Stress-Constrained Topology Optimization for Lattice Materials

Synonyms

Cellular materials; Homogenization; Stress constraint; Topology optimization

Definitions

In the context of architected materials, cellular solids indicate porous materials with cells that fill the space in either two or three dimensions, such as in foams. A lattice material is a subclass of them, where order governs cell arrangement and solid struts form a reticulated framework of cells, such as in a truss-system. Homogenization is a mathematical theory for materials with periodic or quasi-periodic microstructures, made of two or more constituent solids, such as in a composite, or by a single material with voids, such as in a cellular solid. Homogenization treats a periodic material as homogenized medium with effective properties calculated from a limited portion of it, the Representative Volume Element. Topology optimizationis a structural optimization method that enables optimal material distribution within a given domain, subject to volume and possibly other constraints,...

Appendices

Appendix: Sensitivity Calculation of the Stiffness Tensor of the Lattice

To compute the sensitivity of the structure compliance, the derivatives of the stiffness tensor are to be computed for the entire domain microstructure. This appendix presents the derivation of the stiffness tensor sensitivity of the domain with respect to the filtered density $\tilde{\rho}$. A four-node iso-parametric quadrilateral element (Hughes 1989) is used to discretize the finite domain.

The direct stiffness approach is used to find the global stiffness tensor K, where the structure domain is discretized into small elements with elemental stiffness matrix K_e, calculated before assembly, expressed as:

$$\begin{aligned} {K}_e\left(\tilde{\rho}\right)&=\underset{A^e}{\varint }{B}^T{E}^H\left(\tilde{\rho}\right) BdA\\&=\underset{xy}{\iint }{B}^T{E}^H\left(\tilde{\rho}\right)B\ dxdy \end{aligned}$$

(34)

where x and y are the global coordinates, B is the strain-displacement matrix, E^H is the homogenized elastic tensor of each element e described in the “Mechanical Properties of Lattice Materials” section, and A^e is the area of the element in the global coordinates. Since the strain-displacement matrix is independent of the design variables, the derivatives of the elemental stiffness matrix with respect to filtered density, $\tilde{\rho}$, can be expressed as follows:

$$\begin{aligned} \frac{\partial {K}_e\left(\tilde{\rho}\right)}{\partial {\tilde{\rho}}_e}&=\underset{A^e}{\varint }{B}^T\frac{\partial {E}^H\left(\tilde{\rho}\right)}{\partial {\tilde{\rho}}_e} BdA\\ &=\underset{xy}{\iint }{B}^T\frac{\partial {E}^H\left(\tilde{\rho}\right)}{\partial {\tilde{\rho}}_e}B\ dxdy \end{aligned}$$

(35)

Here, the problem is assumed under plane stress conditions with a unit element thickness. Each mesh element corresponds to a rectangular unit cell. The elastic stiffness tensor E^H of a general orthotropic material can thus be expressed for each unit cell as a function of its filtered density, $\tilde{\rho}$, as:

$$\begin{aligned} &{E}^H\left(\tilde{\rho}\right)\\ &\quad=\left[\begin{array}{ccc}\frac{E_{xx}\left(\tilde{\rho}\right)}{1-{\upsilon}_{xy}\left(\tilde{\rho}\right){\upsilon}_{yx}\left(\tilde{\rho}\right)}& \frac{\upsilon_{yx}\left(\tilde{\rho}\right){E}_{xx}\left(\tilde{\rho}\right)}{1-{\upsilon}_{xy}\left(\tilde{\rho}\right){\upsilon}_{yx}\left(\tilde{\rho}\right)}& 0\\ {}\frac{\upsilon_{xy}\left(\tilde{\rho}\right){E}_{yy}\left(\tilde{\rho}\right)}{1-{\upsilon}_{xy}\left(\tilde{\rho}\right){\upsilon}_{yx}\left(\tilde{\rho}\right)}& \frac{E_{yy}\left(\tilde{\rho}\right)}{1-{\upsilon}_{xy}\left(\tilde{\rho}\right){\upsilon}_{yx}\left(\tilde{\rho}\right)}& 0\\ {}0& 0& {G}_{xy}\left(\tilde{\rho}\right)\end{array}\right] \end{aligned}$$

(36)

Since asymptotic homogenization is used to calculate the elastic constants of the unit cell across a range of filtered density (Fig. 2), the elastic tensor of each element can be written as a function of the element filtered density and then used to evaluate the derivative of the elastic tensor with respect to the filtered density. For a four-node quadrilateral element with four Gauss points, the elemental stiffness tensor Eq. (34) and its derivatives (35) can be rewritten by using the Gauss quadrature rule for area integration as:

$$\begin{aligned} {K}_e\left(\tilde{\rho}\right)&=\underset{A^e}{\varint }{B}^T{E}^H\left(\tilde{\rho}\right) BdA\\ &=\underset{xy}{\iint }{B}^T{E}^H\left(\tilde{\rho}\right)B\ dxdy\\ &=\underset{st}{\iint }{B}_{st}^T\left(s,t\right){E}^H\left(\tilde{\rho}\right){B}_{st}\left(s,t\right)\\ &\left|J\right| dsdt =\sum \limits_{i=1}^2\sum \limits_{j=1}^2{w}_i{w}_j{B}_{st}^T\left({s}_i,{t}_j\right){E}^H\\ &\left(\tilde{\rho}\right){B}_{st}\left({s}_i,{t}_j\right)\ \left|J\right| \end{aligned}$$

(37)

$$ {\displaystyle \begin{array}{l}\frac{\partial {K}_e\left(\tilde{\rho}\right)}{\partial \tilde{\rho}}=\underset{A^e}{\varint }{B}^T\frac{\partial {E}^H\left(\tilde{\rho}\right)}{\partial {\tilde{\rho}}_e} BdA\\ \quad=\underset{xy}{\iint }{B}^T\frac{\partial {E}^H\left(\tilde{\rho}\right)}{\partial {\tilde{\rho}}_e}B\ dxdy\\ \quad =\underset{st}{\iint }{B}_{st}^T\left(s,t\right)\frac{\partial {E}^H\left(\tilde{\rho}\right)}{\partial {\tilde{\rho}}_e}{B}_{st}\left(s,t\right)\ \left|J\right| dsdt\\ \quad=\sum \limits_{i=1}^2\sum \limits_{j=1}^2{w}_i{w}_j{B}_{st}^T\left({s}_i,{t}_j\right)\frac{\partial {E}^H\left(\tilde{\rho}\right)}{\partial {\tilde{\rho}}_e}{B}_{st}\\ \quad\left({s}_i,{t}_j\right)\ \left|J\right|\end{array}} $$

(38)

where s and t are the natural coordinates. Once the stiffness matrix derivatives are calculated for each element using (38), the global stiffness derivatives $\partial K\left(\tilde{\rho}\right)/\partial {\tilde{\rho}}_e$ can be assembled to calculate the compliance sensitivity vector $\partial C\left(\tilde{\rho}\right)/\partial {\tilde{\rho}}_e$ for the whole cellular domain (Eq. 25). The sensitivity analysis is then implemented, under the optimization scheme described in the “Methodology” section, to seek the optimum relative density distribution that achieves the objective and satisfies the constraints.

Cross-References

• Density Based Topology Optimization

• Homogenization and Structural Optimization

• Graded Cellular Materials

• Topology Optimization with Stress Constraints