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The purpose of this study is to solve the inverse homogenization problem, or so-called material design problem, using the topological derivative concept.

The optimal topology is obtained through a relaxed formulation of the problem by replacing the characteristic function with a continuous design variable, so-called density variable. The constitutive tensor is then parametrized with the density variable through an analytical interpolation scheme that is based on the topological derivative concept. The intermediate values that may appear in the optimal topologies are removed by penalizing the perimeter functional.

The optimization process benefits from the intermediate values that provide the proposed method reaching to solutions that the topological derivative had not been able to find before. In addition, the presented theory opens the path to propose a new framework of research where the topological derivative uses classical optimization algorithms.

The proposed methodology allows us to use the topological derivative concept for solving the inverse homogenization problem and to fulfil the optimality conditions of the problem with the use of classical optimization algorithms. The authors solved several material design examples through a projected gradient algorithm to show the advantages of the proposed method.

The purpose of this paper is to compare between two methods of volume control in the context of topological derivative-based structural optimization of Kirchhoff plates.

The compliance topology optimization of Kirchhoff plates subjected to volume constraint is considered. In order to impose the volume constraint, two methods are presented. The first one is done by means of a linear penalization method. In this case, the penalty parameter is the coefficient of a linear term used to control the amount of material to be removed. The second approach is based on the Augmented Lagrangian method which has both, linear and quadratic terms. The coefficient of the quadratic part controls the Lagrange multiplier update of the linear part. The associated topological sensitivity is used to devise a structural design algorithm based on the topological derivative and a level-set domain representation method. Finally, some numerical experiments are presented allowing for a comparative analysis between the two methods of volume control from a qualitative point of view.

The linear penalization method does not provide direct control over the required volume fraction. In contrast, through the Augmented Lagrangian method it is possible to specify the final amount of material in the optimized structure.

A strictly simple topology design algorithm is devised and used in the context of compliance structural optimization of Kirchhoff plates under volume constraint. The proposed computational framework is quite general and can be applied in different engineering problems.

The purpose of this paper is to predict the yield locus of porous ductile materials, evaluate the impact of void geometry and compare the computational results with existing analytical models.

A computational homogenization strategy for the definition of the elasto-plastic transition is proposed. Representative volume elements (RVEs) containing single-centred ellipsoidal voids are analysed using three-dimensional finite element models under the geometrically non-linear hypothesis of finite strains. Yield curves are obtained by means of systematic analysis of RVEs considering different kinematical models: linear boundary displacements (upper bound), boundary displacement fluctuation periodicity and uniform boundary traction (lower bound).

The influence of void geometry is captured and the reduction in the material strength is observed. Analytical models usually overestimate the impact of void geometry on the yield locus.

This paper proposes an alternative criterion for porous ductile materials and assesses the accuracy of analytical models through the simulation of three-dimensional finite element models under geometrically non-linear hypothesis.

The purpose of this paper is to assess the Gurson yield criterion for porous ductile metals.

A finite element procedure is used within a purely kinematical multi‐scale constitutive modelling framework to determine estimates of extremal overall yield surfaces. The RVEs analysed consist of an elastic‐perfectly plastic von Mises type matrix under plane strain conditions containing a single centered circular hole. Macroscopic yield surface estimates are obtained under three different RVE kinematical assumptions: linear boundary displacements (an upper bound); periodic boundary displacement fluctuations (corresponding to periodically perforated media); and, minimum constraint or uniform boundary traction (a lower bound).

The Gurson criterion predictions fall within the bounds obtained under relatively high void ratios – when the bounds lie farther apart. Under lower void ratios, when the bounds lie close together, the Gurson predictions of yield strength lie slightly above the computed upper bounds in regions of intermediate to high stress triaxiality. A modification to the original Gurson yield function is proposed that can capture the computed estimates under the three RVE kinematical constraints considered.

Assesses the accuracy of the Gurson criterion by means of a fully computational multi‐scale approach to constitutive modelling. Provides an alternative criterion for porous plastic media which encompasses the common microscopic kinematical constraints adopted in this context.

The purpose of this paper is to present knowledge in estimating yield surfaces of heterogeneous media by use of homogenization, especially where the macroscopic behaviour is driven by weak interfaces between phase constituents.

A computational homogenization procedure is used to determine the yield surface of a Representative Volume Element (RVE) that contains a fully debonded inclusion embedded within ideally plastic matrix, whereby the interface is modelled by a Coulomb type friction law.

The macroscopic behaviour of the RVE is shown to coincide an RVE with a hole for expanding loads, whereas for compressive loads, it was shown to approach an RVE with a fully bonded inclusion.

The present paper builds on Gurson’s work in estimating macroscopic yield surfaces of heterogeneous materials. The work is novel in the sense that there had been no previous publications discussing influence of weak interfaces on yield surfaces.

Topology optimization of structures under self-weight loading is a challenging problem which has received increasing attention in the past years. The use of standard formulations based on compliance minimization under volume constraint suffers from numerous difficulties for self-weight dominant scenarios, such as non-monotonic behaviour of the compliance, possible unconstrained character of the optimum and parasitic effects for low densities in density-based approaches. This paper aims to propose an alternative approach for dealing with topology design optimization of structures into three spatial dimensions subject to self-weight loading.

In order to overcome the above first two issues, a regularized formulation of the classical compliance minimization problem under volume constraint is adopted, which enjoys two important features: (a) it allows for imposing any feasible volume constraint and (b) the standard (original) formulation is recovered once the regularizing parameter vanishes. The resulting topology optimization problem is solved with the help of the topological derivative method, which naturally overcomes the above last issue since no intermediate densities (grey-scale) approach is necessary.

A novel and simple approach for dealing with topology design optimization of structures into three spatial dimensions subject to self-weight loading is proposed. A set of benchmark examples is presented, showing not only the effectiveness of the proposed approach but also highlighting the role of the self-weight loading in the final design, which are: (1) a bridge structure is subject to pure self-weight loading; (2) a truss-like structure is submitted to an external horizontal force (free of self-weight loading) and also to the combination of self-weight and the external horizontal loading; and (3) a tower structure is under dominant self-weight loading.

An alternative regularized formulation of the compliance minimization problem that naturally overcomes the difficulties of dealing with self-weight dominant scenarios; a rigorous derivation of the associated topological derivative; computational aspects of a simple FreeFEM implementation; and three-dimensional numerical benchmarks of bridge, truss-like and tower structures.

The present paper aims to explore a computational homogenisation procedure to investigate the full geometric representation of yield surfaces for isotropic porous ductile media. The effects of cell morphology and imposed boundary conditions are assessed. The sensitivity of the yield surfaces to the Lode angle is also investigated in detail.

The microscale of the material is modelled by the concept of Representative Volume Element (RVE) or unit cell, which is numerically simulated through three-dimensional finite element analyses. Numerous loading conditions are considered to create complete yield surfaces encompassing high, intermediate and low triaxialities. The influence of cell morphology on the yield surfaces is assessed considering a spherical cell with spherical void and a cubic RVE with spherical void, both under uniform strain boundary condition. The use of spherical cell is interesting as preferential directions in the effective behaviour are avoided. The periodic boundary condition, which favours strain localization, is imposed on the cubic RVE to compare the results. Small strains are assumed and the cell matrix is considered as a perfect elasto-plastic material following the von Mises yield criterion.

Different morphologies for the cell imply in different yield conditions for the same load situations. The yield surfaces in correspondence to periodic boundary condition show significant differences compared to those obtained by imposing uniform strain boundary condition. The stress Lode angle has a strong influence on the geometry of the yield surfaces considering low and intermediate triaxialities.

The exhaustive computational study of the effects of cell morphologies and imposed boundary conditions fills a gap in the full representation of the flow surfaces. The homogenisation-based strategy allows us to further investigate the influence of the Lode angle on the yield surfaces.

In the paper an approach for crack nucleation and propagation phenomena in brittle plate structures is presented.

The Francfort–Marigo damage theory is adapted to the Kirchhoff and Reissner–Mindlin plate bending models. Then, the topological derivative method is used to minimize the associated Francfort–Marigo shape functional. In particular, the whole damaging process is governed by a threshold approach based on the topological derivative field, leading to a notable simple algorithm.

Numerical simulations are driven in order to verify the applicability of the proposed method in the context of brittle fracture modeling on plates. The obtained results reveal the capability of the method to determine nucleation and propagation including bifurcation of multiple cracks with a minimal number of user-defined algorithmic parameters.

This is the first work concerning brittle fracture modeling of plate structures based on the topological derivative method.

The purpose of this paper is to use symmetry conditions for the reduction of computing times in problems involving finite element‐based multi‐scale constitutive models of nonlinear heterogeneous media.

Two types of representative volume element (RVE) symmetry often found in practice are considered: staggered‐translational and point symmetry. These are analyzed under three types RVE of kinematical constraints: periodic boundary fluctuations (typical of periodic media), linear boundary displacements (which gives an upper bound for the macroscopic stiffness) and the minimum kinematical constraint (corresponding to uniform boundary tractions and providing a lower bound for the macroscopic stiffness).

Numerical examples show that substantial savings in computing times are achieved by taking advantage of such symmetries. These are particularly pronounced in fully coupled two‐scale analyses, where the macroscopic equilibrium problem is solved simultaneously with a large number of microscopic equilibrium problems at Gauss‐point level. Speed‐up factors in excess of seven have been found in such cases, when both symmetry conditions considered are present at the same time.

This paper extends the original considerations of Ohno et al. to account for other RVE kinematical constraints, namely, the linear boundary displacement and the minimum kinematical constraint (or uniform boundary traction model). Provides a more precise assessment of the impact of the use of such symmetries on computing times by means of numerical examples. In addition, for completeness, the direct enforcement of such constraints within a Newton‐based finite element solution procedure for the RVE equilibrium problem is detailed in the paper.

The purpose of this study is sensitivity analysis of the L²-norm and H¹-seminorm of the solution of a diffusive–convective–reactive problem to topological changes of the underlying material.

The topological derivative method is used to devise a simple and efficient topology design algorithm based on a level-set domain representation method.

Remarkably simple analytical expressions for the sensitivities are derived, which are useful for practical applications including heat exchange topology design and membrane eigenvalue maximization.

The topological asymptotic expansion associated with a diffusive–convective–reactive equation is rigorously derived, which is not available in the literature yet.