Integral Equation Method via Domain Decomposition and Collocation for Scattering Problems

1995 ◽  
Vol 62 (1) ◽  
pp. 186-192 ◽  
Author(s):  
Xiaogang Zeng ◽  
Fang Zhao
Keyword(s):  
Domain Decomposition ◽  
Large Scale ◽  
Three Dimensional ◽  
Integral Equation Method ◽  
Computational Effort ◽  
Boundary Integral ◽  
Infinite Domain ◽  
Scattering Problems ◽  
Exterior Region ◽  
Plane Incident Wave

In this paper an exterior domain decomposition (DD) method based on the boundary element (BE) formulation for the solutions of two or three-dimensional time-harmonic scattering problems in acoustic media is described. It is known that the requirement of large memory and intensive computation has been one of the major obstacles for solving large scale high-frequency acoustic systems using the traditional nonlocal BE formulations due to the fully populated resultant matrix generated from the BE discretization. The essence of this study is to decouple, through DD of the problem-defined exterior region, the original problem into arbitrary subproblems with data sharing only at the interfaces. By decomposing the exterior infinite domain into appropriate number of infinite subdomains, this method not only ensures the validity of the formulation for all frequencies but also leads to a diagonalized, blockwise-banded system of discretized equations, for which the solution requires only O(N) multiplications, where N is the number of unknowns on the scatterer surface. The size of an individual submatrix that is associated with a subdomain may be determined by the user, and may be selected such that the restriction due to the memory limitation of a given computer may be accommodated. In addition, the method may suit for parallel processing since the data associated with each subdomain (impedance matrices, load vectors, etc.) may be generated in parallel, and the communication needed will be only for the interface values. Most significantly, unlike the existing boundary integral-based formulations valid for all frequencies, our method avoids the use of both the hypersingular operators and the double integrals, therefore reducing the computational effort. Numerical experiments have been conducted for rigid cylindrical scatterers subjected to a plane incident wave. The results have demonstrated the accuracy of the method for wave numbers ranging from 0 to 30, both directly on the scatterer and in the far-field, and have confirmed that the procedure is valid for critical frequencies.

The treatment of lubrication forces in boundary integral equations

2006 ◽  
Vol 462 (2067) ◽  
pp. 855-881 ◽  
Author(s):  
Andrea Alberto Mammoli
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equation ◽  
Large Scale ◽  
Boundary Integral Equations ◽  
Hydrodynamic Lubrication ◽  
Integral Equation Method ◽  
Computational Effort ◽  
Boundary Integral ◽  
Spherical Particles ◽  
Fluid Behaviour

Despite their many advantages over other numerical methods, boundary integral formulations still fail to provide accurate predictions of mesoscale motion in dense suspensions of rigid particles because the nearly singular flow between surfaces in close proximity cannot be resolved accurately. A procedure for incorporating analytical solutions for the lubrication flow within a large-scale boundary integral equation method is shown. Although the method is applied to the case of spherical particles, in conjunction with the completed double layer boundary integral equation, it can be developed further to treat more complex geometries and can be adapted to other numerical techniques. In contrast to other apparently similar approaches, the present method does not resort to effective medium approximations, and in principle retains all the advantages typical of boundary integral approaches. The framework also allows for forces other than those due to hydrodynamic lubrication between particles, provided that they are a linear function of the relative velocity or at least can be linearized; for example, forces due to sub-continuum fluid behaviour or forces resulting from surface chemistry. It is shown using several benchmarks that the relative motion between two particles in various flows is captured accurately, both statically and dynamically, in situations where uncorrected simulations fail. Moreover, the computational effort is reduced substantially by the application of the method.

scholarly journals The Numerical Analysis of the Flow on the Smooth and Nonsmooth Boundaries by IBEM/DBIEM

2019 ◽  
Vol 2019 ◽  
pp. 1-14
Author(s):  
Gang Xu ◽  
Guangwei Zhao ◽  
Jing Chen ◽  
Shuqi Wang ◽  
Weichao Shi
Keyword(s):  
Boundary Value ◽  
Three Dimensional ◽  
Boundary Surface ◽  
Tangential Velocity ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Surface Shape ◽  
Normal Vector ◽  
Computational Domain ◽  
Nonsmooth Boundary

The value of the tangential velocity on the Boundary Value Problem (BVP) is inaccurate when comparing the results with analytical solutions by Indirect Boundary Element Method (IBEM), especially at the intersection region where the normal vector is changing rapidly (named nonsmooth boundary). In this study, the singularity of the BVP, which is directly arranged in the center of the surface of the fluid computing domain, is moved outside the computational domain by using the Desingularized Boundary Integral Equation Method (DBIEM). In order to analyze the accuracy of the IBEM/DBIEM and validate the above-mentioned problem, three-dimensional uniform flow over a sphere has been presented. The convergent study of the presented model has been investigated, including desingularized distance in the DBIEM. Then, the numerical results were compared with the analytical solution. It was found that the accuracy of velocity distribution in the flow field has been greatly improved at the intersection region, which has suddenly changed the boundary surface shape of the fluid domain. The conclusions can guide the study on the flow over nonsmooth boundaries by using boundary value method.

Adaptivity and Three Dimensional Hierarchical Boundary Elements for Rigid Acoustic Scattering Problems

1995 ◽  
Author(s):  
Steven J. Newhouse ◽  
Ian C. Mathews
Keyword(s):  
Boundary Element ◽  
Acoustic Analysis ◽  
Three Dimensional ◽  
Acoustic Scattering ◽  
Error Estimator ◽  
Infinite Domain ◽  
Scattering Problems ◽  
Effective Form ◽  
Mesh Density ◽  
Acoustic Scattering Problems

Abstract The boundary element method is an established numerical tool for the analysis of acoustic pressure fields in an infinite domain. There is currently no well established method of estimating the surface pressure error distribution for an arbitrary three dimensional body. Hierarchical shape functions have been used as a highly effective form of p refinement in many finite and boundary element applications. Their ability to be used as an error estimator in acoustic analysis has never been fully exploited. This paper studies the influence of mesh density and interpolation order on several acoustic scattering problems. A hierarchical error estimator is implemented and its effectiveness verified against the spherical problem. A coarse cylindrical mesh is then refined using the new error estimator until the solution has converged. The effectiveness of this analysis is shown by comparing the error indicators derived during the analysis to the solution generated from a very fine cylindrical mesh.

Rigorous accounting diffraction on non-plane gratings irradiated by non-planar waves

2021 ◽  
Author(s):  
Leonid I. Goray
Keyword(s):  
Plane Wave ◽  
Wave Diffraction ◽  
Plane Waves ◽  
Three Dimensional ◽  
Integral Equation Method ◽  
Transverse Magnetic ◽  
Incident Field ◽  
Boundary Integral ◽  
Cylindrical Waves ◽  
Plane Wave Diffraction

Abstract The modified boundary integral equation method (MIM) is considered a rigorous theoretical application for the diffraction of cylindrical waves by arbitrary profiled plane gratings, as well as for the diffraction of plane/non-planar waves by concave/convex gratings. This study investigates two-dimensional (2D) diffraction problems of the filiform source electromagnetic field scattered by a plane lamellar grating and of plane waves scattered by a similar cylindrical-shaped grating. Unlike the problem of plane wave diffraction by a plane grating, the field of a localised source does not satisfy the quasi-periodicity requirement. Fourier transform is used to reduce the solution of the problem of localised source diffraction by the grating in the whole region to the solution of the problem of diffraction inside one Floquet channel. By considering the periodicity of the geometry structure, the problem of Floquet terms for the image can be formulated so that it enables the application of the MIM developed for plane wave diffraction problems. Accounting of the local structure of an incident field enables both the prediction of the corresponding efficiencies and the specification of the bounds within which the approximation of the incident field with plane waves is correct. For 2D diffraction problems of the high-conductive plane grating irradiated by cylindrical waves and the cylindrical high-conductive grating irradiated by plane waves, decompositions in sets of plane waves/sections are investigated. The application of such decomposition, including the dependence on the number of plane waves/sections and radii of the grating and wave front shape, was demonstrated for lamellar, sinusoidal and saw-tooth grating examples in the 0th & –1st orders as well as in the transverse electric and transverse magnetic polarisations. The primary effects of plane wave/section partitions of non-planar wave fronts and curved grating shapes on the exact solutions for 2D and three-dimensional (conical) diffraction problems are discussed.

scholarly journals A Domain Decomposition Method for Hybrid Shell Vector Element with Boundary Integral Method

2012 ◽  
Vol 2012 ◽  
pp. 1-6 ◽  
Author(s):  
Lin Lei ◽  
Jun Hu ◽  
Hao-Quan Hu
Keyword(s):  
Domain Decomposition ◽  
Domain Decomposition Method ◽  
Three Dimensional ◽  
Shell Element ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Thin Coating ◽  
Surface Integral ◽  
Vector Element ◽  
Element Method

For the conducting body coated with thin-layer material, plenty of fine meshes are required in general. In this paper, shell vector element (SVE) is used for modeling of thin coating dielectric. Further, a domain decomposition (DD) method for hybrid shell vector element method boundary integral (SVE-BI) is proposed for analysis of electromagnetic problem of multiple three-dimensional thin-coating objects. By this method, the whole computational domains are divided into sub-SVE domains and boundary element domains. With shell element, not only the unknowns are far less than the one by traditional vector element method, but only surface integral is required. The DDM framework used for hybrid SVE-BI also enhances the computational efficiency of solving scattering from multiple coating objects greatly. Finally, several numerical examples are presented to prove the accuracy and efficiency of this DDM-SVE-BI method.

Large-scale 3D inversion of marine magnetotelluric data: Case study from the Gemini prospect, Gulf of Mexico

Geophysics ◽  
2011 ◽  
Vol 76 (1) ◽  
pp. F77-F87 ◽  
Author(s):  
Michael S. Zhdanov ◽  
Le Wan ◽  
Alexander Gribenko ◽  
Martin Čuma ◽  
Kerry Key ◽  
...  
Keyword(s):  
Gulf Of Mexico ◽  
Large Scale ◽  
Three Dimensional ◽  
Integral Equation Method ◽  
Computer Code ◽  
Hydrocarbon Exploration ◽  
Parallel Computer ◽  
3D Inversion ◽  
Magnetotelluric Data ◽  
Geoelectrical Structures

Three-dimensional magnetotelluric (MT) inversion is an emerging technique for offshore hydrocarbon exploration. We have developed a new approach to the 3D inversion of MT data, based on the integral equation method. The Tikhonov regularization and physical constraint have been used to obtain a stable and reasonable solution of the inverse problem. The method is implemented in a fully parallel computer code. We have applied the developed method and software for the inversion of marine MT data collected by the Scripps Institution of Oceanography (SIO) in the Gemini prospect, Gulf of Mexico. The inversion domain was discretized into 1.6 million cells. It took nine hours to complete 51 iterations on the 832-processor cluster with a final misfit between the observed and predicted data of 6.2%. The inversion results reveal a resistive salt structure, which is confirmed by a comparison with the seismic data. These inversion results demonstrate that resistive geoelectrical structures like salt domes can be mapped with reasonable accuracy using the 3D inversion of marine MT data.

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