R-function Theory for Bending Problem of Shallow Spherical Shells with Polygonal Boundary

The governing differential equations of the bending problem of simply supported shallow spherical shells on Winkler foundation are simplified to an independent equation of radial deflection. The independent equation of radial deflection is decomposed to two Laplace operators by intermediate variable. The R-function theory is applied to describe a shallow spherical shell on Winkler foundation with concave boundary, and then a quasi-Green’s function is established by using the fundamental solution and the normalized boundary equation. The quasi-Green’s function satisfies the homogeneous boundary condition of the problem. The Laplace operators of the problem are reduced to two simultaneous Fredholm integral equations of the second kind by the Green’s formula. The singularity of the kernel of the integral equation is eliminated by choosing a suitable form of the normalized boundary equation. The integral equations are discretized into the homogeneous linear algebraic equations to proceed numerical computing. The singular term in the discrete equation is eliminated by the integral method. Some numerical examples are given to verify the validity of the proposed method in calculating simple boundary conditions and polygonal boundary conditions. A comparison with the ANSYS finite element (FEM) solution shows a good agreement, and it demonstrates the feasibility and efficiency of the present method.

Toeplitz operators on Bergman spaces of polygonal domains

We study the boundedness of Toeplitz operators with locally integrable symbols on Bergman spaces Ap(Ω), 1 < p < ∞, where Ω ⊂ ℂ is a bounded simply connected domain with polygonal boundary. We give sufficient conditions for the boundedness of generalized Toeplitz operators in terms of ‘averages’ of symbol over certain Cartesian squares. We use the Whitney decomposition of Ω in the proof. We also give examples of bounded Toeplitz operators on Ap(Ω) in the case where polygon Ω has such a large corner that the Bergman projection is unbounded.

APPLICATION OF COLLOCATION BEM FOR AXISYMMETRIC TRANSMISSION PROBLEMS IN ELECTRO- AND MAGNETOSTATICS

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.