Stress-constrained continuum topology optimization: a new approach based on elasto-plasticity

Abstract

A new approach for generating stress-constrained topological designs in continua is presented. The main novelty is in the use of elasto-plastic modeling and in optimizing the design such that it will exhibit a linear-elastic response. This is achieved by imposing a single global constraint on the total sum of equivalent plastic strains, providing accurate control over all local stress violations. The single constraint essentially replaces a large number of local stress constraints or an approximate aggregation of them – two common approaches in the literature. A classical rate-independent plasticity model is utilized, for which analytical adjoint sensitivity analysis is derived and verified. Several examples demonstrate the capability of the computational procedure to generate designs that challenge results from the literature, in terms of the obtained stiffness-strength-weight trade-offs. A full elasto-plastic analysis of the optimized designs shows that prior to the initial yielding, these designs can sustain significantly higher loads than minimum compliance topological layouts, with only a minor compromise on stiffness.

Appendix

In this appendix we present a numerical verification of the adjoint sensitivity analysis procedure. Implementing this procedure can be a somewhat cumbersome task, so we believe this verification can prove useful for readers who are not well-acquainted with such procedures. Furthermore, accurate and efficient sensitivity analysis for elasto-plastic response is still a rather open issue, as discussed in a recent publication (Kato et al. 2015). In the following, results of the adjoint computations are compared to numerical derivatives computed by forward finite differences.

We consider a small problem of a symmetric clamped beam, where the symmetric half is modeled with a finite element mesh of 2 ×2 square bi-linear elements. A downwards vertical displacement is prescribed at the top right corner. Two separate loading situations are considered, see Fig. 13 for the problem setup: 1) A point load at the top right corner; and 2) A distributed load at the right edge. The first case is easier to implement because the equations for the global adjoint vectors in (23) and (25) take a simple form when the force is applied only at the prescribed DOF. However, the second case is much more useful, especially in the particular application considered in this article: It is necessary to distribute the applied load over several adjacent nodes because the numerical solution with a point load will inherently include stress concentrations.

Fig. 13

Problem setup for verification of the adjoint sensitivity analysis. Left: point load; right: distributed load

The material and optimization parameters are given in Table 5. The density $\overline {x}_{e}$ in all four elements is set to 0.8. The prescribed displacement of 0.01 is applied within 10 equal increments. Convergence of each increment is assumed when the relative norm of the residual forces is below 10⁻⁶. For the finite difference check, the perturbation value is set to ${\Delta } \overline {x}_{e} = 10^{-6}$. We compare the design sensitivities of two critical quantities in the context of the current application: 1) The end-compliance at the prescribed DOF, $g_{ec} = -\theta _{N} \hat {{f}^{p}} {{u}_{N}^{p}}$, where the superscript p denotes the prescribed DOF; and 2) The sum of plastic strains in the whole domain at the final equilibrium state, $g_{ps} = {\sum }_{e=1}^{N_{e}} {\sum }_{k=1}^{N_{GP}} {\kappa ^{ek}_{N}}$.

Table 5 Material and optimization parameters used in the validation of the adjoint sensitivity analysis

The comparisons between the derivatives computed by the adjoint procedure to those obtained by finite differences are presented in Tables 6 and 7 for the point load and distributed load, respectively. It can be seen that the design sensitivities are practically identical, thus verifying the derivation and the implementation of the adjoint procedure. The nonlinear response of both test cases is presented in Fig. 14, in terms of load-displacement curves at the prescribed DOF and equivalent plastic strain. From the tables it can be seen that even elements that are in the elastic regime contribute to the sum of plastic strains, in two opposite modes – i.e. the addition of material can either increase or decrease the overall plastic strain, whereas it always has a stiffening effect on compliance. Finally, the analysis and sensitivity analysis were repeated with 30 and 50 displacement increments. Practically identical results were obtained for the nonlinear respones as well as their design sensitivities.

Fig. 14

Nonlinear response of the test cases used for verification of the sensitivity analysis. a Load factor vs. prescribed displacement, point load; b Load factor vs. prescribed displacement, distributed load; c Equivalent plastic strains, point load; d Equivalent plastic strains, distributed load

Table 6 Verification of the sensitivity analysis, 4 element domain with a point load

Table 7 Verification of the sensitivity analysis, 4 element domain with a distributed load