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Numerical modeling of rock deformation 07 :: FEM in 2D

Numerical modeling of rock deformation 07 :: FEM in 2D. www.structuralgeology.ethz.ch/education/teaching_material/numerical_modeling Fallsemester 2011 Thursdays 10:15 – 12:00 NO D11 & NO CO1 Marcel Frehner marcel.frehner@erdw.ethz.ch , NO E3 Assistant: Jonas Ruh, NO E69. Goals of today.

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Numerical modeling of rock deformation 07 :: FEM in 2D

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  1. Numerical modeling of rock deformation07 :: FEM in 2D www.structuralgeology.ethz.ch/education/teaching_material/numerical_modeling Fallsemester 2011 Thursdays 10:15 – 12:00 NO D11 & NO CO1 Marcel Frehner marcel.frehner@erdw.ethz.ch, NO E3 Assistant: Jonas Ruh, NO E69

  2. Goals of today • Repeat the finite element derivation from last time • Do everything equally for 2D • Weak form • Finite element approximation • Galerkin method • Assemble the global stiffness matrix and force vector

  3. The model equation 1D This equation is a second order, inhomogeneous, constant-coefficient linear ordinary differential equation. 2D This equation is a second order, inhomogeneous, constant-coefficient linear partialdifferential equation.

  4. FEM: The weak form – 1 1D 2D Multiplication withtest functions Spatial integration

  5. FEM: The weak form – 2 1D 2D Integration by parts

  6. FEM: The weak form – 3 1D 2D Drop boundary terms Original equation

  7. The finite element approximation – 1 1D 2D • Requirements for the shape functions: • Equal 1 at one nodal point;equal 0 at all other nodal points. • Sum of all shape functions has to be 1 in the whole finite element. • Shape functions can approximate functions of the same or lower order. Linear 1D Shape functions

  8. Linear 2D shape functions

  9. The FEM where the shape functions are identical to the test (weight) functions is called Galerkin method after Boris Galerkin. The finite element approximation – 2 1D 2D Use the FEM approximation in the weak form

  10. The finite element approximation – 3 1D 2D Final equation

  11. The final equation in 2D

  12. The finite element approximation – 4 1D 2D Solve derivatives and integrals analytically

  13. Stiffness matrix and force vector in 2D

  14. Programming the finite element method in 2D – 1 • Which nodes belongto which element? • Introduce anindexing matrix: Example: 6 finite elements; 12 nodal points

  15. Programming the finite element method in 2D – 2 Local system for element 2 Check in EL_N where to findelement 2 in the global matrix Add the local equations tothe global system of equations 1 2 3 4 5 6

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