Ruben Sevilla

Tutorial on Hybridizable Discontinuous Galerkin (HDG) for Second-Order Elliptic Problems

The HDG is a new class of discontinuous Galerkin (DG) methods that shares favorable properties with classical mixed methods such as the well known Raviart–Thomas methods. In particular, HDG provides optimal convergence of both the primal and the dual variables of the mixed formulation. This property enables the construction of superconvergent solutions, contrary to other popular DG methods. In addition, its reduced computational cost, compared to other DG methods, has made HDG an attractive alternative for solving problems governed by partial differential equations. A tutorial on HDG for the numerical solution of second-order elliptic problems is presented. Particular emphasis is placed on providing all the necessary details for the implementation of HDG methods.

HDG-NEFEM with degree adaptivity for stokes flows

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.

Influence of periodically fluctuating material parameters on the stability of explicit high-order spectral element methods

This paper aims at studying the influence of material heterogeneity on the stability of explicit time marching schemes for the high-order spectral element discretisation of wave propagation problems. A periodic fluctuation of the density and stiffness parameters is considered, where the period is related to the characteristic element size of the mesh. A new stability criterion is derived analytically for quadratic and cubic one-dimensional spectral elements in heterogeneous materials by using a standard Von Neumann analysis. The analysis presented illustrates the effect of material heterogeneity on the stability limit and also reveals the origin of instabilities that are often observed when the stability limit derived for homogeneous materials is adapted by simply changing the velocity of the wave to account for the material heterogeneity. Several extensions of the results derived for quadratic and cubic one-dimensional spectral elements are discussed, including higher order approximations, different period-icity of the material parameters and higher dimensions. Extensive numerical results demonstrate the validity of the new stability limits derived for heterogeneous materials with periodic fluctuation. Finally numerical examples of the * Corresponding author stability for randomly fluctuating material properties are also presented, discussing the applicability of the theoretical limits derived for material properties with periodic fluctuation.

A Superconvergent HDG Method for Stokes Flow with Strongly Enforced Symmetry of the Stress Tensor

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.

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

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.

A CONSISTENTLY LINEARISED SOLID MECHANICS BASED MESH DEFORMATION TECHNIQUE FOR HIGH ORDER CURVED ELEMENTS

In this work, a novel solid mechanics-based mesh deformation technique for high order curved elements is presented. The technique falls under the a posteriori curved mesh generation category, where higher order nodes are placed on a linear mesh and the geometry is then deformed to conform to the exact CAD boundary. In contrast to the existing a posteriori approaches in the literature such as the techniques based on the inclusion of residual stresses, parametrised and varying material constants, regularisation and smoothing of curved meshes, in this work, a rather consistent solid mechanics approach is followed. This implies, that the underlying Euler-Lagrange equations that need to be solved for, emerge from an energy principle, with well-defined internal energies constructed for an hyperelastic system, which are subsequently, consistently linearised. Depending on the geometrical parameterisation, the approach guarantees better mesh quality and lower condition number for the system of equations. Furthermore, due to the introduction of independent invariants emanating from fibre, surface and volume mappings, the essential mesh distortion measures are encoded in the formulation. The paper proves that for for two-dimensional elements such as triangles and quadrilaterals, not all the distortion measures can be independent. An example of materially instable internal energy is provided to pinpoint the importance of a consistent formulation in the context of highly stretched boundary layer meshes.

Numerical Simulation from Medical Images: Accurate Integration by Means of the Cartesian Grid Finite Element Method

Nowadays, when it comes to generation of patient-specific Finite Element model, there are two main alternatives. On the one hand, it is possible to generate geometrical models through segmentation, whereupon FE models would be obtained using standard mesh generators. On the other hand, we can create a Cartesian grid of uniform hexahedra in which the elements fit each pixel/voxel perfectly. In both cases, geometries will take part during the analysis either as complete models, in the first case, or as auxiliary entities, to apply boundary conditions properly for instance, in the second case. In any case, once the geometrical entities have been obtained from the medical image, the efficient generation of an accurate Finite Element model for numerical simulation in not trivial. The aim of this paper is to propose an efficient integration strategy, using Cartesian meshes, of 3D geometries defined by parametric surfaces, i.e. NURBS, obtained from medical images.