38204699 Finite Elemente

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Lecture Notes

Theory of the Finite Element Method

Oliver Heidbach, June 2004

Beta Version 0.97

Contents1. Introduction 3

2. Stating the problem 7

2.1. Constitutive equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.1. Linear elastic rheology . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2. Linear viscous rheology . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2. Equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1. The Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . 112.2.2. Equation of equilibrium . . . . . . . . . . . . . . . . . . . . . . . 11

2.3. Mathematical classification of linear partial differential equations of 2ndorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4. Physical classification of linear partial differential equations of 2nd order . 13

3. Approximation of the PDE solutions 16

3.1. The Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2. Proof of the Variational Principle . . . . . . . . . . . . . . . . . . . . . . 173.3. 2-D example for the Variational Principle . . . . . . . . . . . . . . . . . . 193.4. Ritz Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5. 2-D example for the Ritz Method . . . . . . . . . . . . . . . . . . . . . . 213.6. 1D-Example for the Ritz Method with numbers . . . . . . . . . . . . . . 233.7. Weighted residuals and Galerkin’s Method . . . . . . . . . . . . . . . . . 243.8. 2-D example for the Galerkin Method . . . . . . . . . . . . . . . . . . . . 26

4. Numerical solution - The Finite Element Method 28

4.1. First step into the world of Finite Elements . . . . . . . . . . . . . . . . 284.1.1. Transformation for a linear 1-D element . . . . . . . . . . . . . . 314.1.2. Transformation of the derivatives for a linear 1-D element . . . . 33

4.2. 1-D Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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5. Finite Element Method - Engineering approach 38

5.1. Engineering approach for the 1-D example . . . . . . . . . . . . . . . . . 38

6. Special topics on the Finite Element Method 41

6.1. Do we need shape functions? . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.2. More definitions of Finite Elements . . . . . . . . . . . . . . . . . . . . . 416.2.1. 1D Finite Element with a quadratic polynomial . . . . . . . . . . 416.2.2. Linear transformation for Triangle Elements . . . . . . . . . . . . 426.2.3. 2-D triangular elements with a linear polynomial . . . . . . . . . 43

6.3. Classification of the various element types . . . . . . . . . . . . . . . . . 44

A. Mathematical rules and equations 45

A.1. Positive definite differential operator . . . . . . . . . . . . . . . . . . . . 45A.2. Calculus rules for the first variation . . . . . . . . . . . . . . . . . . . . . 45A.3. Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.4. Gauss integral equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46A.5. Stokes integral equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 46A.6. Stress definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46A.7. Strain tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

B. List of useful books and papers 50

B.1. Books and paper quoted in the text . . . . . . . . . . . . . . . . . . . . . 50B.2. Books on FEM for further reading . . . . . . . . . . . . . . . . . . . . . . 50B.3. Papers on FE-model in Geoscience . . . . . . . . . . . . . . . . . . . . . 51

C. English - German dictionary of mathematical expressions 57

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1. Introduction

These notes are for students from geoscience who are interested in a theoretical back-ground of the Finite Element Methode (FEM) for geodynamic modeling. Due to thewide range of related topics which are involved in this type of modeling (tectonics, rhe-

ology, linear algebra, function analysis etc.) some basic knowledge and understandingin these fields is assumed.

The FEM is not a straight forward method to explain. There is unfortunately the needof lots of mathematics. However, with these notes I do not intend to cover the methodcompletely nor do I try to be precise in the sense of a strict and pure mathematicalapproach. But these shortcomings have to be accepted in order to make the principlesof the method as transparent as possible.

For a more advanced introduction of the FEM lots of books are recommended in thereference list in the appendix. These notes only provide you with the basics of theFEM. They do not give a complete overview and are therefore not an excuse for a cross

reading in professional textbooks, as for instance in the ”FEM-bible” by Zienckiewicand Tayler (1994a and 1994b) where you will find more than 1400 pages in two volumes!This will give you an idea of the complexity and strength of the method. More goodtextbooks with different approaches to the FEM are Altenbach and Sacharov (1982),Kammel et al. (1990) and Schwarz (1991). In these lecture notes the aim is to draw apicture of the main steps creating a geodynamic model with FEM. Easy understandabletwo-dimensional problems using the partial differential equation for the equilibrium of forces (Poisson-equation) with linear elastic rheology will support and accompany thetheory.

In most cases the geoscientist will use a commercial FEM-code, the so-called solver(e.g. MSC MARC, ABAQUS, ANSYS, NASTRAN), which is, unfortunately, a blackbox. Alternatively you can use an open public code, which is to a certain extent writtenby another geoscientist (e.g. Bird and Kong, 1994). These are often poorly documentedand limited to a specific type of problems. A further development is always a very time-consuming task. The alternative is using a commercial FE-software package, which isoften provided with a graphical user interface, containing the a pre- and a post-processor(e.g. MSC MENTAT, PATRAN, ABAQUS CAE, Hypermesh von Altair) and the solver.The pre-processor is needed in order to construct the geometry of the model and itsdiscretization, the post-processor is needed for the visualization of the model results.The core of the package is always the solver, which contains the mathematical algorithmfor the numerical solution procedures (numerical integration, calculation of the inverse

matrix etc.). The reliability of such a software package is high (also the original prices),but the user is always tempted to use the software in a non appropriate way. Solvabilityor mathematical convergence of the numerical problem does not imply that the resultrepresents the observed process, i.e. the process which has to be modeled. The qualityand plausibility strongly depends upon the assumptions and the model input providedand defined by the user. Therefore the results have to be controlled and critically assessedby the user.

Simulation of an observed physical process is a sequence of consecutive steps. The

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first one is always the mapping of the physical process on a model process. The nextones are: finding a solution and a time-spatial description which often implies numericalmethods. An analytical solution is in the most cases not available. The basic idea of anumerical approximation of the (not available) analytical solution is the transformationof the continuous model description into a discrete one. The equation system will then

be solved by a computer with the chosen algorithm (FEM, Finite Differences, BoundaryElement Method ....) Therefore, the simulation, i.e. the modeling procedure can bedivided into three main steps:

• Definition of the model process and its geometry

• Discretization of the continuum

• Application of a numerical algorithm

Therefore the next chapter after this introduction describes the basic differential equationwe have in geoscience and its classification into several groups. Also, some formulas

and statements of rheology will be given. The third chapter covers the mathematicaltreatment of the differential equations before the numerical algorithm of FEM can beapplied. The following chapter four gives the introduction into the FEM itself. In theappendix you will find some useful explanations, some mathematical rules which are usedin the text and an English-German translation of the main ”mathematical” vocabulariesfor your convenience.

But before starting with the mathematics, I would like to encourage a discussionon the term ”model” before we enter into the world of maths. I strongly recommendto start any discussion about a ”model” and its results with a short definition anddescription of the main features of the model type (simplifications, assumptions, type

of boundary conditions, used algorithm and also the visualization procedure) beforediscussing the (geo-)physical meaning, value or result of the model. This will preventlots of misunderstandings. Setting up a common language is always essential and veryhelpful. There is a wide range and definition of the term ”model” or extended terms like”kinematic model” or ”physical model” in all branches of Geoscience. The following isan incomplete list of different model types:

• Tectonic model

Looking at faults (active and inactive ones) and taking into consideration geophys-ical and geological observations and measurements (gravity, heat flow, seismicity,geological maps, fluvial patterns etc.) a tectonic model can be developed. It does

not necessarily give a value as a result (compare with the kinematic model whichcomes somewhat close to this approach). It is more of an integrative approachwhich attempts to bring into accordance with a wide range of observed phenom-ena. (e.g. Sperner et al., 2001)

• Kinematic model

A model which only looks for translation and rotation of rigid plates as e.g.NUVEL-1A from deMets et al. (1994). No internal deformation of the lithosphericplate is assumed, i.e. a rigid body (Euklid) is applied.

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• Dynamic model

A difficult term and I have, so far, found three different possibilities of an expla-nation from literature: The first uses the term dynamic in models where forces areboundary conditions (e.g. slab pull, ridge push). The second definition is accord-ing to the time-dependence of the model. A pure elastic or elastic-plastic model is

completely time independent and therefore not dynamic, but a model with viscousrheology is time-dependent and can therefore be called dynamic. A third strictdefinition says, that only models which imply acceleration can be called dynamic.In summary: be careful if somebody talks about a ”dynamic model”. I dare notdecide which definition is the correct one.

• Mathematical model

Often used in geodesy where a vast amount of observation data of the field variablewith a high accuracy is available. The mathematical model describes the field byreducing the error towards the observed values (e.g. least square method). This”mathematical model” does not necessarily imply any physics. It aims only at the”perfect” mathematical description of the observed field variable data (see figure1).In general the model parameter in a geodetic (mathematical) model is estimatedthrough the observed values (field

variable) by inversion (e.g. least square method, where the model parameter arethe unknown coefficients of the mathematical model). In geophysics and geologythe model parameters are chosen in order to get the field variable as a result of themodel (forward modeling). Of course inverse models are also applied in geologyand in geophysics and vice versa forward model are applied in geodesy.

Figure 1: Sketch for the forward and inverse modeling procedure.

• Forward model

This term is used for all modeling approaches where the physical properties andthe boundary conditions of the model are given. The field variable, e.g. the

displacement field, is calculated from the model. In figure (1) this would be themodeling procedure from the right side to the left. The result of the model is thencompared with observed values.

• Inverse model

In contrast to the forward model in the inverse model the field variable has beenobserved (e.g. gravity or displacement by GPS) and the model parameters arevaried in order to find the best fit for the observed field variable. Accordingto figure (1) this would be the procedure from the left to the right side. The

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parameters of this mathematical model are varied as long as the average error isminimized.

• Numerical model

Any model which uses a numerical (discrete) method due to the lack of availabil-

ity of an analytical (continuous) solution. Methods are Finite Differences, FiniteVolumes, Finite Elements, Boundary Element etc. .

• Physical model or analogue model

This is a model which you can actually touch. For instance a sandbox modelconsisting of sirup, plasticine and sand. But this term can also refer to yoursimplification (your model) of the observed phenomena in nature. E.g. back arcspreading behind subduction zones is observed and the physical model which couldgive an explanation is the roll-back of the subduction zone. This physical modeldoes not need to consist of formulas, but is only a picture in your mind. Thishas to be done if you want to convert your physical model into a mathematical

formulation in order to solve it e.g. with an numerical algorithm.

• Static model

This term is often used in the commercial FEM software packages. This definitionis based on the type of solving algorithm used in the model. If the problem is pureelastic it is called ”static model”, because the equation of the static equilibrium of forces (accelerations are neglected) has to be solved.

This list is definitely not complete and you will probably find lots of other deviatingdescription in other text books, but it might help you to keep the problem in mindwhenever you find yourself in a discussion about your model with another geoscientist.

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2. Stating the problem

2.1. Constitutive equations

The following constitutive equations are taken from the textbook ”Rheology of the

Earth” by Ranalli (1995). Rheology is Greek1

and can be translated with ”scienceof flow”. Hence rheology describes the deformation (flow) of matter under the presenceof forces. The two most prominent linear rheological laws, also called constitutive equa-tions, are Hooke’s law for elastic deformation and Newton’s law for viscous deformation.They relate the stress tensor σij to the deformation tensor ij. A combination of thesetwo can describe many of the deformation processes observed in the lithosphere.

The stress tensor σij describes the stress state of any point in a continuous body.The strain tensor kl describes the deformation, i.e. the change of displacement withspace within a continuous body. The relation between the applied forces (or the appliedstresses, since stress is measured in force per area with the measure unit Pascal) isdescribed by rheological laws.

2.1.1. Linear elastic rheology

The theoretical model of perfect elasticity postulates that the components of strain atany point within an elastic medium are homogeneous linear functions of the componentsof stress. The most general description is therefore

σij = C ijklkl . (1)

where C ijkl represent the 81 elastic parameter. Assuming isotropy, i.e. the physicalproperties do not depend on the direction, this number can be reduced to the two Lame

constants λ and µ. With δ the Kronecker symbol and the deviatoric part

ij of the straintensor, Hooke law for an isotropic material can be stated as

σij = λkkδij + 2µij . (2)

Hooke’s law is time-independent, i.e. the reaction of the pure elastic material isinstantaneous. As long as the applied forces do not change, the deformation will notchange either. The stress tensor can be divided into two parts: The diagonal componentsof the stress tensor (σ11, σ22 and σ33) which describes the lithostatic pressure (see figure2) and the shear components (off-diagonal components). Since only this deviatoric part(denoted by σ) of the stress tensor produces deformation2 it is convenient to extractthis part and we obtain

σij = 2µij (3)

1”Παντα ρι - everything flows”. This statement is from Herakleitos of Ephesos around 500 v. Chr.,who was one of the first natural philosophers trying to explain the processes in nature)

2This statement is somewhat tricky and implies that a pure volume change of a material is notcalled deformation. When σ11 = σ22 = σ33 is applied to rock material, the volume will decreaseassuming the rock is compressible. But this ”deformation” is not of interest in geodynamic processes.Therefore the term ”deformation” is mostly used in the context of deformation due to deviatoricstresses which change the shape of the rock

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Instead of the Lame constants the Young modulus E and the Poisson number ν is oftenused:

E =µ(3λ + 2µ)

λ + µ

ν = λ2(λ + µ)

µ =E

2(1 + ν )(4)

where µ is called the shear modulus often given as G. The range of E and G for Granitefor example is in the order of 10-100 GPa (depends on temperature, pressure, textureetc.) whereas ν is around 0.3 (maximum is 0.5 for an incompressible material as, forinstance, water).

2.1.2. Linear viscous rheology

A perfect linear viscous material (linar Newton fluid) can be stated as

σij = − pδij + λ θδij + 2ηij . (5)

where p is the pressure, θ the change in volume, λ a material constant and η is theviscosity. Since a viscous deformation process is time-dependent the Newton law statesthe relationship between the applied forces and the deformation rate. When the materialis assumed to be incompressible (θ = 0) the linear Newton law becomes

σij = − pδij + 2ηij

For the shear component σij of the stress tensor σij we therefore obtain

σij = 2ηij . (6)

The dot over the strain and shear strain tensor is the time derivative which gives usthe strain rate tensor and the shear strain rate tensor respectively. The strain tensor isdefined as

ij =1

2∂ui

∂x j+

∂u j

∂xi(7)

and for the strain rate tensor we receive

ij =∂

∂t

1

2

∂ui

∂x j+

∂u j

∂xi

=1

2

∂vi

∂x j+

∂v j

∂xi

(8)

where v is the velocity.

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By using these two shape functions the straight line can be represented as a linearcombination of these two shape functions as shown in figure (9).

φ(ξ) = N 1 φ(x j) + N 2 φ(x j+1)

= (1 − ξ) φ(x j) + ξ φ(x j+1) (99)

= (1 − ξ) u j + ξ u j+1

where u j and u j+1 are the values of the function u and the approximation uh at thenodes j and j + 1. Inserting the given values from figure (9) for φ(x j) and φ(x j+1) wecan write for expression (99)

φ(ξ) = (1 − ξ) 2 + ξ 3

⇒ φ(ξ = 0) = 2

φ(ξ = 1) = 3

The linear approximation in x coordinates for the given values is then

φ(x) = 1 + 0, 5x (100)

and in ξ coordinates

φ(ξ) = 2 + ξ (101)

0 1

u(x)

4 u2

x

1

2

3

u1

x =2 x =5 x =8 x =12432

u3

u4

uh(x)

e1 e3e2node1 node4node3node2

u(x)

Figure 10: Subdivision of a bar into three elements with two nodes each. The exact so-lution u(x) is approximated with linear segments. The values of the unknowncoefficients αi are the values of the function u(x) = uh(x) at the nodes. Thispreserves the continuity of the piecewise linear approximation uh(x).

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4.1.2. Transformation of the derivatives for a linear 1-D element

For the example in the next chapter we also need the derivatives of the transformationformula (95) and some definitions:

x = x j + (x j+1

− x j) ξ

⇒dx

dξ= x j+1 − x j

:= L j

⇒ dx = L jdξ (102)

where L j is the length of the subarea (the finite element). With the chain rule forderivatives we receive:

˜φ

j[ξ(x)] =

dξ(x)

dx

dφ j [ξ(x)]

dξ

= L−1 j

dφ j

dξ(103)

4.2. 1-D Example

Let us now consider the following differential equation of second order

d

dxh(x)

du

dx+ f (x) = 0 (104)

where f (x) is the function for the body forces and the h(x) describes the spatial variationof the material property (e.g. the Young Modulus E ). The differential equation is definedwithin the interval

x ∈ [a, b]

u(a) = ua and u(b) = ub

According to the variation principal we receive the functional

I (u) = ba

12

h(x)

dudx

2

−f (x)u

dx . (105)

Let us now subdivide the area in subareas

a = x1 < x2 < ... < xm+1 = b

D j := [x j, x j+1]; j = 1,...,m

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and uh(x) the approximation of the wanted solution u(x) Expression (105) can now bestated as the sum over the subinterval D j

I (uh) =m

j=1 Dj1

2h(x)

duh

dx 2

−f (x)uhdx (106)

and for each subinterval D j we can then write

I j(uh) =

xj+1xj

1

2h(x)

duh

dx

2

−f (x)uh

dx (107)

Mapping of the approximation function

uh(x) = φ(x) = α0 + α1x (108)

onto a standard interval [0, 1] in the ξ coordinates space and inserting the approximation

uh(ξ) = φ(ξ) = α0 + α1ξ (109)

into expression (107) by using expressions (102) and (103) gives

I j(αi) =

10

1

2h(ξ)

L−1 j

dφ j

dξ

2

−f (ξ)φ j

L jdξ

=1

2L j

10

h(ξ)

dφ j

dξ

2

dξ − L j

10

f (ξ)φ jdξ (110)

Assuming that the intervals D j are small enough the functions f (x) and h(x) is constantwithin the interval. Inserting the abbreviations

f =1

2[f (x j) + f (x j+1)] (111)

h =1

2[h(x j) + h(x j+1)] (112)

into expression (110) we receive

I j(αi) =h

2L j

10

dφ j

dξ

2

dξ − f L j

10

φ jdξ

= h2L j

α21 − f L j

12

(2α0 + α1) (113)

with 10

dφ j

dξ

2

dξ =

10

α21dξ

= α21 (114)

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and 10

φ jdξ =

10

(α0 + α1ξ)dξ

=

1

2(2α0 + α1) . (115)

When u j is the value of the continuous function uh at the node j with its coordinatesx j with j = 1, 2,....,m the following relations are valid:

u j = uh(x j)

= φ j(x j)

= φ j(ξ = 0)

= α0 (116)

u j+1 = uh(x j+1) = φ j(x j+1)

= φ j(ξ = 1)

= α0 + α1 (117)

Stating expressions (116) and (117) in matrix givesα0

α1

=

1 0

−1 1

u j

u j+1

:= p u( j) (118)

For α21 from expression (114) the matrix form is

α21 =

α0

α1

T 0 00 1

α0

α1

= u( j)T

pT

0 00 1

p

u( j)

= u( j)T

1 −1−1 1

u( j)

:= u( j)T

K u( j)

(119)

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where K is the so called the stiffness matrix since here the physical properties are stored.For 0.5(2α0 + α1) from expression (115) the matrix form is

1

2(2α0 + α1) =

α0

α1

T 1

2

21

(120)

= u( j)T pT 12

21

= u( j)T

1 −10 1

1

2

21

= u( j)T

0.50.5

:= u( j)T b

where b is defined as

b := 0.5

0.5

(121)

With the following two definitions

ω j :=h

L j

(122)

γ j := f L j (123)

we can express equations (113) as

I j = u( j)T 1

2ω j Ku( j)−u( j)T (γ j b) (124)

Looking for a minimum of I j the first variation leads to11

δI j =∂I j

∂ u( j)= 0 (125)

⇒ ω j (Ku( j)) − γ j b = 0 (126)

As an example we assume that we have four elements. In matrix form we receive thefollowing equations for each element12.

e1 : ω1

1 −1 0 0 0

−1 1 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

u1

u2

u3

u4

u5

= 12

γ 1γ 1000

(127)

11The unknown variables are now the values of uj at the nodes (nodal values). Therefore the variationhas to be made towards uj .

12In order to be able to combine the single equations into a big set of equations the vectors u(j) arefilled up to a vector u with the missing u1 · · · uj−1, uj+2 · · · um+1and the matrixes K are filled withzeros in order to bring to the correct dimension of the problem which is a (m + 1 × m + 1)-matrix.

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e2 : ω2

0 0 0 0 00 1 −1 0 00 −1 1 0 00 0 0 0 00 0 0 0 0

u1

u2

u3

u4

u5

= 12

0γ 2γ 200

(128)

e3 : ω3

0 0 0 0 00 0 0 0 00 0 1 −1 00 0 −1 1 00 0 0 0 0

u1

u2

u3

u4

u5

= 1

2

00γ 3γ 30

(129)

e4 : ω4

0 0 0 0 00 0 0 0 0

0 0 0 0 00 0 0 1 −10 0 0 −1 1

u1

u2

u3

u4

u5

= 12

00

0γ 4γ 4

(130)

Combining these four expressions for the four elements through superposition we receivefor the global stiffness matrix K(g)

K(g) =

ω1 −ω1 0−ω1 ω1 + ω2 −ω2

−ω2 ω2 + ω3 −ω3

−ω3 ω3 + ω4 −ω4

0 −ω4 ω4

(131)

and

b(g) =1

2(γ 1, γ 1 + γ 2, γ 2 + γ 3, γ 3 + γ 4, γ 4)T (132)

u(g) = (u1, u2, u3, u4, u5)T (133)

This leads to the following set of equations.

K(g)u(

g) = b(

g) (134)

The difference to the Ritz approach to the problem is obvious. The global stiffnessmatrix K(g), which is the coefficient matrix, has now a so called bandwidth structure,i.e. only in the diagonal of the matrix and some parallel lines we have numbers. Mostof the matrix consists of zeros which makes it very easy to invert the matrix and solvethe problem. The term ”stiffness matrix” expresses that in this matrix the physicalbehaviour of the model is stored.

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5. Finite Element Method - Engineering approach

5.1. Engineering approach for the 1-D example

Another possibility to approach the Finite Element Method is to derive the equations

received in the previous chapter by looking at an easy example. This is again a one-dimensional bar which is elongated along the x axis as shown in figure (11). This bar isdivided into three segments of the length L1, L2, L3 (Finite Elements) with three differentYoung’s Modules E 1, E 2, E 3.

x

σ =10 MPa

0 4

E

0

1

2 31

E2 E3

L 1L 2 L 3

Figure 11: The bar is fixed on the left hand side at the wall. It is divided into threeelements e1, e2, e3 of the length L1, L2, L3. Material properties within theelements are the Young’s Modules E 1, E 2, E 3. The bar is pulled by a surfaceforce of σ0 = 10M P a at its right side.

Figure (12) shows the Element e2 with its displacements u2 and surface force F 2 acting

on its left side and the displacements u3 and surface force F 3 acting on its right side.

u2

E2

F2

L2

F3

u3

E2

Figure 12: Sketch of the element e2 with the length L2 with the surface forces F 2, F 3 anddisplacements u2, u3 at its left and right hand side.E 2 is the Young’s modulus.

The Hooke’s law in each individual Finite Element can be written as

σ = E i (135)

=du

dx=

ui − ui+1

Li

38



where Li is the length of each element. According to Hooke’s law the surface force shownin figure (11) acting on the left-hand-side and the right-hand-side respectively can bewritten as

F 2 = σ2 =E 2(u2 − u3)

L2

(136)

F 3 = σ3 =E 2(u3 − u2)

L2

(137)

These two expressions can also be stated in matrix form and we receive

E 2

L2

1 −1

−1 1

u2

u3

=

F 2F 3

⇔ K(2)u(2) = b(2) (138)

K(2) = k

(2)

11−k

(2)

12−k

(2)21 k

(2)22

=

E 2

L2

1 −1−1 1

(139)

For the elements e1 and e2 we can write in the same way

E 1

L1

1 −1

−1 1

u1

u2

=

F 1F 2

⇔ K(1)u(1) = b(1) (140)

K(1) =

k(1)11 −k

(1)12

−k(1)21 k

(1)22

=

E 1

L1

1 −1

−1 1

(141)

E 3

L3

1 −1

−1 1

u3

u4

=

F 3F 4

⇔ K(3)u(3) = b(3) (142)

K(3) =

k(3)11 −k

(3)12

−k(3)21 k

(3)22

=

E 3

L3

1 −1

−1 1

(143)

where for each element K(e) is the symmetric, quadratic element stiffness matrix, u(e)

the (displacement) vector of the nodal value (or in short node vector) and b(e)

the vectorof the acting surface forces. By looking at node i which connects Element ei and ei+1we receive according to the equilibrium of the surface forces:

F i − F (e)i − F

(e+1)i = 0 (144)

39



Fi

Fi

(e) Fi

(e+1)

ei ei+1

Figure 13: Sketch of the node i which connects the elements ei and ei+1. The surface

force F i at the node i has to be in equilibrium with the surface forces F (e)i

and F (e+1)i .

For the four nodes we then receive four equations

F (1)1 = F 1 ⇔ k

(1)11 u1 − k

(1)12 u2 = F 1 (145)

F (1)2 + F

(2)2 = F 2 ⇔ −k

(1)21 u1 + k

(1)22 u2 + k

(2)11 u2 − k

(2)12 u3 = F 2 (146)

F (2)3 + F

(3)3 = F 3 ⇔ −k

(2)21 u2 + k

(2)22 u3 + k

(3)11 u3 − k

(3)12 u4 = F 3 (147)

F (3)4 = F 4 ⇔ −k

(3)21 u3 − k

(3)22 u42 = F 4 (148)

These equations can be summarized in a global assembly and we receive the followingglobal Finite Element equation in matrix form

k(1)11 −k(1)

12 0 0

−k(1)21 −k

(2)22 + k

(2)11 −k

(2)12 0

0 −k(2)21 −k

(2)22 + k

(3)11 −k

(3)12

0 0 −k(3)21 k

(3)22

u1

u2

u3

u4

=

F 1F 2F 3F 4

(149)

This can be written as

K(g) u(g) = b(g) (150)

where K(g) is the global stiffness matrix, u(g) the global displacement vector and b(g)

the global load vector.

40



6. Special topics on the Finite Element Method

6.1. Do we need shape functions?

The term ”shape function” is used for a different description of the approximation func-

tion for an element. But it is not compulsory to use shape functions explicitly in orderto define a Finite Element. This is shown in the example in chapter 4 .2 where no shapefunction was used for the development of the Finite Element equation.

However, in more complicated elements of higher order in 3-D it is much more conve-nient to integrate the equations when the approximation function within the element isdescribed as a linear combination of the shape functions. The properties of the shapefunction are:

1. They can have the values one and zero.2. They are only defined within the element. Outside the element their value is zero.

6.2. More definitions of Finite Elements6.2.1. 1D Finite Element with a quadratic polynomial

By choosing a quadratic polynomial for the approximation function uh(x) we receive athird node, i.e. a third unknown coefficient, for the 1D Finite Element.

ξ0 10.5

node 1 node 2 node 3

Figure 14: 1-D element with a quadratic polynomial, i.e. an element with three nodes.

φ(ξ) = α0 + α1ξ + α2ξ2 (151)

From figure (14) we can express the values of the function uh(x) at the nodal points as

u1 = uh(x1) = φ j(x1) = φ1(ξ = 0) (152)

= α0

u2 = uh(x2) = φ j(x2) = φ2(ξ = 0.5)

= α0 + 0.5α1 + 0.25α2u3 = uh(x3) = φ j(x3) = φ3(ξ = 1)

= α0 + α1 + α2

Solving this for the αi and writing the three expressions in matrix form we receive α0

α1

α2

=

1 0 0

−3 4 −12 −4 2

u1

u2

u3

(153)

41



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10. 2

0

0.2

0.4

0.6

0.8

1

1.2

N1

N2

N3

Figure 15: Graphs of the three shape functions for a 1-D element with a quadratic poly-nomial

The three shape functions are

N 1(ξ) = 1 − 3ξ + 2ξ2 (154)

N 2(ξ) = 4ξ − 4ξ2 (155)

N 3(ξ) = −ξ + 2ξ2 (156)

and uh(ξ) can the be expressed as a linear combination of the shape functions

uh(ξ) = N 1 u1 + N 2 u2 + N 3 u3 (157)

The three shape function N 1, N 2, N 3 are drawn in figure (15).

6.2.2. Linear transformation for Triangle Elements

Before we can have a look at the definitions for the triangle element, we have to statethe mapping function for an arbitrary triangle T i in the xi coordinate system with thenodal points P 1(x

(1)1 , x

(1)2 ), P 2(x

(2)1 , x

(2)2 ) and P 3(x

(3)1 , x

(3)2 ) onto a standard unit triangle

T 0 in the ξi coordinate system (see figure 16).The transformation formula for the mapping from the xi coordinate system into the

ξi coordinate system is

x1 = x(1)1 + [x

(2)1 − x

(1)1 ]ξ1 + (x

(3)1 − x

(1)1 )ξ2 (158)

x2 = x(1)2 + [x

(2)2 − x

(1)2 ]ξ1 + (x

(3)2 − x

(1)2 )ξ2 (159)

42



ξP0 1

1

1

P2

P3

1

2ξ

x

P

0

1

P2

P3

1

2x

Figure 16: Mapping of an arbitrary triangle T i in the xi coordinate system with thenodal points P 1(x

(1)1 , x

(1)2 ), P 2(x

(2)1 , x

(2)2 ) and P 3(x

(3)1 , x

(3)2 ) onto a standard unit

triangle T 0 in the ξi coordinate system.

6.2.3. 2-D triangular elements with a linear polynomial

By choosing a linear polynomial for the approximation function uh(x) we receive for the2D Triangular Element three nodes, i.e. three unknown coefficients αi (see figure 16):

φ(ξ) = α0 + α1ξ1 + α2ξ2 (160)

Again we can express the values of the function uh(x) at the nodal points as

u1 = uh(x1i ) = φ1(ξ1 = 0, ξ2 = 0) (161)

= α0

u2 = uh(x2i ) = φ2(ξ1 = 1, ξ2 = 0)

= α0 + α1

u3 = uh(x3i ) = φ3(ξ1 = 0, ξ2 = 1)

= α0 + α2

Solving this for the αi and writing the three expression in matrix form we receive α0

α1

α2

=

1 0 0

−1 1 0−1 0 1

u1

u2

u3

(162)

The three shape functions are

N 1(ξ1, ξ2) = 1 − ξ1 − ξ2 (163)

N 2(ξ1, ξ2) = ξ1 (164)

N 3(ξ1, ξ2) = ξ2 (165)

and uh(ξ) can be expressed as a linear combination of the shape functions

uh(ξ1, ξ2) = N 1 u1 + N 2 u2 + N 3 u3 (166)

Shape function N 1 is drawn as an example in figure (17).

43



ξP0 1

1

1

P2

P3

1

2ξ

φ(ξ )i

Figure 17: Shape function N 1 for a Triangular Element with a linear polynomial.

6.3. Classification of the various element types

According to space

1-D e.g. line, bar, cantilever2-D e.g. triangle, quad3-D e.g. Tetraeder, Hexaeder

According to the degree of the polynomial in 1-D

α0 + α1x (linear)α0 + α1x + α2x2 (quadratic)α0 + α1x + α2x2 + α3x3 (cubic)

(167)

According to completeness of the polynomial and properties

Isoparametric: all elements where their transformation functionpreserves the geometrical shape

Lagrange: complete polynomial

Serendipity: incomplete polynomial

Conform: every element which fulfills continuity of the node variables at the nodes

For example, if a quad

element with a complete bi-quadratic polynomial is given it belongs to the Lagrangeclass. A quad element with a bi-quadratic polynomial belongs to the Serendipity class.

44



A. Mathematical rules and equations

A.1. Positive definite differential operator

Under certain conditions there is a relationship between a given differential equation

and a variational formulation. Mathematically spoken: If a linear differential opera-tor is positive definite, the differential equation can always be stated as a variationalformulation. If for the inner product

< v, w > =

V

v · w dV (168)

the following equations are true

< L · v,w > = < v, L · w > ⇒ L is self-adjoint, i.e. symmetric

< L · v, v > > 0 ⇒ L is elliptic (169)

the differential operator L is positive definite (e.g., the differential operator Laplace and the differential operator Nabla operator are positive definite).

A.2. Calculus rules for the first variation

The calculus rules for the first variation are

δ(u + v) = δu + δv

δ(u) = (δu)

δ(u · v) = v · δu + u · δv (170)

Example from equation (35) in order to receive equation (36)

δ

∂u

∂x

2= δ

∂u

∂x

∂u

∂x

=∂u

∂xδ

∂u

∂x+ δ

∂u

∂x

∂u

∂x

= 2

∂u

∂xδ

∂u

∂x

= 2∂u

∂x

∂

∂x(δu)

A.3. Integration by parts

By integrating the product rule of differentiation one gets the integration by parts V

(u · v) =

V

u · v +

V

v · u

⇔

V

u · v =

V

(u · v) −

V

u · v (171)

45

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The strike at the brackets in (u · v) denotes the spatial derivation. Example in order toreceive equation (37)

V

a1∂u

∂x1

∂

∂x1

(δu)

dV (172)

u = a1∂u

∂x1

and v =∂

∂x1

(δu) (173)

⇒ u =∂

∂x1

a1

∂u

∂x1

v = δu

⇒

V

a1

∂u

∂x1

∂

∂x1(δu)

dV = (174)

V

∂

∂x1

a1

∂u

∂x1

δu

dV −

V

∂

∂x1

a1

∂u

∂x1

δu

dV (175)

A.4. Gauss integral equation

The integral sentence of Gauss (divergence of a vector field) converts a volume integralto a closed surface integral.

V

div u dV =

S

(u · n) dS (176)

A.5. Stokes integral equation

The integral sentence of Stokes (rotation of a vector field) converts a surface integralinto a closed line integral integral.

S

(rotu · n) dS =

C

u · dC (177)

For a plane case we receive

rotu · n =∂u2

∂x1

−∂u1

∂x2

. (178)

A.6. Stress definitions

The stress tensor can be transformed into a diagonal matrix. The remaining diagonalcomponents are the principal stresses. σ1, σ2 and σ3. For these it is defined that

σ1 > σ2 > σ3 . (179)

46

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In Geoscience compressive stresses are positive. In engineering it is just the other wayaround!

The three invariants of the stress tensor expressed with the principal stresses are

I 1 = σ1 + σ2 + σ3

I 2 = −(σ2σ3 + σ3σ1 + σ1σ2)I 3 = σ1σ2σ3 (180)

The mean stress (pressure) is

σm = p =1

3(σ1 + σ2 + σ3) (181)

The differential stress is defined as

σd = σ1 − σ3 (182)

and the deviatoric stress as

σij = σij − σm (183)

In the presence of pore fluids, the pore fluid pressure σf , reduces the normal stresses.The effective normal stress is defined as

σneff = σn − σf (184)

The brittle or plastic strength σy is a threshold value above which the rock material can

not resist the stresses. It breaks (yields).

σymax=

1

2(σ1 − σ3) = C + µσn (185)

where µ is the friction and C the cohesion. Both values are different depending if theyield strength is prescribed for existing faults (Amonton’s law) or newly created fractures(Mohr-Coulomb criteria). For the friction on existing faults Byerlee (1966) determinedC and µ in the laboratory with

σymax=

1

2(σ1 − σ3) = 0.85σn (186)

when pressure is below 200 MPa (ca. 8 km depth) and

σymax=

1

2(σ1 − σ3) = 60M P a + 06σn (187)

for deeper layers. In the literature this is stated as the Byerlee law.

47



A.7. Strain tensor

The strain tensor describes the deformation (Volume change and shape change) of abody.

ij =1

2∂ui

∂x j +∂u j

∂xi

(188)

After transformation into the principal system the three remaining diagonal componentsare the principal strains 1, 2 and 3 and it is defined

1 > 2 > 3 (189)

The three invariants of the strain tensor expressed in the principal strains are

E 1 = 1 + 2 + 3

E 2 =−

(23 + 31 + 12)E 3 = 123 (190)

x2

0

P

P'

Q'

Q

D'

D

u i + d u i

u i

x1

x3

Figure 18: Displacement of two points P and Q of a body D towards the points P andQ of the new shape of the body D.

In order to deduce the deformation tensor and extraction of the non deforming parts

of translation and rotation two points P and Q according figure (18) of the body Dare considered. After deformation towards P and Q the body has the new shape andposition D. The displacement of the two points P and Q is expressed through thedisplacement vectors ui and ui + dui. In order to express ui(x) at an arbitrary location,

48



ui(x) in written in a Tayler expansion at point P .

ui(Q) =∞n=0

(Q − P )nuni (P )

n!

= ui(P ) +∂u

i∂x j

P

δxi

= ui(P ) +1

2

∂ui

∂x j+

∂u j

∂xi

−∂u j

∂xi

+∂ui

∂x j

δxi

= ui(P ) +1

2

∂ui

∂x j−

∂u j

∂xi

δxi +

1

2

∂ui

∂x j+

∂u j

∂xi

δxi (191)

The first term is the translation, the second the describes the antisymmetric rotationtensor and the third is the deformation tensor. The symmetric deformation tensor canbe split into an isotropic part which is a pure volume change and shear part which

reflects the shape change of the body.

49



B. List of useful books and papers

B.1. Books and paper quoted in the text

• Altenbach, J. and Sacharov, A.S., 1982. Die Methode der Finiten Elemente in der

Festkorpermechanik. VEB Fachbuchverlag, Leipzig.

• Bird, P. and Kong, X., 1994. Computer simulations of California tectonics confirmvery low strength of major faults. Geol. Soc. of Am. Bull., 106: 159-174.

• DeMets, C., Gordon, R.G., Argus, D.F. and Stein, S., 1990. Current Plate Mo-tions. Geophys. J. Int., 101: 425-478.

• DeMets, C., Gordon, R.G., Argus, D.F. and Stein, S., 1994. Effect of recent revi-sions to the geomagnetic reversal time scale on estimates of current plate motions.Geophys. Res. Lett., 21/20: 2191-2194.

• Kammel, G., Franeck, H. and Recke, H.-G., 1990. Einfuhrung in die Methode derfiniten Elemente. Carl Hanser Verlag, Munchen.

• Ranalli, G., 1995. Rheology of the Earth. Chapman and Hall, London.

• Schwarz, H.R., 1991. Methode der finiten Elemente. Teubner, Stuttgart.

• Sperner, B., F. Lorenz, et al. (2001). ”Slab break-off - abrupt cut or gradualdetachment? New insights form the Vrancea Region (SE Carpathians, Romania).”Terra Nova 13: 172-179.

•Stuwe, K., Geodynamik der Lithosphare, Springer, Berlin, 2000.

• Turcotte, D.L., and G. Schubert, Geodynamics, Cambridge University Press, Cam-bridge, 2002.

• Zienkiewicz, O.C. and Tayler, R.L., 1994a. The Finite Element Method, Volume 1:Basic Formulations and Linear Problems, Fourth Edition. McGraw Hill, London.

• Zienkiewicz, O.C. and Tayler, R.L., 1994b. The Finite Element Method, Volume 2:Solid and Fluid Mechanics, Dynamics and Non-linearity, Fourth Edition. McGrawHill, London.

B.2. Books on FEM for further reading

• Bathe, K.J., 1982. Finite Elemente Methode. Springer Verlag, Berlin.

• Braess, D., 1992. Finite Elemente. Springer Verlag, Berlin.

• Braun, J. and Sambridge, M., 1995. A numerical method for solving partial dif-ferential equations on highly irregular evolving grids. Nature, 376: 655-660.

50



• Clough, R.W., 1960. The Finite Element in Plane Stress Analysis, Proc. 2nd.A.S.C.E. Conf. on Electronic Computation, Pitsburgh, pp. 114-123.

• Courant, R., 1943. Variational Methods for the Solution of Problems of Equilib-rium and Vibrations. Bull. Am. Math. Soc., 49.

• Cuvelier, C., Segal, A. and van Steenhoven, A.A., 1986. Finite Element Methodsand Navier-Stokes Equations. D. Reidel Publishing Company, Dordrecht, Holland.

• Meissner, U. and Menzel, A., 1989. Die Methode der finiten Elemente. SpringerVerlag, Berlin.

• Nasitta, K. and Hagel, H., 1992. Finite Elemente. Mechanik, Physik und nicht-lineare Prozesse. Springer Verlag, Berlin.

• Ritz, W., 1909. Uber eine neue Methode zur Losung gewisser Variationsproblemeder mathematischen Physik. J. Reine Angew. Math., 135.

• Tornig, W., 1985. Numerische Losung von partiellen Differentialgleichungen derTechnik. Teubner, Stuttgard.

• van Kan, J.J.I.M. and Segal, A., 1995. Numerik partieller Differentialgleichungenfur Ingenieure. Teubner, Stuttgart.

B.3. Papers on FE-model in Geoscience

• Albarello, D., Mantovani, E. and Viti, M., 1997. Finite Element Modeling of therecent-present deformation pattern in the Calabrian arc and surrounding regions.

Annali Geofisica.

• Bassi, G., Sabadini, R. and Rebai, S., 1997. Modern tectonic regime in the Tyrrhe-nian area: observations and models. Geophys. J. Int., 129: 330-346.

• Becker, T.W., Faccenna, C., O’Connell, R.J. and Giardini, D., 1999. The de-velopment of slabs in the upper mantle: insights from numerical and laboratoryexperiments. J. Geophys. Res., 104: 15207-15226.

• Bird, P. and Piper, K., 1980. Plane-stress finite-element models of tectonic flow insouthern California. Phys. Earth Planet. Int., 21: 158-175.

• Bird, P., 1998. Testing hypotheses on plate-driving mechanisms with global litho-sphere models including topography, thermal structure, and faults. J. Geophys.Res., 103: 10115-10129.

• Bott, M.H.P., 1990. Stress distribution and plate boundary force associated withcollision mountain ranges. Tectonophysics, 182: 193-209.

51



• Bott, M.P.H., 1991. Ridge push and associated plate interior stress in normal andhot spot regions. Tectonophysics, 200: 17-32.

• Burov, E., Jaupart, C. and Mareschal, J.C., 1998. Large-scale crustal hetero-geneities and lithospheric strength in cratons. Earth. Planet. Sc. Lett., 164:

205-219.

• Casas, A.M., Simon, J.L. and Seron, F.J., 1992. Stress Deflection in a TectonicCompressional Field: A Model for the Northwestern Iberian Chain, Spain. J.Geophys. Res., 97: 7183-7192.

• Chery, J., Bonneville, A., Vilotte, J.P. and Yuen, D., 1991. Numerical modelingof caldera dynamical behaviour. Geophys. J. Int., 105: 365-379.

• Christensen, U.R., 1992. An Eulerian Technique for Thermomechanical Modelingof Lithospheric Extension. J. Geophys. Res., 97/B2: 2015-2036.

• Cianetti, S., Gasperini, P., Boccaletti, M. and Giunchi, C., 1997. Reproducing thevelocity and stress fields in the Agean region. Geophys. Res. Lett., 24: 2087-2090.

• Cloetingh, S. and Wortel, R., 1985. Regional Stress Field of the Indian Plate.Geophys. Res. Lett.: 77-80.

• Cloetingh, S., Eldholm, O., Larsen, B.T., Gabrielsen, R.H. and Sassi, W., 1994.Dynamics of extensional basin formation and inversion: introduction. Tectono-physics, 240.

• Conrad, C.P. and Hager, B.H., 1999. The effects of Plate Bending and Fault

Strength at Subduction Zones on Plate Dynamics. J. Geophys. Res.

• Denlinger, R.P., 1992. A Model for Large Scale Plastic Yield of the Gorda Defor-mation Zone. J. Geophys. Res., 97: 15415-15423.

• Gasperini, P. and Sabadini, R., 1990. Finite-element modeling of lateral viscosityheterogeneities and post-glacial rebound. Tectonophysics, 179: 141-149.

• Geiss, E., 1987b. Die Lithosphare im mediterranen Raum. Ein Beitrag zu Struk-tur, Schwerefeld und Deformation, Deutsche Geodatische Kommission, Munchen,115 pp pp.

• Giunchi, C., Sabadini, R., Boschi, E. and Gasperini, P., 1996. Dynamic models of subduction: geophysical and geological evidence in the Tyrrhenian Sea. Geophys.J. Int., 126: 555-578.

• Golke, M., Cloetingh, S. and Fuchs, K., 1994. Finite-Element modeling of pull-apart basin formation. Tectonophysics, 240: 45-57.

52



• Grunthal, G. and Stromeyer, D., 1992. The recent Crustal Stress Field in CentralEurope: Trajectories and Finite Element Modeling. J. Geophys. Res., 97: 11805-11820.

• Heidbach, O. and Drewes, H., 2000. The velocity and strain field in the Mediter-

ranean from deformation models, Boletin ROA, NO.3/2000, San Fernando, Spain,Sept. 2000, pp. 4 pp.

• Heidbach, O., 2000. Der Mittelmeerraum - Numerische modelierung der Litho-spharendynamik im Vergleich mit Ergebnissen aus der Satellitengeodasie, BeckscheVerlagsbuchhandlung, Munchen, 98 pp pp.

• Heidbach, O. and Drewes, H., 2003. 3-D Finite Element model of major tectonicprocesses in the Eastern Mediterranean. In: N. D. (Editor), New insigths instructural interpretation and modeling. Spec. Publs. 212, Geol. Soc., London,

• Henk, A., 1997. Gravitational orogenic collapse vs plate-boundary stresses: a

numerical modeling approach to the Permo-Carboniferous evolution of CentralEurope. Geol. Rundsch., 86: 39-55.

• Houseman, G.A. and Molnar, P., 1997. Gravitational (Rayleigh-Taylor) insta-bility of a layer with non-linear viscosity and convective thinning of continentallithosphere. Geophys. J. Int., 128: 125-150.

• Houseman, G.A. and Gubbins, D., 1997. Deformation of subducted oceanic litho-sphere. Geophys. J. Int., 131: 535-551.

• Jentzsch, G. and Jahr, T., 1995. Der Tiefenbau des Harzes aus Untersuchungen

des Schwerefeldes. Nova Acta Leopoldina, 291: 169-190.

• Liu, G. and Ranalli, G., 1998. A finite element algorithm for modeling the subsi-dence and thermal history of extensional basins. J. Geodynamics, 26.

• Lundgren, P., Saucier, F., Palmer, R. and Langon, M., 1995. Alaska crustal defor-mation: Finite element modeling constrained by geologic and very long baselineinterferometry data. J. Geophys. Res., 100/B11: 22,033-22,045.

• Lundgren, P., Giardini, D. and Russo, R.M., 1998. A geodynamic framework foreastern Mediterranean kinematics. Geophys. Res. Letts., 25: 4007-4010.

• Lynch, D.H. and Morgan, P., 1990. Finite-element models of continental extension.Tectonophysics, 174: 115-135.

• Marotta, A.M. and Sabadini, R., 1995. The style of the Tyrrhenian subduction.Geophys. Res. Lett.: 747-750. Marquart, G. and

• Schmeling, H., 1989. Topography and geoid undulations caused by small-scaleconvection beneath lithosphere of variable elastic thickness. Geophys. J. Int., 97:511-527.

53



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C. English - German dictionary of mathematical

expressions

acceleration Beschleunigungapproximation Naherungassumption Annahmeboundary condition Randbedingungboundary value problem Randwertproblemcircular length Bogenlangecontinuity Stetigkeitcontinuous function stetige Funktiondefect Residuumdeviatoric stress deviatotrische Spannungdifferentiability Differenzierbarkeitdifferential stress differentielle Spannung

dilatation Dehnungellipse Ellipseequilibrium of forces Kraftegleichgewichtequation of motion Bewegungsgleichungshape function Form-Funktionfunctional Funktionalhyperbola Hyperbelinitial value problem Anfangswertproblemintegrand Integrandintegration by parts partielle Integration

mean stress mittlere Spannungone-differentiable einfache Differenzierbarkeitordinary gewohnlichparabola Parabelpiecewise differentiable stuckweise differenzierbarpressure Druckprincipal stress Hauptspannungquadrature Quadraturresidual Residuumsmoothness Glattheit

space Raumspatial raumlichstationary stationar, ortsfeststeady state stationarer Zustandsteady flow stationarer Flussstrain Deformation, VerformungTaylor expansion Taylor Entwicklungunambiguousness Eindeutigkeitweighing function Gewichtsfunktionyield strength Bruchfestigkeit, Scherfestigkeit

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38204699 Finite Elemente