Preconditioned conjugate gradient method for three-dimensional non-convex enclosure geometries with diffuse and grey surfaces
Introduction
All substances continuously emit electromagnetic radiation by virtue of the molecular and atomic agitation associated with the internal energy of the material. In the equilibrium state, this internal energy is in direct proportion to the temperature of the substance. The emitted radiant energy can range from radio waves, which can have wavelength of miles, to cosmic rays with wavelength of >10−14 m. Heat radiation plays a very significant role in our modern technology. It has to be taken into account if the temperature on a visible surface of the system is high enough or if other heat transfer mechanisms are not present (like in a vacuum, for example). Apart from some simple cases like a convex radiation body and known irradiation from infinity, we have to take into account the radiative heat exchange between different parts of the surface of our system. The heat radiative exchange phenomena is governed by a boundary integral equation. Concerning this integral equation, we are aware of some independent works: In [5], [6] a boundary element method was implemented for two-dimensional enclosures to obtain a direct numerical solution for the integral equation, however, this leads to quite high error bounds. In [11], [12] two-dimensional convex and non-convex geometries have been considered and some solution methods for the discrete heat equation have been compared. Some higher-order numerical methods for solving the radiosity equation in the convex case are defined and analysed in [2], [3].
Our main emphasis in this current work is the radiosity equation defined for a three-dimensional non-convex enclosure geometry with diffuse and grey surfaces. In such geometries the presence of the shadow zones must be taken into considerations due to their direct contribution to the calculation of the visibility function. For the numerical simulation of the integral equation we use the boundary element method based on the Galerkin discretization scheme. Consequently, the conjugate gradient algorithm with preconditioning has been implemented for a three-dimensional non-convex geometry (we take an aperture for example) to calculate the outgoing flux. This method has proved to be very efficient for this case.
Section snippets
The formulation of the problem
We consider a three-dimensional non-convex enclosure with a diffuse and grey surface Γ as shown in Fig. 1. We say that a surface is diffuse as emitter (reflector) if it emits (reflects) heat uniformly in every direction. For a grey surface the emissivity and reflectivity are independent of the wavelength (color) of the radiation. Thus, only the total intensity of radiation and not its spectrum is needed in a heat balance model. On Γ we assume, for simplicity, that the temperature is given.
Under
Boundary element method and Galerkin discretization
For the application of the boundary element method in 3D, let τn={Δk}k=1n denotes a sequence of triangulations of the surface Γ for some sequences of integers n→∞ and let h be the mesh size. We choose an integer r⩾0 and with χn we denote the set of all functions φ which are piecewise polynomials of degree ⩽r in the parametrization variables as discussed in [1]. There are no continuity restrictions to the functions in χn. The dimension of χn is n·fr, where n is the number of elements in τn and
Numerical example
The mass matrix Mn and the right-hand side fn in Eq. (3.5) can either be calculated analytically exact for special geometries or numerical integration is carried out. To keep the numerical integration error small, we handle the weak singularity of the integral kernel in the case of a non-smooth boundary by employing double partial integration, see [8], [9], [14], [15].
Since we are considering a non-convex geometry, the main problem is the efficient detection of the shadow zones to calculate the
Acknowledgements
The first author of this paper would like to express his sincere thanks and appreciation to Prof. Dr. Gerald Warnecke for the support he received during his stay at the Institute of Analysis and Numerics at Magdeburg University-Germany.
The second author would like to thank Prof. Dr. W.L. Wendland who gave her the possibility and the support to work in the area of heat radiation as member of the Sonderforschungsbereich SFB 259 “Hochtemperaturprobleme rückkehrfähiger Raumtransportsysteme” at the
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