# Stiffness optimization of non-linear elastic structures

## Abstract

Our paper revisits stiffness optimization of non-linear elastic structures. Due to the non-linearity, several possible stiffness measures can be identified and in this work conventional compliance, i.e. secant stiffness designs are compared to tangent stiffness designs. The optimization problem is solved by the method of moving asymptotes and the sensitivities are calculated using the adjoint method. And for the tangent cost function it is shown that although the objective involves the third derivative of the strain energy an efficient formulation for calculating the sensitivity can be obtained. Loss of convergence due to large deformations in void regions is addressed by using a fictitious strain energy such that small strain linear elasticity is approached in the void regions. We formulate a well-posed topology optimization problem by using restriction which is achieved via a Helmholtz type filter. The numerical examples provided show that for low load levels, the designs obtained from the different stiffness measures coincide whereas for large deformations significant differences are observed.

- Authors:

- Lund Univ. (Sweden). Division of Solid Mechanics
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Design and Optimization

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1416502

- Report Number(s):
- LLNL-JRNL-731767

Journal ID: ISSN 0045-7825; TRN: US1800942

- Grant/Contract Number:
- AC52-07NA27344

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Computer Methods in Applied Mechanics and Engineering

- Additional Journal Information:
- Journal Volume: 330; Journal Issue: C; Journal ID: ISSN 0045-7825

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 42 ENGINEERING; topology optimization; stiffness optimization; finite strains; non-linear elasticity

### Citation Formats

```
Wallin, Mathias, Ivarsson, Niklas, and Tortorelli, Daniel.
```*Stiffness optimization of non-linear elastic structures*. United States: N. p., 2017.
Web. doi:10.1016/j.cma.2017.11.004.

```
Wallin, Mathias, Ivarsson, Niklas, & Tortorelli, Daniel.
```*Stiffness optimization of non-linear elastic structures*. United States. doi:10.1016/j.cma.2017.11.004.

```
Wallin, Mathias, Ivarsson, Niklas, and Tortorelli, Daniel. Mon .
"Stiffness optimization of non-linear elastic structures". United States.
doi:10.1016/j.cma.2017.11.004. https://www.osti.gov/servlets/purl/1416502.
```

```
@article{osti_1416502,
```

title = {Stiffness optimization of non-linear elastic structures},

author = {Wallin, Mathias and Ivarsson, Niklas and Tortorelli, Daniel},

abstractNote = {Our paper revisits stiffness optimization of non-linear elastic structures. Due to the non-linearity, several possible stiffness measures can be identified and in this work conventional compliance, i.e. secant stiffness designs are compared to tangent stiffness designs. The optimization problem is solved by the method of moving asymptotes and the sensitivities are calculated using the adjoint method. And for the tangent cost function it is shown that although the objective involves the third derivative of the strain energy an efficient formulation for calculating the sensitivity can be obtained. Loss of convergence due to large deformations in void regions is addressed by using a fictitious strain energy such that small strain linear elasticity is approached in the void regions. We formulate a well-posed topology optimization problem by using restriction which is achieved via a Helmholtz type filter. The numerical examples provided show that for low load levels, the designs obtained from the different stiffness measures coincide whereas for large deformations significant differences are observed.},

doi = {10.1016/j.cma.2017.11.004},

journal = {Computer Methods in Applied Mechanics and Engineering},

number = C,

volume = 330,

place = {United States},

year = {Mon Nov 13 00:00:00 EST 2017},

month = {Mon Nov 13 00:00:00 EST 2017}

}

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