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Finite Element Analysis

Finite Element Analysis. Mathematical Background. http://www.kmitl.ac.th/~kpbumroo/12019801. Summary from Previous Lecture. Finite Element Computations Domain discretisation Element type and shape function Weak form and Galerkin WRM Element assemble Boundary conditions

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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. Summary from Previous Lecture • Finite Element Computations • Domain discretisation • Element type and shape function • Weak form and Galerkin WRM • Element assemble • Boundary conditions • Solving algebraic equations 12019801 Finite Element Analysis

  3. Summary from Previous Lecture local global nodal values local • Finite Element Computations • Domain discretisation • Element type and shape function 12019801 Finite Element Analysis

  4. Summary from Previous Lecture local • Finite Element Computations • Weak form and Galerkin WRM 12019801 Finite Element Analysis

  5. Summary from Previous Lecture local global • Finite Element Computations • Element assemble 12019801 Finite Element Analysis

  6. Summary from Previous Lecture global • Finite Element Computations • Boundary conditions • Neuman boundary conditions • Dirichlet boundary conditions • Algebraic Equation Solvers • Direct solver • Gaussian elimination with pivoting • LU decomposition • Choleski decomposition • Iterative solver • Gauss-Siedel • Jacobi • Conjugate gradient 12019801 Finite Element Analysis

  7. Numerical Integration Formulas • Newton-Cotes formulas • The Trapezoidal rule • The first of Newton-Cotes formulas 12019801 Finite Element Analysis

  8. Numerical Integration Formulas The composite trapezoidal rule 12019801 Finite Element Analysis

  9. Numerical Integration Formulas The composite trapezoidal rule – MatLab M-file 12019801 Finite Element Analysis

  10. Numerical Integration Formulas • Simpson’s rules • Simpson’s 1/3 rule • Simpson’s 3/8 rule 12019801 Finite Element Analysis

  11. Numerical Integration of Functions • Romberg integration – based on Richardson’s extrapolation • Combine two numerical integrations from trapezoidal rule (with different h) for improving accuracy 12019801 Finite Element Analysis

  12. Numerical Integration of Functions Romberg integration 12019801 Finite Element Analysis

  13. Numerical Integration of Functions Romberg integration – MatLab M-file 12019801 Finite Element Analysis

  14. Gaussian Quadrature Gaussian Quadrature 12019801 Finite Element Analysis

  15. Gaussian Quadrature Gauss-Legendre 12019801 Finite Element Analysis

  16. Gaussian Quadrature Change of interval 12019801 Finite Element Analysis

  17. Gaussian Quadrature Two-dimensional integrals 12019801 Finite Element Analysis

  18. Partial Differential Equations Partial derivative Partial differential equation = an equation involving partial derivatives of an unknown function of two or more independent variables 12019801 Finite Element Analysis

  19. Partial Differential Equations Linear Non-Linear Partial differential equation Linear – if it is linear in the unknown function and all its derivatives 12019801 Finite Element Analysis

  20. Partial Differential Equations Linear second-order equations Classification: 12019801 Finite Element Analysis

  21. Partial Differential Equations • Elliptic equations • Laplace equation • Parabolic equations • Heat conduction equation • Hyperbolic equations • Wave equation 12019801 Finite Element Analysis

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