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Multi-grid reduced-order topology optimization

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Abstract

Additive printing allows the “single step” production of virtually any complex mechanical component. However, the manufacturing process involves a layer-by-layer deposition of material, which leads to an anisotropic mechanical behavior of the whole component. This would then entail a very fine 3D model to simulate the mechanical performance accurately. This simulation would also need to be integrated within an iterative design process in order to obtain the most efficient design. Both reasons explain the prohibitive number of calculations needed, which is currently beyond the capacities of existing software and computers. Recent research papers have opened promising pathways for integrating model reduction techniques within the overall topology optimization process. However, these approaches still present challenges such as choosing the minimum number and appropriate selection of the snapshots required to get accurate simulations. In this work, we present a methodology in the combined field of reduced-order modeling and topology optimization. The key idea consists of projecting the higher dimensional system of equations onto a lower dimensional space with the reduced basis vectors constructed using Proper Orthogonal Decomposition (POD). This reduced basis is updated in an incremental “on-the-fly” manner using alternatively costly high-fidelity and more rapid lower fidelity simulation snapshots. The variable-fidelity resolutions of successive approximations to the global system of equations are then integrated into the topology optimization process. The approaches are tested and computational savings and precision are compared, using both minimum compliance and compliant mechanism design benchmark problems.

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Funding

This study was supported by the National Natural Science Foundation of China (Grant No. 11620101002 and Grant No. 11972166) and the Fundamental Research Funds for the Central Universities (Grant No. 310201911cx029).

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Authors and Affiliations

  1. **’an Key Laboratory of Scientic Computation and Applied Statistics Department of Applied Mathematics, School of Mathematics and Statistics, Northwestern Polytechnical University, **’an, 710072, People’s Republic of China

    Manyu **ao & Dongcheng Lu

  2. Laboratoire Roberval FRE UTC-CNRS 2012, Alliance Sorbonne Universités, Université de Technologie de Compiègne, Compiègne, France

    Piotr Breitkopf

  3. Laboratoire de Génie Civil et Génie Mécanique EA 3913, Institut National des Sciences Appliquées de Rennes, Rennes, France

    Balaji Raghavan

  4. State IJR Center of Aerospace Design and Additive Manufacturing, Northwestern Polytechnical University, **’an, People’s Republic of China

    Weihong Zhang

  5. Department of Civil Engineering, National Institute of Technology Silchar, Silchar, India

    Subhrajit Dutta

Authors
  1. Manyu **ao
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  2. Dongcheng Lu
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  3. Piotr Breitkopf
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  4. Balaji Raghavan
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  5. Weihong Zhang
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Correspondence to Manyu **ao.

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Replication of results

The source codes in this work are an evolution of the 88-line Matlab code and 169-line code, according to the proposed methodology, along with the definition of test cases, which allow to reproduce the numerical results presented in this paper. These codes could be made available on request by emailing the corresponding author.

Appendix

Appendix

In order to determine the global interpolation matrix P, we need to number the degrees of freedom (DOF) as shown in Figure 26.

In Fig. 26, the numbering sequence is from top to bottom and from left to right, which is corresponding to the number order of DOF in the 99-line and 88-line code.

Figure 27 shows an incomplete matrix, which is actually the embodiment of interpolation of each element in the whole. Row and column indexes indicate the weight location in the global matrix. In the end, by assembling the interpolation weights of each element, we obtain all global interpolation matrices between adjacent layers of meshes.

Fig. 27
figure 27

global interpolation matrix Ω2 →Ω1

Full size image

Here

$$ \begin{array}{@{}rcl@{}} &&a=s \\ &&b=s+2(nelyy+1) \end{array} $$
(23)
$$ \begin{array}{@{}rcl@{}} &&c=2s-1+2nely(t-1)/2(nelyy+1) \\ &&d=2s-1+2nely(t-1)/2(nelyy+1)+2(nely+1) \\ &&e=2s-1+2nely(t-1)/2(nelyy+1)+4(nely+1) \end{array} $$
(24)
$$ \begin{array}{@{}rcl@{}} &&s=1,3,5,\cdots,2nelyy+t-1 \\ &&t=1,1+2(nelyy+1),1+4(nelyy+1),\cdots,2nelxx(nelyy+1) \end{array} $$
(25)
$$ \begin{array}{@{}rcl@{}} &&nelxx=nelx/2,nelyy=nely/2 \end{array} $$
(26)

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**ao, M., Lu, D., Breitkopf, P. et al. Multi-grid reduced-order topology optimization. Struct Multidisc Optim 61, 1–23 (2020). https://doi.org/10.1007/s00158-020-02570-y

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