By Heinz W. Engl, Ewald Lindner (auth.), a. Univ. Prof. Dipl.-Ing., Dr. Heinz W. Engl, o. Univ. Prof. Dr. Hansjörg Wacker, Dipl.-Ing. Dr. Walter Zulehner (eds.)

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4. 4o z z Fig. 4 Let PL(Sh) be the set of piecewise linear and continuous functions on the triangulation. The vertices of the triangles are denoted by Pi, i = 1, ... ,M. To each vertex Pi there is an associated basis function pi ~ PL(Sh) with 0 .. lJ These basis functions form a basis for the space PL(Sh). 3 suggests the following choice for E~, A,B,C. rdrdz sh 0} 41 Remark: E~ C H~ (Sh x The condition p(O,z) 0 guarantees that [0,2n[) fork= A,B. The basis functions of PL(Sh) naturally generate a basis k for Eh.

4). Transformation to the body system For technical reasons the whole problem is expressed in terms of the coordinate system fixed in the rigid body. The transformation of a point x, a functional h and a vector function H is given by 28 x' = D(t)T(x-R(t)), h' (x' ,t) h(x,t), H' (x' ,t) D(t)TH(x,t), where the prime indicates the corresponding representation in the body system. This transformation leads to the following system of equations in the body system (for simplicity the primes are dropped) : N with I~.

2 The Mathematical Model and its Properties Usually, phase change problems like ours are modelled as Stefan problems . There, a heat equation with usually different coefficient functions is valid in two regions (representing the solid and the liquid phases), which are separated by a free boundary, the solidification front. From the vast literature about Stefan problems, we quote [14], [15], [4], [19], [30], [3], [5]. 53 In our problem, the soldification does not take place abruptly at a specific temperature.

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