Finite Element Method of Multiscale Type Basic Ideas and - - PowerPoint PPT Presentation
Finite Element Method of Multiscale Type Basic Ideas and Applications Alexandre L. Madureira Laborat orio Nacional de Computa c ao Cient fica (LNCC) Brazil Joint work with Leopoldo Franca (University of Colorado, US) Lutz
Alexandre L. Madureira Laborat´
c˜ ao Cient´ ıfica (LNCC) – Brazil Joint work with Leopoldo Franca (University of Colorado, US) Lutz Tobiska (Otto-von-Guericke University Magdeburg, Germany) Fr´ ed´ eric Valentin (LNCC, Brazil) US–South America Workshop, August 3, 2004
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⇒ Problems in heterogeneous materials
domains with rough boundary
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In general we decompose the solution as: usolution = umacro + umicro
resolving the microscale features.
uMsF EM = ulinear + uMs where ulinear is piecewise linear, and uMs brings information about the microscales.
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Consider the problem Lε u = f in Ω, u = 0
and its weak formulation: find u ∈ H1
0(Ω) such that
a(u, v) = (f, v) for all v ∈ H1
0(Ω).
Here, Ω is a polygon, ε > 0 is a small parameter, and (f, v) =
fv dx
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Example I: Thermal problem Lε u := − div
and a(u, v) =
Example II: Reaction–diffusion problem Lε u := −ε ∆ u + u, and a(u, v) =
ε grad u · grad v + uv dx.
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Consider a partition of the domain Ω into finite elements, and the associated enriched space Vh := V1 ⊕ B, where
0(Ω) is the space of piecewise linear or bilinear functions
0(Ω) is the space of “bubbles”, functions that vanish
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The method consists in finding uh ∈ Vh = V1 ⊕ B where a(uh, v) = (f, v) for all v ∈ Vh. Writing uh = u1 + ub implies a(u1 + ub, v1) = (f, v1) for all v1 ∈ V1, a(u1 + ub, vb) = (f, vb) for all vb ∈ B. Hence, the second equation holds elementwise: a(u1 + ub, vb)|K = (f, vb)|K for all vb ∈ H1
0(K),
for every element K.
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The bubble is the strong solution of the local problem Lε ub = − Lε u1 + f in K, ub = 0
Write ub = T(− Lε u1 + f) and do static condensation: a(u1 + ub, v1) = (f, v1) = ⇒ a(u1 + T(− Lε u1 + f), v1) = (f, v1) = ⇒ a(u1 − T Lε u1, v1) = (f, v1) − a(Tf, v1) = ⇒ a
for all v1 ∈ V1,
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We can see this formulation as a Parameter Free Stabilized Method: Search for u1 ∈ V1 where a(u1, v1) − a(T Lε u1, v1) = (f, v1) − a(Tf, v1) for all v1 ∈ V1.
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We can see this formulation as a numerical upscaling procedure: Search for u1 ∈ V1 where a∗(u1, v1) =< f ∗, v1 > for all v1 ∈ V1, and a∗(u1, v1) = a((I −T Lε)u1, v1), < f ∗, v1 >= (f, v1)−a(Tf, v1). Multiscale interpretation:
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We can see this formulation almost like a Petrov–Galerkin Method: If {ψi} is a basis of V1, and u1 = N
i=1 uiψi, then N
uia((I − T Lε)ψi, ψj) = (f, ψj) − a(Tf, ψj) = ⇒
N
uia(λi, ψj) = (f, ψj) − a(Tf, ψj), where λi = (I − T Lε)ψi. Hence, Lε λi = 0 in K, λi = ψi
The basis functions of the trial space solve the operator locally, and the test functions remain the same.
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The RFB strategy works pretty well for second order PDEs with
reaction–diffusion equation.
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Example: Consider the domain Ω = (0, 1) × (0, 1) and the problem −10−6 ∆ u + u = 1 in Ω, u = 0
y 1 u = 0 u = 0 1 x u = 0 u = 0
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The standard piecewise linear Galerkin aproximation is given by
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
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The RFB aproximation is given by
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4
and the spurious oscillations are still there.
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the functions in the enriched space to be linear over the edges, and hence they are unable to capture the boundary layer effects.
and impose that the basis functions solve the operator inside each element, and solve an ODE over each edge. This ODE is defined using a “1D restriction” of the original operator.
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We want uh ∈ Uh such that a(uh, vh) = (f, vh) for all vh ∈ Vh
values determined by edge restrictions of the governing differential operator
Therefore we start out with a Petrov-Galerkin setting.
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After some formalism (similar to RFB), we gather that the nodal values ui solve
N
uia(θi, ψj) = (f, ψj) − a(Tf, ψj), where Lε θi = 0 in each element. To determine θi use the boundary condition −ε∂ssθi + σθi = 0
θi = 1 at the ith node, θi = 0 at the other nodes, and s is the variable running along the edge. We do have analitic expressions for θi.
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In the simplest case (a square): θ(x, y) = sinh
2εh(1 − x/h)
2εh(1 − y/h)
2εh
2εh
Typical basis functions θ for ε = 1.0:
–1 –0.5 0.5 1 x –1 –0.5 0.5 1 y 0.2 0.4 0.6 0.8 1
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Typical basis functions θ for ε = 0.1:
–1 –0.5 0.5 1 x –1 –0.5 0.5 1 y 0.2 0.4 0.6 0.8 1
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Typical basis functions θ for ε = 10−3:
–1 –0.5 0.5 1 x –1 –0.5 0.5 1 y 0.2 0.4 0.6 0.8 1
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Example I: Consider Ω = (0, 1) × (0, 1), f = 1 with u = 0 on ∂Ω, and ε = 10−6:
y 1 u = 0 u = 0 1 x u = 0 u = 0
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Linear Galerkin:
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
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RFB:
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4
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New Formulation
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4
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NACA Example: Let f = 0, and u = 0 on the outer boundary and u = 1 in the inner boundary (ε = 10−6):
MESH%anticlipinit
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Isovalues of the solutions by Galerkin method:
GALERKIN METHOD%anticlipinit 29
Isovalues of the solutions by the enriched method:
NEW ENRICHED METHOD%anticlipinit
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The new method captures the boundary layer accurately. Zoom of isovalues:
NEW ENRICHED METHOD - ZOOM 31
Solution profile:
GALERKIN METHOD
0.1 0.3 0.5
0.15 0.3 0.45 0.6 0.75 0.9 1.05 UNUSUAL METHOD
0.1 0.3 0.5
0.15 0.3 0.45 0.6 0.75 0.9 1.05 NEW ENRICHED METHOD
0.1 0.3 0.5
0.15 0.3 0.45 0.6 0.75 0.9 1.05
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accurately problems where multiple scales play a significant role
zero edge condition
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