a tdnns-feti method for compressible [0.5ex] and almost ... · reenforcedfacet-wiseby lagrange...
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Johann Radon Institute for Computational and Applied Mathematics
A TDNNS-FETI method for compressibleand almost incompressible elasticity
.. Astrid Pechstein1 & Clemens Pechstein2
1 Institute of Computational MechanicsJohannes Kepler University, Linz (A)
2 Johann Radon Institute for Computationaland Applied Mathematics (RICAM)
Austrian Academy of Sciences, Linz (A)
..
DD22, Lugano, September 16, 2013
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Johann Radon Institute for Computational and Applied Mathematics
Overview
TDNNS (Tangential Displacement – Normal-Normal Stress)
Motivation of TDNNS-FETI
TDNNS-FETI Method
Theory
Numerical Results
, Page 2/29
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Johann Radon Institute for Computational and Applied Mathematics
Overview
TDNNS (Tangential Displacement – Normal-Normal Stress)
Motivation of TDNNS-FETI
TDNNS-FETI Method
Theory
Numerical Results
, TDNNS (Tangential Displacement – Normal-Normal Stress) Page 3/29
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Johann Radon Institute for Computational and Applied Mathematics
Linear elasticity
Ω ⊂ Rd , d = 2 or 3
−divσ = f in Ω
σ = D ε(u) in Ω, where ε(u) = 12 (∇u +∇>u)
u = uD on ΓD
σn = tN on ΓN
Hooke’s law:
D ε =E
(1 + ν)ε+
Eν
(1 + ν)(1− 2ν)tr(ε)I
for d = 2: plane strain or plane stress
, TDNNS (Tangential Displacement – Normal-Normal Stress) Page 4/29
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Johann Radon Institute for Computational and Applied Mathematics
TDNNS Formulation A. Pechstein & J. Schoberl ’11, ’12
Mixed TDNNS finite element (P1, here 2D)
TangentialcontinuousDisplacement(Nedelec II)
σu symmetricNormal-NormalcontinuousStress
Discrete variational formulation∑T
∫T
(divσ + f) · v dx +∑F
∫F
[[σnt ]] · vt ds = g2 ∀v
∑T
∫T
(D−1σ − ε(u)) : τ dx +∑F
∫F
[[un]]τnn ds = g1 ∀τ
Stability & Convergence:I Discrete inf-sup stabilityI Optimal error estimates
But:
I Saddle point system
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Johann Radon Institute for Computational and Applied Mathematics
Hybridized TDNNS Formulation
Normal-normal continuity of stresses is broken andreenforced facet-wise by Lagrange multipliers,these approximate normal displacements.
unσu
Benefits:I algebraically equivalent to previous system
I stresses local static condensationI condensed system is SPD
KU = F U =
[uun
]
, TDNNS (Tangential Displacement – Normal-Normal Stress) Page 6/29
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Johann Radon Institute for Computational and Applied Mathematics
Hybridized TDNNS Formulation
Normal-normal continuity of stresses is broken andreenforced facet-wise by Lagrange multipliers,these approximate normal displacements.
unσu
Benefits:I algebraically equivalent to previous systemI stresses local static condensationI condensed system is SPD
KU = F U =
[uun
], TDNNS (Tangential Displacement – Normal-Normal Stress) Page 6/29
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Johann Radon Institute for Computational and Applied Mathematics
Why use TDNNS?
Single framework, stable w.r.t.I volume locking (more later on)I shear locking
DG-like method, lives up to the motto
Relax as much as you can, but never lose stability!
, TDNNS (Tangential Displacement – Normal-Normal Stress) Page 7/29
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Johann Radon Institute for Computational and Applied Mathematics
Overview
TDNNS (Tangential Displacement – Normal-Normal Stress)
Motivation of TDNNS-FETI
TDNNS-FETI Method
Theory
Numerical Results
, Motivation of TDNNS-FETI Page 8/29
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Johann Radon Institute for Computational and Applied Mathematics
Motivation I – Solver
Astrid’s PhD thesis ’09
Additive Schwarz preconditioner for high order (hybridized) system
I lowest order block (direct)
I overlapping blocks associated to edges, facets, elements
I optimal in h and polynomial degree
But up to now no scalable lowest order TDNNS-solver available!
, Motivation of TDNNS-FETI Page 9/29
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Johann Radon Institute for Computational and Applied Mathematics
Motivation II – Symbolic Study
Victorita Dolean, DD20 plenary talk DD on PDE level
T. Cluzeau, V. Dolean, F. Nataf, and A. Quadrat ’12
two subdomains, Γ = symmetry axis
Ω Ω(1) (2)Γ
I scalar PDE −∆u = f
coupling pair (u, ∂u∂n )
Neumann-Neumann preconditioner is exact
I elasticity −div(σ(u)) = f
coupling pair (u,σn(u))
?
, Motivation of TDNNS-FETI Page 10/29
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Johann Radon Institute for Computational and Applied Mathematics
Motivation II – Symbolic Study
Victorita Dolean, DD20 plenary talk DD on PDE level
T. Cluzeau, V. Dolean, F. Nataf, and A. Quadrat ’12
two subdomains, Γ = symmetry axis
Ω Ω(1) (2)Γ
I scalar PDE −∆u = f
coupling pair (u, ∂u∂n )
Neumann-Neumann preconditioner is exact
I elasticity −div(σ(u)) = f
coupling pair (u,σn(u))
Neumann-Neumann preconditioner is NOT exact!
, Motivation of TDNNS-FETI Page 10/29
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Johann Radon Institute for Computational and Applied Mathematics
Motivation II – Symbolic Study
Victorita Dolean, DD20 plenary talk DD on PDE level
T. Cluzeau, V. Dolean, F. Nataf, and A. Quadrat ’12
two subdomains, Γ = symmetry axis
Ω Ω(1) (2)Γ
I scalar PDE −∆u = f
coupling pair (u, ∂u∂n )
Neumann-Neumann preconditioner is exact
I elasticity −div(σ(u)) = f
coupling pair
([ut
σnn(u)
],
[σnt(u)
un
])Neumann-Neumann preconditioner is exact
symbolic approachSmith factorizationOreMorphisms
, Motivation of TDNNS-FETI Page 10/29
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Johann Radon Institute for Computational and Applied Mathematics
Existing Work – FETI for mechanics
Compressible elasticity
I Farhat & Roux ’94 etc. – FETI (original)
I Farhat, Lesoinne, LeTallec, Pierson & Rixen ’01 – FETI-DP (original)
I Dostal, Horak & R. Kucera ’06 – TFETI
I Klawonn & Widlund ’06 – FETI-DP (3D coarse space)
I Klawonn, Rheinbach & Widlund ’08 – FETI-DP (2D, irregular interface)
Incompressible Stokes
I Li & Widlund ’06 – BDDC (primal pressure dofs, quasi-optimal)
I Kim & Lee ’10 – FETI-DP (no primal pressure dofs, suboptimal)
Almost incompressible elasticity
I Dohrmann & Widlund ’09 – OS (“full” coarse space)
I Dohrmann & Widlund ’09 – hybrid OS-IS (reduced coarse space)
I Pavarino, Widlund & Zampini ’10 – BDDC (reduced coarse space)
I Gippert, Rheinbach & Klawonn ’12 – FETI-DP unresolved incompressible inclusions
Your paper here& even more at this conference!
We won’t compare TDNNS-FETI with the above FETI(-DP) methodsbut show what one can do when one has chosen TDNNS as discretization.
, Motivation of TDNNS-FETI Page 11/29
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Johann Radon Institute for Computational and Applied Mathematics
Overview
TDNNS (Tangential Displacement – Normal-Normal Stress)
Motivation of TDNNS-FETI
TDNNS-FETI Method
Theory
Numerical Results
, TDNNS-FETI Method Page 12/29
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Johann Radon Institute for Computational and Applied Mathematics
one-level (T)FETI for hybridized TDNNS
Saddle point formulationK(1) 0 B(1)>
. . ....
0 K(N) B(N)>
B(1) . . . B(N) 0
U(1)
...
U(N)
λ
=
F(1)
...
F(N)
0
4 multipliers per edge (2D)
Kernel property
ker(K(i)) = range(R(i)) rigid body modes
Elimination of U(i) by generalized inverse K(i)† AA[BK†B −GG> 0
] [λξ
]=
[de
]G := BR
U = K†(F− B>λ) + Rξ
Projection method
PQ := I−QG(G>QG)−1G>
, TDNNS-FETI Method Page 13/29
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Johann Radon Institute for Computational and Applied Mathematics
One-level (T)FETI with scaled Dirichlet preconditioner
Solve (PCG) PQ>BK†B>λ = PQ
>d λ ∈ range(PQ)
Preconditioner PQ BDSB>D S = diag(S(i))
Yet to be specified:
I scalings in BD = [D(1)B(1)| . . . |D(N)B(N)]
I matrix Q
, TDNNS-FETI Method Page 14/29
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Johann Radon Institute for Computational and Applied Mathematics
Choice of Scalings – 2DScalings in BD
diagonal entry of D(i) at multiplier linking (i , j)
12
E (i)
E (i)+E (j)
K(i)diag,max
K(i)diag,max+K
(j)diag,max
multiplicity scaling coefficient scaling A stiffness scaling A
Choices of QI Q = II Q = M−1 = BDSB>DI (Qdiag)m,ij = min(E (i),E (j))
h(ij)H(ij) or using scaled stiffness entries
Assumption on degrees of freedome
e e1 2
t
∫ev · t ds 1
|e|
∫e(v · t)s ds |e| vn|e(e1) |e| vn|e(e2)
, TDNNS-FETI Method Page 15/29
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Johann Radon Institute for Computational and Applied Mathematics
Overview
TDNNS (Tangential Displacement – Normal-Normal Stress)
Motivation of TDNNS-FETI
TDNNS-FETI Method
Theory
Numerical Results
, Theory Page 16/29
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Johann Radon Institute for Computational and Applied Mathematics
Condition Number Estimate
Assumptions
I d = 2
I triangular mesh technical
I E|Ω(i) ≈ E (i) = const
I subdomains = compounds of coarse elements can be relaxed
I subdomain meshes quasi-uniform can be relaxed
Theorem (AP & CP)
For coefficient or stiffness scaling, and Q = M−1 or Q = Qdiag
κ(T)FETI ≤ C (1 + log(Hh ))2
C independent of H, h, and jumps in E .
, Theory Page 17/29
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Johann Radon Institute for Computational and Applied Mathematics
Discrete norms
Energy norm is equivalent to the DG-like (semi)norm
|v, vn|2H(ε,Ω,Th) :=∑T
(‖ε(v)‖2
L2(T )+ h−1
T ‖v · n− vn‖2L2(∂T )
)or (via Korn-type inequality, Brenner ’04)
|v, vn|2H1(Ω,Th) :=∑T
(|v|2
H1(T )+ h−1
T ‖v · n− vn‖2L2(∂T )
)“L2-norm”
‖v, vn‖2L2(Ω,T h) := ‖v‖2
L2(Ω) +∑T
h ‖vn‖2L2(∂T )
, Theory Page 18/29
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Johann Radon Institute for Computational and Applied Mathematics
Two Transformations
X
Y
nodal FETDNNS
(v, vn) v
Features
I preserve continuous functions, in particular rigid body motionsI H1-stable |X (v, vn)|H1(Ω) . |v, vn|H1(Ω,Th) |Y v|H1(Ω,Th) . |v|H1(Ω)
I L2-stable ‖X (v, vn)‖L2(Ω) . ‖v, vn‖L2(Ω,Th) ‖Y v‖L2(Ω,Th) . ‖v‖L2(Ω)
I . involves only shape-regularity constantsI also works subdomain-wise / patch-wise
, Theory Page 19/29
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Johann Radon Institute for Computational and Applied Mathematics
Main Idea of constructing v := X (v, vn)
average with neighbors
v = v
v = vt + vn n
|v, vn|2H1(Ω,Th) =∑T
|v|2H1(T ) + h−1T ‖v · n− vn‖2
L2(∂T )
, Theory Page 20/29
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Johann Radon Institute for Computational and Applied Mathematics
Lift of Existing Tools
Cut-off Estimate (Edge Lemma)
|ΘE(ij)(v, vn)|2H1(Ω(i),Th). (1 + log(H
(i)
h(i) ))2 ‖v, vn‖2H1(Ω(i))
Discrete Subdomain Extension
|E(i→j)(v, vn)|2H1(Ω(j),Th). |v, vn|2H1(Ω(i),Th)
‖E(i→j)(v, vn)‖2L2(Ω(j),Th)
. ‖v, vn‖2L2(Ω(i),Th)
Ω
ε(ij)
(i)Ω
(j)
, Theory Page 21/29
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Johann Radon Institute for Computational and Applied Mathematics
Overview
TDNNS (Tangential Displacement – Normal-Normal Stress)
Motivation of TDNNS-FETI
TDNNS-FETI Method
Theory
Numerical Results
, Numerical Results Page 22/29
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Johann Radon Institute for Computational and Applied Mathematics
1) Compressible Case, Homogeneous Material
plane strain, ν = 0.3, E = 2 · 105 N/mm2
64 subdomains, multiplicity scaling, Q = I
ΓD
FETI TFETIH/h cond iter cond iter
2 15.37 22 3.24 154 19.82 28 5.34 208 25.00 34 8.43 25
16 30.03 38 11.84 2932 35.19 42 15.59 3364 40.65 47 19.65 36
168 coarse dofs 192 coarse dofs
0.1
1
10
100
1000
10000
1 10 100
TFETIFETI
H/hCPU
(sequential)
PCG tolerance 10−8 (relative)
, Numerical Results Page 23/29
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Johann Radon Institute for Computational and Applied Mathematics
2) Compressible Case, Heterogeneous Material
ν = 0.3E1 = 101 N/mm2 E2 = 2 · 105 N/mm2
TFETI (similar for FETI)
ΓD
E
E
1
2
A
1
10
100
1000
10000
100000
1 10 100 1000
mult, Q=Icoeff, Q=M
coeff, Q=diagstiff, Q=diag
cond
H/h
0.1
1
10
100
1000
10000
1 10 100 1000
mult, Q=Icoeff, Q=M
coeff, Q=diagstiff, Q=diag
CPU
H/hA
, Numerical Results Page 24/29
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Johann Radon Institute for Computational and Applied Mathematics
Almost Incompressible Case – Stabilization
Consistent stabilization Astrid’s PhD thesis ’09
∑T
∫T
(divσ + f) · v dx +∑F
∫F
[[σnt ]] · vt ds = g2 ∀v
∑T
∫T
(D−1σ − ε(u)) : τ dx +∑F
∫Fτnn[[un]] ds +
+∑T
h2T
∫T
(divσ + f) · divτ dx = g1 ∀τ
In hybridized form penalizes volume changes: +λ
|T |
∣∣∣ ∫∂T
vn ds∣∣∣2
Stability & Convergence uniform for ν → 1/2
, Numerical Results Page 25/29
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Johann Radon Institute for Computational and Applied Mathematics
3) Almost Incompressible Case
ν1 ∈ [0.3, 0.49999] ν2 = 0.3E1 = 101 N/mm2 E2 = 2 · 105 N/mm2
TFETI (similar for FETI)
ΓD
E
E
1
2 ν2
1ν
H/h = 8
A
1
10
100
1000
10000
100000
1e-06 1e-05 0.0001 0.001 0.01 0.1 1
coeff, Q=diagstiff, Q=diagcoeff, Q=M
stiff, Q=M
cond
12 − ν1
1
10
100
1e-06 1e-05 0.0001 0.001 0.01 0.1 1
coeff, Q=diagstiff, Q=diagcoeff, Q=M
stiff, Q=M
CPU
12 − ν1
A, Numerical Results Page 26/29
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Johann Radon Institute for Computational and Applied Mathematics
4) Irregular Subdomains
ν = 0.3 E = 2 · 105 N/mm2
64 subdomains, multiplicity scaling, Q = I H/h ≈ 32
regular FETI TFETIcond iter cond iter
cont. P1 23.8 31 7.0 23TDNNS 35.2 42 15.6 33
METIS FETI TFETIcond iter cond iter
cont. P1 75.2 49 34.1 43TDNNS 113.6 67 90.7 64
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Johann Radon Institute for Computational and Applied Mathematics
Summary
One-level (T)FETI method for hybridized TDNNS discretization
I Formulation, choice of scalings (2D)I Analysis for 2D compressible caseI Numerical results reveal good parameter choices for ν → 1/2
Benefits
I Very simple coarse space (RBM)I Usual robustness propertiesI Additional robustness for ν → 1/2
Ongoing work
I Analysis of incompressible caseI 3DI Deluxe scalingI Anisotropic domains
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Johann Radon Institute for Computational and Applied Mathematics
References
A. Pechstein & C. PechsteinA FETI method for a TDNNS discretization of plane elasticitywww.numa.uni-linz.ac.at/publications/List/2013/2013-05.pdf
Astrid Sinwel. A New Family of Mixed Finite Elements for Elasticitywww.numa.uni-linz.ac.at/Teaching/PhD/Finished/sinwel-diss.pdf
A. Pechstein & J. Schoberl. Tangential-displacement and normal-normalstress continuous mixed finite elements for elasticityMath. Models Methods Appl. Sci. 21(8):1761–1782, 2011
A. Pechstein & J. Schoberl. Anisotropic mixed finite elements for elasticityInternat. J. Numer. Methods Engrg. 90(2):196–217, 2012
S. C. Brenner. Korn’s inequalities for piecewise H1 vector fieldsSIAM J. Numer. Anal. 41(1):306–324, 2003
Thanks for your attention!
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