Second order sensitivity analysis of heat conduction problems

In the paper the selected problems of sensitivity analysis application in the numerical modeling of heat conduction processes are discussed. The model of heat transfer bases on the Fourier equation supplemented by the geometrical, physical, boundary and initial conditions. In the first part of the paper the problems for which the second order sensitivity V (x, t) = 0, while in the next part the problems for which the second order sensitivity can be taken into account are presented, at the same time the direct approach of sensitivity analysis is used. On the stage of numerical computations the finite difference method [1] is applied. 1. Governing equations The transient temperature field in the solid domain is determined by the following energy equation ( ) ( ) Q t x T t t x T c x + ∇ = ∂ ∂ Ω ∈ , , : 2 λ (1) where T (x, t) is the temperature, c is the volumetric specific heat, λ is the thermal conductivity, Q is the capacity of internal heat sources, x, t denote the spatial co-ordinates and time. Let us assume that on the external surface of Ω the condition is general form, this means ( ) ( ) 0 , , , : 0 =       ∂ ∂ Φ Γ ∈ t x T n t x T x (2) is given, ∂(⋅)/∂n denotes the normal derivative. For t = 0 the initial temperature is known ( ) ( ) x T x T t 0 0 , : 0 = = (3) 2. Sensitivity model independent of the basic model At first the sensitivity of the problem discussed with respect to the boundary temperature Tb (Dirichlet problem) will be analyzed. According to the rules of Please cite this article as: Romuald Szopa, Jarosław Siedlecki, Wioletta Wojciechowska, Second order sensitivity analysis of heat conduction problems, Scientific Research of the Institute of Mathematics and Computer Science, 2005, Volume 4, Issue 1, pages 255-263. The website: http://www.amcm.pcz.pl/ R. Szopa, J. Siedlecki, W. Wojciechowska 256 direct approach [2-5] the Fourier equation and the boundary initial conditions must be differentiated with respect to Tb. So using the Schwarz theorem one obtains ( ) ( ) ( ) ( )        = = = Γ ∈ ∇ = ∂ ∂ Ω ∈ 0 0 , : 0 1 , : , , :