Abstract
The complex variable boundary element method (CVBEM) provides solutions of partial differential equations of the Laplace and Poisson type. Because the CVBEM is based upon convex combinations from a basis set of functions that are analytic throughout the problem domain, boundary, and exterior of the problem domain union boundary (except along branch cuts), both the real and imaginary parts of the CVBEM approximations satisfy the Laplace equation, leaving the modeling error reduction effort to be that of fitting the problem boundary conditions. In this paper, the approximate boundary approach is used to depict the goodness of fit between the CVBEM results and the problem boundary conditions. The approximate boundary is the locus of points where the CVBEM approximation function meets the values of the problem boundary conditions. Because of the collocation method, the approximate boundary necessarily intersects the problem boundary at least at the collocation points specified on the problem boundary. Consequently, adding nodes and collocation points on the problem boundary results in reducing the departure between the approximate boundary and the true problem boundary. Thus, the approximate boundary is developed by tracking level curves from the real and/or imaginary parts of the CVBEM approximation function. Published by Elsevier Ltd.
Highlights
The complex variable boundary element method (CVBEM) provides solutions of partial differential equations of the Laplace and Poisson type
Domain methods like finite element method (FEM) and finite difference methods (FDM) do not have an equivalent modeling error display such as is provided by the CVBEM approximation function as applied to the approximation boundary approach used in this paper
The approximate boundary is a geometric construct of an alternate boundary to the true problem boundary, where the CVBEM approximation function achieves the actual problem boundary condition values
Summary
Domain methods like FEM and FDM do not have an equivalent modeling error display such as is provided by the CVBEM approximation function as applied to the approximation boundary approach used in this paper. Mathematica will be used for its internal graphical interface capabilities by way of MATlink, a Mathematica application module for seamless two-way communication and data transfer with MATLAB This allows CVBEM matrix solutions to be developed by MATLAB whereas the plots of potential function and stream function isocontours are obtained using Mathematica. Isocontours may be developed from the CVBEM approximation using Mathematica, which correspond to level curves of the boundary condition values between collocation points for use in tracking the approximate boundary for the target problem. One develops the approximate boundary (by adding or adjusting locations of nodes and/or collocation points) until the maximum departure from the true boundary is less than the construction tolerance for the prototype
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