Material interpolation schemes in topology optimization

Abstract

In topology optimization of structures, materials and mechanisms, parametrization of geometry is often performed by a grey-scale density-like interpolation function. In this paper we analyze and compare the various approaches to this concept in the light of variational bounds on effective properties of composite materials. This allows us to derive simple necessary conditions for the possible realization of grey-scale via composites, leading to a physical interpretation of all feasible designs as well as the optimal design. Thus it is shown that the socalled arti®cial interpolation model in many circumstances actually falls within the framework of microstructurally based models. Single material and multi-material structural design in elasticity as well as in multi-physics problems is discussed. IntroductionThe area of computational variable-topology shape design of continuum structures is presently dominated by methods which employ a material distribution approach for a ®xed reference domain in the spirit of the so-called`homogenization method' for topology design, [1]. That is, the geometric representation of a structure is similar to a grey-scale rendering of an image, in discrete form corresponding to a raster representation of the geometry. This concept has proven very powerful, but it does involve a number of dif®culties. One is the issue of existence of solutions, another the issue of solution method. Here, the notion of physical models for grey' material is of great importance, and it is these interpolation schemes and their relation to characterizations of composite materials which are the themes in the following.In many applications, the optimal topology of a structure should consist solely of a macroscopic variation of one material and void, meaning that the density of the structure is given by a``0±1'' integer parametrization (often called a black-and-white design). Unfortunately, this class of optimal design problems is ill-posed in that, for example, nonconvergent, minimizing sequences of admissible designs with ®ner and ®ner geometrical details can be found, see [2,3]. Existence of black-and-white solutions can be achieved by con®ning the solution space to limit the complexity of the admissible designs, making the designs dependent on the choice of parameters in the geometrical constraint. Such a restriction of the design space can be accomplished in a number of ways, e.g. by enforcing an upper bound on the perimeter of the

Topology optimization of acoustic–structure interaction problems using a mixed finite element formulation

“…Positive values between 3 and 6 are used for n . In [34] the physical realizability of material microstructures with elastic properties corresponding to different interpolation schemes was proven for linear elasticity problems. In the present acoustic-structural formulation the question of physical realizability is not easy to answer since microscopic structural-acoustic response depends on length-scale and excitation frequency, however, it is reasonable to assume that there exist porous media with properties obeying the RAMP interpolation assuming that the microstructural scale is much smaller than the acoustic wave length.…”

Topology optimization of continuum structures with stress constraints and uncertainties in loading

“…In Equation 1, the von Mises yield criterion is adopted as material failure constraint, for all points of the design domain, and the equilibrium configuration is represented through the stationarity of the total potential energy Π( ) of the system. 43 The SIMP model 44 to material parameterization is used in this formulation.…”

Topology Design of Structures, Materials and Mechanisms — Status and Perspectives

“…Complementing the use of the homogenization method, where anisotropie composites are apriori accepted as part of the design space, a popular method to model material properties whieh are isotropie at intermediate densities is the so-called penalized, proportional fictitious material SIMP-model (SIMP: Solid Isotropie Material with Penalization), see for example [8] for an overview. In this model a continuous variable p is introduced, with 0 :S p :S 1, resembling a density of material by the fact that the volume of the structure is evaluated as (8) The relation between this density and the material tensor Cijkl in the equilibrium analysis is written as (9) where the given material has stiffness given by The interpolation final design has density zero and one in all points, this design is a blackand-white design for which the performance has been evaluated with a correct physical model.…”

“…In this model a continuous variable p is introduced, with 0 :S p :S 1, resembling a density of material by the fact that the volume of the structure is evaluated as (8) The relation between this density and the material tensor Cijkl in the equilibrium analysis is written as (9) where the given material has stiffness given by The interpolation final design has density zero and one in all points, this design is a blackand-white design for which the performance has been evaluated with a correct physical model. For problems where the volume constraint is active, experience shows that optimization does actually result in such designs if one chooses the power p sufficiently big (in order to obtain true '0-1' designs, p 2: 3 is usually required) .…”

“…Instead we refer to existing literat ure containing rather comprehensive surveys, see e.g., [7,8,31]. Moreover, note that reference is mostly made to recent papers that include bibliographies useful for on overview of the area.…”