The Finite Element Method for Computational Structural...

The Finite Element Method for ComputationalStructural Mechanics

Martin Kronbichler

Applied Scientific Computing (Tillämpad beräkningsvetenskap)

January 29, 2010

Martin Kronbichler (TDB) FEM for CSM January 29, 2010 1 / 31

What is structural mechanics about?I A mathematician’s viewpoint:

Solve the (elliptic) PDE3X

j,k=1

λ∂2uj

∂xk∂xj+

3Xj=1

µ

„∂2ui

∂x2j

+∂2uj

∂xi∂xj

«+ fi = 0, i = 1, 2, 3,

for the variable u = [u1, u2, u3] on a certain computational domain Ωsubject to some boundary conditions.

I An engineer’s viewpoint:Body deformed by loads.Describe deformation andstructural failure (body“breaks down”).Pictures from open source finite element

library deal.II, http://www.dealii.org


Module scope

I Get a basic understanding of the mathematical modeling in structuralmechanics

I Perform some easy CSM simulations with the commercial softwarepackage Comsol

I Understand and relate the theoretical model behind simulation results


Schedule

Time Room

Lecture Jan 29, 13–15 2345

Lab 1, group 1 (Comsol intro, assignment questions) Feb 2, 8–10 2315D

Lecture Feb 2, 13–15 2345

Lab 1, group 2 (Comsol intro, assignment questions) Feb 4, 10–12 2315D

Lab 2, group 1 (assignment questions) Feb 5, 10–12 2315D

Lab 2, group 2 (assignment questions) Feb 5, 15–17 2315D

Seminar (presentation of results) Feb 8, 15–17 2345


Module outline and literature

1. General overview over linear elasticity (today)I O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method.

Volume 1: The Basis. Butterworth-Heinemann, Oxford, 2000.Chapters 1, 2, 3, 4.

2. Exemplary elasticity problem and discretization: The Plane StressProblem (Feb 2)

I C.A. Felippa’s lecture notes on “The Plane Stress Problem”, Chapter14. Available online athttp://www.colorado.edu/engineering/CAS/courses.d/IFEM.d

3. Going beyond Plain Stress: A survey on how to pursue generalproblems in structural mechanics (Feb 8, < 0.5 h)

I O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method.Volume 2: Solid Mechanics. Butterworth-Heinemann, Oxford, 2000.

I M.A. Crisfield. Non-linear finite element analysis of solids andstructures: essentials. Vol. 1. John-Wiley & Sons, 1991.


Introduction, history

History of structural mechanics



Truss structures [sv. fackverk]

E. g. a frame structure

I Modeled by “assembling” elementary“beam elements” [sv. bjälke] (if individ-ual beams not too long)

I Beam element: 12 degrees of free-dom/element (in the most general 3Dcase); 3 coordinates + 3 angles per beamend point

θ1

θ2

u1

u2

I Shape of the deformed beam described by a cubic (so-called Hermite)polynomial (4 parameters) in each of the 3 space coordinates

I Calculate displacement of beam parameters: → A discrete modelI balance forces and moments at joints [sv. länk]I solve sparse linear system



Beams

I Shape of long beams not accuratelydescribed by above model

I Idea: line up several elementary beamelements

I Enforce continuity for positions andangles at interfaces between elements(i.e. couple parameters)

I A mathematical continuum model: thediscrete elements are not “physical”



I Above ideas very old (∼1860)I 1950’s: triangular and rectangular elements developed for analysis of

e. g. airplane wingsI Idea by engineers in structural mechanics (at the same time when

computers became available)I Berkeley professor R. W. Clough coined the term “finite element”

I Later recognized as a variant of an old technique (“Ritz method”)I FEM as a general method for discretization of PDEs developed in the

60–70s (mathematics, scientific computing).I Mathematical foundation already in a paper by Courant (1942),

unknown by Clough et al.!



I Above ideas very old (∼1860)I 1950’s: triangular and rectangular elements developed for analysis of

e. g. airplane wingsI Idea by engineers in structural mechanics (at the same time when

computers became available)I Berkeley professor R. W. Clough coined the term “finite element”I Later recognized as a variant of an old technique (“Ritz method”)I FEM as a general method for discretization of PDEs developed in the

60–70s (mathematics, scientific computing).I Mathematical foundation already in a paper by Courant (1942),

unknown by Clough et al.!


Linear elasticity

The basics of linear elasticity


Linear elasticity

The general model for 3D linear static elasticityΩ: reference configurationf : body forceg: surface traction

Γ0: fixed (clamped) surfaceΓ1: surface subject to load; a “free”

surface has zero load

Hand-drawn picture goes here

ProblemUnder the steady loads f and g, determine

(i) the equilibrium displacement field u(x) with components[u(x)] =

(u1(x), u2(x), u3(x)

)Twith respect to the given reference configuration and

(ii) the stresses in the body.


Linear elasticity

The general model for 3D linear static elasticityΩ: reference configurationf : body forceg: surface traction

Γ0: fixed (clamped) surfaceΓ1: surface subject to load; a “free”

surface has zero load

Hand-drawn picture goes hereProblemUnder the steady loads f and g, determine

(i) the equilibrium displacement field u(x) with components[u(x)] =

(u1(x), u2(x), u3(x)

)Twith respect to the given reference configuration and

(ii) the stresses in the body.


Linear elasticity Forces and stresses

Forces

Two types of forces: body and contact forcesBody force: e. g. gravity, magnetic forces. If f is a force density (in

N/m3), the force (in N) on a given volume V is: F =∫Vf dΩ

Contact force: force between parts of material (inside or on the surface ofthe body)

S: a surface with a normal nForce on S from outside depends bothon position x ∈ S and on normal n:

F S =∫St(n,x) ds =

∫Sσ(x) · n ds

t(n,x): the Cauchy stress vectorσ(x): the stress tensor

n(x)

t(n,x)

S



The stress tensor components:

[σ] =

σ11 σ12 σ13

σ21 σ22 σ23

σ31 σ32 σ33

, σT = σ

I Symmetric matrix.Spectral theorem: 3 real eigenvalues, there is an orthogonal set ofeigenvectors

I The eigenvalues: principal stressesI The orthogonal eigenvectors: the principal directions of stressI Choose coordinate system aligned with the principal directions of

stressI Stress tensor becomes diagonalI Forces are directed orthogonal to the coordinate planes



Stress state examples

(i) Simple extension (uniaxial tension)

I Cylindrical specimen, arbitrarycross section with area A,

I Pulling at each side with evenlydistributed force f

I Stress tensor constant withcomponents

[σ] =

f/A 0 00 0 00 0 0

ff

x1

x2



(ii) Simple shear

I Plate specimenI Pulling along top and bottom

surface (area A) with evenlydistributed force f

I Stress tensor constant withcomponents

[σ] =

0

0

0 0 0

Hand-drawn picture goes here



(iii) Hydrostatic stress

I Spherical ball specimenI Body at the exposure of gas at

pressure pI Stress tensor constant with

components

[σ] = −p

1 0 00 1 00 0 1

p

x1

x2



Relevance of stress

I Stress is the variable that enters force balance, see slide 22.I Stress is a model to predict structural failure due to excessive loads,

which is a key task of mechanical engineering:choose the size of component parts such that the maximumstress for a given material (e.g. structural steel) is not exceededanywhere in the body (often add safety factors)



Stress analysis and visualization

I Stress tensor: 6 independent components ([σ] symmetric)I How to analyze & visualize stress field?I Concept 1: look at the Frobenius norm of the tensor

‖σ‖2F =3∑i=1

3∑j=1

σ2ij

Can show: ‖σ‖F independent of choice of coordinate system forcomponents σij

I However, ‖σ‖F not useful as a criterium for predicting structuralfailure (specimen breakdown)



Stress analysis II

I Usually: material very resistant to pure pressure loading (compression,expansion)

I Decompose stress tensor into an isotropic (pressure-like,volume-changing) σI and a deviatoric part σD:

σ =13

trσI︸︷︷︸σI

+(σ − 1

3trσI

)︸︷︷︸

σD

I The von Mises stress is defined as

σv = ‖σD‖F =∥∥∥∥σ − 1

3trσI

∥∥∥∥F

,

(Frobenius norm of the deviatoric part of the stress tensor)



Examples:I Simple extension:

[σ] =

σ 0 00 0 00 0 0

, [σI ] =σ

3

1 0 00 1 00 0 1

, [σD] =σ

3

2 0 00 −1 00 0 −1

so σv = σ

√2/3 ≈ 0.82σ.

I Simple shear:

[σ] =

0 σ 0σ 0 00 0 0

, [σI ] =

0 0 00 0 00 0 0

, [σD] =

0 σ 0σ 0 00 0 0

so σv = σ.

I Hydrostatic stress:

[σ] = −σ

1 0 00 1 00 0 1

, [σI ] = −σ

1 0 00 1 00 0 1

, [σD] =

0 0 00 0 00 0 0

so σv = 0.



Why stresses are relevant, revisited

I Model to predict structural failure due to excessive load:base criterion on von Mises stress – works well for steel, aluminium,copper, gold (“ductile” [sv. töjbar, duktil] materials)

I Von Mises criteria not appropriate to predict failure by crackpropagation (“fatigue” [sv. materialutmattning])

I For cracks, better to use criteria based on maximum principal stress


Linear elasticity Elasticity equations

Elasticity equations: integral and differential formFundamental axiom in mechanics: total forces on each volume V ⊂ Ωare in balance∫

∂Vσ · n ds +

∫Vf dΩ = 0 ∀V ⊂ Ω, (Integral form)

Divergence theorem (Gauss) ⇒∫V

(∇ · σ + f) dΩ = 0 ∀V ⊂ Ω,

thus,−∇ · σ = f in Ω,

u = 0 on Γ0,

σ · n = g on Γ1,

(Differential form)

where

[∇ · σ]i =3∑j=1

∂

∂xjσij .



Constitutive law

I Above problem statements incomplete (3 equations + b.c., 6 unknownstresses and 3 unknown displacements)

I Need a constitutive law, a relation between stress and displacement,σ = σ(u)

I Depends on material properties (rubber and steel at the same loadbehave very differently!)

I Assume:I Homogeneous material (same property at each point)I Isotropic material (same property in each direction)

I Small deformations (geometric linear problems)



Thenσ = λI tr(ε(u)) + 2µε(u), (constitutive relation)

where ε (= εT) is the strain tensor [sv. töjning]:

ε(u) =12(∇u+ (∇u)T

), (kinematic relation)

εij(u) =12

(∂ui∂xj

+∂uj∂xi

).

Parameters λ, µ: Lamé coefficients, usually expressed as

µ =E

2(1 + ν), λ =

Eν

(1 + ν)(1− 2ν)

E: Young modulus [sv. elasticitetsmodul]. Ex. around 200 GPa for steelν: Poisson ratio; 0 < ν < 1/2. Ex. around 0.3 for steel, almost 0.5 for

rubber (indicating incompressible material)



Visualization of quantities and equations in structuralmechanics: Tonti diagram

Prescribeddisplacements

u

Displacement (Dirichlet) BCs

u = uon Γ0

Displacements

u

Kinematicrelation:ε(u) = 1

2·`

∇u+ (∇u)T´

Strains ε

Constitutiverelation: ε = Cσ orσ = λI tr(ε) + 2µε

Stresses σ

Equilibrium(balance eq.):∇ · σ + f = 0

Body forces f

Force(Neumann)

BCs:σ · n = g = t

on Γ1

Prescribedtractions t

quantity Unknown fields

quantity Known fields


Energy and FEM

Linear elasticity from an energy point ofview – the weak form as a basis for FEM


Energy and FEM

Equations: energy viewpointAlternative derivation of the equations: energy principle (basis for FEM)

I Elastic energy stored in the body, the elastic strain energy:

S(u) =12

∫Ωε(u) : σ(u) dΩ :=

12

3∑i,j=1

∫Ωεijσij dΩ

cf. the energy stored in a spring: 12ε f (force f , stretched by ε)

I Work exerted on the body by external forces:

W (u) =∫

Ωu · f dΩ +

∫Γ1

u · g ds

I Total potential energy (fundamental axiom in mechanics)

T (u) = S(u)−W (u)


Energy and FEM

The principle of minimal potential energy

Equilibrium: displacement u minimizes total potential energy

T (u) =12

∫Ωε(u) : σ(u) dΩ︸︷︷︸S(u)

−∫

Ωu · f dΩ −

∫Γ1

u · g ds︸︷︷︸−W (u)

I W (u) is linear in uI For linear elastic materials, σ(u) is linear in u and thus S(u)

quadratic in u

For linear, homogeneous, and isotropic material:

T (u) =12

∫Ω

[λ(tr(ε(u))

)2 + 2µε(u) : ε(u)]dΩ −

∫Ω

u · f dΩ −∫

Γ1

u · g ds

=12a(u,u)− l(u)

where a is linear in both arguments and l is a linear form in u.


Energy and FEM

The principle of minimal potential energy

Equilibrium: displacement u minimizes total potential energy

T (u) =12

∫Ωε(u) : σ(u) dΩ︸︷︷︸S(u)

−∫

Ωu · f dΩ −

∫Γ1

u · g ds︸︷︷︸−W (u)

I W (u) is linear in uI For linear elastic materials, σ(u) is linear in u and thus S(u)

quadratic in u

For linear, homogeneous, and isotropic material:

T (u) =12

∫Ω

[λ(tr(ε(u))

)2 + 2µε(u) : ε(u)]dΩ −

∫Ω

u · f dΩ −∫

Γ1

u · g ds

=12a(u,u)− l(u)

where a is linear in both arguments and l is a linear form in u.Martin Kronbichler (TDB) FEM for CSM January 29, 2010 28 / 31

Energy and FEM Mathematics: Variational Principle

Variational ReformulationNow assume that u minimizes T . Let v be an arbitrary admissibledisplacement (satisfying v = 0 on Γ0). Perturb the minimum and define,for t ∈ R,

φ(t) = T (u+ tv) =12a(u+ tv,u+ tv)− l(u+ tv)

=12a(u,u) + l(u) + t

(a(v,u)− l(v)

)+t2

2a(v,v)

φ is convex quadratic in t since a is positive definite (a(v,v) > 0 whenv 6= 0)

Thus, φ is minimized iff φ′(0) = 0, that is,

a(v,u) = l(v) for each admissible v (1)



Relation to linear algebra

A quadratic function in Rn has the same structure as T (u):

J(u) =12uTAu− bTu ↔ 1

2a(u,u)− l(u)

Here, the variational principle reads for A symmetric and positive definite:

J minimized by u ⇔ ∇J(u) = 0 and12∇2J(u) pos. def.

⇔ Au = b, A pos. def.

⇔ vT(Au− b) = 0 for all v ∈ Rn, A pos. def.



Equation (1) in explicit form: the equilibrium displacement u satisfies∫Ω

[λ tr(ε(v)

)tr(ε(u)

)+ µε(v) : ε(u)

]dΩ =

∫Ωv · f dΩ +

∫Γ1

v · g ds

for each admissible v.

Above relation is a variational expression, called the principle of virtualwork in mechanics, and is the basis for the discretization with the FiniteElement Method (FEM)

In FEM, we want to fulfill (1) not for all admissible functions v, but onlyfor those described by a finite-dimensional subset V h of admissibledisplacements.


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