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APPLICATION OF THE ORDINARY LEAST‐SQUARES APPROACH FOR SOLUTION OF COMPLEX VARIABLE BOUNDARY ELEMENT PROBLEMS
Author(s) -
YU GUOQING,
RASMUSSEN TODD C.
Publication year - 1997
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/(sici)1097-0207(19970415)40:7<1281::aid-nme114>3.0.co;2-4
Subject(s) - overdetermined system , mathematics , discretization , mathematical analysis , boundary value problem , boundary element method , identity matrix , matrix (chemical analysis) , ordinary differential equation , boundary (topology) , finite element method , differential equation , eigenvalues and eigenvectors , physics , materials science , quantum mechanics , composite material , thermodynamics
Cauchy's theorem is used to generate a Complex Variable Boundary Element Method (CVBEM) formulation for steady, two‐dimensional potential problems. CVBEM uses the complex potential, w =ϕ+iψ, to combine the potential function, ϕ, with the stream function, ψ. The CVBEM formulation, using Cauchy's theorem, is shown to be mathematically equivalent to Real Variable BEM which employs Green's second identity and the respective fundamental solution. CVBEM yields an overdetermined system of equations that are commonly solved using implicit and explicit methods that reduce the overdetermined matrix to a square matrix by selectively excluding equations. Alternatively, Ordinary Least Squares (OLS) can be used to minimize the Euclidean norm square of the residual vector that arises due to the approximation of boundary potentials and geometries. OLS uses all equations to form a square matrix that is symmetric, positive definite and diagonally dominant. OLS is more accurate than existing methods and can estimate the approximation error at boundary nodes. The approximation error can be used to determine the adequacy of boundary discretization schemes. CVBEM/OLS provides greater flexibility for boundary conditions by allowing simultaneous specification of both fluid potentials and stream functions, or their derivatives, along boundary elements. © 1997 by John Wiley & Sons, Ltd.