Numerical quadrature for singular and near-singular integrals of boundary element method and its applications in large-scale acoustic problems

The numerical quadrature methods for dealing with the problems of singular and near-singular integrals caused by Burton-Miller method are proposed, by which the conventional and fast multipole BEMs （boundary element methods） for 3D acoustic problems based on constant elements are improved. To solve the problem of singular integrals, a Hadamard finite-part integral method is presented, which is a simplified combination of the methods proposed by Kirkup and Wolf. The problem of near-singular integrals is overcome by the simple method of polar transformation and the more complex method of PART （Projection and Angular ＆ Radial Transformation）. The effectiveness of these methods for solving the singular and near-singular problems is validated through comparing with the results computed by the analytical method and/or the commercial software LMS Virtual.Lab. In addition, the influence of the near-singular integral problem on the computational precisions is analyzed by computing the errors relative to the exact solution. The computational complexities of the conventional and fast multipole BEM are analyzed and compared through numerical computations. A large-scale acoustic scattering problem, whose degree of freedoms is about 340,000, is implemented successfully. The results show that, the near singularity is primarily introduced by the hyper-singular kernel, and has great influences on the precision of the solution. The precision of fast multipole BEM is the same as conventional BEM, but the computational complexities are much lower.

GALERKIN MESHLESS METHODS BASED ON PARTITION OF UNITY QUADRATURE

Numerical quadrature is an important ingredient of Galerkin meshless methods. A new numerical quadrature technique, partition of unity quadrature （PUQ）,for Galerkin meshless methods was presented. The technique is based on finite covering and partition of unity. There is no need to decompose the physical domain into small cell. It possesses remarkable integration accuracy. Using Element-free Galerkin methods as example, Galerkin meshless methods based on PUQ were studied in detail. Meshing is always not required in the procedure of constitution of approximate function or numerical quadrature, so Galerkin meshless methods based on PUQ are “truly” meshless methods.

A Kind of Algorithm on B-Spline Wavelet-BEM

B-Spline wavelet-BEM numerical algorithm is presented. To avoid to treating singular integrals in wavelet-BEM, a method of putting source points out of the domain is used and discussed. Meanwhile, two higher effective numerical quadrature formulae are suggested. Finally, an example in mechanics is given and numerical results show that this method is effective. In addition, this method can be extended to manipulate problems, especially, with singularity.

Efficient and accurate numerical-projection of electromagnetic multipoles for scattering objects

In this paper,we develop an efcient and accurate procedure of electromagnetic multipole decomposition by using the Lebedev and Gaussian quadrature methods to perform the numerical integration.Firstly,we briefy review the principles of multipole decomposition,highlighting two numerical projection methods including surface and volume integration.Secondly,we discuss the Lebedev and Gaussian quadrature methods,provide a detailed recipe to select the quadrature points and the corresponding weighting factor,and illustrate the integration accuracy and numerical efciency(that is,with very few sampling points)using a unit sphere surface and regular tetrahedron.In the demonstrations of an isotropic dielectric nanosphere,a symmetric scatterer,and an anisotropic nanosphere,we perform multipole decomposition and validate our numerical projection procedure.The obtained results from our procedure are all consistent with those from Mie theory,symmetry constraints,and fnite element simulations.

MATHEMATICAL ANALYSIS FOR QUADRILATERAL ROTATED Q_1 ELEMENT Ⅲ: THE EFFECT OF NUMERICAL INTEGRATION

This is the third part of the paper for the rotated Q1 nonconforming element on quadrilateral meshes for general second order elliptic problems. Some optimal numerical formulas are presented and analyzed. The novelty is that it includes a formula with only two sampling points which excludes even a Q1 unisolvent set. It is the optimal numerical integration formula over a quadrilateral mesh with least sampling points up to now.

QUADRATURE METHODS FOR HIGHLY OSCILLATORY SINGULAR INTEGRALS

We address the evaluation of highly oscillatory integrals,with power-law and logarithmic singularities.Such problems arise in numerical methods in engineering.Notably,the evaluation of oscillatory integrals dominates the run-time for wave-enriched boundary integral formulations for wave scattering,and many of these exhibit singularities.We show that the asymptotic behaviour of the integral depends on the integrand and its derivatives at the singular point of the integrand,the stationary points and the endpoints of the integral.A truncated asymptotic expansion achieves an error that decays faster for increasing frequency.Based on the asymptotic analysis,a Filon-type method is constructed to approximate the integral.Unlike an asymptotic expansion,the Filon method achieves high accuracy for both small and large frequency.Complex-valued quadrature involves interpolation at the zeros of polynomials orthogonal to a complex weight function.Numerical results indicate that the complex-valued Gaussian quadrature achieves the highest accuracy when the three methods are compared.However,while it achieves higher accuracy for the same number of function evaluations,it requires signi cant additional cost of computation of orthogonal polynomials and their zeros.

Nonlinear Vibration of A Loosely Supported Curved Pipe Conveying Pulsating Fluid under Principal Parametric Resonance

In this paper,the nonlinear dynamics of a curved pipe is investigated in the case of principal parametric resonance due to pulsating flow and impact with loose supports.The coupled in-plane and out-of-plane governing equations with the consideration of von Karman geometric nonUnearity are presented and discretized via the differential quadrature method（DQM）.The nonlinear dynamic responses are calculated numerically to demonstrate the influence of pulsating frequency.Finally,the impact is taken into consideration.The influence of clearance on frettingwear damage,such as normal work rate,contact ratio and impact force level,is demonstrated.