eXtended hybridizable discontinuous Galerkin for incompressible flow problems with unfitted meshes and interfaces

The eXtended hybridizable discontinuous Galerkin (X-HDG) method is developed for the solution of Stokes problems with void or material interfaces. X-HDG is a novel method that combines the hybridizable discontinuous Galerkin (HDG) method with an eXtended finite element strategy, resulting in a high-order, unfitted, superconvergent method, with an explicit definition of the interface geometry by means of a level-set function. For elements not cut by the interface, the standard HDG formulation is applied, whereas a modified weak form for the local problem is proposed for cut elements. Heaviside enrichment is considered on cut faces and in cut elements in the case of bimaterial problems. Two-dimensional numerical examples demonstrate that the applicability, accuracy, and superconvergence properties of HDG are inherited in X-HDG, with the freedom of computational meshes that do not fit the interfacesPeer ReviewedPostprint (author's final draft

Elliptic harbor wave model with perfectly matched layer and exterior bathymetry effects

Standard strategies for dealing with the Sommerfeld condition in elliptic mild-slope models require strong assumptions on the wave field in the region exterior to the computational domain. More precisely, constant bathymetry along (and beyond) the open boundary, and parabolic approximations–based boundary conditions are usually imposed. Generally, these restrictions require large computational domains, implying higher costs for the numerical solver. An alternative method for coastal/harbor applications is proposed here. This approach is based on a perfectly matched layer (PML) that incorporates the effects of the exterior bathymetry. The model only requires constant exterior depth in the alongshore direction, a common approach used for idealizing the exterior bathymetry in elliptic models. In opposition to standard open boundary conditions for mild-slope models, the features of the proposed PML approach include (1) completely noncollinear coastlines, (2) better representation of the real unbounded domain using two different lateral sections to define the exterior bathymetry, and (3) the generation of reliable solutions for any incoming wave direction in a small computational domain. Numerical results of synthetic tests demonstrate that solutions are not significantly perturbed when open boundaries are placed close to the area of interest. In more complex problems, this provides important performance improvements in computational time, as shown for a real application of harbor agitation.Peer ReviewedPostprint (author's final draft

Imposing essential boundary conditions in mesh-free methods

Imposing essential boundary conditions is a key issue in mesh-free methods. The mesh-free interpolation does not verify the Kronecker delta property and, therefore, the imposition of prescribed values is not as straightforward as for the finite element method. The aim of this paper is to present a general overview on the existing techniques to enforce essential boundary conditions in Galerkin based mesh-free methods. Special attention is paid to the mesh-free coupling with finite elements for the imposition of prescribed values and to methods based on a modification of the Galerkin weak form. Particular examples are used to analyze and compare their performance in different situations

Coupling finite elements and particles for adaptivity: an application to consistently stabilized convection–diffusion

A mixed approximation coupling finite elements and mesh-less methods is presented. It allows selective refinement of the finite element solution without remeshing cost. The distribution of particles can be arbitrary. Continuity and consistency is preserved. The behaviour of the mixed interpolation in the resolution of the convection-diffusion equation is analyzed

Time accurate consistently stabilized mesh-free methods for convection dominated problems

This is the pre-peer reviewed version of the following article: Huerta, A.; Fernandez, S. Time accurate consistently stabilized mesh-free methods for convection dominated problems. "International journal for numerical methods in engineering", Gener 2003, vol. 56, núm. 9, p. 1225-1242, which has been published in final form at http://www3.interscience.wiley.com/journal/102522343/abstractThe behaviour of high-order time stepping methods combined with mesh-free methods is studied for the transient convection-diffusion equation. Particle methods, such as the element-free Galerkin (EFG) method, allow to easily increase the order of consistency and, thus, to formulate high-order schemes in space and time. Moreover, second derivatives of the EFG shape functions can be constructed with a low extra cost and are well defined, even for linear interpolation. Thus, consistent stabilization schemes can be considered without loss in the convergence rates.Peer ReviewedPostprint (author’s final draft

Coupling finite elements and particles for adaptivity: an application to consistently stabilized convection–diffusion

A mixed approximation coupling finite elements and mesh-less methods is presented. It allows selective refinement of the finite element solution without remeshing cost. The distribution of particles can be arbitrary. Continuity and consistency is preserved. The behaviour of the mixed interpolation in the resolution of the convection-diffusion equation is analyzed.Peer ReviewedPostprint (author’s final draft

Time accurate consistently stabilized mesh-free methods for convection dominated problems

The behaviour of high‐order time stepping methods combined with mesh‐free methods is studied for the transient convection–diffusion equation. Particle methods, such as the element‐free Galerkin (EFG) method, allow to easily increase the order of consistency and, thus, to formulate high‐order schemes in space and time. Moreover, second derivatives of the EFG shape functions can be constructed with a low extra cost and are well defined, even for linear interpolation. Thus, consistent stabilization schemes can be considered without loss in the convergence rates

Numerical integration over 2D NURBS-shaped domains with applications to NURBS-enhanced FEM

This paper focuses on the numerical integration of polynomial functions along non-uniform rational B-splines (NURBS) curves and over 2D NURBS-shaped domains, i.e. domains with NURBS boundaries. The integration of the constant function f=1 is of special interest in computer aided design software and the integration of very high-order polynomials is a key aspect in the recently proposed NURBS-enhanced finite element method (NEFEM). Several well-known numerical quadratures are compared for the integration of polynomials along NURBS curves, and two transformations for the definition of numerical quadratures in triangles with one edge defined by a trimmed NURBS are proposed, analyzed and compared. When exact integration is feasible, explicit formulas for the selection of the number of integration points are deduced. Numerical examples show the influence of the number of integration points in NEFEM computations.Peer ReviewedPostprint (author's final draft

Galerkin discontinuo local aplicado a problemas de ondas acústicas: condiciones de frontera absorbente locales de bajo orden

En este trabajo se presenta una comparación de la solución numérica de la ecuación de Helmholtz utilizando el método de elementos finitos Galerkin Discontinuo Local (LDG) con varias condiciones de contorno absorbentes locales de bajo orden. Se describe brevemente el proceso de discretización y se realizan pruebas con ejemplos clásicos presentes en la literatura.Peer Reviewe

Comparison of high-order continuous and hybridizable discontinuous {G}alerkin methods in incompressible fluid flow problems

© 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/The computational efficiency and the stability of Continuous Galerkin (CG) methods, with Taylor–Hood approximations, and Hybridizable Discontinuous Galerkin (HDG) methods are compared for the solution of the incompressible Stokes and Navier–Stokes equations at low Reynolds numbers using direct solvers. A thorough comparison in terms of CPU time and accuracy for both discretization methods is made, under the same platform, for steady state problems, with triangular and quadrilateral elements of degree . Various results are presented such as error vs. CPU time of the direct solver, error vs. ratio of CPU times of HDG to CG, etc. CG can outperform HDG when the CPU time, for a given degree and mesh, is considered. However, for high degree of approximation, HDG is computationally more efficient than CG, for a given level of accuracy, as HDG produces lesser error than CG for a given mesh and degree. Finally, stability of HDG and CG is studied using a manufactured solution that produces a sharp boundary layer, confirming that HDG provides smooth converged solutions for Reynolds numbers higher than CG, in the presence of sharp fronts.Peer ReviewedPostprint (author's final draft