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Finite Element Analysis

Finite Element Analysis. Mathematical Background. http ://www.kmitl.ac.th/~ kpbumroo/12019801. One-Dimensional Computation Example. Second-Order One-Dimensional Linear Differential Equation Dirichlet boundary conditions Exact solution. Finite Difference Approximations. Forward difference

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Finite Element Analysis

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  1. Finite Element Analysis Mathematical Background http://www.kmitl.ac.th/~kpbumroo/12019801

  2. One-Dimensional Computation Example • Second-Order One-Dimensional Linear Differential Equation • Dirichlet boundary conditions • Exact solution 12019801 Finite Element Analysis

  3. Finite Difference Approximations Forward difference Backward difference Central difference 12019801 Finite Element Analysis

  4. Finite Difference Approximations Forward Backward Central Diagram 12019801 Finite Element Analysis

  5. Taylor Series Complete Series Remainder 12019801 Finite Element Analysis

  6. Accuracy and Precision Diagram 12019801 Finite Element Analysis

  7. Total Numerical Errors Truncation and round-off errors 12019801 Finite Element Analysis

  8. Finite Difference Method Second-order difference 12019801 Finite Element Analysis

  9. Finite Difference Approximation Exact solution: • Second-Order One-Dimensional Linear Differential Equation • With boundary conditions 12019801 Finite Element Analysis

  10. Computations by Finite Element Methods 12019801 Finite Element Analysis Domain discretisation

  11. Computations by Finite Element Methods 12019801 Finite Element Analysis Trial functions

  12. Computations by Finite Element Methods Trial functions, interpolation functions, shape functions, or basis functions 12019801 Finite Element Analysis

  13. Computations by Finite Element Methods • Galerkin method • Orthogonal projection of residual error onto the subspace spanned by test functions summed over the domain • Errors are minimised, leading to the best numerical approximation 12019801 Finite Element Analysis

  14. Computations by Finite Element Methods Integrate by parts 12019801 Finite Element Analysis

  15. Computations by Finite Element Methods Variational equation or weak form 12019801 Finite Element Analysis

  16. Computations by Finite Element Methods Compact notation 12019801 Finite Element Analysis

  17. Computations by Finite Element Methods Stiffness (Diffusion or Viscosity) matrix 12019801 Finite Element Analysis

  18. Computations by Finite Element Methods Force vector Neumann boundary vector 12019801 Finite Element Analysis

  19. Computations by Finite Element Methods • Global node assembly – Stiffness matrix 12019801 Finite Element Analysis

  20. Computations by Finite Element Methods Global node assembly – Stiffness matrix 12019801 Finite Element Analysis

  21. Computations by Finite Element Methods Global node assembly – Stiffness matrix 12019801 Finite Element Analysis

  22. Computations by Finite Element Methods Global node assembly – Force vector 12019801 Finite Element Analysis

  23. Computations by Finite Element Methods Global node assembly – Neumann boundary vector 12019801 Finite Element Analysis

  24. Computations by Finite Element Methods • Global node assembly – Neumann boundary vector • Neumann boundary conditions are not applied 12019801 Finite Element Analysis

  25. Computations by Finite Element Methods • Global algebraic equation • At the global node 2 12019801 Finite Element Analysis

  26. Computations by Finite Volume Methods Finite volume approximation 12019801 Finite Element Analysis

  27. Computations by Finite Volume Methods Finite volume approximation 12019801 Finite Element Analysis

  28. Linear Algebraic Equations Vector form Matrix form General Form 12019801 Finite Element Analysis

  29. Matrix Notation 12019801 Finite Element Analysis

  30. Matrix principal/main diagonal of the matrix Row vector (m = 1) Column vector (n = 1) Square matrix (m = n) 12019801 Finite Element Analysis

  31. Matrix Symmetric matrix Diagonal matrix Identity matrix 12019801 Finite Element Analysis

  32. Matrix Tridiagonal matrix Upper triangular matrix [U] Lower triangular matrix [L] Banded matrix 12019801 Finite Element Analysis

  33. Matrix • Equality of two matrices • Addition and subtraction • Commutative • Associative 12019801 Finite Element Analysis

  34. Matrix Algebra Multiplication by a scalar Matrix multiplication 12019801 Finite Element Analysis

  35. Matrix Algebra Matrix multiplication 12019801 Finite Element Analysis

  36. Matrix Algebra • Matrix Multiplication • Associative • Distributive • Not Commutative 12019801 Finite Element Analysis

  37. Matrix Algebra Inversion of matrix Matrix division  not define [A] is a square and nonsingular matrix 12019801 Finite Element Analysis

  38. Matrix Algebra Transpose Augmentation 12019801 Finite Element Analysis

  39. Matrix Algebra Determinant 12019801 Finite Element Analysis

  40. Matrix and MatLab ;  column operator ’  transpose operator *  matrix multiplication .*  multiply on a element-by-element basis inv(A)  inversion eye(n)  nxn identity matrix [A I]  matrix augmentation 12019801 Finite Element Analysis

  41. Linear Algebraic Equations • System of equation – matrix form • m > n  overdetermined • m < n  underdetermined 12019801 Finite Element Analysis

  42. Linear Algebraic Equations andMatLab • Linear equations • MatLab Codes • x = A\b • x = inv(A)*b 12019801 Finite Element Analysis

  43. Cramer’s Rule Cramer’s Rule 12019801 Finite Element Analysis

  44. Cramer’s Rule Example 12019801 Finite Element Analysis

  45. The Elimination of Unknowns The Elimination of Unknowns 12019801 Finite Element Analysis

  46. Naive Gauss Elimination Naive Gauss Elimination Steps 12019801 Finite Element Analysis

  47. Naive Gauss Elimination Forward elimination 12019801 Finite Element Analysis

  48. Naive Gauss Elimination Back substitution 12019801 Finite Element Analysis

  49. Naive Gauss Elimination DOFOR k = 1, n-1 DOFOR i = k+1, n factor = a(i,k)/a(k,k) DOFOR j = k+1, n a(i,j) = a(i,j) – factor*a(k,j) ENDDO b(i) = b(i) – factor*b(k) ENDDO ENDDO Pseudocode – forward elimination 12019801 Finite Element Analysis

  50. Naive Gauss Elimination x(n) = b(n)/a(n,n) DOFOR i = n – 1, 1, -1 sum = b(i) DOFOR j = i + 1, n sum = sum – a(i,j)*x(j) ENDDO x(i) = sum/a(i) ENDDO Pseudocode – back substitution 12019801 Finite Element Analysis

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