Numerical modeling of rock deformation 06 :: Introduction to the FEM

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Numerical modeling of rock deformation06
Introduction to the FEM

• www.structuralgeology.ethz.ch/education/teaching_m
  aterial/numerical_modeling
• Fallsemester 2011
• Thursdays 1015 1200
• NO D11 NO CO1
• Marcel Frehner
• marcel.frehner_at_erdw.ethz.ch, NO E3
• Assistant Jonas Ruh, NO E69

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Goals of today

• Derive the weak form of the governing equation
• Understand the fundamental principle ofthe
  finite element approximation
• Apply the Galerkin method
• Understand how to put together the global
  stiffness matrix

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The big picture Physical models

• Mechanical framework
• Continuum mechanics
• Quantum mechanics
• Relativity theory
• Molecular dynamics

• Solution technique
• Analytical solution
• Linear stability analysis
• Fourier transform
• Greens function
• Numerical solution
• Finite difference method
• Finite element method
• Spectral methods
• Boundary element method
• Discrete element method

• Constitutive
• Equations
• (Rheology,
• Evolution
• equation)
• Elastic
• Viscous
• Plastic
• Diffusion

• Governing equations
• Energy balance
• Conservation laws
• Differential equations
• Integral equations
• System of (linear) equations

• Solution is valid
• for the applied
• Boundary conditions
• Rheology
• Mechanical framework
• etc

• Closed system of equations
• Boundary and initial conditions
• Heat equation
• Navier-Stokes equation
• Wave equation

Dimensional analysis
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Numerical methods

• There are different numerical methods, e.g.
• Finite difference method (FDM)
• Finite element method (FEM)
• Finite volume method (FVM)
• Spectral method
• Discrete element method (DEM)
• Boundary element method (BEM)
• Advantages of the FEM
• Rectangular grid or unstructured mesh ? Easy to
  fit complex geometries
• Deforming mesh (interface tracking)
• Free surface / topography
• Very flexible
• Well suited for
• All geodynamical problems involving (large)
  deformation
• All problems with complex geometries

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The model equation

• We consider the equation
• This equation is a second order, inhomogeneous,
  constant-coefficient linear ordinary differential
  equation.
• It may represent
• 1D Steady state heat conduction withradiogenic
  heat production(Aa thermal diffusivity
  Bqheat production uT temperature)
• Viscous fluid flow betweenparallel plates (Ah
  viscosityBp pressure drop uv velocity)
• Deflection of a stretched wireunder lateral
  load(As tension Bf lateral loaduw
  deflection)

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Analytical solution

• First integration
• Second integration
• Find the two integrationconstants by
  applyingboundary conditions
• Solution

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FEM The weak form 1

• Assumptions Continuous anddifferentiable
  function u(x) inthe 1D space x0...L
• Multiplication with test functions
• Spatial integration
• Integration by parts

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FEM The weak form 2

• The first terms are the so-called boundary terms,
  i.e., the spatial derivative of function u(x) has
  to be evaluated at the boundaries. They will
  anyway be defined by the boundary conditions (see
  later). Therefore, we do not need to take them
  into consideration.
• This is the weak form of our original equation,
  because it has weaker constraints on the
  differentiability of the solution (i.e., only
  first derivative compared to second derivative).

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The finite element approximation 1

• We approximate function u(x) within the finite
  element xixi1 as a sum of known functions,
  the so-called shape functions, multiplied with
  the values of u(x) at the nodal points, u(xi) and
  u(xi1).

Linear 1D Shape functions

• Requirements for the shape functions
• Equal 1 at one nodal point equal 0 at all other
  nodal points.
• Sum of all shape functions has to be 1 in the
  whole finite element.
• ?Shape functions can approximate functions of the
  same or lower order.

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The finite element approximation 2
Weak form
FEM approximation
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The finite element approximation 3
K Stiffness matrix F Force vector
Solve derivatives and integrals analytically
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The finite element approximation 4

• Local system for element 1
• Local system for element 2
• Global system Sum of all elements

Example 2 finite elements
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Boundary conditions

• Global system withoutboundary conditions
• Implementing boundary conditions u1c1 and u3c2

Example 2 finite elements
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Programming finite elements 1
Example 9 finite elements
Local element

• Which nodes belongto which element?
• Introduce an indexing matrix (EL_N)
• Matrix EL_N assigns to each local element the
  corresponding global nodes.

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Programming the finite element 2
Example 9 finite elements
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5

• Local system for finite element 5

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Check in EL_N where to find element 5 in the
global system
5
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Programming the finite elements 3

• The global stiffness matrix is the sum of all
  local stiffness matrices.
• In our code we will first setup a global
  stiffness matrix, Kglobal, full of zeros. Then we
  will
• loop through the finite elements
• Identify where the local element is positioned
  within the global matrix
• add the local stiffness matrix to the global one
  at the correct position
• Exactly the same is done for the global
  right-hand-side vector Fglobal