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Problemstatement:parametrizedweakform
• Exactparametrized elastodynamic problem
• Initialconditions:• Boundaryconditions:
• Space-timequantityofinterest:
m∂2
ue(x,t;µ)
∂t2,v;µ
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟+ c
∂ue(x,t;µ)
∂t,v;µ
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟+ a u
e(x,t;µ),v;µ( ) = g(t)f (v;µ),
∀v ∈ H01(Ω)( )d ,t ∈ [0,T ],µ ∈ D
ui
e(x,0;µ) = 0;∂ui
e
∂t(x,0;µ) = 0
ui
e x,t;µ( ) = 0, ∀x ∈ ∂ΩD
σij
e x,t;µ( )n̂j = ti, ∀x ∈ ∂ΩN
se(µ) = ue
Γo∫0
T
∫ (x,t;µ)Σ(x,t)dxdt = ℓ0
T
∫ ue(x,t;µ)( )dt
µ → ue(µ)( )→ se(µ)?
Problemstatement…
• Bi/linearforms
• Bilinearforms arecontinuous andcoercive.• Assumeaffineparameterdependenceofthebi/linearforms
m w,v;µ( ) = ρ
Ω∫ vi∂2wi
∂t2dΩ
i∑
c w,v;µ( ) = α
Ω∫ ρviwidΩi∑ + β
∂vi
∂xjΩ∫ Cijkl
∂wk
∂xl
dΩi,j,k,l∑
a w,v;µ( ) =
∂vi
∂xjΩ∫ Cijkl
∂wk
∂xl
dΩi,j,k,l∑
f v;µ( ) = bivi dΩΩ∫i
∑ + viti dΓ∂ΩN∫
i∑
m,a
• “MethodofLines”: spatialdiscretize (FE)+temporaldiscretize(Newmark)
–Discretizethetimespan into–Solve followingellipticsystems
–FEquantityofinterest:
Finiteelementdiscretization
[0,T ] [t
k,tk+1], 0 ≤ k ≤ K −1
(K −1)
A uk+1(µ),v;µ( ) = F(v), ∀v ∈Y h,µ ∈ D, 1 ≤ k ≤ K −1
A uk+1(µ),v;µ( ) =
1
Δt2m(uk+1(µ),v;µ) +
12Δt
c(uk+1(µ),v;µ) +14a(uk+1(µ),v;µ)
F(v) = −1
Δt2m(uk−1(µ),v;µ) +
12Δt
c(uk−1(µ),v;µ)−14a(uk−1(µ),v;µ)
+2
Δt2m(uk(µ),v;µ)−
12a(uk(µ),v;µ) + geq(tk )f (v;µ)
u(µ,t0) = 0;
∂u(µ,t0)
∂t= 0
s(µ) = ℓ
tk
tk+1
∫k=0
K−1
∑ u(x,t;µ)( )dt
Trapezoidalscheme
µ → u(µ)( )→ s(µ)?
• Introduce ;andnestedLagrangian RBspaces
–Galerkin projection:
–Solvethefollowingellipticsystems
–RBquantityofinterest:
RBapproximation:Galerkin projection
YN = span{ζn,1 ≤ n ≤ N}, 1 ≤ N ≤ Nmax
S* = {µ1 ∈ D,µ2 ∈ D,…,µN ∈ D}, 1 ≤ N ≤ Nmax
sN (µ) = ℓ
tk
tk+1
∫k=0
K−1
∑ uN (x,t;µ)( )dt
A uN
k+1(µ),v;µ( ) = F(v), ∀v ∈YN ,µ ∈ D, 1 ≤ k ≤ K −1
uN (µ,tk ) = uN n
n=1
N
∑ (µ,tk ) ζn, ∀ζn ∈YN , 1 ≤ k ≤ K
µ → uN (µ)( )→ sN (µ)?
• DualWeightedResidual(DWR)method
–Solveadditionallyanadjoint problem–Removethesnapshotscausesmallerror– keeptheonescauselargeerror
• Buildoptimalgoal-orientedbasisfunctionsbasedonallPODsnapshots–Useadjoint techniquetobuildoptimallybasisfunctionsbasedonallPODsnapshots
• Wewanttobuildoptimallygoal-orientedbasisfunctionswithoutcomputing/storingallthesnapshots?
Approachestobuildgoal-orientedbasisfunctions?
[Meyeretal.2003][Grepl etal. 2005][Bangerth etal.2001][Bangerth etal.2010]
[Buietal.2007][Willcox etal. 2005]
RB+Greedysamplingstrategy [Rozza,Huynh, Patera 2008]
1)Where:givenasetofsnapshotsthePODspace isdefinedas:
»
»• or,writtenas:
»»
• 2)Projectionerror:
»3)Residual
StandardPOD-Greedyalgorithm
(a) Set
(b) Set
(c) While
(d)
(e)
(f)
(g)
(h)
(i)
(j) end.
YNst = 0
µ*st = µ0
N ≤ Nmaxst
W st = eproj
st (µ*st,tk ),0 ≤ k ≤ K{ }
YN +Mst ←YN
st⊕POD(W st,M )
N ← N + M
µ*st = arg max
µ∈Ξtrain
Δu(µ)
uNst(µ,tk )
Y
2
k=1K∑
⎧
⎨
⎪⎪⎪⎪
⎩⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪
⎭⎪⎪⎪⎪
S*
st ← S*st µ*
st{ }∪
(k) Δu(µ) = Rst(v;µ,tk )
′Y
2
k=1K∑
MW
WM = POD {ξ
1,…, ξM
max
},M( )
{ξk}k=1
Mmax
WM = arg minVM⊂span{ξ1,…,ξMmax
}
1Mmax
infαk∈!M
k=1
Mmax
∑ ξk − αmk
m=1
M
∑ vm
2⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
eproj
k (µ) = uk(µ)− projYNuk(µ)
R(v;µ,tk ) = F(v)−A uNk+1(µ),v;µ( ),1 ≤ k ≤ K −1
[Haasdonk etal.2008][Hoangetal.2013]
Goal-orientedvs.standardPOD-Greedyalgorithm(a) Set
(b) Set
(c) While
(d)
(e)
(f)
(g)
(h)
(i)
(j) end.
YNgo = 0
µ*go = µ0
N ≤ Nmaxgo
W go = eproj
go (µ*go,tk ),0 ≤ k ≤ K{ }
YN +Mgo ←YN
go⊕POD(W go,M )
N ← N + M
µ*
go = arg maxµ∈Ξtrain
Δs(µ)
s !Nst(µ)
⎧⎨⎪⎪
⎩⎪⎪
⎫⎬⎪⎪
⎭⎪⎪
S*
go ← S*go µ*
go{ }∪
(k) Δs(µ) = s !Nst(µ)− sN
go(µ)
Asymptoticoutputerrorestimation
Find !N s.t.∀µ ∈ Ξn
st ⊂ Ξn+1st ⊂ S*
st( )
ηT ≤
Δs(µ)
s(µ)− sNgo(µ)
≤ 2− ηT
(a) Set
(b) Set
(c) While
(d)
(e)
(f)
(g)
(h)
(i)
(j) end.
YNst = 0
µ*st = µ0
N ≤ Nmaxst
W st = eproj
st (µ*st,tk ),0 ≤ k ≤ K{ }
YN +Mst ←YN
st⊕POD(W st,M )
N ← N + M
µ*st = arg max
µ∈Ξtrain
Δu(µ)
uNst(µ,tk )
Y
2
k=1K∑
⎧
⎨
⎪⎪⎪⎪
⎩⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪
⎭⎪⎪⎪⎪
S*
st ← S*st µ*
st{ }∪
(k) Δu(µ) = Rst(v;µ,tk )
′Y
2
k=1K∑
CVprocess
Cross-validationprocess
Numericalexample:3Ddentalimplantmodel
N = 26343
T = 0.001s
Δt = 2×10−6s
K = 500
Ξ
train= 900
• Materialproperties
• Explicitbi/linearforms:
3DDentalimplantmodelproblem…
m(w,v) = ρrwivi dΩΩr
∫i∑
r=1
5
∑
a(w,v;µ) =
∂vi
∂xj
Cijklr ∂wk
∂xl
dΩΩr∫
i,j,k,l∑
r=1,r≠3
5
∑ + µ1∂vi
∂xj
Cijkl3 ∂wk
∂xl
dΩΩ3∫
i,j,k,l∑
c(w,v;µ) = βr
r=1,r≠3
5
∑∂vi
∂xj
Cijklr ∂wk
∂xl
dΩΩr∫
i,j,k,l∑ + µ2µ1
∂vi
∂xj
Cijkl3 ∂wk
∂xl
dΩΩ3∫
i,j,k,l∑
f (v) = viφi dΓΓl
∫i∑
ℓ(v) =
1| Γo |
v1Γo∫ dΓ
µ = E,β( )∈ D ≡ [1×106Pa,25×106 ]×[5×10−6,5×10−5 ]⊂ !P=2
Numericalresults…StandardPOD-Greedyalgorithm
GOPOD-Greedyalgorithm
Numericalresults…Cross-validation(CV)process
AlltrueerrorsofsolutionandQoI
Numericalresults:QoITrueerrorsvs Errorapprox.forcase1Gaussload
Maxandmineffectivities forcase1Gaussload
Numericalresults…
• Computationaltime(onlinestage)
• AllcalculationswereperformedonadesktopIntel(R)Core(TM)[email protected],RAM32GB,64-bitOperatingSystem.
References1. Wang,S.,Liu,G.R.,Hoang,K.C.,&Guo,Y.(2010).Identifiable rangeof
osseointegration ofdentalimplantsthrough resonancefrequencyanalysis.Medicalengineering&physics, 32(10), 1094-1106.
2. Hoang,K.C.,Khoo,B.C.,Liu,G.R.,Nguyen,N.C.,&Patera,A.T.(2013).Rapididentificationofmaterialpropertiesoftheinterfacetissueindentalimplantsystemsusing reducedbasismethod. InverseProblemsinScienceandEngineering, 21(8),1310-1334.
3. Hoang,K.C.,Kerfriden,P.,Khoo,B.C.,&Bordas,S.P.A.(2015).Anefficientgoal-orientedsamplingstrategyusing reducedbasismethod forparametrizedelastodynamic problems. NumericalMethodsforPartialDifferentialEquations,31(2),575-608.
4. Hoang,K.C.,Kerfriden,P.,&Bordas,S.(2015).Afast,certifiedand"tuning-free"two-fieldreducedbasismethod forthemetamodelling ofparametrised elasticityproblems. ComputerMethodsinAppliedMechanicsandEngineering,accepted.
5. Hoang,K.C.,Fu,Y.,&Song, J.H.(2015).Anhp-ProperOrthogonalDecomposition-MovingLeastSquaresapproachformoleculardynamicssimulation.ComputerMethodsinAppliedMechanicsandEngineering,accepted.
