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• The FE technique is optimal in some sense
• “Best approximation” and error estimative
• Establish convergence of the technique
MIXED AND PENALTY METHODSPreliminaries
Instituto Tecnológico de Aeronáutica
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The exact problem may be stated as: find u∈Ssuch that for all w∈V
MIXED AND PENALTY METHODSBest approximation
Γ+= ),(),(),( hwfwuwa
u satisfies the essential (geometric) boundary conditions
w satisfies the corresponding homogeneous essential boundary conditions
Instituto Tecnológico de Aeronáutica
AE-245 5
The approximate problem may be stated as: find uh∈Sh such that for all wh∈Vh
MIXED AND PENALTY METHODSBest approximation
Γ+= ),(),(),( hwfwuw hhhha
uh satisfies the essential (geometric) boundary conditions
wh satisfies the corresponding homogeneous essential boundary conditions
Instituto Tecnológico de Aeronáutica
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Assumptions
MIXED AND PENALTY METHODSBest approximation
1. Sh ⊂ Sand Vh ⊂ V
2. a(•,•), (•,•) and (•,•)Γ are symmetric and bilinear forms
3. a(•,•) and ||•||m define equivalent norms where
( )2/1
...,...,,, ...
Ω+++= ∫
Ω
dwwwwww ljkiljkijijiiimw
Instituto Tecnológico de Aeronáutica
AE-245 7
MIXED AND PENALTY METHODSBest approximation
Theorem: let e = uh – u denote the error in the finite element approximation
hhhh
h
Saa
a
∈−−≤
=
UuUuUee
ew
allfor ),(),( b.
0),( a.
Instituto Tecnológico de Aeronáutica
AE-245 8
MIXED AND PENALTY METHODSBest approximation
Pythagorean theorem: assume Sh = Vh.
),(),(),( eeuuuu aaa hh +=
),(),(
),(),(),(
uuuu
uuuuee
aa
aaahh
hh
≤
−=Consequences:
The approximate solution underestimates the strain energy
Instituto Tecnológico de Aeronáutica
AE-245 9
MIXED AND PENALTY METHODSBest approximation
The principle of minimum potential energy
wuU εε +=
where ε ∈ ℜ. Every member of S can be represented in the form above for some w ∈ V and ε ∈ ℜ.
Instituto Tecnológico de Aeronáutica
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MIXED AND PENALTY METHODSBest approximation
Define potential energy:
Γ−−= ),(),(),(2
1)( hUfUUUU εεεεε aI
The potential energy is stationary if and only if the variational equation is satisfied
The potential energy is minimized at u.
The approximate solution overestimates I
Instituto Tecnológico de Aeronáutica
AE-245 11
MIXED AND PENALTY METHODSEffect of numerical quadrature
Theory developed assumes that all integrals are evaluated exactly
This is a serious limitation for isoparametricelements because of numerical integration
Number of Gaussian points should be carefully determined
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MIXED AND PENALTY METHODSSummary
• In practice results obtained via Galerkinformulation are very good.
• The best approximation property is a guarantee of a reasonable convergence.
• There are problems: what if the best approximation is not so good? That is, what if there is no reasonable uh that approximates u?
Instituto Tecnológico de Aeronáutica
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MIXED AND PENALTY METHODSIncompressible elasticity and Stokes flow
• Motions that preserve volume locally
• After deformation each elementary portion of the medium has the same volume as before deformation
• In elasticity, materials with ν approaching 1/2 are often modeled as incompressible.
Instituto Tecnológico de Aeronáutica
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MIXED AND PENALTY METHODSIncompressible elasticity
)( ,,, ijjiijkkij uuu ++= µδλσ
ννµλ21
2
−= ∞=
→λ
ν 2/1lim
The constitutive equation must be reformulated
)( ,, ijjiijij uup ++−= µδσ
)21)(1( νννλ
−+= E
GE =+
=)1(2 ν
µ
Instituto Tecnológico de Aeronáutica
AE-245 15
MIXED AND PENALTY METHODSIncompressible elasticity
p is the hydrostatic pressure that is an additional variable. The additional equation is the incompressibility condition:
0)(div , == kkuu
Instituto Tecnológico de Aeronáutica
AE-245 16
MIXED AND PENALTY METHODSIncompressible elasticity
Given fi, gi and hi find ui and pressure p such that
)(with
on
on
in 0
in 0
,,
,
,
ijjiijij
hiijij
giii
ii
ijij
uup
hn
gu
u
f
++−=
Γ=Γ=Ω=Ω=+
µδσσ
σ
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MIXED AND PENALTY METHODSStokes flow
Equation identical to the incompressible elasticity situation
u is the velocity of the fluid and µ is the fluid viscosity
Stokes flow governs highly viscous flows
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MIXED AND PENALTY METHODSStokes flow
x
ylid
vx = 1, vy = 0
vx = 0vy = 0
vx = 0vy = 0
vx = 0, vy = 0
-300
-200
-100
0
100
p
0.000.20
0.400.60
0.801.00
x
0.000.20
0.400.60
0.801.00
y
x
y
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.10.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0vx
0.9370.8750.8120.7500.6870.6250.5620.5000.4370.3750.3120.2500.1870.1250.062
-0.000-0.063-0.125-0.188-0.250
x
y
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.10.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0vy
0.6010.5510.5020.4520.4020.3520.3020.2530.2030.1530.1030.0540.004
-0.046-0.096-0.145-0.195-0.245-0.295-0.345
0
2
=⋅∇=∇+∇−
v
bv pν
Instituto Tecnológico de Aeronáutica
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MIXED AND PENALTY METHODSMixed and penalty methods
Matrix FE problem:
fKd =Alternative formulation: minimize
fdKdd
d TT
F −=2
)(
Total potential energy function
Instituto Tecnológico de Aeronáutica
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MIXED AND PENALTY METHODSMixed and penalty methods
One parameter family of displacements:
0) (0
=
+=ε
εε
cdFd
d
cd ε+ whereε ∈ ℜ and c is arbitrary
If F is minimized by d
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MIXED AND PENALTY METHODSMixed and penalty methods
)() (0
fKdccd −=
+=
TFd
d
εε
ε
Minimization:
c is arbitrary fKd =
Instituto Tecnológico de Aeronáutica
AE-245 22
MIXED AND PENALTY METHODSLagrange multiplier method
Consider the function
where
)()(),(* ddd mGFmF +=
0)( =−= gG TQd1d
Instituto Tecnológico de Aeronáutica
AE-245 23
MIXED AND PENALTY METHODSLagrange multiplier method
0)()(), (0
* =−+−+=
++=
glmlmFd
d TQQ
T d1f1Kdccdε
εεε
Minimization:
c is arbitrary
l is arbitrary
=
gmTQ
Q fd01
1K
Interpretation: m is a force that must exist in order to maintain the constraint valid.
Instituto Tecnológico de Aeronáutica
AE-245 24
MIXED AND PENALTY METHODSLagrange multiplier method
Mixed equation:
=
gmTQ
Q fd01
1K
The equation above is mixed since it contains displacement and force unknowns.
The “stiffness” matrix is not positive-definite anymore
Instituto Tecnológico de Aeronáutica
AE-245 25
MIXED AND PENALTY METHODSPenalty method
Approximation to the Lagrange multiplier method
)(2
)()( 2** ddd Gk
FF +=
wherek is a large known number. Minimization yields
QTQQ kgk 1fd11K +=+ )( gdQ
k=
∞→lim
Instituto Tecnológico de Aeronáutica
AE-245 26
MIXED AND PENALTY METHODSPenalty method
gdQk
=∞→
lim
• Approximation of the incompressible case by a slightly compressible formulation
• The larger k the better
• Too large k may lead to numerical instabilities
Instituto Tecnológico de Aeronáutica
AE-245 27
MIXED AND PENALTY METHODSIncompressible elasticity
Formulation that solves both compressible and incompressible elasticity problems.
λ
µδσp
u
uup
ii
ijjiijij
+=
++−=
,
,,
0
)(
Instituto Tecnológico de Aeronáutica
AE-245 28
MIXED AND PENALTY METHODSIncompressible elasticity
Strong form
Given fi, gi and hi find ui and p such that
)( where
on
on
in 0/
in 0
,,
,
,
ijjiijij
hiijij
giii
ii
ijij
uup
hn
gu
pu
f
++−=
Γ=Γ=Ω=+Ω=+
µδσσ
λσ
Instituto Tecnológico de Aeronáutica
AE-245 29
MIXED AND PENALTY METHODSIncompressible elasticity
Weak form
Given fi, gi and hi find ui∈Si and p∈P such that for all wi∈Vi and q∈P
∑ ∫∫
∫∫
= ΓΩ
ΩΩ
Γ+Ω
=Ω+−Ω+
3or 2
1
,,, )/()(2
1
iiiii
iiijijji
hi
dhwdfw
dpuqdww λσ
Instituto Tecnológico de Aeronáutica
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MIXED AND PENALTY METHODSIncompressible elasticity
Weak form: variational equation
Γ+=
+−− ),(),()(div,)),(div(),( hwfwuwuwλp
qpa
∫Ω
Ω++= duucwwa kllkijklijji )(2
1)(
2
1),( ,,,,uw
)( jkiljlikijklc δδδδµ +=
Instituto Tecnológico de Aeronáutica
AE-245 31
MIXED AND PENALTY METHODSIncompressible elasticity
Galerkin formulation
Given f, g and h find uh= vh + gh ∈ Sh and ph∈Ph
such that for all wh∈Vh and qh∈Ph
( )))(div,(),(),(),(
/)(div,)),(div(),(hhhhhh
hhhhhhh
qa
pqpa
ggwhwfw
vwvw
−−+
=+−−
Γ
λ
Instituto Tecnológico de Aeronáutica
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MIXED AND PENALTY METHODSIncompressible elasticity
• Interpolation expressions for ph
• ph must be square integrable; it may be discontinuous across element boundaries
• Wider space of functions allowed for p
• Arbitrary combination of interpolation may lead to poor numerical performance
Instituto Tecnológico de Aeronáutica
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MIXED AND PENALTY METHODSIncompressible elasticity
Continuous bilinear displ.
Discontinuous constant pressure
Continuous bilinear displ.
Continuous bilinear pressure
Continuous bilinear displ.
Discontinuous bilinear pressure
Instituto Tecnológico de Aeronáutica
AE-245 34
MIXED AND PENALTY METHODSIncompressible elasticity
Matrix equation
=
b
fp
d
MG
GKT
K is symmetric and positive-definiteM is symmetric and negative-definiteWhen ν = 1/2, M = 0
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• Problems with mesh locking
• Babu ška-Brezzi stability condition
• Approximate method: constraint count
MIXED AND PENALTY METHODSIncompressible elasticity
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• Mixed formulation use two field variables: displacement and pressure
• Difficulty associated with the selection of the k factor in penalty methods
• Can displacement based formulations satisfactorily model the nearly incompressible problem?
• Use of reduced and selective integration
MIXED AND PENALTY METHODSReduced and selective integration
Instituto Tecnológico de Aeronáutica
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Element stiffness matrix
MIXED AND PENALTY METHODSReduced and selective integration
∫Ω
Ω= djTiij DBBk DDD +=
D
D
Part ofD related to µ
Part ofD related to λijijij kkk +=
Instituto Tecnológico de Aeronáutica
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Element stiffness matrix
MIXED AND PENALTY METHODSReduced and selective integration
k ij is the element stiffness matrix that appears in the mixed formulation in the term a.
Nearly incompressible problem: λ/µ >> 1
Terms in k ij tend to be very large when compared to those in k ij.
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MIXED AND PENALTY METHODSReduced and selective integration
Special treatment must be given to k ij
Reduced integration
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MIXED AND PENALTY METHODSReduced and selective integration
Selective reduced integration:
Reduced integration used only on the λ-term
Uniform reduced integration:
Reduced integration used on both λ- and µ-terms
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MIXED AND PENALTY METHODSReduced and selective integration
Equivalence theorem
• Several mixed elements are equivalent to reduced integration elements, that is, both formulations lead to identical stiffness matrices;
• At Gauss points of the reduced integration the pressure agrees with that of the mixed method;
• Reduced integration is a simple way of achieving performance without the complications of mixed type elements.
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MIXED AND PENALTY METHODSPatch test
The reduced integration method does not fully comply with the Galerkin recipe since integration
is not exact.
How can we assess the performance of elements that violate the Galerkin recipe?
Irons proposed the “patch test”.
Instituto Tecnológico de Aeronáutica
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MIXED AND PENALTY METHODSPatch test
Example: prescribe displacements on boundaries to represent the states 1, x and y.
1 2 3
4
5
6
7 8 9
y
y
x
x
vu
06
05
04
03
102
011
testTests 1 and 2: 1
Tests 3 and 4: x
Tests 5 and 6: y
Test: node 5 behaves
accordingly?
Instituto Tecnológico de Aeronáutica
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MIXED AND PENALTY METHODSRank deficiency
Two-dimensional bilinear elasticity element
∫Ω
×××× Ω= dT83333888 BDBk
Exact 2×2 Gaussian integration k is of rank 5
Reduced 1×1 Gaussian integration k is of rank 3
3 rigid body modes
Instituto Tecnológico de Aeronáutica
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MIXED AND PENALTY METHODSRank deficiency
Spurious zero energy modes
Whenk is of rank 3 there are 5 zero eigenvalues: 3 correspond to the rigid body modes and 2 are spurious zero energy modes.
ξ ξ
η η
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AE-245 46
MIXED AND PENALTY METHODSNonconforming elements
Approximation in the Galerkin recipe requires:
VV
SSh
h
⊂
⊂
The convergence of the FEM cannot be guaranteed if the conditions above are not met. Elements that do not satisfy
Sh⊂Sand Vh⊂V are referred to as incompatible or nonconforming.
Instituto Tecnológico de Aeronáutica
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MIXED AND PENALTY METHODSIncompatible modes element
Motivation: difficulty to represent pure bending
Pure bending
Exact pure bending response
Element in pure bending
Element in pure bending response
Instituto Tecnológico de Aeronáutica
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MIXED AND PENALTY METHODSIncompatible modes element
Exact pure bending:
u = −σ0cxy
v = −σ0c[(h/2)2 − x2]/2
Element in pure bending:
u ≈ xy
v = 0
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MIXED AND PENALTY METHODSIncompatible modes element
Remedy proposed: add quadratic terms
∑∑==
+=6
5
4
1
),(),(),(i
iii
iih NN αdu ηξηξηξ
26
25
1),(
1),(
ηηξ
ξηξ
−=
−=
N
NIncompatible modes
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MIXED AND PENALTY METHODSIncompatible modes element
Terms N5 and N6 are used only to obtain the stiffness matrix but not to obtain the loading.
=
0
f
α
d
kk
kk
ααα
α
d
ddd
dkkα dααα1−−= dd αααα kkkkk 1~ −−=
Effective
matrix