Maharavo Randrianarivony

From Computer Aided Design to wavelet BEM

This paper develops an interface between Computer Aided Design (CAD) and the wavelet Galerkin scheme for boundary integral equations. The key issue is an algorithm that decomposes a technical surface which was generated by CAD tools into a regular collection of parameterized four-sided patches. By a postprocessing step the global continuity of the parametrization is guaranteed. Numerical results are reported to illustrate the approach. In particular, the decomposition techniques are applied to real CAD data which come from IGES files.

Wavelet formulation of the polarizable continuum model

The first implementation of a wavelet discretization of the Integral Equation Formalism (IEF) for the Polarizable Continuum Model (PCM) is presented here. The method is based on the application of a general purpose wavelet solver on the cavity boundary to solve the integral equations of the IEF‐PCM problem. Wavelet methods provide attractive properties for the solution of the electrostatic problem at the cavity boundary: the system matrix is highly sparse and iterative solution schemes can be applied efficiently; the accuracy of the solver can be increased systematically and arbitrarily; for a given system, discretization error accuracy is achieved at a computational expense that scales linearly with the number of unknowns. The scaling of the computational time with the number of atoms N is formally quadratic but a N1.5 scaling has been observed in practice. The current bottleneck is the evaluation of the potential integrals at the cavity boundary which scales linearly with the system size. To reduce this overhead, interpolation of the potential integrals on the cavity surface has been successfully used.

Wavelet BEM on molecular surfaces: solvent excluded surfaces

The present paper is concerned with the rapid solution of a boundary integral equation for the apparent surface charge which arises from solvation continuum models. In order to apply the wavelet Galerkin scheme the molecular surface needs to be represented as a parametric surface consisting of smooth four-sided patches. We develop an algorithm which decomposes a solvent excluded surface into a set of globally continuous four-sided NURBS patches. Numerical experiments are carried out to demonstrate the feasibility and scope of the present approach.

Wavelet BEM on molecular surfaces: parametrization and implementation

The present paper is dedicated to the rapid solution of boundary integral equations arising from solvation continuum models. We apply a fully discrete wavelet Galerkin scheme for the computation of the apparent surface charge on van der Waals or solvent accessible surfaces. The molecular surface is described in parametric form as a set of four-sided spherical patches. Each patch is exactly represented as rational Bézier surface. Numerical results are reported to illustrate the approach.

Preparation of CAD and Molecular Surfaces for Meshfree Solvers

Raw CAD and molecular data need to be prepared before they can be used in meshfree numerical solvers. After quickly summarizing the different steps required for that processing, we will focus on the practical realization of global continuity. In particular, we will examine error bounds about chord length approximation in which two procedures can yield errors: sample interpolation, length estimation. We will report eventually on some practical results where the initial data are acquired from IGES files for CAD models and from PDB files for molecular ones.

On transfinite interpolations with respect to convex domains

We propose an approach for constructing a transfinite interpolation where the domain of definition is a convex polytope. For multifaceted domains which are not of tensor product type, it is difficult to directly generalize the usual approach of transfinite interpolation which blends opposite faces and which then substracts that by a mixed term. Therefore, we suggest a short formula which uses topologic entities of the convex domain. Our formula uses some projection operator onto the faces of the polytope. Both representations (V-setting and H-setting) of a polytope are used. We show also that the transfinite interpolation is stable under affine transformations. As a supplement to the theoretical demonstrations, we show some interesting practical illustrations.

On space enrichment estimator for nonlinear Poisson-Boltzmann

We consider the mathematical aspect of the nonlinear Poisson-Boltzmann equation which physically governs the ionic interaction between solute and solvent media. The presented a-posteriori estimates can be computed locally in a very efficient manner. The a-posteriori error is based upon hierarchical space enrichment which ensures its efficiency and reliability. A brief survey of the solving of the nonlinear system resulting from the FEM discretization is reported. To corroborate the analysis, we report on a few numerical results for illustrations. We numerically examine some values of the constants encountered in the theoretical study.

PARALLEL PROCESSING OF ANALYTICAL POISSON-BOLTZMANN USING HIGHER ORDER FEM

We focus on the efficient parallel processing of meshes from biomolecular data so that they can be subsequently used for FEM simulation. All mesh processing about refinements and coherent indexations are applied in parallel. The simplices of the mesh are needed for the application in FEM having higher polynomial degrees. For biomolecular data, the only inputs are the atom coordinates, the van der Waals radii and the probe radius. Our principal goal is to obtain data which are well balanced among the different processors. To corroborate the parallel mesh processing, we show some application to FEM simulation. For that, we consider the parallel FEM of the linearized PoissonBoltzmann problem. We compare the FEM numerical results with known theoretical prediction to validate the accuracy of the parallel implementation.

Treatment of general domains in two space dimensions in a Partition of Unity Method

This paper is concerned with the approximate solution of partial differential equations using meshfree Galerkin methods on arbitrary domains in two space dimensions which we assume to be given as CAD/IGES data. In particular we focus on the particle-partition of unity method (PPUM) yet the presented technique is applicable to most meshfree Galerkin methods. The basic geometric operation employed in our cut-cell type approach is the intersection of a planar NURBS patch and an axis-aligned rectangle. Note that our emphasis is on the speed of the clipping operations since these are invoked frequently while trying to attain a small number of patches for the representation of the intersection. We present some first numerical results of the presented approach.