Jean-François Remacle

A computational approach to handle complex microstructure geometries

In multiscale analysis of components, there is usually a need to solve microstructures with complex geometries. In this paper, we use the extended finite element method (X-FEM) to solve scales involving complex geometries. The X-FEM allows one to use meshes not necessarily matching the physical surface of the problem while retaining the accuracy of the classical finite element approach. For material interfaces, this is achieved by introducing a new enrichment strategy. Although the mesh does not need to conform to the physical surfaces, it needs to be fine enough to capture the geometry of these surfaces. A simple algorithm is described to adaptively refine the mesh to meet this geometrical requirement. Numerical experiments on the periodic homogenization of two-phase complex cells demonstrate the accuracy and simplicity of the X-FEM

An Adaptive Discontinuous Galerkin Technique with an Orthogonal Basis Applied to Compressible Flow Problems

We present a high-order formulation for solving hyperbolic conservation laws using the discontinuous Galerkin method (DGM). We introduce an orthogonal basis for the spatial discretization and use explicit Runge--Kutta time discretization. Some results of higher order adaptive refinement calculations are presented for inviscid Rayleigh--Taylor flow instability and shock reflection problems. The adaptive procedure uses an error indicator that concentrates the computational effort near discontinuities.

A quadrature-free discontinuous Galerkin method for the level set equation

A quadrature free, Runge-Kutta discontinuous Galerkin method (QF-RK-DGM) is developed to solve the level set equation written in a conservative form on two- and tri-dimensional unstructured grids. We show that the DGM implementation of the level set approach brings a lot of additional benefits as compared to traditional ENO level set realizations. Some examples of computations are provided that demonstrate the high order of accuracy and the computational efficiency of the method.

A stabilized finite element method using a discontinuous level set approach for the computation of bubble dynamics

A novel numerical method for solving three-dimensional two phase flow problems is presented. This method combines a quadrature free discontinuous Galerkin method for the level set equation with a pressure stabilized finite element method for the Navier Stokes equations. The main challenge in the computation of such flows is the accurate evaluation of surface tension forces. This involves the computation of the curvature of the fluid interface. In the context of the discontinuous Galerkin method, we show that the use of a curvature computed by means of a direct derivation of the level set function leads to inaccurate and oscillatory results. A more robust, second-order, least squares computation of the curvature that filters out the high frequencies and produces converged results is presented. This whole numerical technology allows to simulate a wide range of flow regimes with large density ratios, to accurately capture the shape of the deforming interface of the bubble and to maintain good mass conservation.

Robust untangling of curvilinear meshes

This paper presents a technique that allows to untangle high-order/curvilinear meshes. The technique makes use of unconstrained optimization where element Jacobians are constrained to lie in a prescribed range through moving log-barriers. The untangling procedure starts from a possibly invalid curvilinear mesh and moves mesh vertices with the objective of producing elements that all have bounded Jacobians. Bounds on Jacobians are computed using the results of Johnen et al. (2012, 2013) [1,2]. The technique is applicable to any kind of polynomial element, for surface, volume, hybrid or boundary layer meshes. A series of examples demonstrate both the robustness and the efficiency of the technique. The final example, involving a time explicit computation, shows that it is possible to control the stable time step of the computation for curvilinear meshes through an alternative element deformation measure.

Geometrical validity of curvilinear finite elements

In this paper, we describe a way to compute accurate bounds on Jacobian determinants of curvilinear polynomial finite elements. Our condition enables to guarantee that an element is geometrically valid, i.e., that its Jacobian determinant is strictly positive everywhere in its reference domain. It also provides an efficient way to measure the distortion of curvilinear elements. The key feature of the method is to expand the Jacobian determinant using a polynomial basis, built using Bezier functions, that has both properties of boundedness and positivity. Numerical results show the sharpness of our estimates.

Aspects of discontinuous Galerkin methods for hyperbolic conservation laws

We review several properties of the discontinuous Galerkin method for solving hyperbolic systems of conservation laws including basis construction, flux evaluation, solution limiting, adaptivity, and a posteriori error estimation. Regarding error estimation, we show that the leading term of the spatial discretization error using the discontinuous Galerkin method with degree p piecewise polynomials is proportional to a linear combination of orthogonal polynomials on each element of degrees p and p+1. These are Radau polynomials in one dimension. The discretization errors have a stronger superconvergence of order O(h2p+1), where h is a mesh-spacing parameter, at the outflow boundary of each element. These results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors in regions where solutions are smooth.We present the results of applying the discontinuous Galerkin method to unsteady, two-dimensional, compressible, inviscid flow problems. These include adaptive computations of Mach reflection and mixing-instability problems.

Parallel Algorithm Oriented Mesh Database

Abstract.In this paper, we present a new point of view for efficiently managing general parallel mesh representations. Taking as a slarting point the Algorithm Oriented Mesh Database (AOMD) of [1] we extend the concepts to a parallel mesh representation. The important aspects of parallel adaptivity and dynamic load balancing are discussed. We finally show how AOMD can be effectively interfaced with mesh adaptive partial differential equation solvers. Results of the calculation of an elasticity problem and of a transient fluid dynamics problem involving thousands of mesh refinements, and load balancings are finally presented.

Efficient Discontinuous Galerkin Methods for solving acoustic problems

I Introduction 2II Mathematical model 2A Finite and infinite elements, frequency-domain approach . . . . . . . . . . . . . . . . . . . . . 21 Governing equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Free field boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Coupling with acoustic ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4B RK-DGM approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Implementation of the space Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Optimization of the computation of surface fluxes . . . . . . . . . . . . . . . . . . . . 75 Convergence properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10IIIValidation 12A Test case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12B Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12IVLarge-scale applications 15A Parallel implementation of the DG method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15B Test case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15C Test case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17V Conclusions 18VIAcknowledgments 18

High-order discontinuous Galerkin schemes on general 2D manifolds applied to the shallow water equations

An innovating approach is proposed to solve vectorial conservation laws on curved manifolds using the discontinuous Galerkin method. This new approach combines the advantages of the usual approaches described in the literature. The vectorial fields are expressed in a unit non-orthogonal local tangent basis derived from the polynomial mapping of curvilinear triangle elements, while the convective flux functions are written is the usual 3D Cartesian coordinate system. The number of vectorial components is therefore minimum and the tangency constraint is naturally ensured, while the method remains robust and general since not relying on a particular parametrization of the manifold. The discontinuous Galerkin method is particularly well suited for this approach since there is no continuity requirement between elements for the tangent basis definition. The possible discontinuities of this basis are then taken into account in the Riemann solver on inter-element interfaces. The approach is validated on the sphere, using the shallow water equations for computing standard atmospheric benchmarks. In particular, the Williamson test cases are used to analyze the impact of the geometry on the convergence rates for discretization error. The propagation of gravity waves is eventually computed on non-conventional irregular curved manifolds to illustrate the robustness and generality of the method.