Stress-based bi-directional evolutionary topology optimization for structures with multiple materials

Abstract

Both multi-material and stress-based topology optimization problems have been extensively investigated. However, there are few studies on the stress-based topology optimization of multi-material structures. Hence, this work proposes a novel topology optimization method for minimizing the maximum von Mises stress of structures with multiple materials under volume constraints. An extended Bi-directional Evolutionary Structural Optimization (BESO) method based on discrete variables which can mitigate the well-known stress singularity problem is adopted. The global von Mises stress is established with the p-norm function, and the adjoint sensitivity analysis is derived. Two benchmark numerical examples are investigated to validate the effectiveness of the proposed method. The effects of key parameters including p-norm, sensitivity and density filter radii on the optimized results and the stress distributions are discussed. The influence of varying mesh densities on the optimized topologies are investigated in comparison with the multi-material stiffness maximization design. The topological results, for multi-material stress design, indicate that the maximum stress can be reduced compared with multi-material stiffness design. It concludes that the proposed approach can achieve a reasonable design that effectively controls the stress level and reduces the stress concentration effect at the critical stress areas of multi-material structures.

Appendix: Sensitivity analysis

According to Eq. (11), the derivative of the objective function with respect to the topological variables can be derived as

$$ \frac{{\partial \sigma_{PN} }}{\partial x_i^j } = \sigma_{PN}^{1 - p} \left( {\sum_{i = 1}^N {\sigma_{vm,i}^{p - 1} \frac{{\partial \sigma_{vm.i} }}{\partial x_i^j }} } \right),\quad i = 1,2,...,N,\quad j = 1,2 $$

(23)

According to Eq. (8), one obtains

$$ \frac{{\partial \sigma_{vm,i} }}{{\partial {{\varvec{\sigma}}}_i }} = \frac{1}{2}\left( {{{\varvec{\sigma}}}_i^T {\varvec{V\sigma }}_i } \right)^{ - {1 / 2}} 2{{\varvec{\sigma}}}_i^T {{\mathbf{V}}} = \sigma_{vm,i}^{ - 1} {{\varvec{\sigma}}}_i^T {{\mathbf{V}}} $$

(24)

In consideration of Eq. (24), and by the chain rule, the derivative of the von Mises stress with respect to the topological variable can be obtained as

$$ \frac{{\partial \sigma_{vm,i} }}{\partial x_i^j } = \frac{{\partial \sigma_{vm,i} }}{{\partial {{\varvec{\sigma}}}_i }}\frac{{\partial {{\varvec{\sigma}}}_i }}{\partial x_i^j } = \sigma_{vm,i}^{ - 1} {{\varvec{\sigma}}}_i^T {{\mathbf{V}}}\frac{{\partial {{\varvec{\sigma}}}_i }}{\partial x_i^j } $$

(25)

In consideration of Eq. (6), Eq. (25) can be further written as

$$ \left\{ {\begin{array}{*{20}c} {\frac{{\partial \sigma_{vm,i} }}{\partial x_i^1 } = \sigma_{vm,i}^{ - 1} {{\varvec{\sigma}}}_i^T {{\mathbf{V}}}\left[ {\begin{array}{*{20}c} {\left( {\overline{p}\left( {x_i^1 } \right)^{\overline{p} - 1} \left( {x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_1^0 {{\mathbf{BL}}}_i {{\mathbf{U}}} + \left( {x_i^1 } \right)^{\overline{p}} \left( {x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_1^0 {{\mathbf{BL}}}_i \frac{{\partial {{\mathbf{U}}}}}{\partial x_i^1 }} \right)} \\ { + \left( {\overline{p}\left( {x_i^1 } \right)^{\overline{p} - 1} \left( {1 - x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_2^0 {{\mathbf{BL}}}_i {{\mathbf{U}}} + \left( {x_i^1 } \right)^{\overline{p}} \left( {1 - x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_2^0 {{\mathbf{BL}}}_i \frac{{\partial {{\mathbf{U}}}}}{\partial x_i^1 }} \right)} \\ \end{array} } \right]} \\ {\frac{{\partial \sigma_{vm,i} }}{\partial x_i^2 } = \sigma_{vm,i}^{ - 1} {{\varvec{\sigma}}}_i^T {{\mathbf{V}}}\left[ {\begin{array}{*{20}c} {\left( {\overline{p}\left( {x_i^1 } \right)^{\overline{p}} \left( {x_i^2 } \right)^{\overline{p} - 1} {{\mathbf{D}}}_1^0 {{\mathbf{BL}}}_i {{\mathbf{U}}} + \left( {x_i^1 } \right)^{\overline{p}} \left( {x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_1^0 {{\mathbf{BL}}}_i \frac{{\partial {{\mathbf{U}}}}}{\partial x_i^2 }} \right)} \\ { + \left( {\left( {x_i^1 } \right)^{\overline{p}} \left( {1 - x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_2^0 {{\mathbf{BL}}}_i \frac{{\partial {{\mathbf{U}}}}}{\partial x_i^2 } - \overline{p}\left( {x_i^1 } \right)^{\overline{p}} \left( {1 - x_i^2 } \right)^{\overline{p} - 1} {{\mathbf{D}}}_2^0 {{\mathbf{BL}}}_i {{\mathbf{U}}}} \right)} \\ \end{array} } \right]} \\ \end{array} } \right. $$

(26)

where the matrix ${{\mathbf{L}}}_i$ gathers the nodal displacements of the of the $i$th element from the global displacement vector, as ${{\mathbf{u}}}_i = {{\mathbf{L}}}_i {{\mathbf{U}}}$.

By differentiating both sides of the state balance equation Eq. (12), one obtains

$$ \frac{{\partial {{\mathbf{U}}}}}{\partial x_i^j } = - {{\mathbf{K}}}^{ - 1} \frac{{\partial {{\mathbf{K}}}}}{\partial x_i^j }{{\mathbf{U}}} $$

(27)

Thus, Eq. (26) can be further written as

$$ \left\{ {\begin{array}{*{20}c} {\frac{{\partial \sigma_{vm,i} }}{\partial x_i^1 } = \sigma_{vm,i}^{ - 1} {{\varvec{\sigma}}}_i^T {{\mathbf{V}}}\left[ {\begin{array}{*{20}c} {\left( {\overline{p}\left( {x_i^1 } \right)^{\overline{p} - 1} \left( {x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_1^0 {{\mathbf{BL}}}_i {{\mathbf{U}}} - \left( {x_i^1 } \right)^{\overline{p}} \left( {x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_1^0 {{\mathbf{BL}}}_i {{\mathbf{K}}}^{ - 1} \frac{{\partial {{\mathbf{K}}}}}{\partial x_i^1 }{{\mathbf{U}}}} \right)} \\ { + \left( {\overline{p}\left( {x_i^1 } \right)^{\overline{p} - 1} \left( {1 - x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_2^0 {{\mathbf{BL}}}_i {{\mathbf{U}}} - \left( {x_i^1 } \right)^{\overline{p}} \left( {1 - x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_2^0 {{\mathbf{BL}}}_i {{\mathbf{K}}}^{ - 1} \frac{{\partial {{\mathbf{K}}}}}{\partial x_i^1 }{{\mathbf{U}}}} \right)} \\ \end{array} } \right]} \\ {\frac{{\partial \sigma_{vm,i} }}{\partial x_i^2 } = \sigma_{vm,i}^{ - 1} {{\varvec{\sigma}}}_i^T {{\mathbf{V}}}\left[ {\begin{array}{*{20}c} {\left( {\overline{p}\left( {x_i^1 } \right)^{\overline{p}} \left( {x_i^2 } \right)^{\overline{p} - 1} {{\mathbf{D}}}_1^0 {{\mathbf{BL}}}_i {{\mathbf{U}}} - \left( {x_i^1 } \right)^{\overline{p}} \left( {x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_1^0 {{\mathbf{BL}}}_i {{\mathbf{K}}}^{ - 1} \frac{{\partial {{\mathbf{K}}}}}{\partial x_i^2 }{{\mathbf{U}}}} \right)} \\ { - \left( {\overline{p}\left( {x_i^1 } \right)^{\overline{p}} \left( {1 - x_i^2 } \right)^{\overline{p} - 1} {{\mathbf{D}}}_2^0 {{\mathbf{BL}}}_i {{\mathbf{U}}} + \left( {x_i^1 } \right)^{\overline{p}} \left( {1 - x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_2^0 {{\mathbf{BL}}}_i {{\mathbf{K}}}^{ - 1} \frac{{\partial {{\mathbf{K}}}}}{\partial x_i^2 }{{\mathbf{U}}}} \right)} \\ \end{array} } \right]} \\ \end{array} } \right. $$

(28)

Substituting Eq. (28) into Eq. (23), one obtains

$$ \left\{ {\begin{array}{*{20}c} {\frac{{\partial \sigma_{PN} }}{\partial x_i^1 } = \sigma_{PN}^{1 - p} \sum_{i = 1}^N {\sigma_{vm,i}^{p - 2} {{\varvec{\sigma}}}_i^T {{\mathbf{V}}}\left[ {\begin{array}{*{20}c} {\left( {\overline{p}\left( {x_i^1 } \right)^{\overline{p} - 1} \left( {x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_1^0 {{\mathbf{BL}}}_i {{\mathbf{U}}} - \left( {x_i^1 } \right)^{\overline{p}} \left( {x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_1^0 {{\mathbf{BL}}}_i {{\mathbf{K}}}^{ - 1} \frac{{\partial {{\mathbf{K}}}}}{\partial x_i^1 }{{\mathbf{U}}}} \right)} \\ { + \left( {\overline{p}\left( {x_i^1 } \right)^{\overline{p} - 1} \left( {1 - x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_2^0 {{\mathbf{BL}}}_i {{\mathbf{U}}} - \left( {x_i^1 } \right)^{\overline{p}} \left( {1 - x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_2^0 {{\mathbf{BL}}}_i {{\mathbf{K}}}^{ - 1} \frac{{\partial {{\mathbf{K}}}}}{\partial x_i^1 }{{\mathbf{U}}}} \right)} \\ \end{array} } \right]} } \\ {\frac{{\partial \sigma_{PN} }}{\partial x_i^1 } = \sigma_{PN}^{1 - p} \sum_{i = 1}^N {\sigma_{vm,i}^{p - 2} {{\varvec{\sigma}}}_i^T {{\mathbf{V}}}\left[ {\begin{array}{*{20}c} {\left( {\overline{p}\left( {x_i^1 } \right)^{\overline{p}} \left( {x_i^2 } \right)^{\overline{p} - 1} {{\mathbf{D}}}_1^0 {{\mathbf{BL}}}_i {{\mathbf{U}}} - \left( {x_i^1 } \right)^{\overline{p}} \left( {x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_1^0 {{\mathbf{BL}}}_i {{\mathbf{K}}}^{ - 1} \frac{{\partial {{\mathbf{K}}}}}{\partial x_i^2 }{{\mathbf{U}}}} \right)} \\ { - \left( {\overline{p}\left( {x_i^1 } \right)^{\overline{p}} \left( {1 - x_i^2 } \right)^{\overline{p} - 1} {{\mathbf{D}}}_2^0 {{\mathbf{BL}}}_i {{\mathbf{U}}} + \left( {x_i^1 } \right)^{\overline{p}} \left( {1 - x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_2^0 {{\mathbf{BL}}}_i {{\mathbf{K}}}^{ - 1} \frac{{\partial {{\mathbf{K}}}}}{\partial x_i^2 }{{\mathbf{U}}}} \right)} \\ \end{array} } \right]} } \\ \end{array} } \right. $$

(29)

The solution of the adjoint problem:

$$ {\varvec{K\lambda }} = \sum_{i = 1}^N {\sigma_{vm,i}^{p - 2} \left[ {\left( {\left( {x_i^1 } \right)^{\overline{p}} \left( {x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_1^0 {{\mathbf{BL}}}_i } \right) + \left( {\left( {x_i^1 } \right)^{\overline{p}} \left( {1 - x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_2^0 {{\mathbf{BL}}}_i } \right)} \right]^T {{\mathbf{V}}}} {{\varvec{\sigma}}}_i $$

(30)

is the adjoint vector, as

$$ {{\varvec{\lambda}}}^T { = }\sum_{i = 1}^N {\sigma_{vm,i}^{p - 2} } {{\varvec{\sigma}}}_i^T {{\mathbf{V}}}\left[ {\left( {\left( {x_i^1 } \right)^{\overline{p}} \left( {x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_1^0 {{\mathbf{BL}}}_i } \right) + \left( {\left( {x_i^1 } \right)^{\overline{p}} \left( {1 - x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_2^0 {{\mathbf{BL}}}_i } \right)} \right]{{\mathbf{K}}}^{ - 1} $$

(31)

Thus, the sensitivity numbers Eq. (29) can be further simplified as

$$ \left\{ {\begin{array}{*{20}c} {\frac{{\partial \sigma_{PN} }}{\partial x_i^1 }{ = }\sigma_{PN}^{1 - p} \sum_{i = 1}^N {\sigma_{vm,i}^{p - 2} {{\varvec{\sigma}}}_i^T {{\mathbf{V}}}\left[ {\overline{p}\left( {x_i^1 } \right)^{\overline{p} - 1} \left( {x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_1^0 {{\mathbf{BL}}}_i {{\mathbf{U}}} + \overline{p}\left( {x_i^1 } \right)^{\overline{p} - 1} \left( {1 - x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_2^0 {{\mathbf{BL}}}_i {{\mathbf{U}}}} \right]} - \sigma_{PN}^{1 - p} {{\varvec{\lambda}}}^T \frac{{\partial {{\mathbf{K}}}}}{\partial x_i^1 }{{\mathbf{U}}}} \\ {\frac{{\partial \sigma_{PN} }}{\partial x_i^2 }{ = }\sigma_{PN}^{1 - p} \sum_{i = 1}^N {\sigma_{vm,i}^{p - 2} {{\varvec{\sigma}}}_i^T {{\mathbf{V}}}\left[ {\overline{p}\left( {x_i^1 } \right)^{\overline{p}} \left( {x_i^2 } \right)^{\overline{p} - 1} {{\mathbf{D}}}_1^0 {{\mathbf{BL}}}_i {{\mathbf{U}}} - \overline{p}\left( {x_i^1 } \right)^{\overline{p}} \left( {1 - x_i^2 } \right)^{\overline{p} - 1} {{\mathbf{D}}}_2^0 {{\mathbf{BL}}}_i {{\mathbf{U}}}} \right]} - \sigma_{PN}^{1 - p} {{\varvec{\lambda}}}^T \frac{{\partial {{\mathbf{K}}}}}{\partial x_i^2 }{{\mathbf{U}}}} \\ \end{array} } \right. $$

(32)

Recalling Eq. (5), the derivatives of the stiffness matrices with respect to topological variables can be obtained as

$$ \frac{{\partial {{\mathbf{K}}}}}{\partial x_i^j } = \sum_{i = 1}^N {{{\mathbf{L}}}_i^T \left( {\left( {x_i^1 } \right)^{\overline{p}} \left( {\left( {x_i^2 } \right)^{\overline{p}} \frac{{\partial {{\mathbf{k}}}_i^1 }}{\partial x_i^j } + \left( {1 - x_i^2 } \right)^{\overline{p}} \frac{{\partial {{\mathbf{k}}}_i^2 }}{\partial x_i^j }} \right)} \right){{\mathbf{L}}}_i } = \left( {x_i^1 } \right)^{\overline{p}} {{\mathbf{L}}}_i^T \left( {\left( {x_i^2 } \right)^{\overline{p}} {{\mathbf{k}}}_i^{1,\left( 0 \right)} + \left( {1 - x_i^2 } \right)^{\overline{p}} {{\mathbf{k}}}_i^{2,\left( 0 \right)} } \right){{\mathbf{L}}}_i $$

(33)

where ${{\mathbf{k}}}_i^j$ and ${{\mathbf{k}}}_i^{j,\left( 0 \right)}$ are the elemental stiffness matrix and the elemental stiffness corresponding to solid material of the $i$th element for the $j$th material, which can be respectively written as

$$ \left\{ {\begin{array}{*{20}c} {{{\mathbf{k}}}_i^j = x_i^j \int_{\Omega_i } {{{\mathbf{B}}}^T {{\mathbf{D}}}_j^0 {{\mathbf{B}}}d\Omega_i } } \\ {{{\mathbf{k}}}_i^{j,\left( 0 \right)} = \int_{\Omega_i } {{{\mathbf{B}}}^T {{\mathbf{D}}}_j^0 {{\mathbf{B}}}d\Omega_i } } \\ \end{array} } \right. $$

(34)

where $\Omega_i$ denotes the domain of the $i$th element.

Substituting Eq. (33) into Eq. (32), the sensitivities can be eventually evaluated as

$$ \begin{aligned} \frac{{\partial \sigma_{PN} }}{\partial x_i^1 } & { = }\sigma_{PN}^{1 - p} \sum_{i = 1}^N {\sigma_{vm,i}^{p - 2} {{\varvec{\sigma}}}_i^T {{\mathbf{V}}}\left[ {\overline{p}\left( {x_i^1 } \right)^{\overline{p} - 1} \left( {x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_1^0 {{\mathbf{Bu}}}_i + \overline{p}\left( {x_i^1 } \right)^{\overline{p} - 1} \left( {1 - x_i^2 } \right)^{\overline{p}} {{\mathbf{D}}}_2^0 {{\mathbf{Bu}}}_i } \right]} \\ & \quad - \sigma_{PN}^{1 - p} {{\varvec{\lambda}}}_i^T \left( {x_i^1 } \right)^{\overline{p}} \left( {\left( {x_i^2 } \right)^{\overline{p}} {{\mathbf{k}}}_i^{1,\left( 0 \right)} + \left( {1 - x_i^2 } \right)^{\overline{p}} {{\mathbf{k}}}_i^{2,\left( 0 \right)} } \right){{\mathbf{u}}}_i \\ \end{aligned} $$

(35)

$$ \begin{aligned} \frac{{\partial \sigma_{PN} }}{\partial x_i^2 } & { = }\sigma_{PN}^{1 - p} \sum_{i = 1}^N {\sigma_{vm,i}^{p - 2} {{\varvec{\sigma}}}_i^T {{\mathbf{V}}}\left[ {\overline{p}\left( {x_i^1 } \right)^{\overline{p}} \left( {x_i^2 } \right)^{\overline{p} - 1} {{\mathbf{D}}}_1^0 {{\mathbf{Bu}}}_i - \overline{p}\left( {x_i^1 } \right)^{\overline{p}} \left( {1 - x_i^2 } \right)^{\overline{p} - 1} {{\mathbf{D}}}_2^0 {{\mathbf{Bu}}}_i } \right]} \\ & \quad - \sigma_{PN}^{1 - p} {{\varvec{\lambda}}}_i^T \left( {x_i^1 } \right)^{\overline{p}} \left( {\left( {x_i^2 } \right)^{\overline{p}} {{\mathbf{k}}}_i^{1,\left( 0 \right)} + \left( {1 - x_i^2 } \right)^{\overline{p}} {{\mathbf{k}}}_i^{2,\left( 0 \right)} } \right){{\mathbf{u}}}_i \\ \end{aligned} $$

(36)

where ${{\varvec{\lambda}}}_i$ is the vector of the adjoint nodal values of the $i$th element.