### 2003-17

by Qatanani, N., Schulz, M..

**Series:** 2003-17, Preprints

- MSC:
- 45B05 Fredholm integral equations
- 65R20 Integral equations
- 65F10 Iterative methods for linear systems
- 65N38 Boundary element methods

**Abstract:**

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 also 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 implementes. 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.

**Keywords:**

heat radiation, Fredholm integral equation, boundary element method, iterative methods, Galerkin scheme.