Triple reciprocity method for unknown function's domain integral in boundary integral equation

Boundary Element Analysis of Cracks in Shear Deformable Plates and Shells. Topics in Engineering, Vol 43

This paper is concerned with a dual reciprocity boundary element method (DRBEM) applied to the transient heat conduction problem of anisotropic materials. The integral equation formulation employs the fundamental solution of Laplace equation for isotropic materials, and hence from the anisotropic part of the governing differential equation a domain integral arises in the boundary integral equation. This domain integral is transformed into boundary integrals by using new radial basis functions. Furthermore, time derivative is approximated by the finite difference method, and the domain integral also appears from this approximation. Such a domain integral at each time step is also transformed into boundary integrals via the same dual reciprocity method. The details of the proposed DRBEM are presented, and a computer code is developed for two-dimensional problems. The usefulness of the present method is demonstrated through comparison of the results obtained by the computer code with the analytical ones.

The radial integration method for evaluation of domain integrals with boundary-only discretization

The approach within the overlapping partial domain method has been considered in this paper. Proposed approach allows solving diffraction problems in which a system of integral equations for unknown partial domain functions cannot be reduced to a single integral equation for one unknown function. The electromagnetic wave diffraction problems on an infinite phased array antenna and on a cascaded junction of rectangular waveguides are considered.

A boundary element method without internal cells for solving viscous flow problems

A precise integration boundary element method for solving transient heat conduction problems

Purpose3D steady heat conduction analysis considering heat source is conducted on the fundamental of the fast multipole method (FMM)-accelerated line integration boundary element method (LIBEM).Design/methodology/approachDue to considering the heat source, domain integral is generated in the traditional heat conduction boundary integral equation (BIE), which will counteract the well-known merit of the BEM, namely, boundary-only discretization. To avoid volume discretization, the enhanced BEM, the LIBEM with dimension reduction property is introduced to transfer the domain integral into line integrals. Besides, owing to the unsatisfactory performance of the LIBEM when it comes to large-scale structures requiring massive computation, the FMM-accelerated LIBEM (FM-LIBEM) is proposed to improve the computation efficiency further.FindingsAssuming N and M are the numbers of nodes and integral lines, respectively, the FM-LIBEM can reduce the time complexity from O(NM) to about O(N+ M), and a full discussion and verification of the advantage are done based on numerical examples under heat conduction.Originality/value(1) The LIBEM is applied to 3D heat conduction analysis with heat source. (2) The domain integrals can be transformed into boundary integrals with straight line integrals by the LIM. (3) A FM-LIBEM is proposed and can reduce the time complexity from O(NM) to O(N+ M). (4) The FM-LIBEM with high computational efficiency is exerted to solve 3D heat conduction analysis with heat source in massive computation successfully.

Volume integration in the hypersingular boundary integral equation

We review here some recent results concerning the Cauchy problem for various incompressible models of Fluid Mechanics. Let us first recall the basic models: in order to avoid technicalities with boundary conditions, we shall consider the simple case of periodic boundary conditions. In other words, all unknown functions will be periodic on R 2 or on R 3 with a period T i in each variable x i (i= 1,2 or i = 1, 2, 3). Next, if u = u (x,t) denote the vectorfield corresponding to the velocity field, the incompressibility condition reads $$div u = 0.$$ The Euler Equation corresponds to the additional equation $$\frac{{\partial u}}{{\partial t}} + (u \cdot \nabla )u + \nabla p = 0$$ Where p is thr pressure field, another unknown (scalar) function that can be thought as a Lagrange multiplier corresponding to the constraint(1).

The Barycentric Lagrange interpolation with non-uniformly spaced Chebyshev interpolation nodes has been modified and re-configured in a new formula to solve Volterra integral equations of the second kind. To avoid the application of the collocation methods necessary to transform the solution into an equivalent linear system, the unknown function is interpolated using the given modified Barycentric Lagrange polynomials and is substituted twice into the integral equation, while the given data function is expanded into Maclaurin polynomial of the same degree as the interpolant unknown function. Thus, the presented technique yields substantially higher accuracy and faster convergence of the solutions. The obtained results of the five illustrated examples show that the numerical solutions converge to the exact solutions, particularly for unknown analytic functions, and strongly converge to the exact ones for all values of including the endpoints of the integration domain regardless of whether the kernel and data functions are algebraic or not.

Buckling analysis of shear deformable shallow shells by the boundary element method

Analytically-integrated radial integration BEM for solving three-dimensional transient heat conduction problems

Adaptive control for output feedback nonlinear systems in the presence of modeling errors

The complexity and nonlinearity of reservoir simulation equations make it possible to apply a great number of linearization techniques. The SPE comparison projects and some other papers compare models for various problem classes. These papers generally compare the results and the speed of runs. The main object of this paper is to examine and compare the models based on the numerical background of the linearization techniques. A general comparison method is proposed that can be applied to analyze these techniques. Recently the proposed comparison method has been applied to examine IMPES and IMPEM techniques. Three primary sets of the two basic types (the "saturation/phase concentration"-type and the "overall density"-type) have been examined. It has been revealed that the direct IMPES and IMPEM methods result in exactly the same pressure distribution, while the saturations and the masses are different. By applying the primary set transformation techniques, the saturation and mass distributions of the direct IMPES and IMPEM techniques can be transformed from one into the other. The fully implicit methods are compared for a 3P black oil system, using 1D radial grid. The relationships, and convergence behavior of the implicit techniques are discussed. An analytical approximation of the implicit part of the Jacobian matrix is suggested for "overall density" type primary sets. An object-oriented C++ program has been used to compare the linearization techniques. Introduction The governing equations of reservoir simulators consist of a set of nonlinear partial differential equations coupled with algebraic, i.e. constraint equations. The number of unknown functions of the models can be quite large, depending on the complexity of the fluid—rock system described. This possibility has yielded a number of equivalent modeling techniques, and these apply different groups of unknown variables as primary sets. The widely used IMPES technique was developed in the early sixties. This technique separates the parabolic and hyperbolic parts of the model equations to obtain an equation that has only one unknown function: the pressure. After the solution of this equation implicitly for the pressure, the saturations can be calculated explicitly from the mass conservation equations. This direct sequential technique was originally developed to simulate black oil models. All the coefficients of a black oil model depend on pressure and saturations; thus, it was a natural choice to select pressure and saturations as primary variables of IMPES black oil models. Since the seventies, there has been a growing demand to use compositional models. In compositional cases the number of model equations (unknown functions) can be numerous, and the nonlinearity of the governing equations is more severe than in black oil models. The coefficients of the model equations, besides the pressure, and the saturations, depend on the phase composition, too. In the seventies a number of IMPES-type techniques were developed to simulate compositional models, but all these models used iterative techniques. It was in the early eighties, when a method was published, which was suitable for separating the pressure equation correctly from the set of governing differential equations. Thus, a direct sequential method could be also derived for compositional cases, too. Due to the explicit calculation of the saturations (concentration), the IMPES type models have time step limits. Implicit models, besides the pressure, also calculate other variables implicitly. These models allow greater time steps, but the drawback is the greater computing time requirement (and numerical dispersion). There have been many linearization technique for handling nonlinearities, and creating implicit schemes. The fully implicit methods are often applied besides or together with the IMPES type methods. A fully implicit technique was proposed for two-phase system at the end of sixties. The first fully implicit compositional model was published at the end of the seventies. A number of fully implicit models have been published since then. Because of the considerable computing requirement of an iteration step, to achieve a fast convergence is especially important in compositional cases. P. 87^

In a general integral geometry problem, there are given the integrals of an unknown function over certain manifolds. The traditional statement of the problem consists in determining the integrand. We consider the case of an underdetermined problem when the unknown functions depend on a greater number of variables than the given integrals. These situations appear in a few applied problems when a rather complicated mathematical model is used and no a priori information is available. For overcoming the lack of appropriate data, we pose the problem of finding part of the information unknown; namely, we search only for the discontinuity surfaces of the integrand. The corresponding uniqueness theorem is proved. The present paper finalizes our studies into the case of integration over one-dimensional manifolds. In the previous articles we considered similar problems in the case of integration over straight lines. In this paper the same result is proved for the integration of unknown functions over unknown curves.