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FEM Approach for One-Phase Solidification Problem via Parabolic Variational Inequalities

This paper discusses the Finite Element Method (FEM) approach to solve a one-phase solidification problem using parabolic variational inequalities. It covers the derivation of a complementarity system, application of FEM, and stability analysis.

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FEM Approach for One-Phase Solidification Problem via Parabolic Variational Inequalities

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  1. One-phase Solidification Problem: FEM approach via Parabolic Variational Inequalities Ali Etaati May 14th 2008

  2. Content: One-phase Stefan problem-solidification example Derivation of a complementarity system Parabolic variational inequality Finite element method

  3. One-Phase Stefan problem

  4. Solidification: • (T: final time)

  5. Energy balance (Stefan condition) at • Solidification (cont’d): • Heat conduction equation (in the solid) • Freezing temperature • (L is the latent heat)

  6. Solidification (cont’d): • Initial and Boundary conditions (temperature distribution): • Initial enthalpy:

  7. Freezing index: • Then:

  8. Linear complementarity system: • a.e. in

  9. Parabolic Variational Inequalities: • Suppose: • Define: • Let • where • and

  10. Parabolic Variational Inequalities (cont’d) • Find with • such that • and for almost all and is such that: • for all with a.e. in (0,T). • Problem 1:

  11. Parabolic Variational Inequalities (cont’d) • Find with such that • and • for all . • Problem 2:

  12. Parabolic Variational Inequalities (cont’d) • Consider a solution of Problem 2, for any and • Equivalence: • then • We obtain the solution for Problem 1. • Clearly a solution of Problem 1 solves Problem 2.

  13. Parabolic Variational Inequalities (cont’d) • a.e. in • Theorem • Solution to Problem 2 satisfies the linear complementarity system:

  14. Parabolic Variational Inequalities (cont’d) • Proof • For any non-negative and so, from Problem 2: • a.e. in • Which implies that

  15. Parabolic Variational Inequalities (cont’d) • Proof (cont’d) • Now let Then for any • for sufficiently small so that • a.e. in • Conversely, by noting that if satisfies the complementarity system, then, for • Hence • It is then clear that w solves Problem 2.

  16. Finite Element approximation (FEM) • find such that • For all • : Space of continuous functions which are linear on each element and which vanish on the boundaries. • : a piecewise linear basis function. • General Discretisation by FEM for Problem 1: • (for all interior points)

  17. Finite Element approximation (cont’d) If f is continuous: (for all ) (for all ) Otherwise: In any case:

  18. Time marching of the discrete system are nodal vectors. : is a symmetric positive definite matrix which causes the problem (*) to have unique solution. where

  19. FEM; Stability • We may take in (**), then • for all • Let • and let’s assume: • Or equivalently that • The complementarity problems are equivalent to • (**)

  20. FEM; Stability (cont’d) • holds when , there is a constant C, independent of space- and time-steps such that: • Stability theorem: • Providing the stability condition

  21. FEM; Stability (cont’d) • for which is bounded by • Lemma: • The explicit method • is stable under the following conditions

  22. Reference • Weak and variational methods for moving boundary problems, C M Elliott & J R Ockendon.

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