1 / 9

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.

binah
Download Presentation

Numerical modeling of rock deformation 10 :: Numerical integration

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related