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• Tensor representation of non-linear models using cross approximations

  J. Aguado, D. Borzacchiello, K. Kollepara, F. Chinesta, A. Huerta

  Journal of scientific computing (2019). Vol. 81 (1), pp. 22-47

  
  

  Abstract

  Tensor representations allow compact storage and efficient manipulation of multi-dimensional data. Based on these, tensor methods build low-rank subspaces for the solution of multi-dimensional and multi-parametric models. However, tensor methods cannot always be implemented efficiently, specially when dealing with non-linear models. In this paper, we discuss the importance of achieving a tensor representation of the model itself for the efficiency of tensor-based algorithms. We investigate the adequacy of interpolation rather than projection-based approaches as a means to enforce such tensor representation, and propose the use of cross approximations for models in moderate dimension. Finally, linearization of tensor problems is analyzed and several strategies for the tensor subspace construction are proposed. This is a post-peer-review, pre-copyedit version of an article published in Journal of scientific computing

  Abstract
  Tensor representations allow compact storage and efficient manipulation of multi-dimensional data. Based on these, tensor methods build low-rank subspaces for the solution [...]

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• A simple microstructural viscoelastic model for flowing foams

  R. Ibañez, A. Scheuer, E. Abisset-Chavanne, F. Chinesta, A. Huerta, R. Keunings

  International journal of material forming (2018). Vol. 12 (2), pp. 295-306 (Preprint)

  
  

  Abstract

  The numerical modelling of forming processes involving the flow of foams requires taking into account the different problem scales. Thus, in industrial applications a macroscopic approach is suitable, whereas the macroscopic flow parameters depend on the cellular structure: cell size, shape, orientation, etc. Moreover, the shape and orientation of the cells are induced by the flow. A fully microscopic description remains useful to understand the foam behaviour and the topological changes induced by the cell elongation or distortion, however, from an industrial point of view, microscopic simulations remain challenging to address practical applications involving flows in complex 3D geometries. In this paper, we propose a viscoelastic flow model where the foam microstructure is represented from suitable microstructure descriptors whose evolution is governed by the macroscopic flow kinematics. This is a post-peer-review, pre-copyedit version of an article published in International journal of material forming.

  Abstract
  The numerical modelling of forming processes involving the flow of foams requires taking into account the different problem scales. Thus, in industrial applications a macroscopic [...]

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• A locking-free face-centred finite volume (FCFV) method for linear elastostatics

  R. Sevilla, M. Giacomini, A. Huerta

  Computers and Structures (2018). Vol. 212, pp. 43-57

  
  

  Abstract

  A face-centred finite volume (FCFV) method is proposed for linear elastostatic problems. The FCFV is a mixed hybrid formulation, featuring a system of first-order equations, that defines the unknowns on the faces (edges in two dimensions) of the mesh cells. The symmetry of the stress tensor is strongly enforced using the well-known Voigt notation and the displacement and stress fields inside each cell are obtained by means of explicit formulas. The resulting FCFV method is robust and locking-free in the nearly incompressible limit. Numerical experiments in two and three dimensions show optimal convergence of the displacement and the stress fields without any reconstruction. Moreover, the accuracy of the FCFV method is not sensitive to mesh distortion and stretching. Classical benchmark tests including Kirch’s plate and Cook’s membrane problems in two dimensions as well as three dimensional problems involving shear phenomenons, pressurised thin shells and complex geometries are presented to show the capability and potential of the proposed methodology.

  Abstract
  A face-centred finite volume (FCFV) method is proposed for linear elastostatic problems. The FCFV is a mixed hybrid formulation, featuring a system of first-order equations, [...]

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• HDG-NEFEM with degree adaptivity for stokes flows

  R. Sevilla, A. Huerta

  Journal of scientific computing (2018). Vol. 77 (3), pp. 1953-1980 (Preprint)

  
  

  Abstract

  The NURBS-enhanced finite element method (NEFEM) combined with a hybridisable discontinuous Galerkin (HDG) approach is presented for the first time. The proposed technique completely eliminates the uncertainty induced by a polynomial approximation of curved boundaries that is common within an isoparametric approach and, compared to other DG methods, provides a significant reduction in number of degrees of freedom. In addition, by exploiting the ability of HDG to compute a postprocessed solution and by using a local a priori error estimate valid for elliptic problems, an inexpensive, reliable and computable error estimator is devised. The proposed methodology is used to solve Stokes flow problems using automatic degree adaptation. Particular attention is paid to the importance of an accurate boundary representation when changing the degree of approximation in curved elements. Several strategies are compared and the superiority and reliability of HDG-NEFEM with degree adaptation is illustrated.

  Abstract
  The NURBS-enhanced finite element method (NEFEM) combined with a hybridisable discontinuous Galerkin (HDG) approach is presented for the first time. The proposed technique [...]

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• A superconvergent HDG method for stokes flow with strongly enforced symmetry of the stress tensor

  M. Giacomini, A. Karkoulias, R. Sevilla, A. Huerta

  Journal of scientific computing (2018). Vol. 77 (3), pp. 1679-1702 (Preprint)

  
  

  Abstract

  This work proposes a superconvergent hybridizable discontinuous Galerkin (HDG) method for the approximation of the Cauchy formulation of the Stokes equation using same degree of polynomials for the primal and mixed variables. The novel formulation relies on the well-known Voigt notation to strongly enforce the symmetry of the stress tensor. The proposed strategy introduces several advantages with respect to the existing HDG formulations. First, it remedies the suboptimal behavior experienced by the classical HDG method for formulations involving the symmetric part of the gradient of the primal variable. The optimal convergence of the mixed variable is retrieved and an element-by-element postprocess procedure leads to a superconvergent velocity field, even for low-order approximations. Second, no additional enrichment of the discrete spaces is required and a gain in computational efficiency follows from reducing the quantity of stored information and the size of the local problems. Eventually, the novel formulation naturally imposes physical tractions on the Neumann boundary. Numerical validation of the optimality of the method and its superconvergent properties is performed in 2D and 3D using meshes of different element types.

  Abstract
  This work proposes a superconvergent hybridizable discontinuous Galerkin (HDG) method for the approximation of the Cauchy formulation of the Stokes equation using same degree [...]

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• A superconvergent hybridisable discontinuous Galerkin method for linear elasticity

  R. Sevilla, M. Giacomini, A. Karkoulias, A. Huerta

  Int. J. Numer. Meth. Engng. (2018). Vol. 116 (2), pp. 91-116

  
  

  Abstract

  The first superconvergent hybridisable discontinuous Galerkin method for linear elastic problems capable of using the same degree of approximation for both the primal and mixed variables is presented. The key feature of the method is the strong imposition of the symmetry of the stress tensor by means of the well known and extensively used Voigt notation, circumventing the use of complex mathematical concepts to enforce the symmetry of the stress tensor either weakly or strongly. A novel procedure to construct element by element a superconvergent postprocessed displacement is proposed. Contrary to other hybridisable discontinuous Galerkin formulations, the methodology proposed here is able to produce a superconvergent displacement field for low-order approximations. The resulting method is robust and locking-free in the nearly incompressible limit. An extensive set of numerical examples is utilised to provide evidence of the optimality of the method and its superconvergent properties in two and three dimensions and for different element types

  Abstract
  The first superconvergent hybridisable discontinuous Galerkin method for linear elastic problems capable of using the same degree of approximation for both the primal and [...]

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• An upwind cell centred Total Lagrangian finite volume algorithm for nearly incompressible explicit fast solid dynamic applications

  J. Haider, C. Lee, A. Gil, A. Huerta, J. Bonet

  Comput. Methods Appl. Mech. Engrg., (2018). Vol. 340, pp. 684-727

  
  

  Abstract

  The paper presents a new computational framework for the numerical simulation of fast large strain solid dynamics, with particular emphasis on the treatment of near incompressibility. A complete set of first order hyperbolic conservation equations expressed in terms of the linear momentum and the minors of the deformation (namely the deformation gradient, its co-factor and its Jacobian), in conjunction with a polyconvex nearly incompressible constitutive law, is presented. Taking advantage of this elegant formalism, alternative implementations in terms of entropy-conjugate variables are also possible, through suitable symmetrisation of the original system of conservation variables. From the spatial discretisation standpoint, modern Computational Fluid Dynamics code “OpenFOAM” [http://www.openfoam.com/] is here adapted to the field of solid mechanics, with the aim to bridge the gap between computational fluid and solid dynamics. A cell centred finite volume algorithm is employed and suitably adapted. Naturally, discontinuity of the conservation variables across control volume interfaces leads to a Riemann problem, whose resolution requires special attention when attempting to model materials with predominant nearly incompressible behaviour (). For this reason, an acoustic Riemann solver combined with a preconditioning procedure is introduced. In addition, a global a posteriori angular momentum projection procedure proposed in Haider et al. (2017) is also presented and adapted to a Total Lagrangian version of the nodal scheme of Kluth and Després (2010) used in this paper for comparison purposes. Finally, a series of challenging numerical examples is examined in order to assess the robustness and applicability of the proposed methodology with an eye on large scale simulation in future works.

  Abstract
  The paper presents a new computational framework for the numerical simulation of fast large strain solid dynamics, with particular emphasis on the treatment of near incompressibility. [...]

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• A face-centred finite volume method for second-order elliptic problems

  R. Sevilla, M. Giacomini, A. Huerta

  Int. J. Numer. Meth. Engng. (2018). Vol. 115 (8), pp. 986-1014 (preprint)

  
  

  Abstract

  This work proposes a novel finite volume paradigm, the face-centred finite volume (FCFV) method. Contrary to the popular vertex (VCFV) and cell (CCFV) centred finite volume methods, the novel FCFV defines the solution on the mesh faces (edges in 2D) to construct locally-conservative numerical schemes. The idea of the FCFV method stems from a hybridisable discontinuous Galerkin (HDG) formulation with constant degree of approximation, thus inheriting the convergence properties of the classical HDG. The resulting FCFV features a global problem in terms of a piecewise constant function defined on the faces of the mesh. The solution and its gradient in each element are then recovered by solving a set of independent element-by-element problems. The mathematical formulation of FCFV for Poisson and Stokes equation is derived and numerical evidence of optimal convergence in 2D and 3D is provided. Numerical examples are presented to illustrate the accuracy, efficiency and robustness of the proposed methodology. The results show that, contrary to other FV methods, the accuracy of the FCFV method is not sensitive to mesh distortion and stretching. In addition, the FCFV method shows its better performance, accuracy and robustness using simplicial elements, facilitating its application to problems involving complex geometries in 3D.

  Abstract
  This work proposes a novel finite volume paradigm, the face-centred finite volume (FCFV) method. Contrary to the popular vertex (VCFV) and cell (CCFV) centred finite volume [...]

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• Proper generalized decomposition solutions within a domain decomposition strategy

  A. Huerta, E. Nadal Soriano, F. Chinesta

  Int. J. Numer. Meth. Engng. (2019). Vol. 113 (13), pp. 1972-1994 (Preprint)

  
  

  Abstract

  Domain decomposition strategies and proper generalized decomposition are efficiently combined to obtain a fast evaluation of the solution approximation in parameterized elliptic problems with complex geometries. The classical difficulties associated to the combination of layered domains with arbitrarily oriented midsurfaces, which may require in‐plane–out‐of‐plane techniques, are now dismissed. More generally, solutions on large domains can now be confronted within a domain decomposition approach. This is done with a reduced cost in the offline phase because the proper generalized decomposition gives an explicit description of the solution in each subdomain in terms of the solution at the interface. Thus, the evaluation of the approximation in each subdomain is a simple function evaluation given the interface values (and the other problem parameters). The interface solution can be characterized by any a priori user‐defined approximation. Here, for illustration purposes, hierarchical polynomials are used. The repetitiveness of the subdomains is exploited to reduce drastically the offline computational effort. The online phase requires solving a nonlinear problem to determine all the interface solutions. However, this problem only has degrees of freedom on the interfaces and the Jacobian matrix is explicitly determined. Obviously, other parameters characterizing the solution (material constants, external loads, and geometry) can also be incorporated in the explicit description of the solution.

  Abstract
  Domain decomposition strategies and proper generalized decomposition are efficiently combined to obtain a fast evaluation of the solution approximation in parameterized elliptic [...]

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• A multidimensional data-driven sparse identification technique: the sparse proper generalized decomposition

  R. Ibañez, E. Abisset-Chavanne, A. Ammar, D. González, E. Cueto, A. Huerta, J. Duval, F. Chinesta

  Complexity (2018). Vol. 2018, pp. 1-11

  
  

  Abstract

  Sparse model identification by means of data is especially cumbersome if the sought dynamics live in a high dimensional space. This usually involves the need for large amount of data, unfeasible in such a high dimensional settings. This well-known phenomenon, coined as the curse of dimensionality, is here overcome by means of the use of separate representations. We present a technique based on the same principles of the Proper Generalized Decomposition that enables the identification of complex laws in the low-data limit. We provide examples on the performance of the technique in up to ten dimensions.

  Abstract
  Sparse model identification by means of data is especially cumbersome if the sought dynamics live in a high dimensional space. This usually involves the need for large amount [...]

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