ترغب بنشر مسار تعليمي؟ اضغط هنا

Taylor-Duffy Method for Singular Tetrahedron-Product Integrals: Efficient Evaluation of Galerkin Integrals for VIE Solvers

369   0   0.0 ( 0 )
 نشر من قبل M. T. Homer Reid
 تاريخ النشر 2017
  مجال البحث فيزياء
والبحث باللغة English
 تأليف M. T. Homer Reid




اسأل ChatGPT حول البحث

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.



قيم البحث

اقرأ أيضاً

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).
Evaluation of basic integrals over Gaussian functions, traditionally utilized for electronic structure computations on molecules and solids, is discussed in a pedagogical form.
By using Poissons summation formula, we calculate periodic integrals over Gaussian basis functions by partitioning the lattice summations between the real and reciprocal space, where both sums converge exponentially fast with a large exponent. We dem onstrate that the summation can be performed efficiently to calculate 2-center Gaussian integrals over various kernels including overlap, kinetic, and Coulomb. The summation in real space is performed using an efficient flavor of the McMurchie-Davidson Recurrence Relation (MDRR). The expressions for performing summation in the reciprocal space are also derived and implemented. The algorithm for reciprocal space summation allows us to reuse several terms and leads to significant improvement in efficiency when highly contracted basis functions with large exponents are used. We find that the resulting algorithm is only between a factor of 5 to 15 slower than that for molecular integrals, indicating the very small number of terms needed in both the real and reciprocal space summations. An outline of the algorithm for calculating 3-center Coulomb integrals is also provided.
151 - Shuichi Sato 2010
We consider singular integral operators and maximal singular integral operators with rough kernels on homogeneous groups. We prove certain estimates for the operators that imply $L^p$ boundedness of them by an extrapolation argument under a sharp con dition for the kernels. Also, we prove some weighted $L^p$ inequalities for the operators.
219 - Shuichi Sato 2008
We prove certain $L^p$ estimates ($1<p<infty$) for non-isotropic singular integrals along surfaces of revolution. As an application we obtain $L^p$ boundedness of the singular integrals under a sharp size condition on their kernels.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا