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The Finite Element Method for the Analysis ofNon-Linear and Dynamic Systems

Prof. Dr. Eleni Chatzi

Lecture 1 - 20 September, 2017

Institute of Structural Engineering Method of Finite Elements II 1

Course Information

InstructorsProf. Dr. Eleni Chatzi, email: [email protected]. Giuseppe Abbiati, email: [email protected]. Konstantinos Agathos, email: [email protected]

AssistantAdrian Egger, HIL E19.5, email: [email protected]

Course WebsiteLecture Notes and Demos will be posted in:http://www.chatzi.ibk.ethz.ch/education/method-of-finite-elements-ii.html

Performance Evaluation - Final Project (100%)Topic Categories announced on 12.10.2017 - You are free to shape yourown project!

Computer Labs are already be pre-announced and it is advised to bring alaptop along for those sessions

Help us Structure the Course!

Participate in our online survey: http://goo.gl/forms/ws0ASBLXiY


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Course Information

Suggested Reading

“Nonlinear Finite Elements for Continua and Structures”, by T.Belytschko, W. K. Liu, and B. Moran, John Wiley and Sons, 2000

“The Finite Element Method: Linear Static and Dynamic FiniteElement Analysis”, by T. J. R. Hughes, Dover Publications, 2000

“Nonlinear finite element analysis of solids and structures”,Crisfield,M.A., Remmers, J.J. and Verhoosel, C.V., John Wiley Sons, 2012

“Computational methods for plasticity: theory and applications”, DeSouza Neto, E.A., Peric, D. and Owen, D.R., 2011

Lecture Notes by Carlos A. FelippaNonlinear Finite Element Methods (ASEN 6107): http://www.

colorado.edu/engineering/CAS/courses.d/NFEM.d/Home.html


http://www.colorado.edu/engineering/CAS/courses.d/NFEM.d/Home.html

http://www.colorado.edu/engineering/CAS/courses.d/NFEM.d/Home.html

Learning Goals

By the end of the lecture, the goal is to learn:

When is a problem nonlinear & what are types of nonlinearity?

What types of nonlinearity exist?

What are the differences, but also similarities, between linearand nonlinear problems?

How to formulate & solve nonlinear problems

What algorithms to use & what are typical problems?


Learning Goals

What we know so far

We can recognize linearity

As the process following the fundamental rule

output = constant× input

F

L

θ

When is this linear?

when M = EI = EI φw

M

φEI

M=FL


Learning Goals

Beyond Linearity

F

L

θ

What if ?

We have excessive stressesin a given section?w

M

φEI

plasticityM=FL

Material Nonlinearity

F

L

θ

What if ?

We have excessive stressesin a given section?w

M=FL

Fracture Phenomena


Learning Goals

Beyond Linearity

F

L

M=FL

θ

& how does the structure actually deform?w

F

M = ?

ΔLLarge Displacements

F

L

M=FL

θ

& how does the structure actually deform when heated?w

Thermomechanics Problems ©khanacademy.org


Significance of the Lecture

The importance of Modeling

Models are abstract representations of the real world

Simulation models are extensively used in civil engineering practice. Suchmodels allow the user to

understand structural system performance

predict structural behavior

diagnose damage

optimize design, etc


Significance of the Lecture

The importance of Modeling

Models are abstract representations of the real world

Structures may fail because

The models used for their design do not properly represent thecomplexity of the physics

Overly simplified models may not predict response under extreme loads

The values of the input parameters have not been selected properly


Course Outline

FEM II - Lecture Schedule

Introduction - From Linear to Nonlinear Considerations

Block #1

Material Nonlinearity I - Constitutive ModelingMaterial Nonlinearity II - FE Implementation

Computer Lab I

Block #2

Large Deformations I Large Deformations II

Block #3

Dynamics I Nonlinear Dynamics II


Course Outline

FEM II - Lecture Schedule

Computer Lab II

Block #4

Fracture I - General principlesFracture II - The eXtended Finite Element Method

Block #5

Thermomechanics ISpecial Topics - Coupling/Penalty Methods (Lagrange Multipliers),Contact, Imposition of Essential BCs

Computer Lab III


Review of the Finite Element Method (FEM)

FEM: Numerical Technique for approximating the solution of continuoussystems.The MFE is based on the physical discretization of the observed domain,thus reducing the number of the degrees of freedom; and shifting from ananalytical to a numerical formulation.

Continuous Discrete

h2

h1

Permeable Soil

Flow of water

L

Impermeable Rock

dx

dy

q|y

q|x+dx

q|y+dy

q|x

k(∂2φ∂2x + ∂2φ

∂2y

)= 0

Laplace EquationF = KX

Direct Stiffness Method


Review of the Linear Finite Element Method (FEM)

How is the Physical Problem formulated?

A linear elastic system under specific loads and constraints at itsboundaries can be described by the following fundamental equations:

Linear balance (Equilibrium) equation.

Linear constitutive model.

Kinematic Equation, or Static balance equation or, if we considermodal analysis as an equivalent static analysis, at least stationary.

Example of a bridge FE model.


Review of the Linear Finite Element Method (FEM)

The Strong Form of a Linear Problem

On the basis of the previous fundamental euations, a governingdifferential equation, also known as the strong form of the problemmay be extracted:

1 Equilibrium Equationsex. f (x) = σA = q(L− x)⇒dσdx + q/A = 0

2 Constitutive Modelex. σ = Eε

3 Kinematics Relationship

ex. ε =du

dx

x

q

q

L-x

Axial bar Example

f(x)


Examples of Linear Balance Equations

Mechanical

x

q

Axial bar Example

Force balance:

dσ

dx+ P = 0

u : displacement [m]

σ : stress [Pa]

P : body load[Nm3

]

Thermomechanical

φ(x)>0

θ1

θ2

θ(x)

Heat flux balance:

dφ

dx+ R = 0

θ : temperature [K ]

φ : heat flux[Wm2

]R : heat source

[Wm3

]Institute of Structural Engineering Method of Finite Elements II 15

Examples of Linear Constitutive Models

Mechanical

Hooke’s law:

σ = Edu

dx


x : space coordinate [m]dudx : strain

[mm

]σ : stress [Pa]

E : Young modulus [Pa]

Thermomechanical

Fourier’s law:

φ = −k dθdx


x : space coordinate [m]dθdx : temperature grad.

[Km

]φ : heat flux

[Wm2

]k : thermal cond.

[WmK

]


Examples of Problems’ Equations

Mechanical

Force balance:

Ed2u

dx2+ P = 0



P : body load[Nm3

]

Thermomechanical

Heat flux balance:

kd2θ

dx2+ R = 0


k : thermal cond.[WmK

]R : heat source

[Wm3

]


Boundary Value Problems (BVPs)

The Strong Form

Mechanical BVP

Ed2u

dx2+ P = 0

u|0,L = u0,L

dudx |0,L =

du0,L

dx



P : body load[Nm3

]

Thermal BVP

kd2θ

dx2+ R = 0

θ|0,L = θ0,L

dθdx |0,L =

dθ0,L

dx


k : thermal cond.[WmK

]R : heat source

[Wm3

]Institute of Structural Engineering Method of Finite Elements II 18

Boundary Value Problems (BVPs)

The Strong Form

Mechanical BVP

Ed2u

dx2+ P = 0

u|0,L = u0,L

dudx |0,L =

du0,L

dx

FEM Formulation:

Kunu×nu

unu×1

= Funu×1

Thermal BVP

kd2θ

dx2+ R = 0

θ|0,L = θ0,L

dθdx |0,L =

dθ0,L

dx

FEM Formulation:

Kθnθ×nθ

θnθ×1

= Fθnθ×1


Weak Form - 1D FEM

So how to solve the Strong Form

The strong form requires strong continuity on the dependent fieldvariables (usually displacements). Whatever functions define thesevariables have to be differentiable up to the order of the PDE thatexist in the strong form of the system equations. Obtaining theexact solution for a strong form of the system equation is a quitedifficult task for practical engineering problems.

The finite difference method can be used to solve the systemequations of the strong form and obtain an approximate solution.However, this method usually works well for problems with simpleand regular geometry and boundary conditions.

Alternatively we can use the finite element method on a weakform of the system. This weak form is usually obtained throughenergy principles which is why it is also known as variational form.


Weak Form - 1D FEM

From Strong Form to Weak form

Three are the approaches commonly used to go from strong to weakform:

Principle of Virtual Work

Principle of Minimum Potential Energy

Methods of weighted residuals (Galerkin, Collocation, LeastSquares methods, etc)

Visit our Course page on Linear FEM for further details:http://www.chatzi.ibk.ethz.ch/education/

method-of-finite-elements-i.html


http://www.chatzi.ibk.ethz.ch/education/method-of-finite-elements-i.html

http://www.chatzi.ibk.ethz.ch/education/method-of-finite-elements-i.html

Weak Form - 1D FEM

From Strong Form to Weak form - Approach #1

Principle of Virtual Work

For any set of compatible small virtual displacements imposed on the bodyin its state of equilibrium, the total internal virtual work is equal to thetotal external virtual work.The principle of virtual displacements facilitates the derivation ofdiscretized equations:

Ru (u) = Fu∫Ω

δεTσdΩ =

∫Ω

δuTPvolu dΩ +

∫Γ

δuTPsuru ·−→dΓ

where

σ : stress state generated by volume Pvolu and

surface Psuru loads.

δu and δε : compatible variations ofdisplacement u and strain ε fields.


Weak Form - 1D FEM

From Strong Form to Weak form - Approach #3

Consider the following (1D) generic strong form:

d

dx

(c(x)

du

dx

)+ f(x) = 0

− c(0)d

dxu(0) = C1 (Neumann BC) & u(L) = 0 (Dirichlet BC)

Galerkin’s Method

Given an arbitrary weight function w , where

S = u|u ∈ C0, u(l) = 0,S0 = w |w ∈ C0,w(l) = 0C0 is the collection of all continuous functions.Multiplying by w and integrating over Ω∫ l

0

w(x)[(c(x)u′(x))′ + f (x)]dx = 0

[w(0)(c(0)u′(0) + C1] = 0


Weak Form - 1D FEM

Using the divergence theorem (integration by parts) we reduce theorder of the differential:

∫ l

0wg ′dx = [wg ]l0 −

∫ l

0gw ′dx

The weak form is then reduced to the following problem. Also, inwhat follows we assume constant properties c(x) = c = const.

Find u(x) ∈ S such that:

∫ l

0w ′cu′dx =

∫ l

0wfdx + w(0)C1

S = u|u ∈ C0, u(l) = 0S0 = w |w ∈ C0,w(l) = 0


Weak Form

Notes:

1 Natural (Neumann) boundary conditions, are imposed on thesecondary variables like forces and tractions.For example, ∂u

∂y (x0, y0) = u0.

2 Essential (Dirichlet) or geometric boundary conditions, are imposedon the primary variables like displacements.For example, u(x0, y0) = u0.

3 A solution to the strong form will also satisfy the weak form, but notvice versa. Since the weak form uses a lower order of derivatives it canbe satisfied by a larger set of functions.

4 For the derivation of the weak form we can choose any weightingfunction w , since it is arbitrary, so we usually choose one that satisfieshomogeneous boundary conditions wherever the actual solutionsatisfies essential boundary conditions. Note that this does not holdfor natural boundary conditions!


FE formulation: Reminder

Let’s recall the steps to the Galerking approach

1 Step #1: - Discretization

2 Step #2: - Aprroximate u(x)

3 Step #3: - Aprroximate w(x)

4 Step #4: - Substitute in the weak form w(x)

5 Step #5: - Obtain the discrete system matrices


FE formulation: Discretization

How to derive a solution to the weak form?

Step #1:Follow the FE approach:Divide the body into finite elements, e, connected to each otherthrough nodes.

𝑥1𝑒 𝑥2𝑒

𝑒

Then break the overall integral into a summation over the finiteelements:

∑e

[∫ xe2

xe1

w ′cu′dx −∫ xe2

xe1

wfdx − w(0)C1

]= 0


1D FE formulation: Galerkin Method

Step #2: Approximate the continuous displacement field u(x) using adiscrete equivalent:

Galerkin’s method assumes that the approximate (or trial) solution, u, canbe expressed as a linear combination of the nodal point displacements ui ,where i refers to the corresponding node number.

u(x) ≈ uh(x) =∑i

Ni (x)ui = N(x)u

where bold notation signifies a vector and Ni (x) are the shape functions.In fact, the shape function can be any mathematical formula that helps usinfer (interpolate) what happens at points that lie within the nodes of themesh.In the 1-D case that we are using as a reference, Ni (x) are defined as 1stdegree polynomials indicating a linear interpolation.


1D FE formulation: Galerkin’s Method

Shape function Properties:

Bounded and Continuous

One for each node

Nei (xej ) = δij , where

δij =

1 if i = j0 if i 6= j

The shape functions can be written as piecewise functions of the xcoordinate:

Ni (x) =

x − xi−1

xi − xi−1, xi−1 ≤ x < xi

xi + 1− x

xi+1 − xi, xi ≤ x < xi+1

0, otherwise

This is not a convenient notation.Instead of using the global coordinatex , things become simplified whenusing coordinate ξ referring to thelocal system of the element (see page25).



Step #3: Approximate w(x) using a discrete equivalent:

The weighting function, w is usually (although not necessarily) chosen tobe of the same form as u

w(x) ≈ wh(x) =∑i

Ni (x)wi = N(x)w

i.e. for 2 nodes:N = [N1 N2] u = [u1 u2]T w = [w1 w2]T

Alternatively we could have a Petrov-Galerkin formulation, where w(x) isobtained through the following relationships:

w(x) =∑i

(Ni + δhe

σ

dNi

dx)wi

δ = coth(Pee

2)− 2

Peecoth =

ex + e−x

ex − e−x



Step #4: Substituting into the weak formulation and rearrangingterms we obtain the following in matrix notation:∫ l

0

w ′cu′dx −∫ l

0

wfdx − w(0)C1 = 0⇒∫ l

0

(wTNT )′c(Nu)′dx −∫ l

0

wTNT fdx −wTN(0)TC1 = 0

Since w, u are vectors, each one containing a set of discrete valuescorresponding at the nodes i , it follows that the above set of equations canbe rewritten in the following form, i.e. as a summation over the wi , uicomponents (Einstein notation):

∫ l

0

(∑i

uidNi (x)

dx

)c

∑j

wjdNj(x)

dx

dx

−∫ l

0

f∑j

wjNj(x)dx −∑j

wjNj(x)C1

∣∣∣∣∣∣x=0

= 0



This is rewritten as,

∑j

wj

[∫ l

0

(∑i

cuidNi (x)

dx

dNj(x)

dx

)− fNj(x)dx + (Nj(x)C1)|x=0

]= 0

The above equation has to hold ∀wj since the weighting function w(x) isan arbitrary one. Therefore the following system of equations has to hold:∫ l

0

(∑i

cuidNi (x)

dx

dNj(x)

dx

)− fNj(x)dx + (Nj(x)C1)|x=0 = 0 j = 1, ..., n

After reorganizing and moving the summation outside the integral, thisbecomes:

∑i

[∫ l

0

cdNi (x)

dx

dNj(x)

dx

]ui =

∫ l

0

fNj(x)dx + (Nj(x)C1)|x=0 = 0 j = 1, ..., n



We finally obtain the following discrete system in matrix notation:

Ku = f

where writing the integral from 0 to l as a summation over thesubelements we obtain:

K = AeKe −→ Ke =

∫ xe2

xe1

[dN(x)

dx

]TcdN(x)

dxdx =

∫ xe2

xe1

BT cBdx

f = Aefe −→ fe =

∫ xe2

xe1

NT fdx + NTh|x=0

where A is not a sum but an assembly, and , x denotesdifferentiation with respect to x .

In addition, B =dN(x)

dxis known as the strain-displacement matrix.


Transitioning to the Nonlinear Case

How to treat the Nonlinear Case?

• Linear Problems

• Stiffness matrix K is constant

• If the load is doubled, displacement is doubled, too

• Superposition is possible

• Nonlinear Problems

• How to find d for a given F?

⋅ = =or ( )K d F P d F

+ = +α = α = α

1 2 1 2( ) ( ) ( )( ) ( )

P d d P d P dP d P d F

= ≠( ) , (2 ) 2P d F P d F

F

ddi

KT

2F

2dd

KF

source: Nam-Ho Kim, EGM6352, Lecture NotesInstitute of Structural Engineering Method of Finite Elements II 33

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