An-Najah Staff

The numerical treatment of boundary integral equations in the form of boundary element methods has became very popular and powerful tool for engineering computations of boundary value problems, in addition to finite difference and finite element methods. Here, we present some of the most important analytical and numerical aspects of the boundary integral equation. The concept of the principle symbol allows the characterization of the boundary integral equation whose variational formulation on the boundary provides there a Gãrding inequality. Therefore, the Galerkin method can be analyzed similarly to the domain finite element methods providing asymptotic convergence if the number of grid points increases. These asymptotic error analysis will be presented in details. To illustrate the efficiency of the Galerkin boundary element method we consider as an numerical experiment the strongly elliptic boundary integral equation with the logarithmic single layer potential. Consequently, we use the Gaussian elimination method as a direct solver and the conjugate gradient iteration to solve the positive definite linear system.  A comparison is drawn between these methods.

Our Main Concern In This Paper Is The Numerical Simulation Of The Heat Radiation Exchange In A Three-Dimensional Non-Convex Enclosure Geometry With A Diffuse And Grey Surface. This Physical Phenomena Is Governed By A Boundary Integral Equation Of The Second Kind. Due To The Non-Convexity Of The Enclosure The Presence Of The Shadow Zones Must Be Taken Into Account In The Heat Radiation Analysis. For That Purpose We Have Developed A Geometrical Algorithm To Provide An Efficient Detection Of These Shadow Zones That Are Needed To Calculate The Visibility Function. For The Discretization Of The Boundary Integral Equation We Have Used The Boundary Element Method Based On The Galerkin–Bubnov Scheme. The System Of Linear Equations Which Subsequently Arise Has Been Solved By The Conjugate Gradient Method With Preconditioning. To Demonstrate The High Efficiency Of This Method A Numerical Experiment Has Been Constructed For Non-Convex Geometry; The Heat Radiation In An Aperture Has Been Considered.

This article deals with the mathematical and the numerical aspects of the Fredholm integral equation of the second kind arising as a result of the heat energy exchange inside a convex and non-convex enclosure geometries. Some mathematical results concerning the integral operator are presented. The Banach fixed point theorem is used to guarantee the existence and the uniqueness of the solution of the integral equation. For the non-convex geometries the visibility function has to be taken into consideration, then a geometrical algorithm is developed to provide an efficient detection of the shadow zones. For the numerical simulation of the integral equation we use the boundary element method based on the Galerkin discretization scheme. Some iterative methods for the discrete radiosity equation are implemented. Several two- and three-dimensional numerical test cases for convex and non-convex geometries are included. This give concrete hints which iterative scheme might be more useful for such practical applications.

This paper gives very significant and up-to-date analytical and numerical results to the three-dimensional heat radiation problem governed by a boundary integral equation. There are two types of enclosure geometries to be considered: convex and nonconvex geometries. The properties of the integral operator of the radiosity equation have been thoroughly investigated and presented. The application of the Banach fixed point theorem proves the existence and the uniqueness of the solution of the radiosity equation. For a nonconvex enclosure geometries, the visibility function must be taken into account. For the numerical treatment of the radiosity equation, we use the boundary element method based on the Galerkin discretization scheme. As a numerical example, we implement the conjugate gradient algorithm with preconditioning to compute the outgoing flux for a three-dimensional nonconvex geometry. This has turned out to be the most efficient method to solve this type of problems.

We consider the integral equation arising as a result of heat radiation exchange in both convex and nonconvex enclosures of diffuse grey surfaces. For nonconvex geometries, the visibility function must be taken into consideration. Therefore, a geometrical algorithm has been developed to provide an efficient detection of the shadow zones. For the numerical realization of the Fredholm integral equation, a boundary element method based on Galerkin-Bubnov discretization scheme is implemented. Consequently, multigrid iteration methods, which are closely related to two-grid methods, are used to solve the system of linear equations. To demonstrate the high efficiency of these iterations, we construct some numerical experiments for different enclosure geometries.