The finite element immersed boundary method: model, stability, and numerical results

Transcription

1 Te finite element immersed boundary metod: model, stability, and numerical results Lucia Gastaldi Università di Brescia ttp://dm.ing.unibs.it/gastaldi/ INdAM Worksop, Cortona, September 18, 2006 Joint work wit: Daniele Boffi and Luca Heltai

2 Outline 1 Te immersed boundary metod 2 Matematical formulation 3 Space discretization by finite elements 4 Stability of te discrete problem 5 Numerical results

3 History Introduced by Peskin Flow patterns around eart valves: a digital computer metod for solving te equations of motion, PD Tesis, 1972 Numerical analysis of blood flow in te eart, J. Comput. Pys., 1977 Extended by Peskin and McQueen since 83 to simulate te blood flow in a tree dimensional model of eart and great vessels Review article by Peskin Te immersed boundary metod, Acta Numerica, 2002 Several application in biology, e.g. simulation of a flapping flexible filament in a flowing soap film, insect fligt, computational model of te coclea,....

4 Te immersed boundary metod ω Ω D Ω s (t) Ω fluid and solid domain Ω s (t) deformable structure domain Ω R d wit d = 2, 3 Ω s (t) R m wit m = d, d 1 x Eulerian variables in Ω s Lagrangian variables in D D reference domain u(x, t) fluid velocity X(s, t) position of te elastic body p(x, t) fluid pressure

5 Elastic materials Ω s (t) D X(s, t) Trajectory of a material point: X : D [0, T ] Ω s (t) Deformation gradient: D(s, t) = X (s, t) wit D = det(d) > 0 s Velocity field: u(x, t) = X (s, t) were x = X(s, t) t

6 Caucy stress tensor D P X(s, t) Pt Ω s (t) From conservation of momenta, in absence of external forces, it olds ( ) u ρ u = ρ t + u u = σ in Ω. Caucy stress tensor Incompressible fluid: σ = σ f = pi + µ( u + ( u) T ) Visco elastic materials: σ = σ f + σ s, σ s elastic part of te stress. σ s can be expressed in Lagrangian variables by means of te Piola-Kircoff stress tensor by: σ s nda = PN da for all P t ; P t P so tat P(s, t) = D(s, t) σ s (X(s, t), t)d T (s, t), s D.

7 Matematical formulation { σf in Ω Caucy stress tensor σ = f (t) σ f + σ s in Ω s (t) Virtual work principle ρ uvdx + Ω Ω σ f : vdx = σ s : vdx Ω s (t) for any smoot v wit compact support in Ω. Use te Lagrangian variables in te solid region: ρ uvdx + σ f : vdx = P : s v(x(s, t))ds Ω Ω Next, integrate by parts: (ρ u σ f ) vdx = ( s P) v(x(s, t))ds Ω D D D PN v(x(s, t))da,

8 Matematical formulation (cont d) An Implicit Cange of Variables using te Dirac Delta distribution v(x(s, t)) = v(x)δ(x X(s, t))dx Ω Ω (ρ u σ f ) vdx = ( s P) v(x(s, t))ds D PN v(x(s, t))da D

9 Matematical formulation (cont d) An Implicit Cange of Variables using te Dirac Delta distribution v(x(s, t)) = v(x)δ(x X(s, t))dx Ω Ω (ρ u σ f ) vdx = D D ( s P) vδ(x X(s, t))dxds Ω PN vδ(x X(s, t))dxda Ω

10 Matematical formulation (cont d) An Implicit Cange of Variables using te Dirac Delta distribution v(x(s, t)) = v(x)δ(x X(s, t))dx Ω Ω (ρ u σ f ) vdx = Ω Ω ( s P)δ(x X(s, t))ds vdx D D PNδ(x X(s, t))da vdx

11 Matematical formulation (cont d) Since v is arbitrary, we get ρ u σ f = s Pδ(x X(s, t))ds Inner force density g(x, t) = Transmission force density t(x, t) = D D D D s P(s, t)δ(x(s, t) x)ds. P(s, t)n(s)δ(x(s, t) x)ds. PNδ(x X(s, t))da Remark If m = d 1, D is eiter a curve in 2D or a surface in 3D, ten D =.

12 Problem setting. Four ingredients. 1. Te Navier-Stokes equations Ω: domain containing te fluid and te elastic structure x: Eulerian variables ( ) u ρ t + u u µ u + p = g + t in Ω ]0, T [ u = 0 in Ω ]0, T [ ρ: fluid density (constant) µ: fluid viscosity (constant) u(x, t), p(x, t): velocity and pressure g: inner force density t: transmission force density

13 Problem setting (cont d) 2. Force densities g(x, t) = s P(s, t)δ(x X(s, t))ds, in Ω ]0, T [ D t(x, t) = P(s, t)n(s)δ(x X(s, t))ds, in Ω ]0, T [ D 3. Motion of te immersed structure X (s, t) = u(x(s, t), t) in D ]0, T [ t 4. Initial and boundary conditions u(x, t) = 0 on Ω ]0, T [ u(x, 0) = u 0 (x) in Ω, X(s, 0) = X 0 (s) in D

14 Equivalence wit standard formulation Te definition of g and t implies tat g(x, t) = 0 for x Ω s (t), t(x, t) = 0 for x Ω s (t). Te given Navier-Stokes equations are equivalent to: ( ) u ρ t + u u µ u + p = 0 in (Ω \ Ω s (t)) ]0, T [ ( ) u ρ t + u u µ u + p = g in Ω s (t) ]0, T [ σ + f n+ + σ f n = D 1 PN on Ω s (t) ]0, T [

15 Elastic materials D X(s, t) Ω s (t) Trajectory of a material point X : D [0, T ] Ω s (t) Deformation gradient: D(s, t) = X (s, t) s Wide class of elastic materials are caracterized by: potential energy density W (D(s, t)) Piola-Kircoff stress tensor P(D(s, t)) = W (s, t) D elastic potential energy E (X(t)) = W (D(s, t))ds D ( ) W force density f(s, t) = s (D(s, t)) = s P(D(s, t)) D can be obtained taking te Frécet derivative of E

16 Example - tin fibers (1D) immersed in a fluid (2D) X(b, t) τ X(a, t) τ Γ t P = T τ ( ) X T = ϕ s ; s, t Hooke s law for te boundary tension; τ = X/ s unit tangent X/ s f = s (T τ ) Easiest possible case W (D) == κ 2 D 2, T = κ D, τ = D/ D, f = κ 2 X s 2 κ stiffness of te fiber

17 Variational formulation Lemma: Variational definition of te source term Assume tat, for all t [0, T ], Ω s (t) is C 1 regular and tat P is W 1,. Ten for all t ]0, T [, te force density F = g + t is a distribution function belonging to H 1 (Ω) d defined as follows: for all v H0 1(Ω)d H 1< F(t), v > H0 1= P(D(s, t)) : s v(x(s, t)) ds t ]0, T [. D

18 Variational formulation Navier-Stokes equations ρ d (u(t), v) + a(u(t), v)+b(u(t), u(t), v) ( v, p(t)) dt =< F(t), v > v H0 1(Ω)d ( u(t), q) = 0 q L 2 0 (Ω) were a(u, v) = µ( u, v), b(u, v, w) = ρ(u v, w). < F(t), v >= P(D(s, t)) : s v(x(s, t)) ds, v H0 1 (Ω) d D X (s, t) = u(x(s, t), t) s [0, L] t u(x, 0) = u 0 (x) x Ω, X(s, 0) = X 0 (s) s [0, L],

19 Stability property Lemma - Energy estimate ρ d 2 dt u(t) µ u(t) d E(X(t)) = 0. dt Proof Take v = u, use te divergence free constraint and te equation of motion of X: ρ d dt (u(t), u(t)) + a(u(t), u(t)) = P(D(s, t)) X(s, t) ds D s t Use te definition of te potential energy density W and te elastic potential energy E: W D D (D(s, t)) ( ) X W (s, t) ds = (D(s, t)) D(s, t)ds t s D D t = d W (D(s, t))ds = d (E (X(t))) dt dt D

20 Finite element spaces T subdivision of Ω into elements wit messize x. Let V H0 1(Ω)d and Q L 2 0 (Ω) be finite dimensional. We coose a pair of spaces V and Q wic satisfies te inf-sup condition. Example V = {v H 1 0 (Ω)d : v continuous piecewise biquadratic} Q 2 Q = {q L 2 0 (Ω) : q discontinuous piecewise linear} P 1 S subdivision of D into elements wit messize s. S = {Y C 0 (D; Ω) : Y continuous piecewise linear} Notation T k k = 1,..., M e elements of S s j, j = 1,..., M vertices of S E set of te edges e of S

21 Discrete source term We ave to discretize: < F(t), v >= X piecewise linear D piecewise constant; P depends only on D P piecewise constant. After integration by parts we get D P(D(s, t)) : s v(x(s, t))ds M e M e < F (t), v > = P : s v(x(s, t))ds = PNv(X(s, t))da T k T k k=1 k=1 tat is < F (t), v > = e E e [[P]] v(x(s, t))da [[P]] = P + N + + P N jump of P across e for internal edges [[P]] = PN wen e D

22 Te finite element immersed boundary metod Find (u, p ) : ]0, T [ V Q and X : [0, T ] S suc tat NS ρ d dt (u (t), v) + a(u (t), v) + b (u (t), u (t), v) ( v, p (t)) = [[P]] v(x(s, t))da e E ( u (t), q) = 0 e v V q Q dx i (t) = u (X i (t), t) i = 1,..., M dt u (0) = u 0 in Ω, X i (0) = X 0 (s i ) i = 1,..., M, were b (u, v, w) = ρ 2 ((u v, w) (u w, v)).

23 Stability property Lemma - Semidiscrete energy estimate ρ d 2 dt u (t) µ u (t) d dt E(X (t)) = 0. Proof Te proof is te same as in te continuous case.

24 Fully discrete problem - Backward Euler sceme Notation: t time step, t n = n t, u n u (t n ), X n i X i (t n ). Initial data: u 0 = u 0, X 0 i = X 0 (s i ) i = 1,, m. Backward Euler sceme - BE Find (u n+1, p n+1 ) V Q and X n+1 S, suc tat < F n+1, v > = e E e LNS ( u n+1 u n ρ, v t [[P]] n+1 v(x n+1 (s, t))da v V ; ) + a(u n+1, v) ( v, p n+1 ) =< F n+1, v > ( u n+1, q) = 0 q Q ; v V X n+1 i X n i t = u n+1 (X n+1 i ) i = 1,, M.

25 Fully discrete problem - Modified Backward Euler sceme Notation: t time step, t n = n t, u n u (t n ), X n i X i (t n ). Initial data: u 0 = u 0, X 0 i = X 0 (s i ) i = 1,, m. Modified backward Euler sceme - MBE Step 1. < F n, v > = [[P]] n v(x n (s, t))da v V ; e E Step 2. find (u n+1, p n+1 ) V Q, suc tat ( u n+1 u n ρ t LNS e ), v + a(u n+1, v) + b (u n+1, u n+1, v) ( v, p n+1 ) =< F n, v > v V ( u n+1, q) = 0 q Q ; Step 3. X n+1 i X n i t = u n+1 (X n i) i = 1,, M.

26 Discrete Energy Estimate Set H iαjβ = 2 W D iα D jβ (D) Assumption Tere exist κ min > 0 and κ max > 0 s.t. for all tensors E κ min E 2 E : H : E κ max E 2 Artificial Viscosity Teorem Let u n, pn and Xn be a solution to te FE-IBM, ten ρ ( u n+1 2 t 2 0 u n ) (µ + µa ) u n ( [ ] E X n+1 t E [X n ] ) 0 CFL Condition: µ + µ a > 0

27 Artificial Viscosity BE sceme µ a = κ min C (m 2) s (d 1) x t L n+1 MBE sceme µ a = κ max C (m 2) s (d 1) x t L n { } L n := max max T k S s j,s i V (T k ) Xn (s j) X n (s i)

28 Proof BE: ñ = n + 1, MBE: ñ = n Take v = u n+1 in te FE-IBM formulation 1 M e 2 t ( un u n 2 0) + µ u n Pñ : s u n+1 (Xñ )ds k=1 T k M e ( ) X n+1 X n = ds t = 1 t = 1 t Pñ : s k=1 T k M e k=1 M e k=1 T k Pñ : (D n+1 D n )ds T k Pñ : (D n+1 D n ).

29 Proof (cont d) BE: ñ = n + 1, MBE: ñ = n ξ 3 B k s 1 X X (s 1 ) ˆT s 2 T k X (s 2 ) X (T k ) ξ 1 ξ 2 s k X (s k ) Pñ : (D n+1 D n ) (W n+1 W n ) + κ D n+1 D n 2 were κ = κ max for MBE and κ = κ min for BE m D n+1 D n 2 C t 2 s 2 u n+1 (Xñj ) un+1 (Xñk ) 2 u n+1 Concluding: 1 t M e k=1 j=1 (Xñj ) un+1 (Xñk ) 2 C (d 1) x T k Pñ : (D n+1 D n ) C m 2 s t x d 1 Xñj Xñk M e k=1 j=1 u n+1 2 0, ˆT k m X ñ j Xñk u n+1 2 0, ˆT k

30 Main result BE µ a = κ min C (m 2) s (d 1) x t L n+1 MBE µ a = κ max C (m 2) s (d 1) x t L n { } L n := max max T k S s j,s i V (T k ) Xn (s j ) X n (s i ) If we use BE and κ min > 0, ten te metod is unconditionally stable.

31 CFL conditions for MBE µ a = κ max C (m 2) s (d 1) x t L n space dim. solid dim. CFL condition 2 1 L n t C x s 2 2 L n t C x 3 2 L n t L n t Cx/ 2 s

32 Static circle d = 2, m = 1 Energy density: W = κ 1 2 D 2

33 Volume loss Percentage m N = N = N = Optimal coice: s x /2

34 Stability analysis d = 2, m = 1 Plot: µ a /µ VS E[X] + 1/2ρ u 2 0 µ a = κc t s x L n 10 1 k = 2, N = 64, M = 128, dt = E η 10 1 E η k = 2, N = 64, M = 128, dt = k = 2, N = 64, M = 128, dt = E η t κ = 2, N=64, M=128, t = t κ = 2, N=64, M=128, t = t κ = 2, N=64, M=128, t = 0.005

35 Examples of instability

36 Ellipse immersed in a static fluid d = 2, m = 1 Fluid initially at rest: u 0 = 0 ( 0.35 cos(2πs) X 0 (s) = 0.25 sin(2πs) ) s [0, 1], Immersed boundary: time=0dt

37 Surface immersed in a static fluid d = 3, m = 2 Initial immersed boundary Static fluid: u 0 = 0

38 Two-dimensional visco-elastic cell: static case d = 2, m = 2 Initial configuration Anisotropic material W = κ X 2t s 1 2 = κ 2t ( X1 s 1 D = [0, 2πR] [0, t], periodic in s 1 X 0 = 2 ( R(1 + s2 ) cos(s 1 /R) R(1 + s 2 ) sin(s 1 /R) X 2 ) 2 ; P = κ ( X1 s 1 2t 0 0 ) X 2 s 1 s 1 ) Inner force density: s P = κ 2 X t s1 2 = κ t Transmission stress density: PN = s 2 R r

39 Computed pressure Collection of fibers Standard finite element mes

40 Anistropic material II d = 2, m = 2 Collection of fibers W = κ X 2t s 2 2 ; P = κ 2t ( 0 0 X 1 s 2 X 2 s 2 ) Standard finite element mes

41 Two-dimensional visco-elastic cell: dynamic case d = 2, m = 2 Initial configuration X 0 = ( R(1 + s2 ) cos(s 1 /R) R(1 + γ + s 2 ) sin(s 1 /R) ) Collection of fibers Standard finite element mes

42 Conclusions Te Immersed Boundary Metod is extended to te treatment of tick materials modeled by yper-elastic constitutive laws. Te finite element approac is efficient and can easily andle te case of tick materials also. Stability analysis of te space-time discretization is provided.