FEM and XFEM in Continuum Mechanics Joint Advanced

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

Contents 1. Finite Element Method : - problem definition, weak formulation - discretization, numerical integration - linear system of equation - example 2. EXtended Finite Element Method : 3. - similarities and differences in comparsion to the FEM 4. - example 5. - application fields 6.

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

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

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

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

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

Discretization decomposition of the domain into elements : x 1 d 1 x 2 x 3 d 2 d 3 d 2 x 4 x 5 x 6 d 4 d 5 d 6

Shape Functions element–wise approximation for trial and test functions : X 2 1 d 2 d 1 u = u 1 + u 2 shape functions : = -1 =1

Approximation approximation of the displacement u(x, tdef) : d 1 2 1 u d 1 d 2 d 1 u(x, tdef) d 3 d 2 d 4 d 2 d 5 d 6 x

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

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

Numerical Integration transformation in the element domain : numerical integration with Gaussian quadrature : Q 1 Q 2

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

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

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

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

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

Weak formulation FEM : X-FEM :

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

Enrichment adding a discontinuous part to the approximation : X 2 1 enrichment : d 1 d 2 q 1 q 2

Level Set enrichment function :

Linearization nonlinear system of equation : Jacobian matrix :

Numerical Integration partitioning : b a

Simulation of a One-Dimensional Cracked Beam Model : F F • 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 L A

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 • . . .