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Numerical modeling of rock deformation 10 :: Numerical integration

Numerical modeling of rock deformation 10 :: Numerical integration. 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.

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Numerical modeling of rock deformation 10 :: Numerical integration

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  1. Numerical modeling of rock deformation10 :: Numerical integration 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. Motivation • So far, we calculated the spatial derivatives of the shape functions and the integrals analytically. • If we start deforming our elements, this is not possible anymore! • We have to do it numerically!

  3. Two types of elements Global element • Coordinates (x,y) • Coordinates arbitrary (i.e., deformed grid) Local element • Coordinates (x,h) • Coordinatesx=[-1...1] , h=[-1...1]

  4. Two types of elements Global element • Spatial derivatives of shape functions needed here (in matrices KM and G) • Integration boundaries given here. Local element • Shape functions defined here • Numerical integration performed on local element (using Gauss-Legendre quadrature)

  5. Transformation of spatial derivatives Global element • Spatial derivatives of shape functions Local element • Spatial derivatives of shape functions Transformation J is the Jacobian matrix

  6. Transformation of integration boundaries Global element • Integration boundaries Local element • Integration boundaries Transformation J is the Jacobian matrix This is now used for numerical integration

  7. Numerical integration: Gauss-Legendre quadrature • In 1D (n=3): • In 2D (nip=9):

  8. The Jacobian matrix • The Jacobian matrix is defined as • It can be calculated in the finite element code using • Spatial derivatives of the shape functions with respect to the local coordinates • Global coordinates of the element

  9. Online material • 2x lecture notes (ppt and pdf each) • Exercise-sheet • Tutorial for self study: Numerical integration andisoparametric elements Shape functions in 2D • Linear shape functions and their spatial derivatives (Maple-file and pdf) • Quadratic shape functions and their spatial derivatives (Maple-file and pdf) • Matlab-code for plotting the quadratic shape functions Finite-element codes • For 2D elasticity including numerical integration with 4 integration points • For 2D viscous flow including numerical integration with 9 integration points

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