Fuzzy aspects of the boundary element method

<abstract><p>This paper is concerned with obtaining approximate solutions of fuzzy Fredholm integral equations using Picard iteration method and bivariate Bernstein polynomials. We first present the way to approximate the value of the multiple integral of any fuzzy-valued function based on the two dimensional Bernstein polynomials. Then, it is used to construct the numerical iterative method for finding the approximate solutions of two dimensional fuzzy integral equations. Also, the error analysis and numerical stability of the method are established for such fuzzy integral equations considered here in terms of supplementary Lipschitz condition. Finally, some numerical examples are considered to demonstrate the accuracy and the convergence of the method.</p></abstract>

In this paper we present a new, direct computational method for calculating the complex thermal transfer impedances between two separate locations of a given physical structure aimed at the implementation into a field-solver based on the SUNRED (SUccessive Node REDuction) algorithm. We tested the method with a simple 2D example containing 125 internal nodes. For testing the proposed new calculation method multiple combinations of Dirichlet and Neumann type boundary conditions were applied. Also, different types of thermal loads such as prescribed unit-step change in dissipation or temperature were assumed (for time domain transient analysis). The test case was also studied with the assumption of sinusoidal dissipation. Results obtained by the proposed new calculation method and results obtained by conventional simulations differ less than the uncertainty of the traditional solution method. The good agreement enables us to use the balanced truncation method to reduce the order of the transfer functions with low computational cost.

The first section of this chapter collects results from [240] which gives a further contribution to the analysis of the hp-version of the boundary element method (BEM) by presenting a more general result for Dirichlet and Neumann problems than [21] allowing the use of a general geometric mesh refinement on the polygonal boundary Γ. Here as in [240] we prove the exponential convergence of the hp- version of the boundary element method by exploiting only features of the solutions of the boundary integral equations. The key result in this approach is an asymptotic expansion of the solution of the integral equations in singularity functions reflecting the singular behaviour of the solutions near corners of Γ. With such expansions we show that the solutions of the integral equations belong to countably normed spaces. Therefore these solutions can be approximated exponentially fast in the energy norm via the hp- Galerkin solutions of those integral equations. This result is not restricted to integral equations which stem from boundary value problems for the Laplacian but applies to Helmholtz problems as well. Further applications are 2D crack problems in linear elasticity. For numerical experiments with hp-version (BEM) see [165, 340].

In this paper, first, we use the Rich type iterative method for numerical solution of nonlinear fuzzy Fredholm integral equation. The main structure of this method is based on the Richardson iterative method and trapezoidal quadrature formula for Numerical solution of nonlinear fuzzy integral equations. Convergence, error analysis and numerical stability of the proposed method are also discussed.

The Numerical Solution of Integral Equations on the Half-Line

The limiting process that leads to the formulation of hypersingular boundary integral equations is first discussed in detail. It is shown that boundary integral equations with hypersingular kernels are perfectly meaningful even at non-smooth boundary points, and that special interpretations of the integrals involved are not necessary. Careful analysis of the limiting process has also strong relevance for the development of an appropriate numerical algorithm. In the second part, a new general method for the evaluation of hypersingular surface integrals in the boundary element method (BEM) is presented. The proposed method can be systematically applied in any BEM analysis, either with open or closed surfaces, and with curved boundary elements of any kind and order (of course, provided the density function meets necessary regularity requirements at each collocation point). The algorithm operates in the parameter plane of intrinsic coordinates and allows any hypersingular integral in the BEM to be directly transformed into a sum of a double and a one-dimensional regular integrals. Since all singular integrations are performed analytically, standard quadrature formulae can be used. For the first time, numerical results are presented for hypersingular integrals on curved (distorted) elements for three-dimensional problems.

Conjugate gradient methods for the solution of boundary integral equations on a piecewise smooth boundary

We study the numerical solution of the boundary integral equation formulation for the scattering of time-harmonic electromagnetic waves from infinite cylinders. For smooth boundaries of the cylinder cross section we describe a Nystro¨m, a collocation and a Galerkin method based on an approximation by trigonometric polynomials on an equidistant mesh. For smooth data in each of the three methods the convergence is exponential. From the three approaches the Nystro¨m method is the most efficient since it requires the least computational effort. For cross sections with corners we develop a Nystro¨m method on a graded mesh based on the idea of transforming the nonsmooth case to a smooth periodic case via an appropriate substitution. We conclude the paper with some considerations on the corresponding three dimensional problem.

The paper discusses small oscillations of a panel in an incompressible medium. Air can be considered an incompressible medium during modal tests of solar array panels for spacecraft deployed on the ground in a lab environment. A panel is represented as a two-sided boundary surface. Conditions are determined for applicability of the potential motion of the medium. Calculation of the attached mass is reduced to the solution of the Neumann boundary value problem. To solve the boundary value problem, the method of boundary elements is used in the piecewise constant approximation variant, which provides a solution of the hypersingular boundary integral equation. Numerical solutions are obtained for the three fundamental modes of rectangular panels. The obtained numerical values are refined using non-linear Shanks transformation. Dependence of attached mass on panel elongation and the amount of the gap between its fragments is studied. For any in-plane oscillation mode of a panel fragment, the attached mass is determined using the principle of linear superposition. An estimate is given of the effect of the distance from the panel to the wall on the attached mass value. Key words: oscillations, incompressible medium, air, attached mass, rectangular panels, boundary elements method.

<div>This paper mainly focuses on the recent advances in the some approximated methods for solving fuzzy Volterra-Fredholm integral equations, namely, Adomian decomposition method, variational iteration method and homotopy analysis method. We converted fuzzy Volterra-Fredholm integral equation to a system of Volterra-Fredholm integral equation in crisp case. The approximated methods using to find the approximate solutions of this system and hence obtain an approximation for the fuzzy solution of the fuzzy Volterra-Fredholm integral equation. To assess the accuracy of each method, algorithms with Mathematica 6 according is used. Also, some numerical examples are included to demonstrate the validity and applicability</div><div>of the proposed techniques.</div>This paper mainly focuses on the recent advances in the some approximated methods for solvingfuzzy Volterra-Fredholm integral equations, namely, Adomian decomposition method, variational iterationmethod and homotopy analysis method. We converted fuzzy Volterra-Fredholm integral equation to asystem of Volterra-Fredholm integral equation in crisp case. The approximated methods using to find theapproximate solutions of this system and hence obtain an approximation for the fuzzy solution of the fuzzyVolterra-Fredholm integral equation. To assess the accuracy of each method, algorithms with Mathematica 6according is used. Also, some numerical examples are included to demonstrate the validity and applicabilityof the proposed techniques.

We propose new symbolic-numerical algorithms implemented in Maple-Fortran environment for solving the parametric self-adjoint elliptic boundary-value problem (BVP) in a 2D finite domain, using high-accuracy finite element method (FEM) with triangular elements and high-order fully symmetric Gaussian quadratures with positive weights, and no points are outside the triangle (PI type). The algorithms and the programs calculate with the given accuracy the eigenvalues, the surface eigenfunctions and their first derivatives with respect to the parameter of the BVP for parametric self-adjoint elliptic differential equation with the Dirichlet and/or Neumann type boundary conditions on the 2D finite domain, and the potential matrix elements, expressed as integrals of the products of surface eigenfunctions and/or their first derivatives with respect to the parameter. We demonstrated an efficiency of algorithms and program by benchmark calculations of helium atom ground state.

Determination of one unknown coefficient in a two-phase free boundary problem in an angular domain with variable thermal conductivity and specific heat

Solute dispersion in a porous formation is Mathematically expressed by partial differential equation well known as advection-dispersion equation (ADE). The present study deals with the solute transport governing equation in a semi-infinite homogeneous porous formation under linear sorption. A constant background solute concentration is assumed initially throughout the solute transport domain. Dirichlet and Neumann type boundary conditions are considered to examine the solute concentration distribution profile in the semi-infinite porous medium. The analytical and numerical solutions of the model problem are derived by Laplace transform technique and Crank-Nicolson method, respectively. Solute dispersion behaviour is studied for various form of flow velocities. Solutions obtained by analytical and numerical techniques are illustrated graphically with the help of MATLAB software. Also, the numerical solution is compared with the analytical solution and found great similarity between them.

In this paper we present the development of numerical methods for two problems in chemical engineering: adsorption of carbon dioxide into phenyl glycidyl ether and diffusion and reaction processes in a porous catalyst. Both problems are modelled by systems of two coupled nonlinear ordinary differential equations. Dirichlet and Neumann type boundary conditions are considered. We develop the standard homotopy analysis method and the boundary-domain element method to solve for the unknown steady-state concentrations of reactants. The validity of the proposed methods is checked and demonstrated by numerical examples. Convergence rates are compared and the advantages and drawbacks of the proposed methods are discussed and compared to the Mathematica NDSOLVE solver. We show, that the proposed homotopy analysis method features exponential convergence rate and is highly accurate and efficient. As low as 6 terms are needed to reach error norms of 10 − 5 . Furthermore, the boundary-domain integral approach is also capable of solving the same problem very efficiently, using a very coarse computational mesh (21 nodes).

Simulation and evaluation of interval-valued fuzzy linear Fredholm integral equations with interval-valued fuzzy neural network