The Finite Element Method for the Analysis of Non … Finite Element Method for the Analysis of...

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

Prof. Dr. Eleni Chatzi

Dr. Giuseppe Abbiati, Dr. Konstantinos Agathos

Lecture 1 - 21 September, 2017

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Learning Goals

understanding the limits of static linear finite element analysis

understanding the difference between linear and nonlineardifferential operators

introducing the Newton-Raphson method as a fundamental toolfor solving nonlinear algebraic equations

building a big picture of most common nonlinear problemsencountered in engineering

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Linear Differential Operators

Is a mapping acting on elements of a space to produce elements ofthe same space, e.g.: functions f : R → R.

Force balance:

Ed2u

dx2= L(u) = −P

Differential operator:

L(·) = Ed2(·)dx2

The differential operator is linear:

L (au1 + bu2) = aL (u1) + bL (u2)

The superposition principle holds.

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From Differential to Algebraic Equations

Linear differential equation:

L (u) = b, with L (u) =n∑

i=0

a2id2iu

dx2i

d judx j|0,L, j ∈ 0, ..., n − 1 essential BCs (primal quantities)

d judx j|0,L, j ∈ n, ..., 2n − 1 natural BCs (dual quantities)

_Linear algebraic equation:

Ku(n×nu)×(n×nu)

u...

d (n−1)udx(n−1)

(n×nu)×1

= Fu(n×nu)×1

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From Differential to Algebraic Equations

Linear differential equation:

L (u) = b, with L (u) =n∑

i=0

a2id2iu

dx2i

d judx j|0,L, j ∈ 0, ..., n − 1 essential BCs (primal quantities)

d judx j|0,L, j ∈ n, ..., 2n − 1 natural BCs (dual quantities)

_Linear algebraic equation:

Ku(n×nu)×(n×nu)

u...

d (n−1)udx(n−1)

(n×nu)×1

= Fu(n×nu)×1

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Nonlinear Differential Operator

Is a mapping acting on elements of a space to produce elements ofthe same space, e.g.: functions f : R → R.

Force balance (in presence of large strains):

L(u) = Ed2u

dx2+ E

du

dx

d2u

dx2= −P

Differential operator:

L(·) = Ed2(·)dx2

+ Ed(·)dx

d2(·)dx2

The differential operator L is nonlinear:

L(au1 + bu2) = Ed2(au1 + bu2)

dx2+ E

d(au1 + bu2)

dx

d2(au1 + bu2)

dx26=

aL(u1) + bL(u2)

... the superposition principle does no longer apply.Institute of Structural Engineering Method of Finite Elements II 5

Nonlinear Static FEA

The principle of virtual displacements facilitates the derivation ofdiscretized equations:

Ru (u) = Fu∫ΩδεTσdΩ =

∫ΩδuTPvol

u dΩ +

∫ΓδuTPsur

u ·−→dΓ

σ : stress state generated by volume Pvolu and surface Psur

u

loads.

δu and δε : compatible variations of displacement u and strain εfields.

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Nonlinear Static FEA

The principle of virtual displacements facilitates the derivation ofdiscretized equations:

Ru (u) = Fu∫ΩδεTσdΩ =

∫ΩδuTPvol

u dΩ +

∫ΓδuTPsur

u ·−→dΓ

σ : stress state generated by volume Pvolu and surface Psur

u

loads.

δu and δε : compatible variations of displacement u and strain εfields.

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Nonlinear Static FEA

Nonlinear differential equation:

L (u) = b

d judx j|0,L, j ∈ 0, ..., n − 1 essential BCs (primal quantities)

d judx j|0,L, j ∈ n, ..., 2n − 1 natural BCs (dual quantities)

_Nonlinear algebraic equation:

Ru(n×nu)×1

u

...d (n−1)udx(n−1)

(n×nu)×1

= Fu(n×nu)×1

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Nonlinear Static FEA

Nonlinear differential equation:

L (u) = b

d judx j|0,L, j ∈ 0, ..., n − 1 essential BCs (primal quantities)

d judx j|0,L, j ∈ n, ..., 2n − 1 natural BCs (dual quantities)

_Nonlinear algebraic equation:

Ru(n×nu)×1

u

...d (n−1)udx(n−1)

(n×nu)×1

= Fu(n×nu)×1

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Nonlinear Static FEA

The solution of the governing BVs turns into a minimization problemthat can be solved with standard optimization tools:

un = argminun

(Fu (tn)− Ru (un))

Key idea:

Ru (u + ∆u) ≈ Ru (u) + Ku∆u

where Ku = ∇uRu is the so called tangent stiffness matrix.

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The Newton-Raphson Algorithm

Graphical representation of the Newton-Raphson algorithm for themonodimensional case:

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The Newton-Raphson Algorithm

Graphical representation of the Newton-Raphson algorithm for themonodimensional case:

... evaluations of tangent stiffness matrices may be reduced.

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The Modified Newton-Raphson Algorithm

Graphical representation of the Newton-Raphson algorithm for themonodimensional case:

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Material Nonlinearities

(a) elastic (brittle behavior materials); (b) increase of stiffness asload increases (pneumatic structures); (c) commonly observedbehavior (concrete or steel) ©C.A. Felippa: A tour of NonlinearAnalysis.

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Material Nonlinearities

(a) elastic (brittle behavior materials); (b) increase of stiffness asload increases (pneumatic structures); (c) commonly observedbehavior (concrete or steel) ©C.A. Felippa: A tour of NonlinearAnalysis.

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Material Nonlinearities

Hysteresis loop ©Wikipedia

ε = εe + εp

εe : elastic strain, εp : plastic strain.

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Material Nonlinearities

Von Mises’s and Tresca’s yielding surfaces ©Wikipedia

if f (σ, σy ) < 0→ σ = E ε

if f (σ, σy ) = 0→

σ = E (ε− εp), if f (σ, σy (α)) = 0

εp = λ∇fα = λp

f = 0

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Large Strain

Force balance:

dσ

dx+ P = 0

Constitutive model:

σ = Eε

Strain measure:

ε =du

dx→ linear

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Large Strain

Force balance:

dσ

dx+ P = 0

Constitutive model:

σ = Eε

Strain measure:

ε =du

dx+

1

2

(du

dx

)2

→ Nonlinear!

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Large displacements of a rigid beam

Small displacements

Equilibrium equation:

Fl = cϕ

Large displacements

Equilibrium equation:

Fl cos (ϕ) = cϕ

Example taken from: ”Nonlinear Finite Element Methods” by P. Wriggers,Springer, 2008

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Large displacements of a rigid beam

Small displacements

Equilibrium equation:

Fl = cϕ

Large displacements

Equilibrium equation:

Fl cos (ϕ) = cϕ

Example taken from: ”Nonlinear Finite Element Methods” by P. Wriggers,Springer, 2008

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Large displacements of a rigid beam

0 20 40 60 80ϕ

0

2

4

6

8

10Fl c

Small displacementsLarge displacements

Force versus rotation

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Examples

Example #1: Forming of a metal profile

Example #2: Large deformation of a race-car tyre

Example #3: Buckling of a truss structure

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Nonlinear Dynamic FEA

Nonlinear differential equation:

L (u, u, u) = bd judx j|0,L, j ∈ 0, ..., n − 1 essential BCs

d judx j|0,L, j ∈ n, ..., 2n − 1 natural BCs

d ju

dx j|t=0, j ∈ 0, ..., n − 1 Initial Conditions (ICs) on essential BCs

_Nonlinear algebraic equation:

Mu(n×nu)×(n×nu)

u...

d (n−1)udx(n−1)

(n×nu)×1

+ Ru(n×nu)×1

u

...d (n−1)udx(n−1)

(n×nu)×1

,

u...

d (n−1)udx(n−1)

(n×nu)×1

= Fu(n×nu)×1

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Nonlinear Dynamic FEA

Nonlinear differential equation:

L (u, u, u) = bd judx j|0,L, j ∈ 0, ..., n − 1 essential BCs

d judx j|0,L, j ∈ n, ..., 2n − 1 natural BCs

d ju

dx j|t=0, j ∈ 0, ..., n − 1 Initial Conditions (ICs) on essential BCs

_Nonlinear algebraic equation:

Mu(n×nu)×(n×nu)

u...

d (n−1)udx(n−1)

(n×nu)×1

+ Ru(n×nu)×1

u

...d (n−1)udx(n−1)

(n×nu)×1

,

u...

d (n−1)udx(n−1)

(n×nu)×1

= Fu(n×nu)×1

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Nonlinear Dynamic FEA

Also in this case, the solution of the governing BVs turns into aminimization problem that can be solved with standard optimizationtools:

un, ˙un, ¨un = argminun,un,un

(Fu (tn)− Ru (un, un)−Muun)

In detail:

A minimization problem must be solved at each load step.

The previous time step solution is taken as starting point.

Apparently the number of unknowns increased ...

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Nonlinear Dynamic FEA: Examples

Example #1: Shake table test of a concrete frame

Example #2: Crash test of a car

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Thermal Stress Analysis

Rail buckling driven by restrained thermal expansion ©Wikipedia

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Linear Thermoelasticity

ε = εe + εT

εe : elastic strain, εT : thermal strain.

εxx =1

E(σxx − ν (σyy + σzz)) + α (T − T0)

εyy =1

E(σyy − ν (σzz + σxx)) + α (T − T0)

εzz =1

E(σzz − ν (σxx + σyy )) + α (T − T0)

εxy =σxy2G

εxz =σxz2G

εyz =σyz2G

α : coef. of thermal expansion; T : abs. temperature; T0 : ref.temperature.

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Linear Thermoelasticity

ε = εe + εT

εe : elastic strain, εT : thermal strain.

εxx =1

E(σxx − ν (σyy + σzz)) + α (T − T0)

εyy =1

E(σyy − ν (σzz + σxx)) + α (T − T0)

εzz =1

E(σzz − ν (σxx + σyy )) + α (T − T0)

εxy =σxy2G

εxz =σxz2G

εyz =σyz2G

α : coef. of thermal expansion; T : abs. temperature; T0 : ref.temperature.

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Linear Thermoelasticity

ε = εe + εT

εe : elastic strain, εT : thermal strain.

εxx =1

E(σxx − ν (σyy + σzz)) + α (T − T0)

εyy =1

E(σyy − ν (σzz + σxx)) + α (T − T0)

εzz =1

E(σzz − ν (σxx + σyy )) + α (T − T0)

εxy =σxy2G

εxz =σxz2G

εyz =σyz2G

α : coef. of thermal expansion; T : abs. temperature; T0 : ref.temperature.

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Thermal Stresses Analysis: Examples

Example #1: Buckling of a railway

Example #2: Spring forging

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Coupled Problems

a) Impact on the containment vessel of a nuclear reactor casing; b) normalload applied at the point of impact; c) geometry of the structure.

Combescure, A., Gravouil, A. (2002). A numerical scheme to couplesubdomains with different time-steps for predominantly linear transientanalysis. Computer Methods in Applied Mechanics and Engineering,191(1112), 11291157.

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Coupled Problems

d) mesh of the structure; e) mesh of the interfaces.

Combescure, A., Gravouil, A. (2002). A numerical scheme to couplesubdomains with different time-steps for predominantly linear transientanalysis. Computer Methods in Applied Mechanics and Engineering,191(1112), 11291157.

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Coupled Problems

Task sequence of the partitioned algorithm.

MA

u uAn+1 + RA

u

(uAn+1, u

An+1

)= FA

u (tn+1) + LAT

Λn+1

MBu uB

n+ jss

+ RBu

(uBn+ j

ss

, uBn+ j

ss

)= FB

u

(tn+ j

ss

)+ LBT

Λn+ jss

LAuAn+ j

ss

+ LB uBn+ j

ss

= 0

Combescure, A., Gravouil, A. (2002). A numerical scheme to couplesubdomains with different time-steps for predominantly linear transientanalysis. Computer Methods in Applied Mechanics and Engineering,191(1112), 11291157.

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Coupled Problems: Examples

Example #1: Bullet impact

Example #1: Fluid-structure interaction

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