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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

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

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

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

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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

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

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

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Discretization decomposition of the domain into elements : d2d2 d1d1 d2d2 d1d1 x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 d4d4 d5d5 d3d3 d6d6

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Shape Functions element–wise approximation for trial and test functions : shape functions : 1 X2X2 d2d2 d1d1 u = u 1 + u 2 = -1= 1

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Approximation approximation of the displacement u(x,t def ) : 1 2 d2d2 d1d1 d2d2 d1d1 d1d1 d2d2 d3d3 d4d4 d5d5 d6d6 u(x,t def )u x

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

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

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

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

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

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

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

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

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Weak formulation FEM : X-FEM :

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

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Enrichment adding a discontinuous part to the approximation : 1 X2X2 d2d2 d1d1 q1q1 q2q2 enrichment :

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Level Set enrichment function :

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Linearization nonlinear system of equation : Jacobian matrix :

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Numerical Integration partitioning : b a

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

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

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