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Register a new userGeneralized Taylor-Duffy Method for Efficient Evaluation of Galerkin Integrals in Boundary-Element Method Computations
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We present a generic technique, automated by computer-algebra systems and available as open-source software cite{scuff-em}, for efficient numerical evaluation of a large family of singular and nonsingular 4-dimensional integrals over triangle-product domains, such as those arising in the boundary-element method (BEM) of computational electromagnetism. To date, practical implementation of BEM solvers has often required the aggregation of multiple disparate integral-evaluation schemes to treat all of the distinct types of integrals needed for a given BEM formulation; in contrast, our technique allows many different types of integrals to be handled by the emph{same} algorithm and the same code implementation. Our method is a significant generalization of the Taylor--Duffy approach cite{Taylor2003,Duffy1982}, which was originally presented for just a single type of integrand; in addition to generalizing this technique to a broad class of integrands, we also achieve a significant improvement in its efficiency by showing how the emph{dimension} of the final numerical integral may often be reduced by one. In particular, if $n$ is the number of common vertices between the two triangles, in many cases we can reduce the dimension of the integral from $4-n$ to $3-n$, obtaining a closed-form analytical result for $n=3$ (the common-triangle case).
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I present an accurate and efficient technique for numerical evaluation of singular 6-dimensional integrals over tetrahedon-product domains, with applications to calculation of Galerkin matrix elements for discretized volume-integral-equation (VIE) solvers using Schaubert-Wilton-Glisson (SWG) and other tetrahedral basis functions. My method extends the generalized Taylor-Duffy strategy---used to handle the singular textit{triangle}-product integrals arising in discretized surface-integral-equation (SIE) formulations---to the tetrahedron-product case; it effects an exact transformation of a singular 6-dimensional integral to an nonsingular lower-dimensional integral that may be evaluated by simple numerical cubature The method is highly general and may---with the aid of automatic code generation facilitated by computer-algebra systems---be applied to a wide variety of singular integrals arising in various VIE formulations with various types of tetrahedral basis function, of which I present several examples. To demonstrate the accuracy and efficiency of my method, I apply it to the calculation of matrix elements for the volume electric-field integral equation (VEFIE) discretized with SWG basis functions, where the method yields 12-digit or higher accuracy with low computational cost---an improvement of many orders of magnitude compared to existing techniques.
We developed a fast numerical algorithm for solving the three dimensional vectorial Helmholtz equation that arises in electromagnetic scattering problems. The algorithm is based on electric field integral equations and is essentially a boundary element method. Nystroms quadrature rule with a triangular grid is employed to linearize the integral equations, which are then solved by using a right-preconditioned iterative method. We apply the fast multipole technique to accelerate the matrix-vector multiplications in the iterations. We demonstrate the broad applications and accuracy of this method with practical examples including dielectric, plasmonic and metallic objects. We then apply the method to investigate the plasmonic properties of a silver torus and a silver split-ring resonator under the incidence of an electromagnetic plane wave. We show the silver torus can be used as a trapping tool to bind small dielectric or metallic particles.
The displacement field for three dimensional dynamic elasticity problems in the frequency domain can be decomposed into a sum of a longitudinal and a transversal part known as a Helmholtz decomposition. The Cartesian components of both the longitudinal and transverse fields satisfy scalar Helmholtz equations that can be solved using a desingularized boundary element method (BEM) framework. The curl free longitudinal and divergence free transversal conditions can also be cast as additional scalar Helmholtz equations. When compared to other BEM implementations, the current framework leads to smaller matrix dimensions and a simpler conceptual approach. The numerical implementation of this approach is benchmarked against the 3D elastic wave field generated by a rigid vibrating sphere embedded in an infinite linear elastic medium for which the analytical solution has been derived. Examples of focussed 3D elastic waves generated by a vibrating bowl-shaped rigid object with convex and concave surfaces are also considered. In the static zero frequency limit, the Helmholtz decomposition becomes non-unique, and both the longitudinal and transverse components contain divergent terms that are proportional to the inverse square of the frequency. However, these divergences are equal and opposite so that their sum, that is the displacement field that reflects the physics of the problem, remains finite in the zero frequency limit.
A monolithic coupling between the material point method (MPM) and the finite element method (FEM) is presented. The MPM formulation described is implicit, and the exchange of information between particles and background grid is minimized. The reduced information transfer from the particles to the grid improves the stability of the method. Once the residual is assembled, the system matrix is obtained by means of automatic differentiation. In such a way, no explicit computation is required and the implementation is considerably simplified. When MPM is coupled with FEM, the MPM background grid is attached to the FEM body and the coupling is monolithic. With this strategy, no MPM particle can penetrate a FEM element, and the need for computationally expensive contact search algorithms used by existing coupling procedures is eliminated. The coupled system can be assembled with a single assembly procedure carried out element by element in a FEM fashion. Numerical results are reported to display the performances and advantages of the methods here discussed.
We propose a weak Galerkin(WG) finite element method for solving the one-dimensional Burgers equation. Based on a new weak variational form, both semi-discrete and fully-discrete WG finite element schemes are established and analyzed. We prove the existence of the discrete solution and derive the optimal order error estimates in the discrete $H^1$-norm and $L^2$-norm, respectively. Numerical experiments are presented to illustrate our theoretical analysis.
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