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 :
- similarities and differences in comparsion to the FEM
- example
- application fields
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 :
x1
x2
x3
d2
d3
x4
x5
x6
d4
d5
d6
d1
d1
d2
Shape Functions
element–wise approximation for trial and test functions :
X2
1
d2
d1
u = u1 + u2
shape functions :
= -1
=1
Approximation
approximation of the displacement u(x,tdef) :
d1
2
1
u
d1
d2
d1
u(x,tdef)
d3
d4
d2
d2
d5
d6
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 :
Q1
Q2
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 :
X2
1
enrichment :
d1
d2
q1
q2
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
• ...