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• Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. Part II: Numerical simulation

  J. Oliver

  International Journal for Numerical Methods in Engineering (1996). Vol. 39 (21), pp. 3601-3623

  
  

  Abstract

  On the basis of the strong discontinuity analysis of standard local stress–strain constitutive equations, a finite element framework for the simulation of strong discontinuities, which belongs to the family of assumed enhanced strain methods, is presented. Taking the standard linear triangle as the underlying element, an additional incompatible mode leads to the formulation of an enriched strain field which is shown to be able to appropriately capture strong discontinuities. The presented numerical simulations show that mesh size and mesh alignment dependencies can be completely removed

  Abstract
  On the basis of the strong discontinuity analysis of standard local stress–strain constitutive equations, a finite element framework for the simulation of strong discontinuities, [...]

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• Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. Part I: fundamentals

  J. Oliver

  International Journal for Numerical Methods in Engineering (1996). Vol. 39 (21), pp. 3575-3600

  
  

  Abstract

  The paper addresses some fundamental aspects about the use of standard constitutive equations to model strong discontinuities (cracks, shear bands, slip lines, etc.) in solid mechanics analyzes. The strong discontinuity analysis is introduced as a basic tool to derive a general framework, in which different families of constitutive equations can be considered, that allows to extract some outstanding aspects of the intended analysis. In particular, a link between continuum and discrete approaches to the strain localization phenomena is obtained. Applications to standard continuum damage and elastoplastic constitutive equations are presented. Relevant aspects to be considered in the numerical simulation of the problem (tackled in Part 2 of the work) are also presented

  Abstract
  The paper addresses some fundamental aspects about the use of standard constitutive equations to model strong discontinuities (cracks, shear bands, slip lines, etc.) in solid [...]

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• Continuum modelling of strong discontinuities in solid mechanics using damage models

  J. Oliver

  Computational Mechanics (1995). Vol. 17 (1-2), pp. 49-61

  
  

  Abstract

  Numerical simulation of strong discontinuities by using standard stress-strain constitutive equations including strain-softening is addressed. The concept of strong discontinuity analysis is introduced and driven, as a matter of example, into a standard continuum damage model. Then, the relevant features that make the constitutive equation compatible with the appearance of strong discontinuities are extracted. Those features are used in the design of a specific finite element approach to the strong discontinuity problem which is placed in the framework of the assumed enhanced strain methods. Numerical simulations show that mesh size and mesh alignment dependencies, typical of some continuum approaches, can be removed.

  Abstract
  Numerical simulation of strong discontinuities by using standard stress-strain constitutive equations including strain-softening is addressed. The concept of strong discontinuity [...]

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• An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids

  J. Oliver, J. Simó, F. Armero

  Computational Mechanics (1993). Vol. 12 (5), pp. 277-296

  
  

  Abstract

  Ket qualitative features of solutions exhibiting strong discontinuities in rate-independent inelastic solids are identified and exploited in the design of a new class of finite element approximations. The analysis shows that the softening law must be re-interpreted in a distributional sense for the continuum solutions to make mathematical sense and provides a precise physical interpretation to the softening modulus. These results are verified by numerical simulations employing a regularized discontinuous finite element method which circumvent the strong mesh-dependence exhibited by conventional methods, without resorting to viscosity or introducing additional ad-hoc parameters. The analysis is extended to a new class of anisotropic rate-independent damage models for brittle materials

  Abstract
  Ket qualitative features of solutions exhibiting strong discontinuities in rate-independent inelastic solids are identified and exploited in the design of a new class of finite [...]

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• A consistent characteristic length for smeared cracking models

  J. Oliver

  International Journal for Numerical Methods in Engineering (1989). Vol. 28 (2), pp. 461-474

  
  

  Abstract

  A numerical scheme for crack modelling by means of continuous displacement fields is presented. In two-dimensional problems a crack is modelled as a limiting case of two singular lines (with continuous displacements, but discontinuous displacement gradients across them) which tend to coincide with each other. An analysis of the energy dissipated inside the band bounded by both lines allows one to obtain an expression for the characteristic length as the ratio between the energy dissipated per unit surface area (fracture energy) and the energy dissipated per unit volume (specific energy) at a point. The application of these mathematical expressions to the finite element discretized medium allow one to obtain a general spatial and directional expression for the characteristic length which guarantees the objectivity of the results with respect to the size of the finite element mesh. The numerical results presented show the reliability of the proposed expressions.

  Abstract
  A numerical scheme for crack modelling by means of continuous displacement fields is presented. In two-dimensional problems a crack is modelled as a limiting case of two singular [...]

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

  A. Ferrer

  International Journal for Numerical Methods in Engineering (2019). Vol. 120 (3), pp. 361-381 (Preprint)

  
  

  Abstract

  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 provided

  Abstract
  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, [...]

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• A consistent relaxation of optimal design problems for coupling shape and topological derivatives

  S. Amstutz, C. Dapogny, A. Ferrer

  Numerische mathematik (2018). Vol. 140 (1), pp. 35-94

  
  

  Abstract

  In this article, we introduce and analyze a general procedure for approximating a ‘black and white’ shape and topology optimization problem with a density optimization problem, allowing for the presence of ‘grayscale’ regions. Our construction relies on a regularizing operator for smearing the characteristic functions involved in the exact optimization problem, and on an interpolation scheme, which endows the intermediate density regions with fictitious material properties. Under mild hypotheses on the smoothing operator and on the interpolation scheme, we prove that the features of the approximate density optimization problem (material properties, objective function, etc.) converge to their exact counterparts as the smoothing parameter vanishes. In particular, the gradient of the approximate objective functional with respect to the density function converges to either the shape or the topological derivative of the exact objective. These results shed new light on the connections between these two different notions of sensitivities for functions of the domain, and they give rise to different numerical algorithms which are illustrated by several experiments

  Abstract
  In this article, we introduce and analyze a general procedure for approximating a ‘black and white’ shape and topology optimization problem with a density optimization [...]

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

  A. Ferrer, J. Cante, J. Hernández, J. Oliver

  International Journal for Numerical Methods in Engineering (2018). Vol. 114 (3), pp. 232-254 (Preprint)

  
  

  Abstract

  The Domain Interface Method (DIM) is extended in this contribution for the case of mixed fields as encountered in multiphysics problems. The essence of the non-conforming domain decomposition technique consists in a discretization of a fictitious zero-thickness interface as in the original methodology and continuity of the solution fields across the domains is satisfied by incorporating the corresponding Lagrange Multipliers. The multifield DIM inherits the advantages of its irreducible version in the sense that the connections between non-matching meshes, with possible geometrically non-conforming interfaces, is accounted by the automatic Delaunay interface discretization without considering master and slave surfaces or intermediate surface projections as done in many established techniques, e.g. mortar methods. The multifield enhancement identifies the Lagrange multiplier field and incorporates its contribution in the weak variational form accounting for the corresponding consistent stabilization term based on a Nitsche method. This type of constraint enforcement circumvents the appearance of instabilities when the Ladyzhenskaya–Babuška–Brezzi (LBB) condition is not fulfilled by the chosen discretization. The domain decomposition framework is assessed in a large deformation setting for mixed displacement/pressure formulations and coupled thermomechanical problems. The continuity of the mixed field is studied in well selected benchmark problems for both mixed formulations and the objectivity of the response is compared to reference monolithic solutions. Results suggest that the presented strategy shows sufficient potential to be a valuable tool in situations where the evolving physics at particular domains require the use of different spatial discretizations or field interpolations.

  Abstract
  The Domain Interface Method (DIM) is extended in this contribution for the case of mixed fields as encountered in multiphysics problems. The essence of the non-conforming [...]

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

  J. Hernández, M. Caicedo, A. Ferrer

  Computer Methods in Applied Mechanics and Engineering (2017). Vol. 313, pp. 687-722 (Preprint)

  
  

  Abstract

  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

  Abstract
  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”), [...]

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

  S. Giusti, A. Ferrer, J. Oliver

  Computer Methods in Applied Mechanics and Engineering (2016). Vol. 311, pp. 134-150

  
  

  Abstract

  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

  Abstract
  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 [...]

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