Martin P. Bendsøe

Generating optimal topologies in structural design using a homogenization method

Optimal shape design of structural elements based on boundary variations results in final designs that are topologically equivalent to the initial choice of design, and general, stable computational schemes for this approach often require some kind of remeshing of the finite element approximation of the analysis problem. This paper presents a methodology for optimal shape design where both these drawbacks can be avoided. The method is related to modern production techniques and consists of computing the optimal distribution in space of an anisotropic material that is constructed by introducing an infimum of periodically distributed small holes in a given homogeneous, i~otropic material, with the requirement that the resulting structure can carry the given loads as well as satisfy other design requirements. The computation of effective material properties for the anisotropic material is carried out using the method of homogenization. Computational results are presented and compared with results obtained by boundary variations.

Optimal shape design as a material distribution problem

Shape optimization in a general setting requires the determination of the optimal spatial material distribution for given loads and boundary conditions. Every point in space is thus a material point or a void and the optimization problem is a discrete variable one. This paper describes various ways of removing this discrete nature of the problem by the introduction of a density function that is a continuous design variable. Domains of high density then define the shape of the mechanical element. For intermediate densities, material parameters given by an artificial material law can be used. Alternatively, the density can arise naturally through the introduction of periodically distributed, microscopic voids, so that effective material parameters for intermediate density values can be computed through homogenization. Several examples in two-dimensional elasticity illustrate that these methods allow a determination of the topology of a mechanical element, as required for a boundary variations shape optimization technique.

Topology Optimization of Continuum Structures with Local Stress Constraints

We introduce an extension of current technologies for topology optimization of continuum structures which allows for treating local stress criteria. We first consider relevant stress criteria for porous composite materials, initially by studying the stress states of the so-called rank 2 layered materials. Then, on the basis of the theoretical study of the rank 2 microstructures, we propose an empirical model that extends the power penalized stiffness model (also called SIMP for Solid Isotropic Microstructure with Penalization for inter-mediate densities). In a second part, solution aspects of topology problems are considered. To deal with the so-called ‘singularity’ phenomenon of stress constraints in topology design, an ϵ-constraint relaxation of the stress constraints is used. We describe the mathematical programming approach that is used to solve the numerical optimization problems, and show results for a number of example applications.

A new approach to variable-topology shape design using a constraint on perimeter

This paper introduces a method for variable-topology shape optimization of elastic structures called theperimeter method. An upper-bound constraint on the perimeter of the solid part of the structure ensures a well-posed design problem. The perimeter constraint allows the designer to control the number of holes in the optimal design and to establish their characteristic length scale. Finite element implementations generate practical designs that are convergent with respect to grid refinement. Thus, an arbitrary level of geometric resolution can be achieved, so single-step procedures for topology design and detailed shape design are possible. The perimeter method eliminates the need for relaxation, thereby circumventing many of the complexities and restrictions of other approaches to topology design.

Optimization methods for truss geometry and topology design

Truss topology design for minimum external work (compliance) can be expressed in a number of equivalent potential or complementary energy problem formulations in terms of member forces, displacements and bar areas. Using duality principles and non-smooth analysis we show how displacements only as well as stresses only formulations can be obtained and discuss the implications these formulations have for the construction and implementation of efficient algorithms for large-scale truss topology design. The analysis covers min-max and weighted average multiple load designs with external as well as self-weight loads and extends to the topology design of reinforcement and the topology design of variable thickness sheets and sandwich plates. On the basis of topology design as an inner problem in a hierarchical procedure, the combined geometry and topology design of truss structures is also considered. Numerical results and illustrative examples are presented.

Shape optimization of structures for multiple loading conditions using a homogenization method

A formulation for shape optimization of elastic structures subject to multiple load cases is presented. The problem is solved using a homogenization method. When compared to the single load solution strategy, it is shown that the more general formulation can produce more stable designs while it introduces little additional complexity.

Topology design of structures

Part I: Topology Design of Discrete Structures. Part II: Discrete Design and Selection Problems. Part III: The Homogenization Method for Topology Design. Part IV: Alternative Methods for Topology Design of Continuum Structures. Part V: Boundary Shape Design Methods. Part VI: Relaxation and Optimal Shape Design. Part VII: Effective Media Theory and Opimal Design. Part VIII: Extending the Scope of Topology Design. Part IX: Topology Design in a Computer-Aided Design Environment. Part X: Aspects of Toplogy Design. Index.

A New Method for Optimal Truss Topology Design

Truss topology optimization formulated in terms of displacements and bar volumes results in a large, nonconvex optimization problem. For the case of maximization of stiffness for a prescribed volume,this paper presents a new equivalent, an unconstrained and convex minimization problem in displacements only, where the function to be minimized is the sum of terms, each of which is the maximum of two convex,quadratic functions. Existence of solutions is proved, as is the convergence of a nonsmooth steepest descent-type algorithm for solving the topology optimization problem. The algorithm is computationally attractive and has been tested on a large number of examples, some of which are presented.

Topology and Generalized Layout Optimization of Elastic Structures

An overview of the method of homogenization to find the optimum layout of a linearly elastic structure is presented. The work discussed here presents a formulation to address the simultaneous optimization of the topology, shape and size of the structure. The discussion includes optimization of plane, plate and three dimensional shell structures.

Equivalent displacement based formulations for maximum strength Truss topology design

Abstract Maximum strength elastic truss structural design is conveniently formulated in terms of displacements and bar volumes. The resulting problem is nonconvex, and for topology design very large, as one seeks the optimal topology as a subset of a large number of potential bars connecting all nodal points of an initially chosen set. In this paper we present a number of equivalent formulations in the displacements only, taking full advantage of the structure of the optimization problem. The equivalent formulations are of min-max type or are quadratic programming problems in the displacements, reducing in some cases even to linear programming problems.