A GENERATIVE DESIGN OPTIMIZATION APPROACH FOR ADDITIVE MANUFACTURING
4 NUMERICAL EXAMPLES
The proposed SVM-based postprocessing approach of SIMP-based TO solutions is demon-strated in this section by considering two three-dimensional benchmarks in different settings.
The approach is implemented in our in-house toolbox TopoBox1 using Matlab and Fortran (mex-files). All stl-files presented in this work are generated using this implementation with TopoBox following the workflow presented in Figure 1.
The first example is a cubic design domain, where four corner of one of the square faces are fixed and one of the opposite edge is subjected to a vertical force at the middle according to
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N. Str¨omberg
Figure 3. The discretization of the design domain is performed using 18350 linear tetrahedral elements of equal size. A TO solution is generated after 50 iterations using the algorithm presented in Section 2, see Figure 3, and this solution is the input to the 1-norm SVM by letting elements with zero density having category yi= −1 and elements with density one having yi=1. The 1-norm SVM problem in (21) is solved usinglinprog of Matlab and the optimal solution consists of 282 support vectors. The corresponding optimal SVM-based boundary in (22) is shown in Figure 3. The volume of the corresponding stl-model is 25.1 percent of the design volume.
Figure 6:3D-printing of the two benchmarks with and without graded lattice structures.
The next example is one of Michell’s benchmarks [16]. The design domain is taken to be a rectangular cuboid using 15565 linear tetrahedral elements. The bottom rectangle is fixed at the corners and a force is applied at the center of this rectangle, see Figure 4. The optimal TO solution shown in Figure 4 is obtained after 50 iterations. This solution is again categorized according to the principle shown in Figure 2. The 1-norm SVM problem in (21) is solved and the solution contains of 130 support vectors. The corresponding SVM-based boundary in (22) is shown in Figure 4. The volume of the corresponding stl-model is 23.7 percent of the design volume.
In conclusion, our discrete optimal SIMP-based TO solutions are now represented by smooth support vector machines. Notice that our SVM-based postprocessing approach applied on
N. Str¨omberg
coarse discrete TO solutions generates continuous smooth representations of the optimal ge-ometries automatically. This is of course most preferable from a computational point of view.
In addition, the SVM in (22) is extremely fast since it is completely defined by the support vectors (summation in (22) is only needed for the support vectors, i.e. over indexi satisfying ui=0) and one additional bias. Thus, we can now start to operate quickly on these surfaces in different manners. Our specific goal is to have a powerful tool that automatically generates stl-files for additive manufacturing. In Figure 5, this is demonstrated by using our two SVMs established above to generate stl-files with graded Schwarz-D lattice structures automatically.
Here, the grading is performed at the supports and the loads, see Figure 5. Thus, the volume of the two optimal design are reduced even further. The volume of the optimal design of the block with graded lattice is now 13.4 percent of the original design domain, and the volume of the Michell structure with graded lattice structure is 14.4 percent of the design domain. Figure 6 shows the corresponding 3D-printed components of the generated stl-files for these two bench-marks with and without graded lattice structures2. A recent paper on graded lattice structures was presented by Panesar et al. [17].
Next, we can set up design of experiments by morphing the SVM-based representation of the TO-concepts. In such manner, detailed design optimization of the TO-concepts by using meta-models can be applied. This is illustrated in Figure 7. More details concerning the morphing and metamodelling of the TO-based concepts can be found in [18]. Some other recent papers on metamodel-based design optimization are given by Str¨omberg in [19, 20].
Figure 7:The generative design approach for additive manufacturing.
2A link to these stl-files can be obtained by sending an e-mail to the author.
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5 CONCLUSIONS
In this paper a new SVM-based approach for automatic postprocessing of SIMP-based TO solutions is presented. By using the SVM-based representation, the TO-based concepts can be modified by morphing and adding lattice structures. Furthermore, design of experiments of these modifications can easily be set up and detailed design optimization by using metamodel can be applied. These steps constitute our proposed generative design optimization approach, see Figure 7.
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