Concurrent topology optimization of multiscale structures with multiple porous materials under random field loading uncertainty

Abstract

Design of multiscale structures is a challenging task due to a vast design space of both materials and structures. Consideration of load uncertainty adds another level of complexity. In this paper, a robust concurrent TO (topology optimization) approach is developed for designing multiscale structures composed of multiple porous materials under random field loading uncertainty. To determine the optimal distribution of the porous materials at the macro/structural scale, our key idea is to employ the discrete material optimization method to interpolate the material properties for multiple porous materials. In addition, for the first time we interpret the interpolation schemes in the existing concurrent TO model of porous material with a clear physical meaning by putting forward a SIMP-like single interpolation scheme. This scheme integrates the SIMP (Solid Isotropic Material with Penalization) at the microscale and PAMP (Porous Anisotropic Material with Penalization) at the macroscale into a single equation. Efficient uncertainty characterization and propagation methods based on K-L expansion and linear superposition are introduced, and several important improvements in objective function evaluation and sensitivity analysis are presented. Improved sensitivity analysis equations are derived for volume preserving filtering, which is employed to deal with numerical instabilities at the macro and micro scales in the robust concurrent TO model. Measures to ensure manufacturability and to improve analysis accuracy and efficiency are devised. 2D and 3D examples demonstrate the effectiveness of the proposed approach in simultaneously obtaining robust optimal macro structural topology and material microstructural topologies.

Appendices

Appendix A - Improved Sensitivity Analysis for Volume Preserving Filter

In this section, we will derive improved sensitivity for topology optimization using volume preserving filtering (Xu et al. 2010). Assuming the design domain is meshed into N finite elements and given pseudo density (design variables) ρ _i of the ith element (i ∈ {1, 2,  ... , N}), after linear density filtering, we obtain density ${\overline{\rho}}_e$ for the eth element as,

$$ {\overline{\rho}}_e={\sum}_{i\in {\Psi}_e}{w}_i{\rho}_i{v}_i/{\sum}_{i\in {\Psi}_e}{w}_i{v}_i, $$

(52)

where v _i represents the volume of the ith element. w _i is the weighting coefficient given by

$$ {w}_{\mathrm{i}}=\mathrm{r}-\left\Vert {x}_{\mathrm{i}}-{x}_{\mathrm{e}}\right\Vert, $$

(53)

where x _i is the center location vector of element i and r is the filter radius. Ψ _e in (52) denotes the neighborhood of element e, which is specified by the elements whose centers are located within r of element e, i.e.

$$ {\Psi}_{\mathrm{e}}\kern0.5em =\left\{\mathrm{i}|\left\Vert {x}_{\mathrm{i}}-{x}_{\mathrm{e}}\right\Vert \le \mathrm{r}\right\}. $$

(54)

Then, ${\overline{\rho}}_e$ is filtered by the volume preserving nonlinear density filter and the filtered/physical density is

$$ {\tilde{\rho}}_e=\left\{\begin{array}{c}\hfill \eta \left[{e}^{-\beta \left(1-{\overline{\rho}}_e/\eta \right)}-\left(1-{\overline{\rho}}_e/\eta \right){e}^{-\beta}\right]\kern9.75em 0\le {\overline{\rho}}_e\le \eta \hfill \\ {}\hfill \left(1-\eta \right)\left[1-{e}^{-\beta \left({\overline{\rho}}_e-\eta \right)/\left(1-\eta \right)}+\left({\overline{\rho}}_e-\eta \right){e}^{-\beta}/\left(1-\eta \right)\right]+\eta \kern1.5em \eta <{\overline{\rho}}_e\le 1\hfill \end{array}\right., $$

(55)

where β is the smooth parameter in the Heaviside function. η is the volume preserving parameter to ensure the volume is the same before and after nonlinear filtering in (55). η is determined by method of bisection meeting the following volume preserving condition,

$$ g\left({\tilde{\rho}}_e,{\overline{\rho}}_e\right)={\sum}_{e=1}^N{\tilde{\rho}}_e{v}_e-{\sum}_{e=1}^N{\overline{\rho}}_e{v}_e=0. $$

(56)

By virtue of the chain rule, the sensitivities of a general performance function f with respect to the design variable ρ _i is obtained as follows in the literature (Xu et al. 2010),

$$ \frac{\partial f}{\partial {\rho}_i}=\sum_{e\in {\Phi}_i}\frac{\partial f}{\partial {\tilde{\rho}}_e}\frac{\partial {\tilde{\rho}}_e}{\partial {\overline{\rho}}_e}\frac{\partial {\overline{\rho}}_e}{\partial {\rho}_i}, $$

(57)

where $\partial {\tilde{\rho}}_e/\partial {\overline{\rho}}_e$ and $\partial {\overline{\rho}}_e/\partial {\rho}_i$ should be deduced from (55) and (52).

We discovered that the numerical sensitivities obtained using (57) usually have some discrepancy with the sensitivities obtained using the finite difference method. To remove this discrepancy, the improved analytical sensitivity analysis should be stated as,

$$ \begin{array}{cc}\hfill \frac{\partial f}{\partial {\rho}_i}=\sum_{e\in {\Phi}_i}\frac{\partial f}{\partial {\tilde{\rho}}_e}\frac{\partial {\tilde{\rho}}_e}{\partial {\overline{\rho}}_e}\frac{\partial {\overline{\rho}}_e}{\partial {\rho}_i}+\sum_{e=1}^N\frac{\partial f}{\partial {\tilde{\rho}}_e}\frac{\partial {\tilde{\rho}}_e}{\partial \eta}\frac{\partial \eta}{\partial {\rho}_i},\hfill & \hfill \left( i\in \left\{1,2,\dots, N\right\}\right),\hfill \end{array} $$

(58)

where ∂η/∂ρ _i can be obtained by taking the derivative of the volume preserving condition dg/dρ _i  = 0 as

$$ \frac{\partial \eta}{\partial {\rho}_i}=\left(\sum_{e\in {\Phi}_i}\left(1-\frac{\partial {\tilde{\rho}}_e}{\partial {\overline{\rho}}_e}\right)\frac{\partial {\overline{\rho}}_e}{\partial {\rho}_i}{v}_e\right)/\left(\sum_{e=1}^N\frac{\partial {\tilde{\rho}}_e}{\partial \eta}{v}_e\right). $$

(59)

Finally for a general performance function f, its sensitivity can be sated as,

$$ \frac{\partial f}{\partial {\rho}_i}=\sum_{e\in {\Phi}_i}\frac{\partial f}{\partial {\tilde{\rho}}_e}\frac{\partial {\tilde{\rho}}_e}{\partial {\overline{\rho}}_e}\frac{\partial {\overline{\rho}}_e}{\partial {\rho}_i}+\sum_{e=1}^N\frac{\partial f}{\partial {\tilde{\rho}}_e}\frac{\partial {\tilde{\rho}}_e}{\partial \eta}\left(\sum_{e\in {\Phi}_i}\left(1-\frac{\partial {\tilde{\rho}}_e}{\partial {\overline{\rho}}_e}\right)\frac{\partial {\overline{\rho}}_e}{\partial {\rho}_i}{v}_e\right)/\left(\sum_{e=1}^N\frac{\partial {\tilde{\rho}}_e}{\partial \eta}{v}_e\right), $$

(60)

which is different from (57).

Appendix B - PCG with Rounding-off Error Correction

For a linear system Ax = kwith A symmetric and positive definite, the following formulas are used in conjugate gradient method:

$$ \begin{array}{cc}\hfill \begin{array}{l}{\mathbf{p}}_0={\mathbf{r}}_0=\mathbf{k}-{\mathbf{Ax}}_0\\ {}{\alpha}_i={\left|{\mathbf{r}}_i\right|}^2/{\mathbf{p}}_i^T{\mathbf{Ap}}_i\\ {}{\mathbf{x}}_{i+1}={\mathbf{x}}_i+{\alpha}_i{\mathbf{p}}_i\\ {}{\mathbf{r}}_{i+1}={\mathbf{r}}_i-{\alpha}_i{\mathbf{Ap}}_i\\ {}{b}_i={\left|{\mathbf{r}}_{i+1}\right|}^2/{\left|{\mathbf{r}}_i\right|}^2\\ {}{\mathbf{p}}_{i+1}={\mathbf{r}}_{i+1}+{b}_i{\mathbf{p}}_i\end{array}\hfill & \hfill, \hfill \end{array} $$

where x ₀ is an arbitrary starting point.

With the correction to remove rounding-off error, the above routine is refined by the following formulas:

$$ \begin{array}{l}{\mathbf{p}}_0={\mathbf{r}}_0=\mathbf{k}-{\mathbf{Ax}}_0,\kern1em {d}_0=1\\ {}{\alpha}_i=\frac{{\left|{\mathbf{r}}_i\right|}^2}{{\mathbf{p}}_i^T{\mathbf{Ap}}_i}\frac{1}{d_i}\\ {}{\mathbf{x}}_{i+1}={\mathbf{x}}_i+{\alpha}_i{\mathbf{p}}_i\\ {}{\mathbf{r}}_{i+1}={\mathbf{r}}_i-{\alpha}_i{\mathbf{Ap}}_i\\ {}{b}_i=\frac{{\left|{\mathbf{r}}_{i+1}\right|}^2}{{\left|{\mathbf{r}}_i\right|}^2}{d}_i\\ {}{\mathbf{p}}_{i+1}={\mathbf{r}}_{i+1}+{b}_i{\mathbf{p}}_i\\ {}{d}_{i+1}=1-{b}_i\frac{{\mathbf{p}}_{i+1}^T{\mathbf{Ap}}_i}{{\mathbf{p}}_{i+1}^T{\mathbf{Ap}}_{i+1}}\end{array} $$

The above is for conjugate gradient method. For preconditioned conjugate gradient method in OOFEM, similar corrections can be incorporated as well.