FEM and X-FEM in Continuum MechanicsFEM and X-FEM in Continuum Mechanics

Joint Advanced StudentJoint Advanced Student

School (JASS) 2006,School (JASS) 2006,

St. Petersburg,St. Petersburg,

Numerical Simulation,Numerical Simulation,

3. April 20063. April 2006

State University St. Petersburg,State University St. Petersburg,

TU MünchenTU München

Ursula MayerUrsula Mayer

ContentsContents

1.1. Finite Element Method :Finite Element Method :

- problem definition, weak formulation- problem definition, weak formulation

- discretization, numerical integration- discretization, numerical integration

- linear system of equation- linear system of equation

- example- example

2.2. EXtended Finite Element Method :EXtended Finite Element Method :

- similarities and differences in comparsion to the FEM- similarities and differences in comparsion to the FEM

- example- example

- application fields - application fields

Linear Momentum EquationLinear Momentum Equation

linear momentum :linear momentum :

displacement :displacement :

density :density :

stress :stress :

material law for linear elasticity :material law for linear elasticity :

Young‘s modulus :Young‘s modulus :

strain :strain :

E

Partial Differential EquationEquation

hyperbolic PDE ( linear wave equation) : hyperbolic PDE ( linear wave equation) :

boundary conditions :boundary conditions :

- Neumann (traction) :Neumann (traction) :

- Dirichlet (displacement):Dirichlet (displacement):

initial conditions :initial conditions :

- displacement :displacement :

- velocitiy : velocitiy :

Weak FormulationWeak Formulation

multiplying with a test function, integrating over the domain :multiplying with a test function, integrating over the domain :

applying Gauss‘s theorem and integration by parts :applying Gauss‘s theorem and integration by parts :

mechanical interpretation : Principle of Virtual Workmechanical interpretation : Principle of Virtual Work

Function SpacesFunction Spaces

function space for trial functions :function space for trial functions :

function space for test functions :function space for test functions :

SummarySummary

• problem definition : constitutive law in linear momentum equation :problem definition : constitutive law in linear momentum equation :

wave equation (hyperbolic PDE) = strong formwave equation (hyperbolic PDE) = strong form

• obtaining the weak form : Principle of Virtual Workobtaining the weak form : Principle of Virtual Work

• definition of the function spaces for trial and test functiondefinition of the function spaces for trial and test function

DiscretizationDiscretization

decomposition of the domain into elements : decomposition of the domain into elements :

d2

d1

d2d1

x1 x2 x3 x4 x5 x6

d4 d5d3 d6

Shape FunctionsShape Functions

element–wise approximation for trial and test functions :element–wise approximation for trial and test functions :

shape functions :shape functions :

1 X2

d2d1

u = u1 + u2

= -1 = 1

ApproximationApproximation

approximation of the displacement u(x,tapproximation of the displacement u(x,tdefdef) :) :

1 2

d2

d1

d2

d1

d1 d2 d3 d4d5 d6

u(x,tdef)u

x

Nonlinear System of EquationsNonlinear System of Equations

inserting the trial and test function in the weak form :inserting the trial and test function in the weak form :

nonlinear system of equationsnonlinear system of equations

mechanical interpretation : Newton‘s first lawmechanical interpretation : Newton‘s first law

Linearization with the Newton-Raphson MethodLinearization with the Newton-Raphson Method

residual :residual :

Taylor-expansion of the residual : Taylor-expansion of the residual :

Jacobian matrix :Jacobian matrix :

iteration step :iteration step :

Numerical IntegrationNumerical Integration

transformation in the element domain : transformation in the element domain :

numerical integration with Gaussian quadrature :numerical integration with Gaussian quadrature :

Q2Q1

Time Integration with the Newmark-beta-methodTime Integration with the Newmark-beta-method

update of displacement, velocity and acceleration :update of displacement, velocity and acceleration :

unconditionally stable for :unconditionally stable for :

SummarySummary

• approximation of the solutionapproximation of the solution

• nonlinear system of equationsnonlinear system of equations

• linearization with Newton-Raphson methodlinearization with Newton-Raphson method

• Gaussian quadrature for domain integralsGaussian quadrature for domain integrals

• time integration with Newmark-beta-methodtime integration with Newmark-beta-method

Simulation of a One-Dimensional BeamSimulation of a One-Dimensional Beam

Model :Model :

• rod is pulled on both sides by rod is pulled on both sides by

constant forces Fconstant forces F

• linear-elastic material lawlinear-elastic material law

• constant intersection A constant intersection A

• one - dimensional simulationone - dimensional simulation

L

FF

A

Introduction to the X-FEMIntroduction 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 fluiddynamics various possibilities of application in mechanics and fluiddynamics

Partial Differential EquationEquation

hyperbolic PDE ( linear wave equation) : hyperbolic PDE ( linear wave equation) :

boundary conditions :boundary conditions :

- Neumann (traction) :Neumann (traction) :

- Dirichlet (displacement):Dirichlet (displacement):

initial conditions :initial conditions :

- displacement :displacement :

- velocitiy : velocitiy :

Weak formulationWeak formulation

FEM :FEM :

X-FEM :X-FEM :

Function SpacesFunction Spaces

function space for trial functions :function space for trial functions :

function space for test functions :function space for test functions :

EnrichmentEnrichment

adding a discontinuous part to the approximation :adding a discontinuous part to the approximation :

1 X2

d2d1

q1 q2enrichment :enrichment :

Level Set Level Set

enrichment function :enrichment function :

Linearization Linearization

nonlinear system of equation : nonlinear system of equation :

Jacobian matrix :

Numerical Integration Numerical Integration

partitioning :partitioning :

b a

Simulation of a One-Dimensional Cracked BeamSimulation of a One-Dimensional Cracked Beam

Model :Model :

• rod is pulled on both sides by rod is pulled on both sides by

constant forces Fconstant forces F

• linear-elastic material lawlinear-elastic material law

• constant intersection A constant intersection A

• one - dimensional simulationone - dimensional simulation

• cracked is introduced according cracked is introduced according

to the to the stress analysisstress analysis

L

FF

A

Applications of the X-FEM and OutlookApplications of the X-FEM and Outlook

Applications: Applications:

• interfaces : solid-solid, fluid-fluid, fluid-structureinterfaces : solid-solid, fluid-fluid, fluid-structure

• dynamic simulation : predefined cracks, interfacesdynamic simulation : predefined cracks, interfaces

• quasi-static simulation : crack propagationquasi-static simulation : crack propagation

Further developments :Further developments :

• crack evolution and propagation in dynamic simulationscrack evolution and propagation in dynamic simulations

• ......