Two-scale topology design optimization of stiffened or porous plate subject to out-of-plane buckling constraint

Abstract

This paper studies maximum out-of-plane buckling load design of thin bending plates for a given amount of material. Two kinds of plates are considered. One is made of periodic homogeneous porous material. Another is uniformly stiffened solid plate. The plate material, thickness, design domain of its middle plane and boundary conditions are given. The pattern of prescribed in-plane external load or displacements along the part of boundaries, which move freely, is given. Both plate topology and micro-structural topology of porous material or stiffener layout are concurrently optimized. The artificial element material densities in both macro and micro-scale are chosen as design variables. The volume preserving nonlinear density filter is applied to obtain the black-white optimum topology and comparison of its different sensitivities is made to show the reason for oscillation during optimization process in Appendix. The new numerical implementation of asymptotic homogenization method (NIAH, Cheng (Acta Mech Sinica 29(4): 550–556, 2013) and Cai (Int J Solids Struct 51(1), 284–292, 2014) is applied to homogenization of periodic plate structures and analytic sensitivity analysis of effective stiffness with respect to the topological design variables in both macro-scale and micro-scale. On basis of that, this paper implements the sensitivity analysis of out-of-plane buckling load by using commercial FEA software and enables the application of gradient-based search algorithm in optimization. Several numerical implementation details are discussed. Three numerical examples are given to show the validity of this method.

Appendix

In the Appendix, we will give an example to show the difference between the sensitivities in Eqs. (30a) and (30b) and the avoidance of oscillation with the latter one. Consider a cantilever beam in Fig. 17, the size and load parameters are H = 80, L = 50, p = 1. The material of the beam is E₀ = 1, v = 0.3. The design domain is discretized into N = 80 × 50 elements. During optimization, the material properties of the ith elements is expressed in the modified SIMP approach as ${E}_i={E}_{\min }+{\tilde{\rho}}_i^3\left({E}_0-{E}_{\min}\right),{E}_{\min }={10}^{-9}$. The optimization problem is the minimum compliance problem and is formulated as

$$ \begin{array}{l}\mathrm{find}\kern0.5em {\rho}_i\left(i=1,2,\dots, N\right)\\ {} \min \kern1em {\mathbf{F}}^T\mathbf{U}\\ {}\mathrm{s}.\mathrm{t}.\kern1.5em \mathbf{K}\mathbf{U}=\mathbf{F}\\ {}\kern3.5em {\displaystyle \sum_{\mathrm{i}=1}^N{\rho}_i{v}_i}/{\displaystyle \sum_{i=1}^n{v}_i}-{V}^{*}\le 0\\ {}\kern3.5em 0.01\le {\rho}_i\le 1\end{array} $$

(27)

Fig. 17

Illustration of a cantilever beam

Where v _i is the volume of the ith element and V* is the prescribed volume fraction that equals 50 % in this problem. The volume preserving nonlinear density filter is given as.

$$ \begin{array}{l}{\tilde{\rho}}_i=\left\{\begin{array}{c}\hfill \eta \left[{e}^{-\beta \left(1-{\overline{\rho}}_i/\eta \right)}-\left(1-{\overline{\rho}}_i/\eta \right){e}^{-\beta}\right]\kern9em 0\le {\overline{\rho}}_i\le \eta \kern2em \hfill \\ {}\hfill \left(1-\eta \right)\left[1-{e}^{-\beta \left({\overline{\rho}}_i-\eta \right)/\left(1-\eta \right)}+\left({\overline{\rho}}_i-\eta \right){e}^{-\beta }/\left(1-\eta \right)\right]+\eta \kern1.75em \eta <{\overline{\rho}}_i\le 1\kern2.25em \hfill \end{array}\right.\\ {}{\overline{\rho}}_i=\frac{{\displaystyle \sum_{j\in {\psi}_i}w\left({\mathbf{x}}_j\right){v}_j{\rho}_j}}{{\displaystyle \sum_{j\in {\psi}_i}w\left({\mathbf{x}}_j\right){v}_j}}\end{array} $$

(28)

Where the filter radius is 1.8 in the linear density filter. The numerical procedure of the minimum compliance problem is listed as follows.

• 1. Construct finite element model and initialize design variables.

• 2. Use the volume preserving nonlinear density filter to compute the physical densities with Eq. (28)

• 3. Solve the structural response under external load and calculate sensitivity information.

• 4. Solve the optimization problem and update design variables using GCMMA.

• 5. Stop at 500th step. Other stop criteria such minimal density or objective function change are not used as we want to show the oscillation when β is large. Using other convergence criteria may cause the iteration stops at an earlier step, where oscillation is not obvious.

During optimization, the volume preserving nonlinear Heaviside density filter is used and the smooth parameter β is updated every 50 steps as

$$ \begin{array}{l}\beta =2\beta \kern3em \left(\varepsilon \le 15\right)\\ {}\beta =\beta +10\kern0.5em \left(\beta >15\right)\end{array} $$

(29)

To demonstrate the effect of the sensitivity in Eq. (30b) on iteration history, three cases are tested. In the first case, the sensitivity in Eq. (30a) is used while η is derived from the volume preserving condition in Eq. (28). In the second case, the sensitivity in Eq. (30b) is used while η is derived from the volume preserving condition in Eq. (28). In the last case, η is set to 0.5, which makes Eqs. (30a) and (30b) no difference.

$$ \frac{\partial f}{\partial {\rho}_i}={\displaystyle \sum_{j\in {\psi}_i}\frac{\partial f}{\partial {\tilde{\rho}}_j}\frac{\partial {\tilde{\rho}}_j}{\partial {\overline{\rho}}_j}}\frac{\partial {\overline{\rho}}_j}{\partial {\rho}_i} $$

(30a)

$$ \frac{\partial f}{\partial {\rho}_i}={\displaystyle \sum_{j\in {\psi}_i}\frac{\partial f}{\partial {\tilde{\rho}}_j}\frac{\partial {\tilde{\rho}}_j}{\partial {\overline{\rho}}_j}}\frac{\partial {\overline{\rho}}_j}{\partial {\rho}_i}+\left({\displaystyle \sum_{j\in {\psi}_i}\frac{\partial \eta }{\partial {\overline{\rho}}_j}\frac{\partial {\overline{\rho}}_j}{\partial {\rho}_i}}\right)\cdot {\displaystyle \sum_{j=1}^N\frac{\partial f}{\partial {\tilde{\rho}}_j}\frac{\partial {\tilde{\rho}}_j}{\partial \eta }} $$

(30b)

$$ \eta =0.5:\kern2.5em \frac{\partial f}{\partial {\rho}_i}={\displaystyle \sum_{j\in {\psi}_i}\frac{\partial f}{\partial {\tilde{\rho}}_j}\frac{\partial {\tilde{\rho}}_j}{\partial {\overline{\rho}}_j}}\frac{\partial {\overline{\rho}}_j}{\partial {\rho}_i} $$

(30c)

The iteration history of the objective function with the three cases is plotted in Figs. 18, 19 and 20.

Fig. 18

Optimization history with sensitivity in Eq. (30a)

Fig. 19

Optimization history with sensitivity in Eq. (30b)

Fig. 20

Optimization history with sensitivity in Eq. (30c) when η = 0.5

It can be seen from Fig. 18 that when β is small, the sensitivity in Eq. (30a) is still valid. However, oscillation emerges and the topology result begins to scatter when β becomes large. In Fig. 19, the oscillation is avoided with the sensitivity in Eq. (30b) and the topology result converges well. In Fig. 20, the sensitivity in Eq. (25) becomes the accurate one when η is fixed and the result also converges well.