Topology and shape optimization for elastoplastic structural response

Abstract

It is a common practice to base both, material topology optimization as well as a subsequent shape optimization on linear elastic response. However, in order to obtain a realistic design, it might be essential to base the optimization on a more realistic physical behavior, i.e. to consider geometrically or/and materially nonlinear effects.

In the present paper, an elastoplastic von Mises material model with linear isotropic hardening/softening for small strains is used. The objective of the design problem is to maximize the structural ductility in the elastoplastic range while the mass in the design space is prescribed. With respect to the specific features of either topology or shape optimization, for example the number of optimization variables or their local–global influence on the structural response, different methods are applied. For topology optimization problems, the gradient of the ductility is determined by the variational adjoint approach. In shape optimization, the derivatives of the state variables with respect to the optimization variables are evaluated analytically by a variational direct approach. The topology optimization problem is solved by an optimality criteria (OC) method and the shape optimization problem by a mathematical programming (MP) method. In topology optimization, a geometrically adaptive procedure is additionally applied in order to increase the efficiency and to avoid artificial stress singularities. The procedures are verified by 2D design problems under plane stress conditions.

Introduction

Up to now, materially and geometrically linear structural response is mainly assumed in structural optimization. However, in order to generate a reliable design by structural optimization the nonlinear structural response, e.g. including buckling or elastoplastic behavior, ought to be considered for both, topology and shape optimization procedures. For example, topology optimization often leads to slender structures which are very sensitive with respect to buckling.

It has to be mentioned that the sensitivity analysis represents the crucial point of the entire optimization procedure. The main effort in order to determine these sensitivities is that for path-dependent problems, the structural sensitivities are also path dependent. These sensitivities can be evaluated numerically by the finite difference method or analytically by the adjoint variable method or the direct differentiation method. In the case of the analytical method, a variational and a discrete approach can be distinguished.

Due to the large number of optimization variables in topology optimization, a variational adjoint approach to determine the structural sensitivities leads to an efficient formulation. For shape optimization problems including elastoplasticity, a variational direct approach is favorable.

Ryu et al. [28] address problems including geometrically and materially nonlinear problems. They mention that for materially nonlinear models, further investigations considering analytical sensitivity expressions are necessary. Vidal and Haber [38] present a method in order to determine the structural sensitivities which is consistent with the implicit methods to integrate the constitutive equations based on the return mapping algorithms for elastoplasticity. A general theory for continuous optimization problems including geometrically and materially nonlinear structural response is also addressed by Tsay and Arora [36] and Kleiber [9]. Tortorelli [35] derives a method evaluating the design sensitivities for elastostatic finite displacement systems.

In material topology optimization, Neves et al. [20] and Maute [13] approximate the geometrically nonlinear response restricting the stability problem to a linearized eigenvalue problem and maximizing the critical buckling load. Bruns and Tortorelli [6] determine the topology for the stiffest structure including large deflections but small strains.

Material topology optimization including material nonlinearities is addressed by Yuge and Kikuchi [41], where the structural stiffness of frame structures is maximized. The algorithm is based on a linear Timoshenko beam theory and an elastoplastic material model with linear work-hardening. Softening and unloading are not considered. Taylor and Logo [33] and Taylor [34] maximize the load factor limiting the maximum energy of truss structures for nonlinear material. Mayer et al. [17] determine the optimum topology of shell structures under dynamic load conditions. The response of a dynamic analysis is based on an elastoplastic material model. However, the sensitivities are approximated for quasi-static conditions. Nonlinear material behavior in topology optimization is reported by Swan and Kosaka [32] for a modified Voigt–Reuss material model and by Maute et al. [16] for the von Mises yield criterion considering the consistent material tangent.

For shape optimization, geometrically nonlinear behavior is considered by Smaoui and Schmit [31] minimizing the structural weight of space trusses by varying the cross-sectional areas and the location of the nodes. Reitinger and Ramm [24] and Polynkin et al. [22] maximize the critical load factor of thin-walled shell structures. Bletzinger [5] determines the optimal shape of membrane structures. Shape optimization including hyperelastic material behavior is applied by Barthold et al. [1].

The problem of design sensitivities for elastoplastic material behavior is addressed by Vidal and Haber [38] and by Barthold and Wiechmann [2] assuming small strains. Wiechmann and Barthold [39] also determine the structural sensitivities for large strain elastoplasticity applying a variational principle.

Since analytical expressions for the sensitivities are cumbersome to derive, Wu and Arora [40] apply a semi-analytical method. Vaz and Hinton [37] also use the semi-analytical approach for the evaluation of the sensitivities to handle materially nonlinear problems for shape optimization.

The optimization of discrete structures, like beams and trusses, including geometrical and material nonlinearities, is addressed by Choi and Santos [7] and Ohsaki and Arora [21].

In the present paper, a combined approach is presented in order to determine the design by topology and shape optimization including material nonlinearities for ductile materials like steel or aluminium. As usual those ductile materials are modelled by an elastoplastic von Mises yield criterion with linear isotropic work-hardening/softening for small strains. In order to take advantage of the structural behavior in the elastoplastic range, the structural ductility is maximized while the mass in the design space is prescribed.

First, the structural layout is determined by material-based topology optimization. The optimization variables are the density of each finite element. Conventional topology optimization procedures with a fixed discretization of the design space often lead only to a rough approximation of the optimum geometry. The contours are described by jagged boundaries which cause stress singularities and artificial yielding. These shortcomings can be overcome by an adaptive procedure adapting smoothly the design space during the topology optimization process (Maute and Ramm [14]). The final layout is obtained by a CAD-oriented shape optimization process considering additional criteria, like displacement or/and stress limits (Maute and Ramm [15]). Here the optimization variables are the control nodes of splines and surface patches.

For the topology optimization problem, a variational adjoint approach is used to determine the sensitivity of the objective. The sensitivities of the displacements for the following shape optimization procedure are determined by the variational direct approach. From these the sensitivities of stresses, objective and constraints can be derived. It should be noted that the kind of controlling parameters for the path following procedure, i.e. following a displacement or load control, influences the scheme of how the sensitivities are determined.

For optimization problems with a large number of optimization variables, as in topology optimization, optimality criteria (OC) methods turn out to be efficient and robust. For a few number of variables, as it is the case in shape optimization, mathematical programming (MP) methods, for example a SQP-algorithm, are favorable.

The sensitivities determined by the above-mentioned procedures are compared to those obtained by a finite difference scheme showing that the present approach works well. The nonlinear structural response is determined by a displacement-controlled algorithm, which is well-suited for elastoplastic problems. The optimization procedures are applied to 2D design problems under plane stress conditions and the results for elastoplastic material are compared to those where only linear elastic material is adopted.