Abstract
This paper considers the numerical solution of boundary integral equations for an exterior transmission problem in a three-dimensional axisymmetric domain. The resulting potential problem is formulated in a meridian plane as the second kind integral equation for a boundary potential and the first kind integral equation for a boundary flux. The numerical method is an axisymmetric collocation with equal order approximations of the boundary unknowns on a polygonal boundary. The complete elliptic integrals of the kernels are approximated by polynomials. An asymptotic kernels behavior is analyzed for accurate numerical evaluation of integrals. A piecewise-constant midpoint collocation and a piecewise-linear nodal collocation on a circular arc and on its polygonal interpolation are used for test computations on uniform meshes. We analyze empirically the influence of the polygonal boundary interpolation to the accuracy and the convergence of the presented method. We have found that the polygonal boundary interpolation does not change the convergence behavior on the smooth boundary for the piecewise-constant and the piecewise-linear collocation.
Highlights
Many problems in electro- and magnetostatics require solution of the Maxwell’s equations in a domain, comprised by media of different properties
This specific formulation is important for problems of control and manipulation of ferrofluid droplets by external magnetic fields, which presents a promising technique for handling samples in biological and chemical systems [15, 18]
The P1-collocation on the polygonal boundary shows a decrease of the convergence order by one if the solid angle is not included into the integral formulation
Summary
Many problems in electro- and magnetostatics require solution of the Maxwell’s equations in a domain, comprised by media of different properties. An axisymmetric collocation boundary element method with the polynomial approximation of the complete elliptic integrals is a well-known technique for solution of the potential problems. An accuracy of this technique, applied to the transmission problem on an approximate polygonal boundary between two media, is in focus of the research. Accurate computations of the boundary unknowns are of high importance for free-surface problems of electroand ferrohydrostatics, where an a-priori unknown moving boundary presents an interface between two media In this case the boundary element method is applied over the boundary, which itself is a numerical solution of the freesurface subproblem at the every iteration step of the solution process, see e.g. in [11]. We compare approximate and exact solutions for the boundary potential and the boundary flux in different norms (Test 1) and compute the accuracy of the boundary field for different values of permeabilities (Test 2)
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