numerical modeling of rock deformation

Numerical modeling of rock deformation: FEM Introduction. Stefan Schmalholz, ETH Zurich

Numerical modeling of rock deformation:

06 FEM Introduction

Stefan [email protected]

NO E 61

AS 2009, Thursday 10-12, NO D 11

mailto:[email protected]


Introduction• We have learned that geodynamic processes can be described with partial

differential equations (PDE’s) derived from the concepts of continuum mechanics.

• These PDE’s describing geodynamic processes are often complicated and non-linear and closed-form analytical solutions do not exist.

• Therefore, the PDE’s are transformed into a system of linear equations which can be solved by a computer.

• The finite element method (FEM) is one method to transform systems of PDE’s into systems of linear equations. Other methods are for example the finite difference method, the finite volume method, the spectral method or the boundary element method.

• In this lecture the basic concept of the FEM is introduced by transforming a simple ordinary equation into a system of linear equations.


u

x0 L

B

Case A)

The model equation

2

2

( ) 0u xA Bx

This equation is a second order, inhomogeneous ordinary differential equation.

Physically it describes for example:

• A) The deflection of a stretched wire under a lateral load (u = deflection, A = tension, B = lateral load)

• B) Steady state heat conduction with radiogenic heat production(u = temperature, A = thermal diffusivity, B = heat production)

• C) Viscous fluid flow between parallel plates(u = velocity, A = viscosity, B = pressure drop)

We consider the equation


The analytical solution

u

x0 L

B

We find the general analytical solution by simple integration

2

2

1

21 2

( ) integrate

( ) integrate

( )2

x

x

u x B dxAx

u x B x C dxx A

Bu x x C x CA

2

1

( 0) 0 0

( ) 02

u x CBLu x L C

A

2( )2 2B BLu x x xA A

x

u

L=10, A=1, B=-1

The two integration constants are found by two boundary conditions

The special solution is


The weak formulation - 1The original equation

2

2

( ) 0u xA Bx

2

2

( )( ) 0u xN x A Bx

Multiply with test function

2

20

2

20

( )( ) 0

( )( ) ( ) 0

L

L

u xN x A B dxx

u xN x A N x B dxx

Integrate spatially

0

0 0

00

( ) ( ) ( )( ) ( ) 0

( ) ( ) ( )( ) ( ) 0

( ) ( ) ( )( ) ( ) 0

L

L L

LL

u x N x u xN x A A N x B dxx x x x

u x N x u xN x A dx A N x B dxx x x x

u x N x u xN x A A N x B dxx x x

Integrate by parts

( )( ),

a b b aab b a a ab bx x x x x x

u xa N x b Ax

The product rule of differentiation


The weak formulation - 2

The first term requires our function u(x) at the boundaries x=0 and x=L and is given by the boundary conditions. Therefore, we do not need to test u(x) at these boundary points.

00

( ) ( ) ( )( ) ( ) 0L

Lu x N x u xN x A A N x B dxx x x

0

( ) ( ) ( ) 0L N x u xA N x B dx

x x

The above equation is the weak form of our original equations, because it has weaker constraints on the differentiability on our solution (only first derivative compared to second derivative in original equation).

The original equation2

2

( ) 0u xA Bx


The finite element approximation - 1We approximate our unknown solution as a sum of known functions, the so-called shape functions, multiplied with unknown coefficients. The shape functions are one at only one nodal point and zero at all other nodal points, so that the coefficient corresponding to a certain shape function is the solution at the node.

ix 1ix 1ix 2ix

( )u x

x

iu1iu


The finite element approximation - 1

1 1 11

( ) ( ) ( ) ( ) ( ) ii i i i i i

i

uu x u N x u N x N x N x

u

TN u

We approximate our unknown solution as a sum of known functions, the so-called shape functions, multiplied with unknown coefficients. The shape functions are one at only one nodal point and zero at all other nodal points, so that the coefficient corresponding to a certain shape function is the solution at the node.

ix 1ix 1ix 1ix

( )u x

x

iu1iu



1 1 11

( ) ( ) ( ) ( ) ( ) ii i i i i i

i

uu x u N x u N x N x N x

u

TN u

1

( ) 1 1ii

i i

x x xN xx x dx

ix 1ix

x

11

( ) ii

i i

x x xN xx x dx

1

dx0

We approximate our unknown solution as a sum of known functions, the so-called shape functions, multiplied with unknown coefficients. The shape functions are one at only one nodal point and zero at all other nodal points, so that the coefficient corresponding to a certain shape function is the solution at the node.

Finite Elementix 1ix 1ix 1ix

( )u x

x

iu1iu


0

( ) ( ) ( ) 0L N x u xA N x B dx

x x


101 1 1

1

01 11

( ) ( )( ) ( ) 0

( ) ( )

( )( )( ) ( )

0( )( )

( ) ( ) (

dx i i ii i

i i i

i

dx i ii i

i ii

i i

N x u N xA N x N x Bdx

N x u N xx x

N xu N xN x N xx A Bdx

u N xN x x xx

N x N x N xx x

1

01 11 1 1

1

1 1 1

) ( )( )

0( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

i i

dx i i

i ii i i i

i i i i

i

ii i i i

N xu N xx x A Bdx

u N xN x N x N x N xx x x x

N x N x N x N xux x x x Adx

uN x N x N x N xx x x x

0 01 1

( )0

( )

0

dx dx i

i

N xBdx

N x

Ku - F

The FEM where the shape functions are identical to the test (weight) functions is called Galerkin method after Boris Galerkin.

11

( ) ( ) ii i

i

uu x N x N x

u


20 0

2 2 20

( ) ( ) 1

0

dx dxi iii

dx

N x N xK Adx A dx

x x dxx dx AA A

dxdx dx dx


2,

2

A A Bdxdx dxA A Bdxdx dx

K F

1

( ) 1 1ii

i i

x x xN xx x dx

ix 1ix

x

11

( ) ii

i i

x x xN xx x dx

1

dx0

1

0 011 1 1

( ) ( ) ( ) ( )( )

, ( )( ) ( ) ( ) ( )

i i i i

dx dx i

ii i i i

N x N x N x N xN xx x x x Adx Bdx

N xN x N x N x N xx x x x

K F

1( ) ( ) 1i iN x N xx x dx

0Ku - F



1

2

3

02 0

0 02

000 0 0

A A Bdxdx dx uA A Bdxudx dx

u

1

2

3

0 0 0 0 00 0

20

02

uA A Bdxudx dx

u BdxA Adx dx

1

2

3

0022 00

0 2

A A Bdxdx dx uA A A u Bdxdx dx dx

u BdxA Adx dx

0 LL/2 dxdx

1 2 31 2

Local system for element 1

Local system for element 2 Global system for sum of all elements



1

2

3

1 0 0 02

00 0 1

uA A A u Bdxdx dx dx

u

1

2

3

0022 00

0 2


u BdxA Adx dx

Global system without boundary conditions

Implementing boundary conditions: u1 and u3 =0

0 LL/2 dxdx

1 2 31 2



1

2

3

1 0 0 02

00 0 1

uA A A u Bdxdx dx dx

u

1

2

3

1

2

3

1 0 0 01 2 1 55 5 5

00 0 1

1 0 0 0 01 5 1 5 12.52 2 2

0 00 0 1

uuu

uuu

1

2

3

0022 00

0 2


u BdxA Adx dx

21 10( 5) 5 5 12.52 2

u x

Global system without boundary conditions

Implementing boundary conditions: u1 and u3 =0

Calculate solution usingL=10 (dx=L/2=5), A=1, B=-1

Analytical solution at x=5

0 LL/2 dxdx

1 2 31 2

2( )2 2B BLu x x xA A


Programming finite elements - 1

11

2 3 4 5 6 7 8 9 102 3 4 5 6 7 8 9

Nine finite elements

Ten nodes

Which nodes belong to what element?Introduce a node matrix!

1 22 33 44 5

5 66 77 88 99 10

123456789

The NODE matrix assigns to each local element the corresponding global node

Global nodesLocal nodes

1 21 2

1 2

1 2 3

4 5 6

1 2

34

1 21 2

34

1 2 5 4

2 3 6 5

12

1 2 3 4

2D excursion

ix 1ix

x1

1 2



Global nodesLocal nodes

11

2 3 4 5 6 7 8 9 102 3 4 5 6 7 8 9

Nine finite elements

Ten nodes

1 2

1

2

2

2

A A Bdxudx dxuA A Bdx

dx dx

Local system for finite element 4

1 22 33 44 5

5 66 77 88 99 10

123456789

1 2

4

5

A A Budx dx

A A udx dx

2

2

dx

Bdx

4 5

4

5



A Adx dx A AA A dx dxdx dx A A

dx dx

A Adx dxA Adx dx

GlobalK

The global stiffness matrix is the sum of all local stiffness matrixes. In our code we will first setup a global stiffness matrix KGlobal full of zeros. We will then 1) loop through the finite elements, 2) identify where the local element is positioned within the global matrix and 3) add the local stiffness matrix at the correct position to the global matrix.

Exactly the same is done for the right hand side vector F.


0

( ) ( ) ( ) 0L N x u xA N x B dx

x x

The FEM Maple code

Adx

Adx

Adx

Adx

B dx

2

B dx

2

> restart;> # FEM # Derivation of small matrix for 1D 2nd order steady state> # Number of nodes per elementnonel := 2:> # Shape functionsN[1] := (dx-x)/dx:N[2] := x/dx:> # FEM approximationu := sum( t[j]*N[j],j=1..nonel):> # Weak formulation of 1D equationfor i from 1 to nonel doEq_weak[i] := int( -A*diff(u,x)*diff(N[i],x) + B*N[i] ,x=0..dx);od:> # Create element matrixes for conductive and transient termsK:=matrix(nonel,nonel,0):F:=matrix(nonel,1,0):for i from 1 to nonel do

for j from 1 to nonel doK[i,j] := coeff(Eq_weak[i] ,t[j]):

od:F[i,1] := -coeff(Eq_weak[i] ,B)*B:

od:> # Display K and Fmatrix(K);matrix(F);


The FEM Matlab codeIn the numerical solution all parameters can be made variable easily, while it gets more difficult to find analytical solutions for variable parameters, e.g., A and B.


FEM applications

Taken from MIT lecture, de Weck and Kim


FEM element types

Taken from unknown web script


FEM applicationsShortening of the continental lithosphere with: (i) viscoelastoplastic rheology, (ii) transient temperature field, (iii) viscous heating and (iv)gravity.


FEM applicationsThree-dimensional viscous folding

numerical modeling of rock deformation

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