Ole Sigmund

Numerical instabilities in topology optimization : A survey on procedures dealing with checkerboards, mesh-dependencies and local minima

In this paper we seek to summarize the current knowledge about numerical instabilities such as checkerboards, mesh-dependence and local minima occurring in applications of the topology optimization method. The checkerboard problem refers to the formation of regions of alternating solid and void elements ordered in a checkerboard-like fashion. The mesh-dependence problem refers to obtaining qualitatively different solutions for different mesh-sizes or discretizations. Local minima refers to the problem of obtaining different solutions to the same discretized problem when choosing different algorithmic parameters. We review the current knowledge on why and when these problems appear, and we list the methods with which they can be avoided and discuss their advantages and disadvantages.

On the Design of Compliant Mechanisms Using Topology Optimization

ABSTRACT This paper presents a method for optimal design of compliant mechanism topologies. The method is based on continuum-type topology optimization techniques and finds the optimal compliant mechanism topology within a given design domain and a given position and direction of input and output forces. By constraining the allowed displacement at the input port, it is possible to control the maximum stress level in the compliant mechanism. The ability of the design method to find a mechanism with complex output behavior is demonstrated by several examples. Some of the optimal mechanism topologies have been manufactured, both in macroscale (hand-size) made in Nylon, and in microscale (<.5mm)) made of micromachined glass.

DESIGN OF MATERIALS WITH EXTREME THERMAL EXPANSION USING A THREE-PHASE TOPOLOGY OPTIMIZATION METHOD

Abstract Composites with extremal or unusual thermal expansion coefficients are designed using a three-phase topology optimization method. The composites are made of two different material phases and a void phase. The topology optimization method consists in finding the distribution of material phases that optimizes an objective function (e.g. thermoelastic properties) subject to certain constraints, such as elastic symmetry or volume fractions of the constituent phases, within a periodic base cell. The effective properties of the material structures are found using the numerical homogenization method based on a finite-element discretization of the base cell. The optimization problem is solved using sequential linear programming. To benchmark the design method we first consider two-phase designs. Our optimal two-phase microstructures are in fine agreement with rigorous bounds and the so-called Vigdergauz microstructures that realize the bounds. For three phases, the optimal microstructures are also compared with new rigorous bounds and again it is shown that the method yields designed materials with thermoelastic properties that are close to the bounds. The three-phase design method is illustrated by designing materials having maximum directional thermal expansion (thermal actuators), zero isotropic thermal expansion, and negative isotropic thermal expansion. It is shown that materials with effective negative thermal expansion coefficients can be obtained by mixing two phases with positive thermal expansion coefficients and void.

Materials with Prescribed Constitutive Parameters : An Inverse Homogenization Problem

Abstract This paper deals with the construction of materials with arbitrary prescribed positive semi-definite constitutive tensors. The construction problem can be called an inverse problem of finding a material with given homogenized coefficients. The inverse problem is formulated as a topology optimization problem i.e. finding the interior topology of a base cell such that cost is minimized and the constraints are defined by the prescribed constitutive parameters. Numerical values of the constitutive parameters of a given material are found using a numerical homogenization method expressed in terms of element mutual energies. Numerical results show that arbitrary materials, including materials with Poissons ratio −1.0 and other extreme materials, can be obtained by modelling the base cell as a truss structure. Furthermore, a wide spectrum of materials can be constructed from base cells modelled as continuous discs of varying thickness. Only the two-dimensional case is considered in this paper but formulation and numerical procedures can easily be extended to the three-dimensional case.

Design of multiphysics actuators using topology optimization : Part II : Two-material structures

This is the second part of a two-paper description of the topology optimization method applied to the design of multiphysics actuators and electrothermomechanical systems in particular. The first paper is focussed on one-material structures, the second on two-material structures. The extensions of the topology optimization method in this part include design descriptions for two-material structures, constitutive modelling of elements with mixtures of two materials, formulation of optimization problems with multiple constraints and multiple materials and a mesh-independency scheme for two-material structures. The application in mind is the design of thermally and electrothermally driven micro actuators for use in MicroElectroMechanical Systems (MEMS). MEMS are microscopic mechanical systems coupled with electrical circuits. MEMS are fabricated using techniques known from the semi-conductor industry. Several of the examples from Part I are repeated, allowing for the introduction of a second material in the design domain. The second material can differ in mechanical properties such as Youngs modulus or electrical and thermal conductivity. In some cases there are significant gains in introducing a second material. However, the gains depend on boundary conditions and relations between the material properties and are in many cases insignificant.

Checkerboard patterns in layout optimization

Effective properties of arrangements of strong and weak materials in a checkerboard fashion are computed. Kinematic constraints are imposed so that the displacements are consistent with typical finite element approximations. It is shown that when four-node quatrilateral elements are involved, these constraints result in a numerically induced, artificially high stiffness. This can account for the formation of checkerboard patterns in continuous layout optimization problems of compliance minimization.

Systematic design of phononic band-gap materials and structures by topology optimization.

Phononic band–gap materials prevent elastic waves in certain frequency ranges from propagating, and they may therefore be used to generate frequency filters, as beam splitters, as sound or vibration protection devices, or as waveguides. In this work we show how topology optimization can be used to design and optimize periodic materials and structures exhibiting phononic band gaps. Firstly, we optimize infinitely periodic band–gap materials by maximizing the relative size of the band gaps. Then, finite structures subjected to periodic loading are optimized in order to either minimize the structural response along boundaries (wave damping) or maximize the response at certain boundary locations (waveguiding).

A new class of extremal composites

Abstract The paper presents a new class of two-phase isotropic composites with extremal bulk modulus. The new class consists of micro geometrics for which exact solutions can be proven and their bulk moduli are shown to coincide with the Hashin–Shtrikman bounds. The results hold for two and three dimensions and for both well- and non-well-ordered isotropic constituent phases. The new class of composites constitutes an alternative to the three previously known extremal composite classes: finite rank laminates, composite sphere assemblages and Vigdergauz microstructures. An isotropic honeycomb-like hexagonal microstructure belonging to the new class of composites has maximum bulk modulus and lower shear modulus than any previously known composite. Inspiration for the new composite class comes from a numerical topology design procedure which solves the inverse homogenization problem of distributing two isotropic material phases in a periodic isotropic material structure such that the effective properties are extremized.

Composites with extremal thermal expansion coefficients

We design three‐phase composites having maximum thermal expansion, zero thermal expansion, or negative thermal expansion using a numerical topology optimization method. It is shown that composites with effective negative thermal expansion can be obtained by mixing two phases of positive thermal expansions with a void phase. We also show that there is no mechanistic relationship between negative thermal expansion and negative Poisson’s ratio.

Multiphase composites with extremal bulk modulus

Abstract This paper is devoted to the analytical and numerical study of isotropic elastic composites made of three or more isotropic phases. The ranges of their effective bulk and shear moduli are restricted by the Hashin–Shtrikman–Walpole (HSW) bounds. For two-phase composites, these bounds are attainable, that is, there exist composites with extreme bulk and shear moduli. For multiphase composites, they may or may not be attainable depending on phase moduli and volume fractions. Sufficient conditions of attainability of the bounds and various previously known and new types of optimal composites are described. Most of our new results are related to the two-dimensional problem. A numerical topology optimization procedure that solves the inverse homogenization problem is adopted and used to look for two-dimensional three-phase composites with a maximal effective bulk modulus. For the combination of parameters where the HSW bound is known to be attainable, new microstructures are found numerically that possess bulk moduli close to the bound. Moreover, new types of microstructures with bulk moduli close to the bound are found numerically for the situations where the aforementioned attainability conditions are not met. Based on the numerical results, several new types of structures that possess extremal bulk modulus are suggested and studied analytically. The bulk moduli of the new structures are either equal to the HSW bound or higher than the bulk modulus of any other known composite with the same phase moduli and volume fractions. It is proved that the HSW bound is attainable in a much wider range than it was previously believed. Results are readily applied to two-dimensional three-phase isotropic conducting composites with extremal conductivity. They can also be used to study transversely isotropic three-dimensional three-phase composites with cylindrical inclusions of arbitrary cross-sections (plane strain problem) or transversely isotropic thin plates (plane stress or bending of plates problems).