J.M. Guedes

Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods

Abstract This paper discusses the homogenization method to determine the effective average elastic constants of linear elasticity of general composite materials by considering their microstructure. After giving a brief theory of the homogenization method, a finite element approximation is introduced with convergence study and corresponding error estimate. Applying these, computer programs PREMAT and POSTMAT are developed for preprocessing and postprocessing of material characterization of composite materials. Using these programs, the homogenized elastic constants for macroscopic stress analysis are obtained for typical composite materials to show their capability. Finally, the adaptive finite element method is introduced to improve the accuracy of the finite element approximation.

Generalized topology design of structures with a buckling load criterion

Material based models for topology optimization of linear elastic solids with a low volume constraint generate very slender structures composed mainly of bars and beam elements. For this type of structure the value of the buckling critical load becomes one of the most important design criteria and so its control is very important for meaningful practical designs. This paper tries to address this problem, presenting an approach to introduce the possibility of critical load control into the topology optimization model.Using the material based formulation for topology design of structures, the problem of optimal structural reinforcement for a critical load criterion is formulated. The stability problem is conveniently reduced to a linearized eigenvalue problem assuming only material effective properties and macroscopic instability modes. The respective optimality criteria are presented by introducing the Lagrangian associated with the optimization problem. Based on this Lagrangian a first-order method is used as a basis for the numerical update scheme. Two numerical examples to validate the developments are presented and analysed.

Numerical modeling of bone tissue adaptation—A hierarchical approach for bone apparent density and trabecular structure

In this work, a three-dimensional model for bone remodeling is presented, taking into account the hierarchical structure of bone. The process of bone tissue adaptation is mathematically described with respect to functional demands, both mechanical and biological, to obtain the bone apparent density distribution (at the macroscale) and the trabecular structure (at the microscale). At global scale bone is assumed as a continuum material characterized by equivalent (homogenized) mechanical properties. At local scale a periodic cellular material model approaches bone trabecular anisotropy as well as bone surface area density. For each scale there is a material distribution problem governed by density-based design variables which at the global level can be identified with bone relative density. In order to show the potential of the model, a three-dimensional example of the proximal femur illustrates the distribution of bone apparent density as well as microstructural designs characterizing both anisotropy and bone surface area density. The bone apparent density numerical results show a good agreement with Dual-energy X-ray Absorptiometry (DXA) exams. The material symmetry distributions obtained are comparable to real bone microstructures depending on the local stress field. Furthermore, the compact bone porosity is modeled giving a transversal isotropic behavior close to the experimental data. Since, some computed microstructures have no permeability one concludes that bone tissue arrangement is not a simple stiffness maximization issue but biological factors also play an important role.

Optimal design of periodic linear elastic microstructures

Abstract This paper presents two computational models to design the periodic microstructure of cellular materials for optimal elastic properties. The material equivalent mechanical properties are obtained through a homogenization model. The two formulations address the problem of finding the optimal representative microstructural element for periodic media that maximizes either the weighted sum of the equivalent strain energy density for specified multiple macroscopic strain fields, or a linear combination of the equivalent mechanical properties. Constraints on material volume fraction and material symmetries are considered. The computational models are established using finite elements and mathematical programming techniques and tested in several numerical examples.

Permeability analysis of scaffolds for bone tissue engineering

Porous artificial bone substitutes, especially bone scaffolds coupled with osteobiologics, have been developed as an alternative to the traditional bone grafts. The bone scaffold should have a set of properties to provide mechanical support and simultaneously promote tissue regeneration. Among these properties, scaffold permeability is a determinant factor as it plays a major role in the ability for cells to penetrate the porous media and for nutrients to diffuse. Thus, the aim of this work is to characterize the permeability of the scaffold microstructure, using both computational and experimental methods. Computationally, permeability was estimated by homogenization methods applied to the problem of a fluid flow through a porous media. These homogenized permeability properties are compared with those obtained experimentally. For this purpose a simple experimental setup was used to test scaffolds built using Solid Free Form techniques. The obtained results show a linear correlation between the computational and the experimental permeability. Also, this study showed that permeability encompasses the influence of both porosity and pore size on mass transport, thus indicating its importance as a design parameter. This work indicates that the mathematical approach used to determine permeability may be useful as a scaffold design tool.

TOPOLOGY OPTIMIZATION OF THREE-DIMENSIONAL LINEAR ELASTIC STRUCTURES WITH A CONSTRAINT ON PERIMETER

Abstract This work presents a computational model for the topology optimization of a three-dimensional linear elastic structure. The model uses a material distribution approach and the optimization criterion is the structural compliance, subjected to an isoperimetric constraint on volume. Usually the obtained topologies using this approach do not characterize a well-defined structure, i.e. it has regions with porous material and/or with checkerboard patterns. To overcome these problems an additional constraint on perimeter and a penalty on intermediate volume fraction are considered. The necessary conditions for optimum are derived analytically, approximated numerically through a suitable finite element discretization and solved by a first-order method based on the optimization problem augmented Lagrangian. The computational model is tested in several numerical applications.

On the prediction of material properties and topology for optimal continuum structures

A new formulation is presented for mathematical modelling to predict material properties for the optimal design of continuum structures. The method is based on an extended form of an already established characterization for continuum design, where the material properties tensor for an arbitrary structural continuum appears as the design variable. The extension is comprised of means to represent an independently specifiedunit relative cost factor, which appears simply as a weighting function in the argument of the isoperimetric (cost) constraint of the original model. A procedure is demonstrated where optimal black/white topology is predicted out of a sequence of solutions to material properties design problems having thisgeneralized cost formulation form. A systematic adjustment is made in the unit relative cost field for each subsequent solution step in the sequence, and at the stage identified with final topology, no more than a small fraction of a percent of the total element area in the system has material property density off the bounding “black” or “white” levels. This technique is effective for the prediction of optimal black/white topology design for design around obstacles of arbitrary shape, as well as the more unusual topology design problems. Results are presented for 2D examples of both types of problem. In addition to the treatment for (the usual) minimum compliance design, an alternate formulation of the design problem is presented as well, one that provides for the prediction of optimum topology with a generalized measure of compliance as the objective.

Optimization of structure and material properties for solids composed of softening material

An existing formulation for the prediction of the optimal material properties tensor for systems made of linear material is extended here to encompass design with nonlinear softening material. The original model for the linear case was stated as a convex, constrained nonlinear programming problem, and this property is preserved in the extension. The development is exemplified for the case of design for minimum global compliance. An extremum problem formulation for elastostatics is incorporated into the design problem, as a convenient way to model the structural analysis of general softening systems. Unlike the case for design with linear material, in the nonlinear problem the form of the solution for optimal material properties, and the associated stress fields as well, evolve with increasing load. Computational results are presented for a two-dimensional example problem, in which a system is designed for different loads while parameters in the material model are held fixed.

Necessary conditions for optimal design of structures with a nonsmooth eigenvalue based criterion

Necessary conditions of optimality for maximizing the buckling load for single or multimodal structures are derived using generalized gradients. The possible design dependence of the pre-buckling displacement is taken into account and implies the appearance of a number of adjoint problems not normally present in, for example, standard vibration problems. The relation to other equivalent forms of the necessary conditions are pointed out and the formulae are demonstrated on simple example problems.

A generic homogenization model for magnetoelectric multiferroics

A generic homogenization modeling framework which incorporates crystallographic domain features is introduced and computationally implemented for magnetoelectric multiferroics of all symmetries. The homogenization, mathematically applicable to heterogeneous media with contrasts in physical properties, replaces the heterogeneity of the multiferroics by an equivalent effective medium with uniform physical characteristics. A statistically representative unit-cell is proposed to encompass all forms of multiferroics and their composites in bulk. The variational formulation of the coupled magneto-electromechanical problem reveals the nature of interaction between mechanical, electrical, and magnetic fields of a multiferroic at a microscopic scale with high resolution. Furthermore, the mathematical homogenization theory of the multiferroic is implemented in finite element method by solving the coupled equilibrium electrical, magnetic, and mechanical fields. A “multiferroic finite element” is conceived for this purpose. The model is applied to a two-phase multiferroic magnetoelectric composite to demonstrate its validity by characterizing the equivalent physical properties.