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Peric D.,University of Swansea | De Souza Neto E.A.,University of Swansea | Feijoo R.A.,Laboratorio Nacional Of Computacao Cientifica Lncc Mct | Partovi M.,University of Swansea | Molina A.J.C.,University of Swansea
International Journal for Numerical Methods in Engineering | Year: 2011

This work describes a homogenization-based multi-scale procedure required for the computation of the material response of non-linear microstructures undergoing small strains. Such procedures are important for computer modelling of heterogeneous materials when the length-scale of heterogeneities is small compared to the dimensions of the body. The described multi-scale procedure relies on a unified variational basis which, apart from the continuum-based variational formulation at both micro- and macroscales of the problem, also includes the variational formulation governing micro-to-macro transitions. This unified variational basis leads naturally to a generic finite element-based framework for homogenization-based multi-scale analysis of heterogenous solids. In addition, the unified variational formulation provides clear axiomatic basis and hierarchy related to the choice of boundary conditions at the microscale. Classical kinematical constraints are considered over the representative volume element: (i) Taylor, (ii) linear boundary displacements, (iii) periodic boundary displacement fluctuations and (iv) minimal constraint, also known as uniform boundary tractions. In this context the Hill-Mandel averaging requirement, which links microscopic and macroscopic stress power, plays a fundamental role in defining the microscopic forces compatible with the assumed kinematics. Numerical examples of both microscale and two-scale finite element simulations of elasto-plastic material with microcavities are presented to illustrate the main features and scope of the described computational strategy. © 2010 John Wiley & Sons, Ltd. Source

Amstutz S.,Universite Ibn Tofail | Novotny A.A.,Laboratorio Nacional Of Computacao Cientifica Lncc Mct
ESAIM - Control, Optimisation and Calculus of Variations | Year: 2011

The topological asymptotic analysis provides the sensitivity of a given shape functional with respect to an infinitesimal domain perturbation, like the insertion of holes, inclusions, cracks. In this work we present the calculation of the topological derivative for a class of shape functionals associated to the Kirchhoff plate bending problem, when a circular inclusion is introduced at an arbitrary point of the domain. According to the literature, the topological derivative has been fully developed for a wide range of second-order differential operators. Since we are dealing here with a forth-order operator, we perform a complete mathematical analysis of the problem. © EDP Sciences, SMAI, 2010. Source

Novotny A.A.,Laboratorio Nacional Of Computacao Cientifica Lncc Mct
Mathematical Methods in the Applied Sciences | Year: 2010

A simple analytical expression for crack nucleation sensitivity analysis is proposed relying on the concept of topological derivative and applied within a two-dimensional linear elastic fracture mechanics theory (LEFM). In particular, the topological asymptotic expansion of the total potential energy together with a Griffith-type energy of an elastic cracked body is calculated. As a main result, we derive a crack nucleation criterion based on the topological derivative and a criterion for determining the direction of crack growth based on the topological gradient. The proposed methodology leads to an axiomatic approach of crack nucleation sensitivity analysis. Copyright © 2010 John Wiley & Sons, Ltd. Source

Novotny A.A.,Laboratorio Nacional Of Computacao Cientifica Lncc Mct
Mechanics Research Communications | Year: 2013

The topological derivative represents the first term of the asymptotic expansion of a given shape functional with respect to the small parameter which measures the size of singular domain perturbations. The topological derivative has been successfully applied in the treatment of problems such as topology optimization, inverse analysis and image processing. In this paper, the calculation of the topological derivative for a general class of shape functionals is presented. In particular, we evaluate the topological derivative of a modified energy shape functional associated to the steady-state heat conduction problem, considering the nucleation of a small circular inclusion as the topological perturbation. Several methods were proposed to calculate the topological derivative. In this paper, the so-called topological-shape sensitivity method is extended to deal with a modified adjoint method, leading to an alternative approach to calculate the topological derivative based on shape sensitivity analysis together with a modified Lagrangian method. Since we are dealing with a general class of shape functionals, which are not necessarily associated to the energy, we will show that this new approach simplifies the most delicate step of the topological derivative calculation, namely, the asymptotic analysis of the adjoint state. © 2013 Elsevier Ltd. Source

Amstutz S.,Laboratoire danalyse non lineaire et geometrie | Giusti S.M.,Laboratorio Nacional Of Computacao Cientifica Lncc Mct | Novotny A.A.,Laboratorio Nacional Of Computacao Cientifica Lncc Mct | De Souza Neto E.A.,University of Swansea
International Journal for Numerical Methods in Engineering | Year: 2010

This paper proposes an algorithm for the synthesis/optimization of microstructures based on an exact formula for the topological derivative of the macroscopic elasticity tensor and a level set domain representation. The macroscopic elasticity tensor is estimated by a standard multi-scale constitutive theory where the strain and stress tensors are volume averages of their microscopic counterparts over a representative volume element. The algorithm is of simple computational implementation. In particular, it does not require artificial algorithmic parameters or strategies. This is in sharp contrast with existing microstructural optimization procedures and follows as a natural consequence of the use of the topological derivative concept. This concept provides the correct mathematical framework to treat topology changes such as those characterizing microstuctural optimization problems. The effectiveness of the proposed methodology is illustrated in a set of finite element-based numerical examples. © 2010 John Wiley & Sons, Ltd. Source

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