Topology optimization of continuum structures with Drucker–Prager yield stress constraints

Abstract

This paper presents an efficient topology optimization strategy for seeking the optimal layout of continuum structures exhibiting asymmetrical strength behaviors in compression and tension. Based on the Drucker–Prager yield criterion and the power-law interpolation scheme for the material property, the optimization problem is formulated as to minimize the material volume under local stress constraints. The ε-relaxation of stress constraints is adopted to circumvent the stress singularity problem. For improving the computational efficiency, a grouped aggregation approach based on the Kreisselmeier–Steinhauser function is employed to reduce the number of constraints without much sacrificing the approximation accuracy of the stress constraints. In conjunction with the adjoint-variable sensitivity analysis, the minimization problem is solved by a gradient-based optimization algorithm. Numerical examples demonstrate the validity of the present optimization model as well as the efficiency of the proposed numerical techniques. Moreover, it is also revealed that the optimal design of a structure with pressure-dependent material may exhibit a considerable different topology from the one obtained with pressure-independent material model.

Introduction

Topology or layout optimization, which aims to determine the best material distribution in a predefined domain, is now regarded as a powerful automated tool in the conceptual design stage of products. Since Bendsøe and Kikuchi [1] first applied the homogenization technique for generating optimal topologies in 1988, topology optimization of continuum structures has been a hot research topic in the field of structural and multidisciplinary optimization for two decades. As a result, a number of milestone papers and monographs (e.g. [2], [3], [4]) have been published. Many numerical techniques (including the solid isotropic microstructures with penalization [5], [6], the evolutionary structural optimization [7], the level set method [8]) have also been developed and applied to a wide variety of practical engineering applications.

An overwhelming majority of topology optimization studies are devoted to distributing a given amount of material for achieving the optimal global behavior, such as the minimum mean compliance or maximum fundamental natural frequency. Obviously, topology optimization settings for global behaviors have an advantage in the aspect of numerical computation because typically only a small number of constraints (more often, a single material volume constraint) need to be considered in the optimization problem. Since most of materials in real structural design problems may suffer from strength failure, topology optimization with stress constraints has also attracted much interest. Earlier efforts on generating optimal configurations of continuum structures subjected to von Mises stress constraints over each individual element can be traced back to late 1990s, see e.g. Cheng and Zhang [9] and Shim and Manoochehri [10]. Since then, the topology optimization with local stress constraints has been studied by Duysinx and Sigmund [11], Duysinx and Bendsøe [12], Pereira et al. [13] and Bruggi [14]. In their studies, several effective relaxation approaches have been introduced in order to avoid the singularity associated with the stress constraints (see [15] and the references therein). On the other hand, dealing with the local stress constraints in real-scale topology optimization problems may be very complicated and time-consuming due to a large number of highly nonlinear stress constraints. For tackling this difficulty, the global stress constraint approach, which was first applied to continuum topology optimization by Yang and Chen [16], Duysinx and Sigmund [11] and recently further developed by e.g. Le et al. [17] and París et al. [18], [19], can be successfully used. Therein, the optimization problem has only one or several global constraints, which gather the effects of all the local stress constraints by using certain aggregation functions. Thus the optimization problem becomes tractable since the number of constraints is drastically reduced.

As the literature survey reveals, the above-mentioned works on topology optimization considering material failure constraints were based on the quadratic von Mises yield criterion. This criterion can well predict failure behavior of metals and thus has been used widely in mechanical engineering. However, many non-metallic materials, such as concrete, rocks, ceramics and polymers, are characterized by increasing shear strength as a result of increasing hydrostatic pressure. Structures made of these pressure-dependent materials would typically exhibit different stress limits in tension and compression. In such a case, the von Mises criterion is not appropriate and the Drucker–Prager criterion [20] defined in terms of stress invariants is available as one of the simplest plasticity yield models. When a material exhibits a failure behavior modeled with the Drucker–Prager yield criterion, it can be referred to as a “Drucker–Prager modeling plasticity material”.

It is noted that a few studies have considered non-metallic material properties in the topology synthesis of continuum structures. Duysinx [21] described a topology optimization method based on Raghava and Ishai equivalent stresses, which are generally used to predict the pressure-dependent behavior of adhesive materials. Demorat and Duvaut [22] extended classical compliance minimization to long-fiber composite membranes by taking the orientation and the fiber density as local design parameters. Querin et al. [23] proposed to modify the optimality criterion approach for topology optimization of truss-like continua that exhibit different material behaviors in tension and compression. To achieve this, they identified all compressive parts of a structure, converted the material properties to orthotropic ones, and then updated the design variables by using the optimality criterion. It is shown in these studies that the optimal topology obtained by considering non-metallic material properties may have significant differences as compared with one obtained under von Mises stress constraints. Therefore, it seems quite obvious that Drucker–Prager criterion should be seriously considered in topology optimization formulations of structures containing so-called Drucker–Prager modeling plasticity materials such as concrete, soil and rock.

Despite the fact that Drucker–Prager yield criterion can be easily integrated into finite element analysis and size optimization techniques [24], it has not been taken into account in existing topology optimization approaches, partly because of the complicated procedure and the high computational cost involved in the solutions. In this study, by using the Drucker–Prager criterion for description of material yielding behavior, a topology optimization method is presented for the layout design of continuum structures with Drucker–Prager modeling plasticity materials. Based on the solid isotropic microstructure with penalization (SIMP) model for linking the material properties and the relative density design variables, the considered problem is formulated as to find the optimal material distribution that minimizes the total material volume under Drucker–Prager yield stress constraints on all elements. The grouped constraints aggregation method, as well as the ε-relaxation technique for regularization of the stress constraints, is employed to make the topology problem numerically tractable. The design sensitivity analysis is implemented with the adjoint variable method. Then the method of moving asymptotes (MMA) [25] is employed to update the design variables. Finally, the validity of the present optimization model and the proposed techniques is demonstrated by topology optimization solutions of four numerical examples.