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Quasi-1D Finite Element Method for Isothermal Flow of Ionized Gas Through a Nozzle

Quasi-1D Finite Element Method for Isothermal Flow of Ionized Gas Through a Nozzle. Robert Lee lee@ee.eng.ohio-state.edu. In FD, one approximates the derivatives by finite differences. In FEM, one approximates the unknown variable by basis functions

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Quasi-1D Finite Element Method for Isothermal Flow of Ionized Gas Through a Nozzle

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  1. Quasi-1D Finite Element Methodfor Isothermal Flow of Ionized GasThrough a Nozzle Robert Lee lee@ee.eng.ohio-state.edu

  2. In FD, one approximates the derivatives by finite differences. In FEM, one approximates the unknown variable by basis functions On similar grids, the matrix equation produced by FEM and FD may actually be the same Comparison of finite differencesversus finite elements

  3. FEM is ideally suited for unstructured grids while FD is applied to structured grids Comparison of finite differencesversus finite elements (cont’d 2) structured unstructured

  4. Method of Weighted Residuals The method of weighted residuals is now the most popular way to obtain an equation for the application of FEM. L is the operator (in our case diff. eq. of interest, and u is the unknown variable. f(x) is the forcing function. Let us define an approximation of u given by Then

  5. Method of Weighted Residuals (cont’d 2) We must try to force R(x) = 0. One possible way called point matching is 0 1

  6. Method of Weighted Residuals (cont’d 3) Point matching is equivalent to the following: We must consider other functions.

  7. The basis functions fj(x)are generated by simple functions defined piecewise (element by element) over the FEM grid. The basis and weighting must be smooth enough such that their derivatives in the weight residual equation exists (assume nth order derivatives), i.e., Finite Element Method (FEM) FEM provides a systematic and very general way of generating the basis functions (usually polynomial approximations. The criteria are:

  8. The basis functions are chosen in such a way that the coefficients defining the unknown quantity are precisely the value of the unknown quantity at the nodes. FEM (cont’d 2)

  9. 1 2 3 4 5 FEM (cont’d 3) Let us consider a 1-D grid Element Node Coordinate

  10. 1 1 2 2 3 3 4 4 5 5 FEM (cont’d 4) Assuming that we want to use the same functions for both the basis and weighting functions, the simplest function is the linear function shown below 1

  11. Application to Conservation of Mass Let

  12. Application to Conservation of Mass (cont’d 2) We evaluate the above expression element by element M is the number of elements Apply integration by parts,

  13. Application to Conservation of Mass (cont’d 3) leads to Note: Most of the endpoint contributions cancel out if A is continuous since r and u are continuous. Consider two elements

  14. Application to Conservation of Mass (cont’d 4) After plugging in approximation of r and u,

  15. Single Element Evaluation Consider integral for one element

  16. Single Element Evaluation (cont’d 2) Let us consider the integral over a single element And ignore the contribution from the boundary

  17. Mapping from Global to Local System m m+1 Global node number 1 Local node number 2 Additional notation:

  18. Element Evaluation in Local System Reordering terms,

  19. Application to Conservation of Momentum After linearization, FD in time, and applying the method of weighted residuals, the expression for a single element is

  20. Application to Conservation of Momentum (cont’d 2) Applying integration by parts, grouping terms, and disregarding B.C.’s,

  21. Application to Conservation of Momentum (cont’d 3) Substituting basis function representation for unknowns and converting to local system,

  22. Local Matrix Build Subscripts associated with indices i j in equations.

  23. Local Matrix Build (cont’d 2) For example,

  24. Local Matrix Build (cont’d 3) For example,

  25. Transferring Local Matrix to Global Matrix Local Node # 1 2 1 2 1 2 1 2 1 2 3 4 5 Global Node # For element

  26. Transferring Local Matrix to Global Matrix (cont’d 2)

  27. Evaluation of Area A Assume A varies linearly within element Wm,

  28. Numerical Integration: Gauss Quadrature Gauss quadrature of order N is given by

  29. Numerical Integration: Gauss Quadrature (cont’d 2) Note: Gauss quadrature is especially well suited for polynomial functions since for large enough N, RN=0. A polynomial of order 2N-1 can be exactly integrated by a quadrature of order N. For our case, we need coordinate transformation,

  30. Numerical Integration: Gauss Quadrature (cont’d 2) The highest order polynomial of interest for our problem is of order 4. We need quadrature of order 3,

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