SIMP-ALL: a generalized SIMP method based on the topological derivative concept

Topology optimization has emerged in the last years as a promising research fieldwith a wide range of applications. One of the most successful approaches, theSIMP method, is based on regularizing the problem and proposing a penaliza-tion interpolation function. In this work, we propose an alternative interpolationfunction, the SIMP-ALL method that is based on the topological derivative con-cept. First, we show the strong relation in plane linear elasticity between theHashin-Shtrikman (H-S) bounds and the topological derivative, providing anew interpretation of the last one. Then, we show that the SIMP-ALL interpo-lation remains always in between the H-S bounds regardless the materials tobe interpolated. This result allows us to interpret intermediate values as realmicrostructures. Finally, we verify numerically this result and we show the con-venience of the proposed SIMP-ALL interpolation for obtaining auto-penalizedoptimal design in a wider range of cases. A MATLAB code of the SIMP-ALLinterpolation function is also providedPeer ReviewedPostprint (published version

Multi-scale topological design of structural materials : an integrated approach

The present dissertation aims at addressing multiscale topology optimization problems. For this purpose, the concept of topology derivative in conjunction with the computational homogenization method is considered. In this study, the topological derivative algorithm, which is clearly non standard in topology optimization, and the optimality conditions are first introduced in order to a provide a better insight. Then, a precise treatment of the interface elements is proposed to reduce the numerical instabilities and the time-consuming computations that appear when using the topological derivative algorithm. The resulting strategy is examined and compared with current methodologies collected in the literature by means of some numerical tests of different nature. Then, a closed formula of the anisotropic topological derivative is obtained by solving analytically the exterior elastic problem. To this aim, complex variable theory and symbolic computation is considered. The resulting expression is validated through some numerical tests. In addition, different anisotropic topology optimization problems are solved to show the macroscopic topological implications of considering anisotropic materials. Finally, the two-scale topology optimization problem is tackled. As a first approach, an structural stiffness increase is achieved by considering the microscopic topologies as design variables of the problem. An alternate direction algorithm is proposed to address the high non-linearities of the problem. In addition, to mitigate the unaffordable time-consuming computations, a reduction technique is presented by means of pre-computing the optimal microscopic topologies in a computational material catalogue. As an extension of the first approach, besides designing the microscopic topologies, the macroscopic topology is also considered as a design variable, leading to even more optimal solutions. In addition, the proposed algorithms are modified in order to obtain manufacturable optimal designs. Two-scale topology optimization examples exhibit the potential of the proposed methodologyAquest treball té com a objectiu abordar els problemes d'optimització de topologia de múltiples escales. Amb aquesta finalitat, es consideren els conceptes de derivada topologia junt amb el mètode d'homogeneïtzació computacional. En aquest estudi, es presenta primer les condicions d'optimalitat i l'algorisme d'optimització utilitzat quan es considera la derivada topològica. A continuació, es proposa un tractament més precís dels elements de la interfície per reduir les inestabilitats numèriques i els elevats càlculs computacionals que apareixen quan s'utilitza l'algorisme de la derivada topològica. L'estratègia resultant s'examina i es compara amb les metodologies actuals, que es poden trobar sovint recollides a la literatura, mitjançant algunes proves numèriques. A més, s'obté una fórmula tancada de la derivada topològica anisotròpica quan es resol analíticament el problema exterior d'elasticitat. Per aconseguir-ho, es considera la teoria de variable complexa i la computació simbòlica. L'expressió resultant es valida a través d'algunes proves numèriques. A més, es resolen diferents problemes d'optimització topològica anisotròpica per mostrar les implicacions de la topològica macroscòpica en considerar materials anisòtrops. Finalment, s'aborda els problemes d'optimització topològica de dues escales. Com a primera estratègia, es considera les topologies microestructurals com a variables de disseny del problema per obtenir un augment de la rigidesa de l'estructura. Es proposa un algoritme de direcció alternada per fer front a les altes no linealitats del problema. A més, per mitigar els elevats càlculs computacionals, es presenta una tècnica de reducció per mitjà d'un precalcul de les topologies microestructural òptimes, que posteriorment són recollides en un catàleg de material. Com a una extensió de la primera estratègia, a més del disseny de les topologies microestructurals, la topologia macroscòpica també es considera com una variable de disseny, obtenint així solucions encara més òptimes. A més, els algoritmes proposats es modifiquen per tal d'obtenir dissenys que poden ser posteriorment fabricats. Alguns exemples numèrics d'optimització topològica de dues escales mostren el potencial de la metodologia proposada.Postprint (published version

Inverse homogenization using the topological derivative

Purpose: The purpose of this study is to solve the inverse homogenization problem, or so-called material design problem, using the topological derivative concept. Design/methodology/approach : 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. Findings: 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. Originality/value: 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.This research was partially supported by Serra Húnter Research Program (Spain), PID-UTN (Research and Development Program of the National Technological University, Argentina) and CONICET (National Council for Scientific and Technical Research, Argentina). The supports of these agencies are gratefully acknowledged.Peer ReviewedPostprint (author's final draft

Dimensional hyper-reduction of nonlinear finite element models via empirical cubature

We present a general framework for the dimensional reduction, in terms of number of degrees of freedom as well as number of integration points (“hyper-reduction”), of nonlinear parameterized finite element (FE) models. The reduction process is divided into two sequential stages. The first stage consists in a common Galerkin projection onto a reduced-order space, as well as in the condensation of boundary conditions and external forces. For the second stage (reduction in number of integration points), we present a novel cubature scheme that efficiently determines optimal points and associated positive weights so that the error in integrating reduced internal forces is minimized. The distinguishing features of the proposed method are: (1) The minimization problem is posed in terms of orthogonal basis vector (obtained via a partitioned Singular Value Decomposition) rather that in terms of snapshots of the integrand. (2) The volume of the domain is exactly integrated. (3) The selection algorithm need not solve in all iterations a nonnegative least-squares problem to force the positiveness of the weights. Furthermore, we show that the proposed method converges to the absolute minimum (zero integration error) when the number of selected points is equal to the number of internal force modes included in the objective function. We illustrate this model reduction methodology by two nonlinear, structural examples (quasi-static bending and resonant vibration of elastoplastic composite plates). In both examples, the number of integration points is reduced three order of magnitudes (with respect to FE analyses) without significantly sacrificing accuracy.Peer ReviewedPostprint (published version

A consistent approximation of the total perimeter functional for topology optimization algorithms

This article revolves around the total perimeter functional, one particular version of the perimeter of a shape O contained in a fixed computational domain D measuring the total area of its boundary ¿O, as opposed to its relative perimeter, which only takes into account the regions of ¿O strictly inside D. We construct and analyze approximate versions of the total perimeter which make sense for general “density functions” u, as generalized characteristic functions of shapes. Their use in the context of density-based topology optimization is particularly convenient insofar as they do not involve the gradient of the optimized function u. Two different constructions are proposed: while the first one involves the convolution of the function u with a smooth mollifier, the second one is based on the resolution of an elliptic boundary-value problem featuring Robin boundary conditions. The “consistency” of these approximations with the original notion of total perimeter is appraised from various points of view. At first, we prove the pointwise convergence of our approximate functionals, then the convergence of their derivatives, as the level of smoothing tends to 0, when the considered density function u is the characteristic function of a “regular enough” shape O ¿ D. Then, we focus on the G-convergence of the second type of approximate total perimeter functional, that based on elliptic regularization. Several numerical examples are eventually presented in two and three space dimensions to validate our theoretical findings and demonstrate the efficiency of the proposed functionals in the context of structural optimization.This work is partly supported by the project ANR-18-CE40-0013 SHAPO, financed by the French Agence Nationale de la Recherche (ANR). S.A. benefitted from the support of the chair "Modeling advanced polymers for innovative material solutions" led by the École Polytechnique and the Fondation de l'École Polytechnique, and sponsored by Arkema.Postprint (published version

Stress minimization for lattice structures. Part I: Micro-structure design

This work is partially supported by the SOFIA project, funded by Bpifrance (Banque Publique d’Investissement). This work has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 833092.Peer ReviewedPostprint (published version

Topological sensitivity analysis in heterogeneous anisotropic elasticity problem: theoretical and computational aspects

The topological sensitivity analysis for the heterogeneous and anisotropic elasticity problem in two-dimensions is performed in this work. The main result of the paper is an analytical closed-form of the topological derivative for the total potential energy of the problem. This derivative displays the sensitivity of the cost functional (the energy in this case) when a small singular perturbation is introduced in an arbitrary point of the domain. In this case, we consider a small disc with a completely different elastic material. Full mathematical justification for the derived formula, and derivations of precise estimates for the remainders of the topological asymptotic expansion are provided. Finally, the influence of the heterogeneity and anisotropy is shown through some numerical examples of structural topology optimization.Peer ReviewedPostprint (author's final draft

Approximating the Basset force by optimizing the method of van Hinsberg et al.

In this work we put the method proposed by van Hinsberg et al. [29] to the test, highlighting its accuracy and efficiency in a sequence of benchmarks of increasing complexity. Furthermore, we explore the possibility of systematizing the way in which the method's free parameters are determined by generalizing the optimization problem that was considered originally. Finally, we provide a list of worked-out values, ready for implementation in large-scale particle-laden flow simulations.Peer ReviewedPostprint (author's final draft

Vademecum-based approach to multi-scale topological material design

The work deals on computational design of structural materials by resorting to computational homogenization and topological optimization techniques. The goal is then to minimize the structural (macro-scale) compliance by appropriately designing the material distribution (microstructure) at a lower scale (micro-scale), which, in turn, rules the mechanical properties of the material. The specific features of the proposed approach are: (1) The cost function to be optimized (structural stiffness) is defined at the macro-scale, whereas the design variables defining the micro-structural topology lie on the low scale. Therefore a coupled, two-scale (macro/micro), optimization problem is solved unlike the classical, single-scale, topological optimization problems. (2) To overcome the exorbitant computational cost stemming from the multiplicative character of the aforementioned multiscale approach, a specific strategy, based on the consultation of a discrete material catalog of micro-scale optimized topologies (Computational Vademecum) is used. The Computational Vademecum is computed in an offline process, which is performed only once for every constitutive-material, and it can be subsequently consulted as many times as desired in the online design process. This results into a large diminution of the resulting computational costs, which make affordable the proposed methodology for multiscale computational material design. Some representative examples assess the performance of the considered approach.Peer ReviewedPostprint (published version

Two-scale topology optimization in computational material design: an integrated approach

In this work, a new strategy for solving multiscale topology optimization problems is presented. An alternate direction algorithm and a precomputed offline microstructure database (Computational Vademecum) are used to efficiently solve the problem. In addition, the influence of considering manufacturable constraints is examined. Then, the strategy is extended to solve the coupled problem of designing both the macroscopic and microscopic topologies. Full details of the algorithms and numerical examples to validate the methodology are provided.Peer ReviewedPostprint (published version