PolyStress: a Matlab implementation for local stress-constrained topology optimization using the augmented Lagrangian method

Abstract

We present PolyStress, a Matlab implementation for topology optimization with local stress constraints considering linear and material nonlinear problems. The implementation of PolyStress is built upon PolyTop, an educational code for compliance minimization on unstructured polygonal finite elements. To solve the nonlinear elasticity problem, we implement a Newton-Raphson scheme, which can handle nonlinear material models with a given strain energy density function. To solve the stress-constrained problem, we adopt a scheme based on the augmented Lagrangian method, which treats the problem consistently with the local definition of stress without employing traditional constraint aggregation techniques. The paper discusses several theoretical aspects of the stress-constrained problem, including details of the augmented Lagrangian-based approach implemented herein. In addition, the paper presents details of the Matlab implementation of PolyStress, which is provided as electronic supplementary material. We present several numerical examples to demonstrate the capabilities of PolyStress to solve stress-constrained topology optimization problems and to illustrate its modularity to accommodate any nonlinear material model. Six appendices supplement the paper. In particular, the first appendix presents a library of benchmark examples, which are described in detail and can be explored beyond the scope of the present work.

Appendices

Appendix A: Library of benchmark examples

We provide a summary of the design problems discussed herein, including a description of the design domain and the names of the Matlab files needed to generate the finite element mesh for each problem.

Table 6 Examples provided with PolyStress

Appendix B: Imposing symmetry using the regularization filter

In a continuum setting, we can impose symmetries to the space of admissible density fields by means of the operator $\mathcal {P}_{s}(\eta (\textbf {x}))$, which maps the design function ηx according to the desired symmetries (e.g., $\mathcal {P}_{s}(\eta (\textbf {x}))=\eta (x_{1},|{x_{2}}|$) if we desire symmetries with respect to the x₁-axis). The symmetrized space of admissible density fields is given by (12) and leads to a regularization mapping of the form $\mathcal {P}(\eta )=(\mathcal {P}_{F}\circ \mathcal {P}_{s})(\eta )$.

Now, in a discretized setting, the symmetrized filter matrix P (i.e., the discrete counterpart of $\mathcal {P}$), becomes:

$$ P_{\ell k}=\frac{w_{\ell k}v_{k}}{\sum\limits_{j=1}^{N}w_{\ell j}v_{j}},~~w_{\ell k}=\max\left( 0,1-\frac{\|\tilde{\textbf{x}}_{\ell}-\tilde{\textbf{x}}_{k}\|_{2}}{R}\right)^{q}, $$

(74)

where $\tilde {\textbf {x}}_{\ell }$ (or $\tilde {\textbf {x}}_{k}$) are given by:

$$ \tilde{\textbf{x}}_{\ell}=\left[x_{1}^{\ell},~|x_{2}^{\ell}|\right]^{T}, $$

(75)

when imposing symmetry about the x₁-axis:

$$ \tilde{\textbf{x}}_{\ell}=\left[|x_{1}^{\ell}|,~x_{2}^{\ell}\right]^{T}, $$

(76)

when imposing symmetry about the x₂-axis, and:

$$ \tilde{\textbf{x}}_{\ell}=\left[|x_{1}^{\ell}|,~|x_{2}^{\ell}|\right]^{T}, $$

(77)

when imposing symmetry about both the x₁-axis and the x₂-axis. The capability of imposing symmetry to the designs through the filter matrix has been added to the PolyFilter routine, which can be found in the electronic supplementary material.

Appendix C: Continuity proof for the bilinear material model

As discussed by Curnier et al. (1994), the specific strain energy function from (59) is continuous in the entire space of infinitesimal strains, $\mathcal {E}$. In fact, according to (59), the material behaves as a linear material with properties (λ_t, μ) when χ(ε) = tr(ε) > 0 (i.e., in the tension sub-domain, $\mathcal {E}_{t}$) and as a linear material with properties (λ_c, μ) when χ(ε) = tr(ε) < 0 (i.e., in the compression sub-domain, $\mathcal {E}_{c}$). Below, we demonstrate that both the strain energy and the stress tensor are continuous in the entire space of infinitesimal strains.

First, we demonstrate that the strain energy density function (59) is continuous in $\mathcal {E}$. According to (59)–(60), if tr(ε) > 0:

$$ W_{0}=W_{0}^{+}=\frac{1}{2}\lambda_{t}\text{tr}^{2}(\boldsymbol{\varepsilon})+\mu\text{tr}(\boldsymbol{\varepsilon}^{2}), $$

(78)

which is a twice continuously differentiable function on $\mathcal {E}_{t}$. Similarly, if tr(ε) < 0:

$$ W_{0}=W_{0}^{-}=\frac{1}{2}\lambda_{c}\text{tr}^{2}(\boldsymbol{\varepsilon})+\mu\text{tr}(\boldsymbol{\varepsilon}^{2}), $$

(79)

which is a twice continuously differentiable function on $\mathcal {E}_{c}$. The strain energy is also continuous across the interface between the tension and the compression sub-domains (i.e., when tr(ε) = 0) because:

$$ W_{0}\left.\right|_{\text{tr}(\boldsymbol{\varepsilon})=0}=W_{0}^{+}\left.\right|_{\text{tr}(\boldsymbol{\varepsilon})=0}=W_{0}^{-}\left.\right|_{\text{tr}(\boldsymbol{\varepsilon})=0}=\mu\text{tr}(\boldsymbol{\varepsilon}^{2}). $$

(80)

Following the same procedure, we now demonstrate that the stress tensor, $\boldsymbol {\sigma }=\frac {\partial W_{0}}{\partial {\varepsilon }}$, is also continuous across the entire strain space. According to (60) and (61)₁, if tr(ε) > 0:

$$ \boldsymbol{\sigma}(\boldsymbol{\varepsilon})=\boldsymbol{\sigma}^{+}=\lambda_{t}\text{tr}(\boldsymbol{\varepsilon})\textbf{I}+2\mu\boldsymbol{\varepsilon}, $$

(81)

which is continuously differentiable on $\mathcal {E}_{t}$. Similarly, if tr(ε) < 0:

$$ \boldsymbol{\sigma}(\boldsymbol{\varepsilon})=\boldsymbol{\sigma}^{-}=\lambda_{c}\text{tr}(\boldsymbol{\varepsilon})\textbf{I}+2\mu\boldsymbol{\varepsilon}, $$

(82)

which is continuously differentiable on $\mathcal {E}_{c}$. Now, across the interface between the tension and the compression sub-domains (i.e., when tr(ε) = 0), we have:

$$ \boldsymbol{\sigma}(\boldsymbol{\varepsilon})\left.\right|_{\text{tr}(\boldsymbol{\varepsilon})=0}=\boldsymbol{\sigma}^{+}\left.\right|_{\text{tr}(\boldsymbol{\varepsilon})=0}=\boldsymbol{\sigma}^{-}\left.\right|_{\text{tr}(\boldsymbol{\varepsilon})=0}=2\mu\boldsymbol{\varepsilon}, $$

(83)

which concludes our proof of continuity. The elasticity tensor in (61)₂ can be discontinuous across the interface between the tension and the compression subdomains, but it is nonetheless piecewise continuous on $\mathcal {E}_{t}$ and $\mathcal {E}_{c}$, respectively. This discontinuity in the elasticity tensor has not shown any adverse effect in terms of numerical instabilities and has not caused convergence issues.

Appendix D: PolyScript

Appendix E: PolyStress

Appendix F: NLFEM