FEM and X-FEM in Continuum Mechanics

1. FEM and X-FEM in Continuum Mechanics Joint Advanced Student School (JASS) 2006, St. Petersburg, Numerical Simulation, 3. April 2006 State University St. Petersburg, TU München Ursula Mayer

2. Contents • Finite Element Method : • - problem definition, weak formulation • - discretization, numerical integration • - linear system of equation • - example • EXtended Finite Element Method : • - similarities and differences in comparsion to the FEM • - example • - application fields

3. E Linear Momentum Equation linear momentum : displacement : density : stress : material law for linear elasticity : Young‘s modulus : strain :

4. Partial Differential Equation hyperbolic PDE ( linear wave equation) : • boundary conditions : • Neumann (traction) : • Dirichlet (displacement): • initial conditions : • displacement : • velocitiy :

5. Weak Formulation multiplying with a test function, integrating over the domain : applying Gauss‘s theorem and integration by parts : mechanical interpretation : Principle of Virtual Work

6. Function Spaces function space for trial functions : function space for test functions :

7. Summary • problem definition : constitutive law in linear momentum equation : • wave equation (hyperbolic PDE) = strong form • obtaining the weak form : Principle of Virtual Work • definition of the function spaces for trial and test function

8. x2 x1 x4 x5 x3 x6 d1 d4 d2 d3 d5 d2 d6 d1 Discretization decomposition of the domain into elements :

9. X2 1 d2 d1 u = u1 + u2 = -1 = 1 Shape Functions element–wise approximation for trial and test functions : shape functions :

10. Approximation approximation of the displacement u(x,tdef) : 2 1 u u(x,tdef) d2 d1 d5 d6 d3 d1 d4 d1 d2 x d2

11. Nonlinear System of Equations inserting the trial and test function in the weak form : nonlinear system of equations mechanical interpretation : Newton‘s first law

12. Linearization with the Newton-Raphson Method residual : Taylor-expansion of the residual : Jacobian matrix : iteration step :

13. Q1 Q2 Numerical Integration transformation in the element domain : numerical integration with Gaussian quadrature :

14. Time Integration with the Newmark-beta-method update of displacement, velocity and acceleration : unconditionally stable for :

15. Summary • approximation of the solution • nonlinear system of equations • linearization with Newton-Raphson method • Gaussian quadrature for domain integrals • time integration with Newmark-beta-method

16. F F A L Simulation of a One-Dimensional Beam • Model : • rod is pulled on both sides by • constant forces F • linear-elastic material law • constant intersection A • one - dimensional simulation

17. Introduction to the X-FEM • method for the treatment of discontinuities (i.e.: interfaces, crack,...) • discontinuous part in the approximation: enrichment function • no remeshing • growth of mass and stiffness matrices • various possibilities of application in mechanics and fluiddynamics

18. Partial Differential Equation hyperbolic PDE ( linear wave equation) : • boundary conditions : • Neumann (traction) : • Dirichlet (displacement): • initial conditions : • displacement : • velocitiy :

19. Weak formulation FEM : X-FEM :

20. Function Spaces function space for trial functions : function space for test functions :

21. Enrichment adding a discontinuous part to the approximation : X2 1 d1 d2 enrichment : q1 q2

22. Level Set enrichment function :

23. Linearization nonlinear system of equation : Jacobian matrix :

24. a b Numerical Integration partitioning :

25. F F A L Simulation of a One-Dimensional Cracked Beam • Model : • rod is pulled on both sides by • constant forces F • linear-elastic material law • constant intersection A • one - dimensional simulation • cracked is introduced according • to the stress analysis

26. Applications of the X-FEM and Outlook • Applications: • interfaces : solid-solid, fluid-fluid, fluid-structure • dynamic simulation : predefined cracks, interfaces • quasi-static simulation : crack propagation • Further developments : • crack evolution and propagation in dynamic simulations • ...