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Comparison between wavelet and fast multipole data sparse approximations for Poisson and kinematics boundary - domain integral equations
Jure Ravnik, Leopold Škerget, Zoran Žunič, 2009, original scientific article

Abstract: The boundary element method applied on non-homogenous partial differential equations requires calculation of a fully populated matrix of domain integrals. This paper compares two techniques: the fast multipole method and the fast wavelet transform, which are used to reduce the complexity of such domain matrices. The employed fast multipole method utilizes the expansion of integral kernels into series of spherical harmonics. The wavelet transform for vectors of arbitrary length, based on Haar wavelets and variable thresholding limit, is used. Both methods are tested and compared by solving the scalar Poisson equation and the velocity-vorticity vector kinematics equation. The results show comparable accuracy for both methods for a given data storage size. Wavelets are somewhat better for high and low compression ratios, and the fast multipole methods gives better results for moderate compressions. Considering implementation of the methods, the wavelet transform can easily be adapted for any problem, while the fast multipole method requires different expansion for each integral kernel.
Keywords: wavelets, fast multipole method, Poisson equation, BEM
Published: 31.05.2012; Views: 1505; Downloads: 73
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Fast single domain-subdomain BEM algorithm for 3D incompressible fluid flow and heat transfer
Jure Ravnik, Leopold Škerget, Zoran Žunič, 2009, original scientific article

Abstract: In this paper acceleration and computer memory reduction of an algorithm for the simulation of laminar viscous flows and heat transfer is presented. The algorithm solves the velocity-vorticity formulation of the incompressible Navier-Stokes equations in 3D. It is based on a combination of a subdomain boundary element method (BEM) and single domain BEM. The CPU time and storage requirements of the single domain BEM are reduced by implementing a fast multipole expansion method. The Laplace fundamental solution, which is used as a special weighting function in BEM, is expanded in terms of spherical harmonics. The computational domain and its boundary are recursively cut up forming a tree of clusters of boundary elements and domain cells. Data sparse representation is used in parts of the matrix, which correspond to boundary-domain clusters pairs that are admissible for expansion. Significant reduction of the complexity is achieved. The paper presents results of testing of the multipole expansion algorithm by exploring its effect on the accuracy of the solution and its influence on the non-linear convergence properties of the solver. Two 3D benchmark numerical examples are used: the lid-driven cavity and the onset of natural convection in a differentially heated enclosure.
Keywords: boundary element method, fast multipole method, fluid flow, heat transfer, velocity-vorticity fomulation
Published: 31.05.2012; Views: 1416; Downloads: 66
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