Philippe Remy Bernard Devloo

PZ: An object oriented environment for scientific programming

An object oriented environment for scientific programming is presented. The environment includes classes for matrix computations (Tmatrix) and a set of classes for the implementation of finite element algorithms (PZ). The PZ environment implements one and two dimensional finite elements with arbitrary orders of interpolation and applicable to a variety of systems of differential equations. The TMatrix environment includes an abstract matrix class interface and a variety of storage patterns and solution methods.

A finite element model for three dimensional hydraulic fracturing

This paper is devoted to the development of a model for the numerical simulation of hydraulic fracturing processes with 3d fracture propagation. It takes into account the effects of fluid flow inside the fracture, fluid leak-off through fracture walls and elastic response of the surrounding porous media. Finite element techniques are adopted for the discretization of the conservation law of fluid flow and the singular integral equation relating the traction and the fracture opening. The discrete model for the singular integral equation is implemented using a stencil matrix structure allowing an efficient implementation of the fluid-structure interaction problem. Newtons method combined with GMRES linear solver are used to solve the resulting nonlinear set of equations. An algorithm for fracture propagation is proposed which is based on the balance of the amount of fluid transported to a certain point with the amount of fluid that could be lost by leak-off. To illustrate the feasibility of the model, we present simulation results for typical operational parameters.

Wavelets and adaptive grids for the discontinuous Galerkin method

Abstract In this paper, space adaptivity is introduced to control the error in the numerical solution of hyperbolic systems of conservation laws. The reference numerical scheme is a new version of the discontinuous Galerkin method, which uses an implicit diffusive term in the direction of the streamlines, for stability purposes. The decision whether to refine or to unrefine the grid in a certain location is taken according to the magnitude of wavelet coefficients, which are indicators of local smoothness of the numerical solution. Numerical solutions of the nonlinear Euler equations illustrate the efficiency of the method.

Object Oriented Tools for Scientific Computing

Abstract.A set of object oriented tools is presented which, when combined, yield an efficient parallel finite element program. Special emphasis is given to details within the concept of the tools which enhance their efficiency. The experience of the author has shown that the design concepts documented are crucial for the efficiency of the issuing code, and that they can easily be incorporated within existing object oriented programs.

Two-dimensional hp adaptive finite element spaces for mixed formulations

One important characteristic of mixed finite element methods is their ability to provide accurate and locally conservative fluxes, an advantage over standard H 1 -finite element discretizations. However, the development of p or h p adaptive strategies for mixed formulations presents a challenge in terms of code complexity for the construction of H ( d i v ) -conforming shape functions of high order on non-conforming meshes, and compatibility verification of the approximation spaces for primal and dual variables ( i n f - s u p condition). In the present paper, a methodology is presented for the assembly of such approximation spaces based on quadrilateral and triangular meshes. In order to validate the computational implementations, and to show their consistent applications to mixed formulations, elliptic model problems are simulated to show optimal convergence rates for h and p refinements, using uniform and non-uniform (non-conformal) settings for a problem with smooth solution, and using adaptive h p -meshes for the approximation of a solution with strong gradients. Results for similar simulations using H 1 -conforming formulation are also presented, and both methods are compared in terms of accuracy and required number of degrees of freedom using static condensation.

Hierarchical high order finite element bases for H ( div ) spaces based on curved meshes for two-dimensional regions or manifolds

The mixed finite element formulation for elliptic problems is characterized by simultaneous calculations of the potential (primal variable) and of the flux field (dual variable). This work focuses on new H ( div ) -conforming finite element spaces, which are suitable for flux approximations, based on curved meshes of a planar region or a manifold domain embedded in R 3 . The adopted methodology for the construction of H ( div ) bases consists in using hierarchical H 1 -conforming scalar bases multiplied by vector fields that are properly constructed on the master element and mapped to the geometrical elements by the Piola transformation, followed by a normalization procedure. They are classified as being of edge or internal type. The normal component of an edge function coincides on the corresponding edge with the associated scalar shape function, and vanishes over the other edges, and the normal components of an internal shape function vanishes on all element edges. These properties are fundamental for the global assembly of H ( div ) -conforming functions locally defined by these vectorial shape functions. For applications to the mixed formulation, the configuration of the approximation spaces is such that the divergence of the dual space and the primal approximation space coincides. Results of verification numerical tests are presented for curved triangular and quadrilateral partitions on circular, cylindrical and spherical regions, demonstrating stable convergence with optimal convergence rates, coinciding for primal and dual variables. Innovative two dimensional Hdiv approximation spaces for curved quadrilateral and triangular elements.Hdiv bases properly constructed on the master element and mapped to the geometrical elements by the Piola transformation, followed by a normalization procedure.Stable convergence with optimal convergence rates, coinciding for primal and dual variables.Study of number of condensed equations as a function of the polynomial order of the shape functions.

Application of a combined continuous–discontinuous Galerkin finite element method for the solution of the Girkmann problem

Abstract In January 2008, the International Association of Computational Mechanics (IACM) invited the scientific community to solve the Girkmann problem by finite element methods. The challenge was to get certain quantities of interest in terms of the approximate solutions within 5% of accuracy of their exact counterparts. For instance, the goal could be to evaluate the bending moment or the shear force at the junction of the shell and the ring. The purpose of the present paper is to solve an axisymmetric solid elastic model for this problem using a continuous finite element method in combination with discontinuous interface elements in the region of interest (named as the DG-FEM method). The numerical results of the p version of the DG-FEM method are presented and discussed. The results are verified with respect to previous results published in the literature showing excellent consistent results.

OOPar: An Object Oriented Environment for Implementing Parallel Algorithms

OOPar is a C++ library which implements an interface to MPI (or other communication library) to offer the user a high level interface for implementing parallel algorithms. Large scale software projects have been parallelized using OOPar and their results will be presented during this presentation. OOPar introduces two concepts to help the programmer in defining a parallel algorithm: Distributed data and tasks which act on the distributed data. Distributed data are objects of classes which can be transmitted between processors. The user of the OOPar library can define his own types by deriving class from the TPZSaveable class and implementing the virtual Read/Write methods. Distributed data objects are managed by an object of type OOPDataManager. Each processor has its OOPDataManager object which administers TPZSaveable objects and the access requests issued by the OOPTask objects. Task objects implement part of a parallel algorithm acting upon and transforming distributed data objects. Task objects are submitted to a TaskManager object. Each processor has a unique TaskManager object. In order to sequence tasks in the proper order, a version is associated with distributed data objects and tasks depend on a distributed data object with a certain version. After transforming a distributed data object, the task object increments the version of the data object.

Error analysis for a new mixed finite element method in 3D

There are different possibilities of choosing balanced pairs of approximation spaces for dual (flux) and primal (pressure) variables; to be used in discrete versions of the mixed finite element method for elliptic problems arising in fluid simulations. Three cases shall be studied for discretized three dimensional formulations, based on tetrahedral, hexahedral, and prismatic meshes. The principle guiding the constructions of the approximation spaces is the property that, the divergence of the dual space and the primal approximation space, should coincide, while keeping the same order of accuracy for the flux variable, and varying the accuracy order of the primal variable. Some cases correspond either to the classic spaces of Raviart-Thomas, Brezzi-Douglas-Marini, Brezzi-Douglas-Fortin-Marini or Nedelec types. A new kind of approximation is proposed by further incrementing the order of some internal flux functions, and matching primal functions at the border fluxes. In this article we develop a unified error analysis for all these space families, and element geometries.

Two dimensional hierarchical mixed finite element approximations with enhanced primal variable accuracy

The purpose of the present paper is to analyse different possibilities of choosing balanced pairs of approximation spaces for dual (flux) and primal (pressure) variables to be used in discrete versions of the mixed finite element method for affine two dimensional meshes. In all space configurations, the principle guiding their construction is the property that the divergence of the dual space and the primal approximation space should coincide, while keeping the same order of accuracy for the flux variable and varying the accuracy order of the primal variable. There is the classic case of BDMk spaces based on triangular meshes and polynomials of total degree k for the dual variable, and k − 1 for the primal variable, showing stable simulations with optimal convergence rates of orders k + 1 and k,respectively. Another case is related to RTk and BDF Mk+1 spaces for quadrilateral and triangular meshes, respectively. It gives identical approximation order k + 1 for both primal and dual variables, an improvement in accuracy obtained by increasing the degree of primal functions to k, and by enriching the dual space with some properly chosen internal shape functions of degree k + 1, while keeping degree k for the border fluxes. A new type of approximation is proposed by further incrementing the order of some internal flux functions to k + 2, and matching primal functions to k + 1 (higher than the border fluxes of degree k). Thus, higher convergence rate of order k + 2 is obtained for the primal variable. Using static condensation, the global condensed system to be solved in all the cases have same dimension (and structure), which is proportional to the space dimension of the border fluxes for each element geometry.