology optimized structure views IV.

techniques in the current paper. Five different filtering techniques were implemented and compared.

The finite element analysis with three types of mesh sizes was performed via direct solver and iterative solver.

The optimized structure was prototyped using 3D printer and the material consumption was relatively less than the solid structure.

6 front right top isometric Figure 3. 3D printed topology optimized structure views IV. CONCLUSION The topology optimization of the MBB structure using the 3D SIMP method was performed under different filtering and solver techniques in the current paper. Five different filtering techniques were implemented and compared. The results show that the sensitivity filter combined with the gray scale filter tends to have the most black-and-white pattern. The finite element analysis with three types of mesh sizes was performed via direct solver and iterative solver. The results represent that the iterative solver is more convenient for large scale problems. The optimized structure was prototyped using 3D printer and the material consumption was relatively less than the solid structure. REFERENCES Liu, Kai, and Andrés Tovar. "An efficient 3D topology optimization code written in Matlab." Structural and Multidisciplinary Optimization 50.6 (2014): Sigmund, Ole. "A 99 line topology optimization code written in Matlab." Structural and Multidisciplinary Optimization 21.2 (2001):

A first laboratory exercise in topology optimization using matlab.

A first laboratory exercise in topology optimization using matlab. Andreas Rietz Department of Mechanical Engineering / Department of Mathematics Linköping University SE-58183 Linköping, Sweden Abstract