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´ ıfica (LNCC) – Brazil Joint work with Leopoldo Franca (University of Colorado, US) Lutz Tobiska (Otto-von-Guericke University Magdeburg, Germany) eric Valentin (LNCC, Brazil) Fr´ ed´ US–South America Workshop , August 3, 2004 1

Multiple Scale Phenomena • PDEs with highly oscillatory coefficients ⇐ ⇒ Problems in heterogeneous materials • Different scales in the domain itself , like plates and shells, or domains with rough boundary • Reaction dominated reaction–diffusion eqtns 2

Decomposition In general we decompose the solution as: u solution = u macro + u micro • Aim of multiscale modeling: macroscopic behaviour without resolving the microscale features. • Multiscale Finite Element Method: decompose u MsF EM = u linear + u Ms where u linear is piecewise linear, and u Ms brings information about the microscales. 3

General Idea Consider the problem L ε u = f in Ω , u = 0 on ∂ Ω , and its weak formulation: find u ∈ H 1 0 (Ω) such that for all v ∈ H 1 a ( u, v ) = ( f, v ) 0 (Ω) . Here, Ω is a polygon, ε > 0 is a small parameter, and � ( f, v ) = fv d x Ω 4

Example I: Thermal problem L ε u := − div � � K ( x, ε ) grad u , and � � � a ( u, v ) = K ( x, ε ) grad u · grad v d x . Ω Example II: Reaction–diffusion problem L ε u := − ε ∆ u + u, and � a ( u, v ) = ε grad u · grad v + uv d x . Ω 5

Residual Free Bubbles (RFB) Consider a partition of the domain Ω into finite elements, and the associated enriched space V h := V 1 ⊕ B, where • V 1 ⊂ H 1 0 (Ω) is the space of piecewise linear or bilinear functions • B ⊂ H 1 0 (Ω) is the space of “bubbles”, functions that vanish over the edges of the finite elements 6

The method consists in finding u h ∈ V h = V 1 ⊕ B where a ( u h , v ) = ( f, v ) for all v ∈ V h . Writing u h = u 1 + u b implies a ( u 1 + u b , v 1 ) = ( f, v 1 ) for all v 1 ∈ V 1 , a ( u 1 + u b , v b ) = ( f, v b ) for all v b ∈ B. Hence, the second equation holds elementwise: for all v b ∈ H 1 a ( u 1 + u b , v b ) | K = ( f, v b ) | K 0 ( K ) , for every element K . 7

The bubble is the strong solution of the local problem L ε u b = − L ε u 1 + f in K, u b = 0 on ∂K. Write u b = T ( − L ε u 1 + f ) and do static condensation: a ( u 1 + u b , v 1 ) = ( f, v 1 ) ⇒ a ( u 1 + T ( − L ε u 1 + f ) , v 1 ) = ( f, v 1 ) = ⇒ a ( u 1 − T L ε u 1 , v 1 ) = ( f, v 1 ) − a ( Tf, v 1 ) = ( I − T L ε ) u 1 , v 1 � � = ⇒ a = ( f, v 1 ) − a ( Tf, v 1 ) for all v 1 ∈ V 1 , 8

First point of view We can see this formulation as a Parameter Free Stabilized Method : Search for u 1 ∈ V 1 where a ( u 1 , v 1 ) − a ( T L ε u 1 , v 1 ) = ( f, v 1 ) − a ( Tf, v 1 ) for all v 1 ∈ V 1 . 9

Second point of view We can see this formulation as a numerical upscaling procedure : Search for u 1 ∈ V 1 where a ∗ ( u 1 , v 1 ) = < f ∗ , v 1 > for all v 1 ∈ V 1 , and a ∗ ( u 1 , v 1 ) = a (( I − T L ε ) u 1 , v 1 ) , < f ∗ , v 1 > = ( f, v 1 ) − a ( Tf, v 1 ) . Multiscale interpretation: • V 1 is the coarse space, seeing only the “macro” properties • V B is the fine space, capturing the small scale features 10

Third point of view We can see this formulation almost like a Petrov–Galerkin Method : If { ψ i } is a basis of V 1 , and u 1 = � N i =1 u i ψ i , then N � u i a (( I − T L ε ) ψ i , ψ j ) = ( f, ψ j ) − a ( Tf, ψ j ) i =1 N � u i a ( λ i , ψ j ) = ( f, ψ j ) − a ( Tf, ψ j ) , where λ i = ( I − T L ε ) ψ i . = ⇒ i =1 Hence, L ε λ i = 0 in K, λ i = ψ i on ∂K, The basis functions of the trial space solve the operator locally, and the test functions remain the same. 11

Upset: The RFB strategy works pretty well for second order PDEs with oscillatory coefficients (Sangalli, 2003), but fails for the reaction–diffusion equation. 12

Example : Consider the domain Ω = (0 , 1) × (0 , 1) and the problem − 10 − 6 ∆ u + u = 1 in Ω , u = 0 on ∂ Ω , y u = 0 1 u = 0 u = 0 0 u = 0 1 x 13

The standard piecewise linear Galerkin aproximation is given by 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 1 0.2 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0 0.2 0.1 0 14

The RFB aproximation is given by 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 1 0.8 0.2 0.6 0.4 0.2 0 0 and the spurious oscillations are still there. 15

So, what goes wrong? • Imposing that the bubbles vanish at each element edge causes the functions in the enriched space to be linear over the edges, and hence they are unable to capture the boundary layer effects. • We use then an idea by Tom Hou and X.H. Wu (JCP, 1997), 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. 16

New idea: enriching the finite element space with local, but not bubble-like function We want u h ∈ U h such that a ( u h , v h ) = ( f, v h ) for all v h ∈ V h 1. Enrich the trial space U h with local solutions with boundary values determined by edge restrictions of the governing differential operator 2. Enrich the test space V h with residual-free bubble functions Therefore we start out with a Petrov-Galerkin setting. 17

After some formalism (similar to RFB), we gather that the nodal values u i solve N � u i a ( θ i , ψ j ) = ( f, ψ j ) − a ( Tf, ψ j ) , i =1 where L ε θ i = 0 in each element. To determine θ i use the boundary condition − ε∂ ss θ i + σθ i = 0 over the edges θ i = 1 at the ith node , θ i = 0 at the other nodes , and s is the variable running along the edge. We do have analitic expressions for θ i . 18

In the simplest case (a square): �� �� � � 1 1 sinh 2 ε h (1 − x/h ) sinh 2 ε h (1 − y/h ) θ ( x, y ) = �� �� � � 1 1 sinh 2 ε h sinh 2 ε h 19

Typical basis functions θ for ε = 1 . 0: 1 0.8 0.6 0.4 0.2 0 –1 –1 –0.5 –0.5 0 0 y x 0.5 0.5 1 1 20

Typical basis functions θ for ε = 0 . 1: 1 0.8 0.6 0.4 0.2 0 –1 –1 –0.5 –0.5 0 0 y x 0.5 0.5 1 1 21

Typical basis functions θ for ε = 10 − 3 : 1 0.8 0.6 0.4 0.2 0 –1 –1 –0.5 –0.5 0 0 y x 0.5 0.5 1 1 22

Numerical Results 23

Example I : Consider Ω = (0 , 1) × (0 , 1), f = 1 with u = 0 on ∂ Ω, and ε = 10 − 6 : y u = 0 1 u = 0 u = 0 0 u = 0 1 x 24

Linear Galerkin: 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 1 0.9 0.2 0.8 0.7 0.6 0.5 0.4 0.3 0 0.2 0.1 0 25

RFB: 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 1 0.8 0.2 0.6 0.4 0.2 0 0 26

New Formulation 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 1 0.9 0.8 0.7 0.6 0.5 0.4 0 0.3 0.2 0.1 0 27

NACA Example : Let f = 0, and u = 0 on the outer boundary and u = 1 in the inner boundary ( ε = 10 − 6 ): MESH%anticlipinit� 28

Isovalues of the solutions by Galerkin method: GALERKIN METHOD%anticlipinit� 29

Isovalues of the solutions by the enriched method: NEW ENRICHED METHOD%anticlipinit� 30

The new method captures the boundary layer accurately. Zoom of isovalues: NEW ENRICHED METHOD - ZOOM 31

Solution profile: GALERKIN METHOD UNUSUAL METHOD NEW ENRICHED METHOD 1.05 1.05 1.05 0.9 0.9 0.9 0.75 0.75 0.75 0.6 0.6 0.6 0.45 0.45 0.45 0.3 0.3 0.3 0.15 0.15 0.15 0 0 0 -0.15 -0.15 -0.15 -0.5 -0.5 -0.5 -0.3 -0.3 -0.3 -0.1 -0.1 -0.1 0.1 0.1 0.1 0.3 0.3 0.3 0.5 0.5 0.5 32

Conclusions • Inovative finite element methods have the potential to solve accurately problems where multiple scales play a significant role • The Residual Free Bubbles approach fails for reaction-diffusion eqtns. The culprit is the restriction that bubbles should vanish on element edges • We propose a new Petrov-Galerkin formulation eliminates the zero edge condition 33