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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
Institute of Structural Engineering Method of Finite Elements II 1
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
Institute of Structural Engineering Method of Finite Elements II 2
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.
Institute of Structural Engineering Method of Finite Elements II 3
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
Institute of Structural Engineering Method of Finite Elements II 4
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
Institute of Structural Engineering Method of Finite Elements II 4
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.
Institute of Structural Engineering Method of Finite Elements II 6
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.
Institute of Structural Engineering Method of Finite Elements II 6
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
Institute of Structural Engineering Method of Finite Elements II 7
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
Institute of Structural Engineering Method of Finite Elements II 7
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.
Institute of Structural Engineering Method of Finite Elements II 8
The Newton-Raphson Algorithm
Graphical representation of the Newton-Raphson algorithm for themonodimensional case:
Institute of Structural Engineering Method of Finite Elements II 9
The Newton-Raphson Algorithm
Graphical representation of the Newton-Raphson algorithm for themonodimensional case:
... evaluations of tangent stiffness matrices may be reduced.
Institute of Structural Engineering Method of Finite Elements II 9
The Modified Newton-Raphson Algorithm
Graphical representation of the Newton-Raphson algorithm for themonodimensional case:
Institute of Structural Engineering Method of Finite Elements II 10
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.
Institute of Structural Engineering Method of Finite Elements II 11
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.
Institute of Structural Engineering Method of Finite Elements II 11
Material Nonlinearities
Hysteresis loop ©Wikipedia
ε = εe + εp
εe : elastic strain, εp : plastic strain.
Institute of Structural Engineering Method of Finite Elements II 12
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
Institute of Structural Engineering Method of Finite Elements II 13
Large Strain
Force balance:
dσ
dx+ P = 0
Constitutive model:
σ = Eε
Strain measure:
ε =du
dx→ linear
Institute of Structural Engineering Method of Finite Elements II 14
Large Strain
Force balance:
dσ
dx+ P = 0
Constitutive model:
σ = Eε
Strain measure:
ε =du
dx+
1
2
(du
dx
)2
→ Nonlinear!
Institute of Structural Engineering Method of Finite Elements II 14
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
Institute of Structural Engineering Method of Finite Elements II 15
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
Institute of Structural Engineering Method of Finite Elements II 15
Large displacements of a rigid beam
0 20 40 60 80ϕ
0
2
4
6
8
10Fl c
Small displacementsLarge displacements
Force versus rotation
Institute of Structural Engineering Method of Finite Elements II 16
Examples
Example #1: Forming of a metal profile
Example #2: Large deformation of a race-car tyre
Example #3: Buckling of a truss structure
Institute of Structural Engineering Method of Finite Elements II 17
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
Institute of Structural Engineering Method of Finite Elements II 18
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
Institute of Structural Engineering Method of Finite Elements II 18
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 ...
Institute of Structural Engineering Method of Finite Elements II 19
Nonlinear Dynamic FEA: Examples
Example #1: Shake table test of a concrete frame
Example #2: Crash test of a car
Institute of Structural Engineering Method of Finite Elements II 20
Thermal Stress Analysis
Rail buckling driven by restrained thermal expansion ©Wikipedia
Institute of Structural Engineering Method of Finite Elements II 21
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.
Institute of Structural Engineering Method of Finite Elements II 22
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.
Institute of Structural Engineering Method of Finite Elements II 22
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.
Institute of Structural Engineering Method of Finite Elements II 22
Thermal Stresses Analysis: Examples
Example #1: Buckling of a railway
Example #2: Spring forging
Institute of Structural Engineering Method of Finite Elements II 23
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.
Institute of Structural Engineering Method of Finite Elements II 24
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.
Institute of Structural Engineering Method of Finite Elements II 24
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.
Institute of Structural Engineering Method of Finite Elements II 24
Coupled Problems: Examples
Example #1: Bullet impact
Example #1: Fluid-structure interaction
Institute of Structural Engineering Method of Finite Elements II 25