Wave propagation in 1D stratified media and Finite Element Method

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Wave propagation in 1-D stratified media and
Finite Element Method

• Achintya Pal
• Institute for Geophysics
• The University of Texas at Austin
• EDGER Forum, February 25-26, 2008

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Outline

• Brief comparison between two broad categories of
  numerical methods
• Finite Difference
• Finite Element
• Introduction of simple types of Boundary Value
  Problems (BVP) in Geophysics
• Correspondence with general structure of Finite
  Element Method (FEM)
• Discussion of basic steps of FEM for solving the
  problem

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Numerical Solution techniques Finite Difference
and Finite Elements

• Finite Difference (FD)
• Discretization of differential operators in
  the exact field equations for the continuum
• Finite Element (FE)
• Discretize the medium itself and numerically
  find a solution to the approximate problem
• FE technique is based on division of
  continuum into finite blocks or elements
  connected to each other at a finite number of
  discrete nodes.

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Historically, the method of Finite Element was
applied for solving complex elasticity,
structural analysis problems in mechanical, civil
and other branches of engineering The problems
share one essential characteristic Mesh
discretization of a continuous domain into a set
of discrete sub-domains
Source Wikipedia
2D mesh for the problem (mesh is denser around
the object of interest)
2D FEM solution for a magneto-static configuration
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Finite Element Method (FEM) applied to
Geophysical problems

• Most attractive feature of FEM is its ability to
  handle complex geometries and boundaries with
  relative ease in 2- or 3-dimensions
• The problem of wave propagation in 1-, 2- or
  3-dimensional media seems naturally amenable to
  solution by FEM
• We take the simplest model
• Horizontally stratified medium
• Homogeneous layers (elements) joined at
  horizontal interfaces (nodes)

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Mapping of Finite Elements to Horizontally
Stratified Medium
O1
O2
O3
O4
Elements
x1
x2
x3
x4
x5
Nodes
0
2
1
3
4
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Objective To find the wave-field at any point
(x,z) within the stratified medium
Layer m-1
Layer m
Layer m1
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Integral Transform Solution
For a horizontally stratified medium
x
Source at zs
(x,z)
z
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Wave number integration technique

• The solution of depth-separated equation for
• along with the boundary conditions B
  0

completely determine the fields over the
stratified medium across the interfaces The total
solution (depth-dependent Greens function)
Particular solution two solutions of
homogeneous equation A and A- are coefficients
to be determined from boundary conditions at
interfaces between layers. Finally, ?(x,z) is
obtained by the following transformation
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Direct Global Matrix Approach (DGM)Schmidt and
Jensen(1985), Schmidt and Tango(1986)

• DGM approach uses a solution of the depth
  separated wave equation with
• Each layer in the stratified media being a
  finite element
• Divide the continuum into finite blocks or
  elements connected to each other at discrete
  nodes
• Exact local solutions for the single elements,
  together with the continuity between elements at
  the nodes lead directly to an exact solution for
  the approximate global problem

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Matching of solution at each interface
For a horizontally stratified medium
x
Source at zs
Layer m
zm
Layer m1
(x,z)
z
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Matching of solution at each interface
Direct Global Matrix approach by some local to
global mapping
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Reduction of 2N coupled equations to (N1)
equations (Robert Geller Tomohiko Hatori)
Instead of the choice of usual solution (SH case)
within the n-th layer
If one chooses
This matching of solution will be shown to have
implication for the Finite Element Method (FEM)
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Mapping of Finite Elements to Horizontally
Stratified Medium
O1
O2
O3
O4
Elements
x1
x2
x3
x4
x5
Nodes
0
2
1
3
4
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Two essential steps to solve a boundary value
problem using FEM

• Rephrase the original Boundary Value Problem
  (BVP) in its weak or variational form
• Discretization the step in which the weak form
  is discretized in a finite dimensional space
• After this step, one has concrete formulae for a
  large but finite dimensional linear problem whose
  solution will approximately solve the original BVP

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Boundary value problems and Finite Element Method
Differential equations Boundary conditions
solve for wave field completely In variational
formulation of boundary value problem, the
boundary conditions at each interface are
automatically satisfied The essential features of
the equation can be written as
How can a function u satisfy the above equation
when u" cannot exist at xxs ? The classical
requirement that u satisfy the differential
equation at every point x is too strong Weak or
variational formulations are designed to
accommodate irregular data and irregular
solutions as well as smooth solutions These are
the formulations used to construct finite element
approximations
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Weak or variational formulation ofBoundary value
problems

• Find a function u such that the differential
  equation, together with the boundary conditions,
  are satisfied in the sense of weighted averages
  

Symmetric weak formulation after integration by
parts
where v are continuous and have first derivatives
such that
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Galerkin Method
Both u and v are expanded in terms of a finite
set of N basis functions
uN is completely determined if the degrees of
freedom ai are found Substituting for uN ,vN and
their derivatives we have
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The choice of basis functions
What kind of basis functions should one
choose? The finite element method provides a
general and systematic technique for constructing
basis functions for Galerkin approximations of
boundary value problems The main idea is that
(1) the basis functions can be defined
piecewise over sub-regions of the entire domain
called finite elements and (2) They can be
chosen to be very simple functions such as linear
or polynomials of low degree
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The choice of basis functions
Basis functions
Derivatives of basis functions
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Weak or variational formulation of laterally
homogeneous SH wave propagation (Geller Hatori,
Geophys. J. Int (1995))
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Review of paper by Geller Hatori(Correspondence
with FE formulation Geophys. J. Int (1995))
fn
zn-1
zn
zn1
Amn is a sparse tridiagonal matrix
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Review of paper by Geller Hatori(Correspondence
with FE formulation Geophys. J. Int (1995))
Shape of the trial or basis functions
f1
f2
f3
f4
f5
w
c2
c3
c4
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Boundary/Continuity Conditions
Continuity of displacement and stress is to be
satisfied at each interface of a layered
medium In FEM, continuity of traction or stress
is automatically satisfied by the weak form
solution without its having to be explicitly
satisfied by each of the trial functions Continuit
y of displacement at each interface is satisfied
by the choice of piecewise continuous trial
functions
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Summary

• Frequency wave-number domain solution for 1-D
  wave propagation in terms of plane wave solution
  solving for 2N coefficients for N interfaces is
  in the mould of finite element method
• It has been shown that with proper choice of
  basis functions, different from the physical
  plane wave solution, the same problem can be
  reduced to solving for N coefficients in finite
  element method satisfying the boundary conditions
• It will be shown further that wave propagation in
  2D/3D media can be treated even in time domain by
  extensions of finite element methods like
  spectral element method by clever choice of basis
  functions

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Thank you

• for your attention

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Weak variational formulation of laterally
homogeneous SH wave propagation (Geller Hatori,
Geophys. J. Int (1995))
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Matching of solution at each interface
Interface m-1
Layer m
Interface m
Layer m1
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Local to Global Mapping
Interface m-1
Layer m
Interface m
Layer m1
(2Nx2) (2x2) (2x2N)
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Topology or Connectivity Matrices
Sm
am
A

Global Matrix
Global Matrix
T4
T1
V
T2
T3
v1
v2
v3
v4




Local Matrix
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Example of finite element solution in two
dimensions
A piecewise linear function of the polygon below
on each triangle of the triangulation
Triangulation of a 15 sided polygonal region