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A 213-line topology optimization code for geometrically nonlinear structures

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Abstract

This paper presents a 213-line MATLAB code for topology optimization of geometrically nonlinear structures. It is developed based on the density method. The code adopts the ANSYS parametric design language (APDL) that provides convenient access to advanced finite element analysis (FEA). An additive hyperelasticity technique is employed to circumvent numerical difficulties in solving the material density-based topology optimization of elastic structures undergoing large displacements. The sensitivity information is obtained by extracting the increment of the element strain energy. The validity of the code is demonstrated by the minimum compliance problem and the compliant inverter problem.

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Funding

This work is supported by the National Natural Science Foundation of China (Grant Nos. 51820105007, U1501247 and 51605166). These support is greatly appreciated. Additionally, the authors thank Dr. K. Svanberg at KTH (Stockholm, Sweden) for providing the MMA code for academic research.

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Authors and Affiliations

  1. Guangdong Key Laboratory of Precision Equipment and Manufacturing Technology, South China University of Technology, Guangzhou, China

    Qi Chen, Xianmin Zhang & Benliang Zhu

Authors
  1. Qi Chen
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  2. Xianmin Zhang
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  3. Benliang Zhu
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Correspondence to Xianmin Zhang.

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Appendices

Appendix A

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figure b

Appendix B

Fig. 10
figure 10

The deformation of the first iteration

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Table 5 The finite difference check of the sensitivity (in J)
Full size table

A finite difference check is performed to validated the approximate sensitivity given in Section 2.3. The optimization problem in Section 4.1 with 2 × 8 elements is considered. The additive hyperelasticity technique is utilized. The external load is applied in one node instead of uniformly distributed over nine nodes (Luo et al. 2015) in Section 4.1. The applied external force is 250 N. The sensitivity in the first optimization iteration is checked. The initial density, \(\tilde {x}_{i}\), of all elements is 0.8. The deformation of the first iteration is shown in Fig. 10.

The approximate sensitivity, \({\partial C}/{\partial {{{\tilde {x}}}_{i}}}\), in (20) is obtained. Then, for each element, we reduce the element density, \(\tilde {x}_{i}\), to 0.795 and calculate the corresponding structure compliance. According to the finite difference method, the estimated sensitivity, \({\partial C_{d}}/{\partial {{{\tilde {x}}}_{i}}}\), is given by the following:

$$ {{\frac{\partial C_{d}}{\partial {{{\tilde{x}}}_{i}}}=\frac{{\Delta} C}{{\Delta} {{{\tilde{x}}}_{i}}}}} $$
(27)

where \({\Delta } {{{\tilde {x}}}_{i}}= 0.795-0.8=-~0.005\), ΔC is the increment of the structure compliance.

\({\partial C}/{\partial {{{\tilde {x}}}_{i}}}\) and \({\partial C_{d}}/{\partial {{{\tilde {x}}}_{i}}}\) for each element are given in Table 5.

The approximate sensitivity \({\partial C}/{\partial {{{\tilde {x}}}_{i}}}\) in (20) is nearly the same as the estimated sensitivity, \({\partial C_{d}}/{\partial {{{\tilde {x}}}_{i}}}\), obtained by the finite difference method. The accuracy of the approximation is validated.

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Chen, Q., Zhang, X. & Zhu, B. A 213-line topology optimization code for geometrically nonlinear structures. Struct Multidisc Optim 59, 1863–1879 (2019). https://doi.org/10.1007/s00158-018-2138-5

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