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# FINITE ELEMENT FORMULATION FOR CONVECTIVE-DIFFUSIVE PROBLEMS WITH SHARP GRADIENTS USING FINITE CALCULUS

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### FINITE ELEMENT FORMULATION FOR CONVECTIVE-DIFFUSIVE PROBLEMS WITH SHARP GRADIENTS USING FINITE CALCULUS

1. 1. FINITE ELEMENT FORMULATION FOR CONVECTIVE-DIFFUSIVE PROBLEMS WITH SHARP GRADIENTS USING FINITE CALCULUS Aleix Valls Tomas International Center for Numerical Methods in Engineering (CIMNE) Modulo C1. Despacho C2. Universidad Politécnica de Cataluña. Campus Norte UPC, 08034 Barcelona, Spain [email_address]
2. 2. Introduction <ul><li>A finite element method (FEM) for convective-diffusive problems presenting sharp gradients of the solution both in the interior of the domain and in boundary layers. </li></ul><ul><li>The Finite Incremental Calculus (FIC) method is based in the solution by the Galerkin FEM of a modified set of governing equations, which includes characteristic length distances </li></ul>Numerical Oscillations FIC Stabilization
3. 3. <ul><li>Example: 1-D case (source less) </li></ul><ul><li>Steady-state convection-diffusive problem </li></ul><ul><li>Taylor expansion second order </li></ul>FIC Governing equations d 2 d d 1 A C B q A q B
4. 4. FIC Governing equations <ul><li>Steady-state convection-diffusive problem </li></ul>= FIC
5. 5. FIC Governing equations <ul><li>Multidimensional case (with source term) </li></ul><ul><li>Steady-state convection-diffusive problem </li></ul><ul><li>Governing equation </li></ul><ul><li>Boundary Conditions </li></ul><ul><li>Where h is characteristic length vector. </li></ul>FIC Stabilization terms
6. 6. Finite element discretization <ul><li>A finite element interpolation of the unknown: </li></ul><ul><li>Application of the Galerkin FE method to Governing equations gives, after integrating by parts term </li></ul><ul><li>The last integral has been expressed as a sum of the elements contributions to allow for interelement discontinuities in the term </li></ul><ul><li>Note that the residual terms have disappeared from the Neumann boundary . This is due to the consistency between the FIC terms. </li></ul>
7. 7. Finite element discretization <ul><li>Integrating by parts the diffusive terms </li></ul><ul><li>In matrix form </li></ul>
8. 8. Finite element discretization <ul><li>Simplifications: </li></ul><ul><ul><li>h constant over the element </li></ul></ul><ul><ul><li>Linear elements </li></ul></ul><ul><ul><li>Q constant </li></ul></ul>
9. 9. Computation of the characteristic length vector <ul><li>For the sake of preciseness the method is explained for 2D problems. </li></ul><ul><li>FIC balance equation in the principal curvature axes of the solution </li></ul><ul><li>For simplicity we consider the 2D sourceless case ( Q = 0) with an isotropic diffusion defined by a constant diffusion parameter k . </li></ul>
10. 10. Computation of the characteristic length vector <ul><li>The FIC balance equation is: </li></ul><ul><li>As and are the principal curvature axes of the solution then </li></ul>
11. 11. Computation of the characteristic length vector <ul><li>Introducing previous simplification we can rewrite FIC equation as (linear elements): </li></ul><ul><li>In matrix form </li></ul>
12. 12. Computation of the characteristic length vector <ul><li>The velocities along the principal curvature axes can be obtained by projecting the Cartesian velocities into the principal curvature axes </li></ul><ul><li>The characteristic length distances are defined as </li></ul><ul><li>where and are typical element dimensions along the principal curvature axes, respectively and and are the corresponding stabilization parameters. </li></ul>
13. 13. Computation of the characteristic length vector <ul><li>Stabilization parameters: </li></ul><ul><li>computed by considering the solution of two uncoupled 1D problems along the principal curvature axes. </li></ul><ul><li>Element dimensions </li></ul>
14. 14. Orthotropic Matrix Diffusion <ul><li>The next step is to transform the problem to global axes x, y </li></ul>FIC governing equations introduce orthotropic diffusion matrix
15. 15. About the FIC method <ul><li>Remark 1: </li></ul><ul><li>Clearly, if the principal curvature direction is parallel to the velocity direction, then </li></ul><ul><li>Where . Note that the method coincides with the standard SUPG approach in this case. </li></ul>
16. 16. About the FIC method <ul><li>Remark 2: </li></ul><ul><li>The global balance diffusion matrix can be also computed from the expression of vector h in global axes as </li></ul>
17. 17. General iterative scheme <ul><li>Step 0 (SUPG step). At each integration point choose , i.e. the gradient direction is taken coincident with the velocity direction. Compute . The expression of the balancing diffusion matrix coincides now precisely with the SUPG form . </li></ul><ul><li>Solve for . </li></ul><ul><ul><li>Verify that the solution is stable. This can be performed by verifying that there are not under or overshoots in the numerical results with respect to the expected physical values. If the SUPG solution is unstable, then implement the following iterative scheme. </li></ul></ul><ul><li>For each iteration: </li></ul><ul><li>Step 1 Compute at the element center. . Then compute and </li></ul><ul><li>Solve for . </li></ul><ul><li>Step 2 Estimate the convergence of the process. We have chosen the following convergence norm. </li></ul><ul><li>where N is the total number of nodes in the mesh and is the maximum prescribed value at the Dirichlet boundary. In above steps the left upper indices denote the iteration number. </li></ul><ul><li>If condition is not satisfied, start a new iteration and repeat steps 1 and 2 until convergence. Indexes 0 and 1 are replaced now by i and i + 1, respectively. </li></ul>
18. 18. Examples <ul><li>Example 1 </li></ul>
19. 19. Example 1 SUPG FIC
20. 20. Example 1 Cut y=0.5 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 X Phi SUPG FIC
21. 21. Example 2
22. 22. Example 2 SUPG FIC
23. 23. Example 2