An interpolation‐based fast multipole method for higher‐order boundary elements on parametric surfaces

In this article, a black-box higher order fast multipole method for solving boundary integral equations on parametric surfaces in three spatial dimensions is proposed. Such piecewise smooth surfaces are the topic of recent studies in isogeometric analysis. Due to the exact surface representation, the rate of convergence of higher order methods is not limited by approximation errors of the surface. An element-wise clustering strategy yields a balanced cluster tree and an efficient numerical integration scheme for the underlying Galerkin method. By performing the interpolation for the fast multipole method directly on the reference domain, the cost complexity in the polynomial degree is reduced by one order. This gain is independent of the application of either (mathcal{H})- or (mathcal{H}^2)-matrices. In fact, several simplifications in the construction of (mathcal{H}^2)-matrices are pointed out, which are a by-product of the surface representation. Extensive numerical examples are provided in order to quantify and qualify the proposed method. In this article, a black-box higher order fast multipole method for solving boundary integral equations on parametric surfaces in three spatial dimensions is proposed. Such piecewise smooth surfaces are the topic of recent studies in isogeometric analysis. Due to the exact surface representation, the rate of convergence of higher order methods is not limited by approximation errors of the surface. An element-wise clustering strategy yields a balanced cluster tree and an efficient numerical integration scheme for the underlying Galerkin method. By performing the interpolation for the fast multipole method directly on the reference domain, the cost complexity in the polynomial degree is reducedby one order. This gain is independent of the application of either H - or H 2- matrices. In fact, several simplificationsin the construction of  H 2 -matrices are pointed out, which are a by-product of the surface representation. Extensive numerical examples are provided in order to quantify and qualify the proposed method.

[1]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[2]  Maharavo Randrianarivony,et al.  From Computer Aided Design to wavelet BEM , 2009, Comput. Vis. Sci..

[3]  C. Schwab,et al.  Boundary Element Methods , 2010 .

[4]  Thomas-Peter Fries,et al.  Fast Isogeometric Boundary Element Method based on Independent Field Approximation , 2014, ArXiv.

[5]  Wolfgang Dahmen,et al.  Compression Techniques for Boundary Integral Equations - Asymptotically Optimal Complexity Estimates , 2006, SIAM J. Numer. Anal..

[6]  Olaf Steinbach,et al.  Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements , 2007 .

[7]  J. Nédélec,et al.  Integral equations with non integrable kernels , 1982 .

[8]  S. Börm.  Efficient Numerical Methods for Non-local Operators , 2010 .

[9]  Ralf Schweizer,et al.  Integral Equation Methods In Scattering Theory , 2016 .

[10]  Helmut Harbrecht,et al.  Comparison of fast boundary element methods on parametric surfaces , 2013 .

[11]  W. Hackbusch.  Integral Equations: Theory and Numerical Treatment , 1995 .

[12]  W. Hackbusch.  A Sparse Matrix Arithmetic Based on $\Cal H$-Matrices. Part I: Introduction to ${\Cal H}$-Matrices , 1999, Computing.

[13]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

[14]  Alessandro Reali,et al.  Studies of Refinement and Continuity in Isogeometric Structural Analysis (Preprint) , 2007 .

[15]  W. Hackbusch,et al.  H 2 -matrix approximation of integral operators by interpolation , 2002 .

[16]  H. Harbrecht,et al.  Wavelet formulation of the polarizable continuum model. II. Use of piecewise bilinear boundary elements. , 2015, Physical chemistry chemical physics : PCCP.

[17]  C. Schwab,et al.  Quadrature for $hp$-Galerkin BEM in ${\hbox{\sf l\kern-.13em R}}^3$ , 1997 .

[18]  C. Schwab,et al.  Quadrature for hp-Galerkin BEM in lR3 , 1997 .

[19]  Helmut Harbrecht,et al.  Solution of free boundary problems in the presence of geometric uncertainties , 2017 .

[20]  Mario Bebendorf,et al.  Approximation of boundary element matrices , 2000, Numerische Mathematik.

[21]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[22]  Maharavo Randrianarivony,et al.  Wavelet BEM on molecular surfaces: solvent excluded surfaces , 2011, Computing.

[23]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[24]  Thomas-Peter Fries,et al.  Isogeometric Boundary Element Method with Hierarchical Matrices , 2014, ArXiv.

[25]  Dieter Lasser,et al.  Grundlagen der geometrischen Datenverarbeitung , 1989 .

[26]  Helmut Harbrecht,et al.  Efficient approximation of random fields for numerical applications , 2015, Numer. Linear Algebra Appl..

[27]  Klaus Gibermann,et al.  Multilevel approximation of boundary integral operators , 2001 .

[28]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[29]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[30]  Maharavo Randrianarivony,et al.  Wavelet BEM on molecular surfaces: parametrization and implementation , 2009, Computing.

[31]  M. G. Duffy,et al.  Quadrature Over a Pyramid or Cube of Integrands with a Singularity at a Vertex , 1982 .

[32]  W. Hackbusch,et al.  On the fast matrix multiplication in the boundary element method by panel clustering , 1989 .