Inverse Homogenization Research Articles

On the reformulation of topology optimization problems as linear or convex quadratic mixed 0–1 programs

We consider equivalent reformulations of nonlinear mixed 0–1 optimization problems arising from a broad range of recent applications of topology optimization for the design of continuum structures and composite materials. We show that the considered problems can equivalently be cast as either linear or convex quadratic mixed 0–1 programs. The reformulations provide new insight into the structure of the problems and may provide a foundation for the development of new methods and heuristics for solving topology optimization problems. The applications considered are maximum stiffness design of structures subjected to static or periodic loads, design of composite materials with prescribed homogenized properties using the inverse homogenization approach, optimization of fluids in Stokes flow, design of band gap structures, and multi-physics problems involving coupled steady-state heat conduction and linear elasticity. Several numerical examples of maximum stiffness design of truss structures are presented.

Optimal design of composite structures for strength and stiffness: an inverse homogenization approach

We introduce a rigorously based numerical method for compliance minimization problems in the presence of pointwise stress constraints. The method is based on new multiscale quantities that measure the amplification of the local stress due to the microstructure. The design method is illustrated for two different kinds of problems. The first identifies suitably graded distributions of fibers inside shaft cross sections that impart sufficient overall stiffness while at the same time adequately control the amplitude of the local stress at each point. The second set of problems are carried out in the context of plane strain. In this study, we recover a novel class of designs made from locally layered media for minimum compliance subject to pointwise stress constraints. The stress-constrained designs place the more compliant material in the neighborhood of stress concentrators associated with abrupt changes in boundary loading and reentrant corners.

Design of maximum permeability material structures

This paper extends recent advances in the topology optimization of fluid flows to the design of periodic, porous material microstructures. Operating in a characteristic base cell of the material, the goal is to determine the layout of solid and fluid phases that will yield maximum permeability and prescribed flow symmetries in the bulk material. Darcy’s law governs flow through the macroscopic material while Stokes equations govern flow through the microscopic channels. Permeability is computed via numerical homogenization of the base cell using finite elements. Solutions to the proposed inverse homogenization design problem feature simply connected pore spaces that closely resemble minimal surfaces, such as the triply periodic Schwartz P minimal surface for 3 − d isotropic, maximum permeability materials.

Homogenization and inverse homogenization for 3D composites of complex architecture

Purpose – To describe the mathematics, mechanics and computer code that are involved in deriving the mechanical properties of a 3D composite material with a complicated internal architecture. To inform the reader how an application programming interface (API) can be used with a commercial FEA code to undertake the task. Finally to validate the process an demonstrate the versatility of the process.Design/methodology/approach – The complex architecture of the composite is imported to an FEA environment and meshed. The special code is written in Pascal that applies six sets of constraints to simulate unit strain vectors on a cell of the composite. After six separate analyses are undertaken, the forces necessary to achieve the boundary constraints are summed to provide stresses and hence the necessary coefficients in the stress to strain relationship for the composite. After global FEA the strains in the homogenized material are used as input to the inverse homogenizer so that stress and strain levels in the ...

Optimizing multifunctional materials: Design of microstructures for maximized stiffness and fluid permeability

Topology optimization is used to systematically design periodic materials that are optimized for multiple properties and prescribed symmetries. In particular, mechanical stiffness and fluid transport are considered. The base cell of the periodic material serves as the design domain and the goal is to determine the optimal distribution of material phases within this domain. Effective properties of the material are computed from finite element analyses of the base cell using numerical homogenization techniques. The elasticity and fluid flow inverse homogenization design problems are formulated and existing techniques for overcoming associated numerical instabilities and difficulties are discussed. These modules are then combined and solved to maximize bulk modulus and permeability in periodic materials with cubic elastic and isotropic flow symmetries. The multiphysics problem is formulated such that the final design is dependent on the relative importance, or weights, assigned by the designer to the competing stiffness and flow terms in the objective function. This allows the designer to tailor the microstructure according to the materials’ future application, a feature clearly demonstrated by the presented results. The methodology can be extended to incorporate other material properties of interest as well as the design of composite materials.

Inverse homogenization and design of microstructure for pointwise stress control

Summary New higher order homogenization results are employed in an inverse homogenization procedure to identify graded microstructures that provide desirable structural response while ensuring stress control near joints or junctions between structural elements. The methodology is illustrated for long cylindrical shafts reinforced with sti cylindrical elastic b ers with generators parallel to the shaft. The local b er geometry can change across the shaft cross section. The methodology is implemented numerically for cross{sectional shapes that possesses reentrant corners typically seen in lap joints and junctions of struts. Graded locally layered microgeometies are identied that provide the required structural rigidity with respect to torsion loading while at the same time mitigating the inuence of stress concentrations at the reentrant corners. Modern design practice increasingly incorporates the use of load bearing components made from composite materials. Composites are now used in structural geometries that involve abrupt dimensional changes within structural components, such as skins connected to ribs, panel reinforcements and junctions of struts. Associated with these geometries are stress concentrations and the potential for failure. In this paper a design strategy is formulated for identifying graded microstructures that can be used to control the local uctuating stresses near stress concentrations induced by rivets, bolts or reentrant corners. Reentrant corners are are typically found in lap joints and near the junction between stieners and panels. The inverse homogenization design method is based upon the formulation of a homogenized design problem expressed in terms of suitable macroscopic quantities that satisfy two requirements: The rst requirement is that the homogenized design problem should be computationally tractable. The second requirement is that the solution of the homogenized design problem must provide the means to explicitly identify graded microstructures that engender suitable structural response while at the same time control local uctuating stresses in regions located near stress concentrations. It is now well known that eectiv e macroscopic constitutive properties relating average stress to average strain can be employed in the numerical design of composite structures

A material optimization model to approximate energy bounds for cellular materials under multiload conditions

This paper describes a computational model, based on inverse homogenization and topology design, for approximating energy bounds for two-phase composites under multiple load cases. The approach allows for the identification of possible single-scale cellular materials that give rise to the optimal bounds within this class of composites. A comparison of the computational results with the globally optimal bounds given via rank-N layered composites illustrates the behaviour for tension and shear load situations, as well as the importance of considering the shape of the basic unit cell as part of the design process.

Designing materials with prescribed elastic properties using polygonal cells

AbstractAn extension of the material design problem is presented in which the base cell that characterizes the material microgeometry is polygonal. The setting is the familiar inverse homogenization problem as introduced by Sigmund. Using basic concepts in periodic planar tiling it is shown that base cells of very general geometries can be analysed within the standard topology optimization setting with little additional effort. In particular, the periodic homogenization problem defined on polygonal base cells that tile the plane can be replaced and analysed more efficiently by an equivalent problem that uses simple parallelograms as base cells. Different material layouts can be obtained by varying just two parameters that affect the geometry of the parallelogram, namely, the ratio of the lengths of the sides and the internal angle. This is an efficient way to organize the search of the design space for all possible single‐scale material arrangements and could result in solutions that may be unreachable using a square or rectangular base cell. Examples illustrate the results. Copyright © 2003 John Wiley & Sons, Ltd.

On Bone Mechanics, Modelling and Optimization

Analysis, evolution and layout in bone mechanics are described as seen from an optimal design and solid mechanics perspective. Anisotropic behaviour and three-dimensional modelling are necessary and focus is therefore put on the aspects of dealing with a large number of material parameters. The tools of homogenization, inverse homogenization and some basic results from optimal structural design are described.

Inverse homogenization for evaluation of effective properties of a mixture

The paper deals with indirect evaluation of the effective thermal or hydraulic conductivity of a random mixture of twodifferent materials from the known effective complexpermittivity of the same mixture. The method is based onderiving information about the microstructure of thecomposite from measurements of its effective properties; wecall this approach inverse homogenization. This structuralinformation is contained in the spectral measure in the Stieltjes representation of the effective complexpermittivity. The spectral measure can be reconstructed fromeffective measurements and used to estimate other effective properties of the same material. We introduce S-equivalence of the geometric structures corresponding tothe same spectral measure, and show that the microstructures of different mixtures can be distinguishedby the homogenized measurements up to the introducedequivalence. We show that the identification problem for thespectral function has a unique solution, however, the problemis extremely ill-posed. Several stabilization techniques arediscussed such as quadratically constrained minimization andreconstruction in the class of functions of boundedvariation. The approach is applicable to porous media,biological materials, artificial composites and otherheterogeneous materials in which the scale of microstructureis much smaller than the wavelength of the electromagneticsignal.

On the inverse homogenization problem of linear composite materials

The standard problem of homogenization consists of a derivation or estimation of the electromagnetic constitutive properties of a composite material. In that approach, the constitutive properties of the constituent materials (as specified by their respective permittivity, permeability, and magnetoelectric dyadics), their mixing ratios, and certain geometrical properties pertaining to the constituents are known. Here, the inverse problem is pursued: What information about the electromagnetic constitutive parameters of one of the constituent materials in a two-component mixture can be extracted from a knowledge of the constitutive properties of the homogenized composite medium and those of the other component material? This approach is called the inverse homogenization problem, and it is studied within the framework of linear homogenization through the Maxwell Garnett and the Bruggeman formalisms. © 2001 John Wiley & Sons, Inc. Microwave Opt Technol Lett 28: 421–423, 2001.

Design of microstructures of viscoelastic composites for optimal damping characteristics

An inverse homogenization problem for two-phase viscoelastic composites is formulated as a topology optimization problem. The effective complex moduli are estimated by the numerical homogenization using the finite element method. Sensitivity analysis shows that the sensitivity calculations do not require the solution of any adjoint problem. The objective function is defined so that the topology optimization problem finds microstructures of viscoelastic composites which exhibit improved stiffness/damping characteristics within the specified operating frequency range. Design constraints include volume fraction, effective complex moduli, geometric symmetry and material symmetry. Several numerical design examples are presented with discussions on the nature of the designed microstructures. From the designed microstructures, it is found that mechanism-like structures and wavy structures are formed to maximize damping while retaining stiffness at the desired level.

Multiphase composites with extremal bulk modulus

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).

A new class of extremal composites

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.

Inverse bounds for microstructural parameters of composite media derived from complex permittivity measurements

Bounds on the volume fraction of the constituents in a two-component mixture are derived from measurements of the effective complex permittivity of the mixture, using the analyticity of the effective property. First-order inverse bounds for general anisotropic materials, as well as second-order bounds for isotropic mixtures, are obtained. By exploiting an analytic representation of the effective complex permittivity, the problem of estimating the structural parameters is reduced to a problem of evaluating the moments and support of a measure containing information about the geometrical structure of the material. Rigorous bounds on the volume fraction are found by inverting first- and second-order (Hashin–Shtrikman) forward bounds on the complex permittivity. The inverse bounds are applied to measurements of the effective complex permittivity of sea ice, which is a three-component mixture of ice, brine and air. The sea ice is treated via the two-component theory applied to a mixture of brine and an ice/air composite. The bounds on the brine volume of sea ice derived from the effective permittivity measurements are in excellent agreement with data from experiments. The inversion of forward bounds on the complex permittivity of composite media provides a basis for a theory of inverse homogenization for recovering microstructural parameters from bulk property measurements. Such results are applicable to problems in remote sensing, medical imaging and non-destructive testing of materials.

Inverse electromagnetic scattering models for sea ice

Inverse scattering algorithms for reconstructing the physical properties of sea ice from scattered electromagnetic field data are presented. The development of these algorithms has advanced the theory of remote sensing, particularly in the microwave region, and has the potential to form the basis for a new generation of techniques for recovering sea ice properties, such as ice thickness, a parameter of geophysical and climatological importance. Moreover, the analysis underlying the algorithms has led to significant advances in the mathematical theory of inverse problems. In particular, the principal results include the following. (1) Inverse algorithms for reconstructing the complex permittivity in the Helmholtz equation in one and higher dimensions, based on layer stripping and nonlinear optimization, have been obtained and successfully applied to a (lossless) laboratory system. In one dimension, causality has been imposed to obtain stability of the solution and layer thicknesses can be obtained from the recovered dielectric profile, or directly from the reflection data through a nonlinear generalization of the Paley-Wiener theorem in Fourier analysis. (2) When the wavelength is much larger than the microstructural scale, the above algorithms reconstruct a profile of the effective complex permittivity of the sea ice, a composite of pure ice with random brine and air inclusions. A theory of inverse homogenization has been developed, which in this quasistatic regime, further inverts the reconstructed permittivities for microstructural information beyond the resolution of the wave. Rigorous bounds on brine volume and inclusion separation for a given value of the effective complex permittivity have been obtained as well as an accurate algorithm for reconstructing the brine volume from a set of values. (3) Inverse algorithms designed to recover sea ice thickness have been developed. A coupled radiative transfer-thermodynamic sea ice inverse model has accurately reconstructed the growth of a thin, artificial sea ice sheet from time-series electromagnetic scattering data.

Evaluation of the Micro-mechanical Behavior of Composite Materials Using an Inverse Homogenization Anslysis

Evaluation of the Micro-mechanical Behavior of Composite Materials Using an Inverse Homogenization Anslysis

Tailoring materials with prescribed elastic properties

Abstract This paper describes a method to design the periodic microstructure of a material to obtain prescribed constitutive properties. The microstructure is modelled as a truss or thin frame structure in 2 and 3 dimensions. The problem of finding the simplest possible microstructure with the prescribed elastic properties can be called an inverse homogenization problem, and is formulated as an optimization problem of finding a microstructure with the lowest possible weight which fulfils the specified behavioral requirements. A full ground structure known from topology optimization of trusses is used as starting guess for the optimization algorithm. This implies that the optimal microstructure of a base cell is found from a truss or frame structure with 120 possible members in the 2-dimensional case and 2016 possible members in the 3-dimensional case. The material parameters are found by a numerical homogenization method, using Finite-Elements to model the representative base cell, and the optimization problem is solved by an optimality criteria method. Numerical examples in two and three dimensions show that it is possible to design materials with many different properties using base cells modelled as truss or frame works. Hereunder is shown that it is possible to tailor extreme materials, such as isotropic materials with Poisson's ratio close to − 1, 0 and 0.5, by the proposed method. Some of the proposed materials have been tested as macro models which demonstrate the expected behaviour.

Tailoring Materials for Specific Needs

This paper describes a method to design the periodic microstructure of a material to obtain specified constitutive parameters. The problem can be called an inverse homogenization problem and is formulated as an optimization problem of finding the microstructure with the lowest possible weight which fulfills the specified behavioural requirements. A full ground structure known from topology optimization of trusses is used as starting guess for the optimization algorithm, which implies that the optimal microstructure of a base cell is found from a truss structure with 120 possi ble members in the 2-dimensional case and 2016 possible members in the 3-dimensional case. The material parameters are found by a numerical homogenization method using finite elements to model the representative base cell, and the optimization problem is solved by an optimality criteria method. Numerical examples in two and three dimensions show that it is possible to design materials with many different properties, including isotropic materials with Poisson's ratio close to — 1 and 0.5, us ing base cells modelled as truss structures. Some of the proposed base cells have been tested as macro models and methods to produce them in practice are discussed.

Materials with prescribed constitutive parameters: An inverse homogenization problem

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 Poisson's 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.