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A Hybrid Newton-Krylov method for stabilized FEMdiscretization of Navier-Stokes equations
Stefano BerroneDIPARTIMENTO DI MATEMATICA, POLITECNICO DI TORINO,CORSO DUCA DEGLI ABRUZZI 24, 10129, TORINO, ITALY
e-mail: [email protected]: http://calvino.polito.it/˜sberrone
Stefania BellaviaDIPARTIMENTO DI ENERGETICA, UNIVERSITA DI FIRENZE,
VIA LOMBROSO 6/17, 50134, FIRENZE, ITALY
e-mail: [email protected]
22nd IFIP TC7 Conference on System Modeling and OptimizationTurin, Italy, July 18-22, 2005
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IFIP 2005 Torino, July 19, 2005
Outline
• Steady state Navier-Stokes Equations and SUPG-FEM discretization.• Inexact Newton-GMRES Methods.• Globally convergent methods: classical backtracking Newton-GMRES Method, Trust Re-
gion methods.• Global convergence enhancement: A combined Linesearch-subspace trust region
method.• Numerical results: Backward Facing Step, Lid Driven Cavity.
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IFIP 2005 Torino, July 19, 2005
Steady-state, incompressible Navier-Stokes Equations
−1
Re4 u + (u · ∇) u + ∇p = f in Ω,
∇ · u = 0 in Ω,
u = 0 on ΓD,
1Re
∂ u
∂n− pn = gN on ΓN,
where:
• Re:the Reynolds number;• Ω ⊂ IR2: ∂Ω = ΓD ∪ ΓN , ΓD ∩ ΓN = ∅ and | ΓD | 6= 0;• n: unit outward normal vector to ∂Ω;• f ∈ [L2(Ω)]2;
• gN ∈ [H12(ΓN)]2.
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IFIP 2005 Torino, July 19, 2005
Finite Element Discretization
• Ω: polygonal domain;• Thh a regular family of partitions of Ω into triangles T (usual conformity and minimal-
angle conditions [Ciarlet 1978]);
• Vh ⊂V = [H10,D(Ω)]2, Qh ⊂Q =
(L2
0(Ω) if | ΓN | = 0
L2(Ω) if | ΓN | > 0two conforming finite element spaces based on the partition Th;
• ΠT f , ΠE gN approximations of f , gN by projections.
With the standard Galerkin formulation, only some choices of the spaces Vh and Qh sat-isfies the discrete inf-sup (Babuska-Brezzi) condition needed for uniqueness of the pressure[Brezzi-Fortin 1991].
This may be avoided by resorting to the Streamline Upwind/Petrov Galerkin (SUP G)formulation [Franca-Frey-Hughes 1992, Franca-Frey 1992].
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IFIP 2005 Torino, July 19, 2005
SUPG-formulationF ind [uh, ph]∈Vh × Qh such that, ∀[vh, qh]∈Vh × Qh :
(Momentum equations)
1Re
(∇ uh, ∇ vh) + ((uh · ∇) uh, vh) − (ph, ∇ · vh) +
+X
T ∈Th
τT
„−
1Re
4 uh +(uh · ∇) uh + ∇ph, (uh · ∇) vh
«T
+
+X
T ∈Th
δT (∇ · uh, ∇ · vh)T =
= (ΠT f, vh) + (ΠE gN , vh)ΓN+
XT ∈Th
τT (ΠT f, (uh · ∇) vh)T ,
(Continuity)
(qh, ∇ · uh) +
+X
T ∈Th
τT
„−
1Re
4 uh +(uh · ∇) uh + ∇ph, ∇qh
«T
=
=X
T ∈Th
τT (ΠT f, ∇qh)T .
For the stabilization parameters τT and δT we essentially follow [Franca-Frey 1992].
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IFIP 2005 Torino, July 19, 2005
Newton Methods for Nonlinear Equations
The discretized steady-state Navier-Stokes equations give rise to a system of nonlinearequations:
F (x) = 0
where x = [uh, ph] is the vector of velocity-pressure values in the grid points.
A classical method for solving this problem is the Newton method:
• Given x0 ∈ IRn, a tolerance tol,• set k = 0,• While ‖F (xk)‖ > tol
– solve F ′(xk)sk = −F (xk),– set xk+1 = xk + sk,– k = k + 1
⇒ Convergence: for this kind of applications finding a good initial guess is almost impossi-ble; difficulties arise also in the context of adaptive methods.
⇒ Efficiency: enhanced by inexact methods.
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IFIP 2005 Torino, July 19, 2005
Inexact Newton Methods
In an Inexact Newton Method, the Newton equation is relaxed to an Inexact Newtoncondition
‖F′(xk)sk + F (xk)‖ ≤ ηk‖F (xk)‖.
• ηk ∈ [0, 1) forcing term. The ultimate rate of convergence of the Inexact methodsdepends on the choice of the ηk’s.
• This naturally allows the use of iterative linear methods: conditions like this are preciselythe small linear residual termination conditions for iterative linear solver.
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IFIP 2005 Torino, July 19, 2005
Inexact Newton-GMRES Methods
GMRES is employed to compute the Inexact Newton step sk:
• Given x0 ∈ IRn, a tolerance tol,• set k = 0,• While ‖F (xk)‖ > tol
– choose ηk ∈ [0, 1)– perform GMRES iterations on F ′(xk)sk = −F (xk)
until‖F ′(xk)sk + F (xk)‖
‖F (xk)‖≤ ηk,
– set xk+1 = xk + sk,– k = k + 1,
• rk = F ′(xk)sk + F (xk) residual;
•‖F ′(xk)sk + F (xk)‖
‖F (xk)‖≤ ηk classical relative residual sopping condition.
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IFIP 2005 Torino, July 19, 2005
Globalization Strategies
(Inexact) Newton methods exhibit local convergence properties. In order to enhance theirrobustness, these methods have to be augmented with a suitable globalization strategy.
Newton-type methods combined with globalization strategies produce a sequence xksuch that at each iteration a sufficient decrease of the merit function
f(x) =12‖F (x)‖2
is ensured, i.e.f(xk+1) < αkf(xk), αk ∈ (0, 1)
aiming to create robust and locally fast algorithms.
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IFIP 2005 Torino, July 19, 2005
Linesearch
• Given a search direction sk, xk+1 is given by
xk+1 = xk + λsk
where the reduction factor λ ∈ (0, 1] is such that a sufficient decrease condition
f(xk + λsk) < (1 − tλ)f(xk)
is reached. A value λ satisfying the condition exists if sk is a descent direction for f(x) inxk, i.e. if
∇f(xk)Tsk < 0.
The backtrack strategy is: set λ = 1 then reduce λ of a given factor θ ∈ (0, 1) until therequired sufficient decrease condition on f is reached. The parameter t is chosen in (0, 1),usually t = 1.0E − 4.
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IFIP 2005 Torino, July 19, 2005
Inexact Newton method with Backtracking (INB)
Let xk, ηmax ∈ (0, 1), t ∈ (0, 1), 0 < θm < θmax < 1 be given.
1. Choose ηk ∈ [0, ηmax].2. Compute sk such that
‖F (xk) + F′(xk)sk‖ ≤ ηk‖F (xk)‖.
3. Perform the INB backtracking strategy:3.1 Set sk = sk, ηk = ηk.3.2. While ‖F (xk + sk)‖ > (1 − t(1 − ηk))‖F (xk)‖ do:
Choose θ ∈ [θm, θmax].Update sk = θsk and ηk = 1 − θ(1 − ηk).
4. Set xk+1 = xk + sk.
[Eisenstat-Walker 1994].
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IFIP 2005 Torino, July 19, 2005
Failures of Newton-GMRES with linesearch
• The Newton and Inexact Newton directions can be nearly orthogonal to the gradient of f
when F ′ is ill conditioned. Moreover this may happen in the Inexact Newton methods if thelinear systems are solved at a too low accuracy:
cos θN = −
sTk ∇f(xk)
‖sk‖‖∇f(xk)‖>
1 − ηk
(1 + ηk)k2(F ′(xk)).
• When the direction is “poor”, in order to obtain a reduction in f a very high number ofbacktracks is required ⇒ this in practice implies a failure of the backtracking strategy.
• This kind of failures has been observed in practical applications, in particular in the solutionof nonlinear systems arising from Navier Stokes equations:
– [Tuminaro-Walker-Shadid 2002]– [Pawlowski-Shadid-Simonis-Walker 2004]
• In this cases the Newton direction should be rejected and new directions should be con-sidered.
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IFIP 2005 Torino, July 19, 2005
Trust-region methods
• Let us consider the quadratic function f(x) = 12‖F (x)‖2 for which ∇f = (F ′)T F
has the same direction as ∇‖F ‖.• The quadratic model mk(p) of f(x) = 1
2‖F (x)‖2 in xk
mk(p) =12‖F
′(xk)p + F (xk)‖2.
• Given a trust-region radius ∆k, the trial direction pk is the solution of the trust-regionproblem
minp∈IRn
mk(p) : ‖p‖ ≤ ∆k
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IFIP 2005 Torino, July 19, 2005
• As ∆k tends to zero, the step pk tends to become parallel to the gradient of f : trustregion methods have the advantage to generate directions that may be stronger descentdirections than sk.
∆∆
∆
p
sk
k
kk
C
N
Dogleg approximate solution
– ∆k: trust region radius,– pC: Cauchy point,– sN
k : Newton step.
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IFIP 2005 Torino, July 19, 2005
Low dimensional subspace trust-region
• When dealing with large scale problems, by the computational point of view it may beconvenient to minimize the quadratic model in low dimensional subspaces Sk ⊂ IRn
minp∈Sk
mk(p) : ‖p‖ ≤ ∆k,
• The solution of the subspace trust region problem is usually not expensive due to thelow dimension of the subspace. The main computational cost lies in the construction of thesubspace Sk.
• Our approach: Sk = spansk, ∇fk. The subspace trust-region problem reduces to atrust-region problem in R2.
• This requires the computation of ∇fk = (F ′(xk))T F (xk), in our context the Jacobianis available.
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IFIP 2005 Torino, July 19, 2005
A combined global method
• We adopt the following strategy:– perform MAXBT backtracks in the inexact direction sk;
– if the sufficient descent condition is not satisfied, the direction sk is rejected;
– a trust region radius ∆k is selected
– a dogleg approximate solution of the subspace trust region problem is employed asthe new trial step.
• Additional cost in computing the new direction pk:– computation of ∇fk = F ′(xk)TF (xk)– dogleg strategy in R2 that is clearly performed at a very low computational cost.
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IFIP 2005 Torino, July 19, 2005
HINM Algorithm (Hybrid Inexact Newton Method)
Given xk, ηmax ∈ (0, 1), ηk ∈ [0, ηmax], t ∈ (0, 1), β1 ∈ (0, 1), α1, α2, ∈ (0, 1),0 < θm < θmax < 1, MAXBT > 0, ∆min > 0 and ∆k > ∆min:
1. Compute sk such that ‖F (xk) + F ′(xk)sk‖ ≤ ηk‖F (xk)‖.2. Apply the INB backtracking strategy and perform at most MAXBT backtracks.3. If ‖F (xk + sk)‖ > (1 − t(1 − ηk))‖F (xk)‖
3.1 Do3.1.1 Compute the approximate dogleg solution sk to argmin
‖s‖≤∆k,s∈Sk
mk(s).
3.1.2 Set ∆k = minα1 ∆k, α2 ‖sk‖.while ρf
k(sk) ≥ β1.4. Set xk+1 = xk + sk.5. Choose ∆k+1 > ∆min.
Here
ρfk(sk) =
f(xk) − f(xk + sk)mk(0) − mk(sk)
is a measure of the agreement between f(x) and its quadratic model m(s) around xk.
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Convergence properties
Theorem. Let r > 0 and L = ∪∞k=0 x ∈ IRn | ‖x − xk‖ ≤ r be a neighborhood
of sequence xk generated by HINM Algorithm. Assume that F ′ is Lipschitz continuousin L, with Lipschitz constant 2γL and ‖F ′(x)‖ is bounded above on L. Then, if F ′
k isinvertible for k ≥ 0, the repeat loop in Step 3 terminates. Further, if there exists a limit pointx∗ of xk such that F ′(x∗) is invertible, then
a) limk→∞ ‖F (xk)‖ = ‖F (x∗)‖ = 0.b) limk→∞ xk = x∗;c) sk = sk, for sufficiently large k.
⇒ Eventually the adopted step is the Inexact Newton step. Then the ultimate rate ofconvergence depends on the choice of the forcing terms, as in classical backtracking InexactNewton methods.
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IFIP 2005 Torino, July 19, 2005
Numerical Results
Comparison between Newton-GMRES with backtrack (INB) and our Hybrid Inexact New-ton Method (HINM).
• Initial guess: x0 = (0, . . . , 0)T ;• GMRES: restart=200, maximal number of iterations 600;• ILU right preconditioning;• Success is declared when ‖F (xk)‖ < 1.0E − 6;• Failure is declared when after 200 iterations ‖F (xk)‖ > 1.0E − 6 holds;
– INB: when after 10 backtracks the sufficient decrease condition is not satisfied;– HINM: when ∆k < 1.0E − 8.
• MAXBT = 4 for HINM;• The backtrack reduction factor θ is chosen using the two point parabolic model.
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IFIP 2005 Torino, July 19, 2005
Forcing terms:
• Large and small constant forcing terms (linear convergence):– ηk = 1.0E−1,– ηk = 1.0E−4.
• Adaptive choice: Select η0 ∈ [0, 1) and chose
ηk =
˛‖F (xk+1)‖ − ‖F (xk) + F ′(xk)sk‖
˛‖F (xk)‖
, k = 1, 2, . . .
with the safeguard: ηk = maxηk, η(1+
√5)/2
k−1 if η(1+
√5)/2
k−1 > 0.1.– Prevent the initial ηk’s from becoming too small far away from a solution.– ηk → 0: superlinear convergence .
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IFIP 2005 Torino, July 19, 2005
Numerical Results: Backward Facing Step
-
6
- l
- L
6
?
h 6
?
Hx
y
u1 = 0; u2 = 0
u1 = 0; u2 = 0
Geometry of the backward facing step
upper eddy
lower eddy
(Re = 600)
a
Eddies’ position for the BFS problem
Numerical experiments performed with Re = 200, 400, 500, 600, 650, 700, 800
Number of Failures: Comparison between the HINM method and the INB methodGrid ηk → 1.0E-1 1.0E-4 Ad. Choice2460 HINM 4 2 4
INB 5 4 46781 HINM 2 4 3
INB 4 4 413200 HINM 1 0 2
INB 2 1 3
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Angle between the chosen sk and −∇f(xk)
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IFIP 2005 Torino, July 19, 2005
Numerical Results: Lid Driven Cavity
-
6
x
y
1
1
u1 =0; u2 =0
u1 =1; u2 =0
u1 =0
u2 =0
u1 =0
u2 =0
Geometry of the lid driven cavity Eddies’ position for the cavity problem
Numerical experiments performed with Re = 2000, 3000, 4000, 5000, 6000, 8000, 10000
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IFIP 2005 Torino, July 19, 2005
Equation Scaling
For this problem we consider the following scaled nonlinear equation
D−1
F (x) = 0
where the constant diagonal matrix D is computed at the first iteration by
Di,i =X
j
|F ′i,j(x0)|.
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IFIP 2005 Torino, July 19, 2005
Comparison between the HINM and the INB, Cavity Number of failuresGrid ηk → 1.0E-1 1.0E-4 Ad. Choice1089 HINM 0 0 4
INB 5 0 6
Performances of the HINM on tests unsolved by the INB, Nodes=1089
Test η Re iterlGMiter
mGMitermax sw ‖F ‖
1 1.0E-1 2000 59 28 43 27 4.127E-72 1.0E-1 4000 27 39 67 2 9.426E-73 1.0E-1 6000 50 60 123 18 6.878E-74 1.0E-1 8000 85 57 150 57 9.928E-75 1.0E-1 10000 139 74 137 108 8.076E-76 choice 1 3000 83 39 59 46 8.340E-77 choice 1 4000 64 39 88 26 7.543E-7
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Node=1089, Re=4000, ηk = 1.0E − 1
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IFIP 2005 Torino, July 19, 2005
Node=1089, Re=6000, ηk = 1.0E − 1
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IFIP 2005 Torino, July 19, 2005
Node=1089, Re=10000, ηk = 1.0E − 1
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