Integration Of Polynomial Functions Research Articles

Computation of Galerkin Double Surface Integrals in the 3-D Boundary Element Method

Many boundary element integral equation kernels are based on the Green's functions of the Laplace and Helmholtz equations in three dimensions. These include, for example, the Laplace, Helmholtz, elasticity, Stokes, and Maxwell's equations. Integral equation formulations lead to more compact, but dense linear systems. These dense systems are often solved iteratively via Krylov subspace methods, which may be accelerated via the fast multipole method. There are advantages to Galerkin formulations for such integral equations, as they treat problems associated with kernel singularity, and lead to symmetric and better conditioned matrices. However, the Galerkin method requires each entry in the system matrix to be created via the computation of a double surface integral over one or more pairs of triangles. There are a number of semi-analytical methods to treat these integrals, which all have some issues, and are discussed in this paper. We present novel methods to compute all the integrals that arise in Galerkin formulations involving kernels based on the Laplace and Helmholtz Green's functions to any specified accuracy. Integrals involving completely geometrically separated triangles are non-singular and are computed using a technique based on spherical harmonics and multipole expansions and translations, which results in the integration of polynomial functions over the triangles. Integrals involving cases where the triangles have common vertices, edges, or are coincident are treated via scaling and symmetry arguments, combined with automatic recursive geometric decomposition of the integrals. Example results are presented, and the developed software is available as open source.

FORTRAN subroutines for mathematical operations on polynomial functions

The paper presents FORTRAN subroutines that can perform mathematical operations involving polynomial functions both in symbolic and in exact numerical modes. These subroutines may be used to perform operations including addition, multiplication, differentiation and integration of polynomial functions. An illustrative example performing various mathematical operations on polynomials has been demonstrated using these subroutines and the results obtained have been found to be highly accurate. The subroutines should be useful to analysts who require exact (or high precision) solutions to problems having a large number of mathematical operations of polynomial functions, such as in analyses of beam-column and plate structures.