Shape preserving design of geometrically nonlinear structures using topology optimization

Abstract

Subparts of load carrying structures like airplane windows or doors must be isolated from distortions and hence structural optimization needs to take such shape preserving constraints into account. The paper extends the shape preserving topology optimization approach from simple linear load cases into geometrically nonlinear problems with practical significance. Based on an integrated deformation energy function, an improved warpage formulation is proposed to measure the geometrical distortion during large deformations. Structural complementary elastic work is assigned as the objective function. The average distortion calculated as the integrated deformation energy accumulated in the incremental loading process is accordingly constrained to obtain warpage control. In the numerical implementation, an energy interpolation scheme is utilized to alleviate numerical instability in low stiffness regions. An additional loading case avoids isolation phenomena. Optimization results show that shape preserving design is successfully implemented in geometrically nonlinear structures by effectively suppressing local warping deformations.

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Funding

Yu Li received financial support from CSC (China Scholarship Council). Fengwen Wang and Ole Sigmund received support from the Villum foundation through the VILLUM Investigator project InnoTop. This work is also supported by the National Key Research and Development Program (2017YFB1102800) and the National Natural Science Foundation of China (11722219, 11620101002, 51790171, 5171101743).

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Correspondence to Yu Li or Jihong Zhu.

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Appendix: Sensitivity analysis of the integrated deformation energy function in shape preserving domains

Appendix: Sensitivity analysis of the integrated deformation energy function in shape preserving domains

From (11), the integrated deformation energy function in the shape preserving domain is calculated with the trapezoidal method

$$\begin{array}{@{}rcl@{}} {\Phi}_{{\Omega}_{s}} &\approx&\frac{1}{2}\left( (\boldsymbol f_{{\Omega}_{s}}^{0}+\boldsymbol f_{{\Omega}_{s}}^{1})^{\mathrm{T}}(\boldsymbol{u}_{{\Omega}_{s}}^{1}-\boldsymbol{u}_{{\Omega}_{s}}^{0})\right.\\ &&+ \left. {\sum}_{i= {2}}^{n-1}(\boldsymbol f_{{\Omega}_{s}}^{{i-1}}+\boldsymbol f_{{\Omega}_{s}}^{{i}})^{\mathrm{T}}(\boldsymbol{u}_{{\Omega}_{s}}^{{i}}-\boldsymbol{u}_{{\Omega}_{s}}^{ {i-1}})\right.\\ &&+ \left. (\boldsymbol f_{{\Omega}_{s}}^{n-1}+\boldsymbol f_{{\Omega}_{s}}^{n})^{\mathrm{T}}(\boldsymbol{u}_{{\Omega}_{s}}^{n}-\boldsymbol{u}_{{\Omega}_{s}}^{n-1}) \right)\\ &=&\frac{1}{2}\left( (\boldsymbol f_{{\Omega}_{s}}^{1^{\mathrm{T}}}\boldsymbol{u}_{{\Omega}_{s}}^{2}-\boldsymbol f_{{\Omega}_{s}}^{2^{\mathrm{T}}}\boldsymbol{u}_{{\Omega}_{s}}^{1})+ \sum\limits_{i= {2}}^{n-1}(\boldsymbol f_{{\Omega}_{s}}^{i^{\mathrm{T}}}\boldsymbol{u}_{{\Omega}_{s}}^{i + 1}\right.\\ &&- \left. \boldsymbol f_{{\Omega}_{s}}^{{i + 1}^{\mathrm{T}}}\boldsymbol{u}_{{\Omega}_{s}}^{i})+ \boldsymbol f_{{\Omega}_{s}}^{n^{\mathrm{T}}}\boldsymbol{u}_{{\Omega}_{s}}^{n} \right) , \end{array} $$
(A.1)

where \(\boldsymbol f_{{\Omega }_{s}}^{i}\) and \(\boldsymbol {u}_{{\Omega }_{s}}^{i}\) are the internal force and displacement vector of the shape preserving domain Ωs in the i th load increment, respectively.

Sensitivity of the integrated deformation energy function in the shape preserving domain with respect to the physical density variable \(\overline \rho _{e}\) is calculated using adjoint sensitivity analysis. The derivation is further expressed as

$$\begin{array}{@{}rcl@{}} \frac{\mathrm d{\Phi}_{{\Omega}_{s}}}{\mathrm d\overline\rho_{e}} &=&\sum\limits_{i = 1}^{n} \left[ \frac{\partial{\Phi}_{{\Omega}_{s}}}{\partial\boldsymbol{u}_{i}}\frac{\partial\boldsymbol{u}_{i}}{\partial\overline\rho_{e}}+ {(\boldsymbol\lambda_{i} )}^{\text T} \left( - {\boldsymbol K_{i}^{\tan}}\frac{\partial\boldsymbol{u}_{i}}{\partial\overline\rho_{e}}+\frac{\partial\boldsymbol r_{i}}{\partial\overline\rho_{e}} \right) \right]\\ &=&\sum\limits_{i = 1}^{n} \left[ \left( \frac{\partial {\Phi}_{{\Omega}_{s}}}{\partial\boldsymbol{u}_{i}}- {(\boldsymbol\lambda_{i} )}^{\text T} {\boldsymbol K_{i}^{\tan}}\right) \frac{\partial\boldsymbol{u}_{i}}{\partial\overline\rho_{e}} +{(\boldsymbol\lambda_{i} )}^{\text T}\frac{\partial\boldsymbol r_{i}}{\partial\overline\rho_{e}} \right],\\ \end{array} $$
(A.2)

where \( {\boldsymbol K_{i}^{\tan }}\) is the symmetrical tangent stiffness matrix for increment i, and \({\partial \boldsymbol r_{i}}/{\partial \boldsymbol {u}_{i}}=- {\boldsymbol K_{i}^{\tan }}\). Thus, the corresponding adjoint load for the i th load increment is expressed as

$$\begin{array}{@{}rcl@{}} {\boldsymbol K_{i}^{\tan}}\boldsymbol\lambda_{i} = \frac{\partial {\Phi}_{{\Omega}_{s}}}{\partial\boldsymbol{u}_{i}}&=& \left\{\begin{array}{lllll} \frac{1}{2}\frac{\partial\left( \boldsymbol f_{{\Omega}_{s}}^{1}\boldsymbol{u}_{{\Omega}_{s}}^{2}-\boldsymbol f_{{\Omega}_{s}}^{2}\boldsymbol{u}_{{\Omega}_{s}}^{1}\right)} {\partial\boldsymbol{u}_{1}} \\ \frac{1}{2}\frac{\partial\left( (\boldsymbol f_{{\Omega}_{s}}^{i-1}\boldsymbol{u}_{{\Omega}_{s}}^{i}-\boldsymbol f_{{\Omega}_{s}}^{i}\boldsymbol{u}_{{\Omega}_{s}}^{i-1})+ (\boldsymbol f_{{\Omega}_{s}}^{i}\boldsymbol{u}_{{\Omega}_{s}}^{i + 1}-\boldsymbol f_{{\Omega}_{s}}^{i + 1}\boldsymbol{u}_{{\Omega}_{s}}^{i})\right)}{\partial\boldsymbol{u}_{i}}\\ \frac{1}{2}\frac{\partial\left( (\boldsymbol f_{{\Omega}_{s}}^{n-1}\boldsymbol{u}_{{\Omega}_{s}}^{n}-\boldsymbol f_{{\Omega}_{s}}^{n}\boldsymbol{u}_{{\Omega}_{s}}^{n-1})+ \boldsymbol f_{{\Omega}_{s}}^{n}\boldsymbol{u}_{{\Omega}_{s}}^{n}\right)}{\partial\boldsymbol{u}_{n}} \end{array}\right.\\ &=& \left\{\begin{array}{llll} \frac{1}{2}\left( {\boldsymbol K_{i}^{\tan}}\boldsymbol{u}_{{\Omega}_{s}}^{2}-\boldsymbol f_{{\Omega}_{s}}^{2}\right) & \text{for } i = 1\\ \frac{1}{2}\left( {\boldsymbol K_{i}^{\tan}}(\boldsymbol{u}_{{\Omega}_{s}}^{i + 1}-\boldsymbol{u}_{{\Omega}_{s}}^{i-1})+ (\boldsymbol f_{{\Omega}_{s}}^{i-1}-\boldsymbol f_{{\Omega}_{s}}^{i + 1})\right) & \text{for } i = 2 \text{to} n-1\\ \frac{1}{2}\left( {\boldsymbol K_{i}^{\tan}}(\boldsymbol{u}_{{\Omega}_{s}}^{n}-\boldsymbol{u} _{{\Omega}_{s}}^{n-1})+ (\boldsymbol f_{{\Omega}_{s}}^{n-1}+\boldsymbol f_{{\Omega}_{s}}^{n})\right) & \text{for } i=n \end{array}\right.. \end{array} $$
(A.3)

By inserting the solutions λi of these linear systems into (A.2), the final sensitivity is found as

$$ \frac{\mathrm d{\Phi}_{{\Omega}_{s}}}{\mathrm d\overline\rho_{e}}= \sum\limits_{i = 1}^{n}{\left( \boldsymbol\lambda_{i} \right)}^{\text T}\frac{\partial\boldsymbol r_{i}}{\partial\overline\rho_{e}} . $$
(A.4)

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Li, Y., Zhu, J., Wang, F. et al. Shape preserving design of geometrically nonlinear structures using topology optimization. Struct Multidisc Optim 59, 1033–1051 (2019). https://doi.org/10.1007/s00158-018-2186-x

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Keywords

  • Shape preserving design
  • Topology optimization
  • Geometrical nonlinearity
  • Integrated deformation energy