An Eulerian Approach for Computing the Finite Time Lyapunov Exponent (FTLE)

Transcription

1 An Eulerian Approach for Computing the Finite Time Lyapunov Exponent (FTLE) Shingyu Leung Department of Mathematics, Hong Kong University of Science and Technology May, Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, / 33

2 Outline Introduction to FTLE ECS vs. LCS LCS through FTLE Lagrangian Formulation to FTLE 3 Part : Eulerian Formulation to FTLE 4 Part : Propagating the FTLE for many time-levels 5 Advantages and Limitations 6 Part 3: FTLE on a Codimension One Manifold 7 Examples 8 Conclusion Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, / 33

3 Introduction to FTLE Eulerian Coherent Structure (ECS) To segment the domain into different regions with similar behavior according to an Eulerian quantity such as the strain, kinetic energy, or the vorticity. Lagrangian Coherent Structure (LCS) To partition the space-time domain into different regions according to a Lagrangian quantity advected along with passive tracers. Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 3 / 33

4 One such Lagrangian quantity is the FTLE, rate of separation between adjacent particles over a finite time interval with an infinitesimal perturbation in the initial location. x (t;x,t ) = u(x(t;x,t ),t) with the initial condition x(t ) = x and u : R d R R d. Flow map Φ : Ω R R R d, Φ(x;t,T) = x(t;x,t ) The leading order in the perturbation δx(t) = δx(),[dφ(x;t,t)] DΦ(x;t,T)δx() Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 4 / 33

5 Denoting (x;t,t) the Cauchy-Green deformation tensor (x;t,t) = [DΦ(x;t,T)] DΦ(x;t,T), the largest strength deformation max δx(t) = λ max [ (x;t,t)] e() = exp[σ T (x,t ) T ] e() δx() FTLE σ T (x,t ) = T ln λ max [ (x;t,t)] forward FTLE if T > and the backward FTLE if T <. LCS: (roughly speaking) Ridge of the FTLE. Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 5 / 33

6 Lagrangian formulation to FTLE Lagrangian approach to FTLE Compute the flow map Φ(x;t,T) by solving the ODE system x (t;x,t ) = u(x(t;x,t ),t) on an initial Cartesian mesh at time t = t. Construct the deformation tensor (x;t,t) by finite differencing the flow map Φ(x i ;t,t) and then find it s largest eigenvalue. Remarks The velocity field might be defined discretely on a mesh. The numerical accuracy when using a high order adaptive time integrator such as ode45 in MATLAB. 3 For computing the FTLE at the next time step t = t = t + t, all rays obtained on the previous time step t = t are all discarded. Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 6 / 33

7 Part : Eulerian formulation to FTLE for one time level Goal: Lagrangian computations Eulerian formulation Define Ψ = (Ψ,Ψ,,Ψ d ) : Ω R R d with the initial condition Ψ(x,t ) = x = (x,x,,x d ). Following the particle trajectory with x = x, any particle identity should be preserved in the Lagrangian framework, i.e. DΨ(x, t) Dt = Ψ(x,t) t +(u )Ψ(x,t) = Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 7 / 33

8 These level set functions defined on a uniform Cartesian mesh give the backward flow map from t = t +T to t = t, i.e. Φ(y;t +T, T) = Ψ(y,t +T) Figure: Lagrangian and Eulerian interpretations of the function Ψ. Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 8 / 33

9 Computing the backward FTLE σ T (x,t +T) Discretize the computational domain Initialize Ψ (x i,y j,t ) = x i and Ψ (x i,y j,t ) = y j 3 Solve the Liouville equations Ψ m t +(u )Ψ m =, m =, for up to t = t K = t +T. 4 Compute the Cauchy-Green deformation tensor with Ψ = (Ψ,Ψ ) T, (x i,y j ;t K, T) = [DΨ(x i,y j,t K )] DΨ(x i,y j,t K ) 5 Determine the backward FTLE at (x i,y j,t +T) σ T (x i,y j,t K ) = T ln λ max [ (x i,y j ;t K, T)] Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 9 / 33

10 Computing the forward FTLE σ T (x,t ) Initialize the level set functions at t = t +T by Ψ(x,t +T) = x. Solve the corresponding level set equations backward in time. 3 Determining the Jacobian of the resulting flow map and then computing the largest eigenvalue of the deformation tensor (x;t,t). 4 The forward FTLE is formed by σ T (x,t ) = T ln λ max [ (x;t,t)]. Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, / 33

11 Part : Propagating the FTLE for many time-levels Goal: Propagate the backward FTLE σ T (x,t +T) forward in time from t +T to T f in order to approximate σ T (x,t) or σ (t t ) (x,t) for t > t +T. Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, / 33

12 Theorem (Theorem 3. in Shadden, Lekien and Marsden 5) The traditional (forward) Lyapunov exponent is constant along trajectories. Also, the (forward) finite-time Lyapunov exponent becomes constant along trajectories for large integration times T. The difference between σ T (x,t ) and σ T (Φ s (x;t ),t +s) for some arbitrary but fixed s, σ T (x,t ) σ T (Φ s (x;t ),t +s) s T max t σ t t (x,t ) = O( T ) The material derivative of the (forward) FTLE is bounded by Dσ T (x,t) Dt = lim σ T (Φ s (x;t),t +s) σ T (x,t) = O( T ) s s If T is large enough, we obtain the approximation Dσ T (x,t) Dt = Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, / 33

13 If T is large enough, we can approximate the solution σ T (x,t) for t > t +T by simply solving the following Liouville equation σ T (x,t) t +u σ T (x,t) = with the initial condition at t = t +T. Let S be the time difference from t = t +T. Since σ T (Φ S (x;t +T),t +T +S) σ T (x,t +T) S ( ) S T max σ t (x,t) = O, t T the error in the approximation is linear in S. Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 3 / 33

14 Advantages and limitations of the Eulerian formulation Timestep constraint Lagrangian: t, stiffness of the velocity field Eulerian: t = O( x), CFL condition Computational Complexity d: the dimension of the domain N = O( x ): number of mesh points in each spacial dimension M = T/ t = O( t ): number of time steps for t = T K = (T f t )/ t = O( t ): number of time steps until t = T f (number of time levels of FTLE to be computed) Lagrangian: O(N d MK) = O( x d t t ) Eulerian: O(N d [(d )M +K +]) = O( x d t ) Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 4 / 33

15 Velocity Field CFD solver Real life measurements Storage for the velocity field Lagrangian approach: O(dN d+ ), N d+ mesh points in the space-time domain and a d-vector per node. Eulerian approach: O(dN d ), N d mesh points in the space domain and a d-vector per node. Storage for the flow map and the FTLE Lagrangian approach: O(d) Eulerian approach: O(dN d ) Numerical dissipation in the Eulerian formulation An order of O(S/T) error is introduced in the approximation Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 5 / 33

16 Part 3: FTLE on a Codimension One Manifold Goal: A simple approach to compute the FTLE on a codimension one manifold. A codimension one manifold M(t) in R d evolving in time under the velocity field ũ : M(t) t R d. A possible approach: triangulate the surface, evolve all vertices according to the flow, compute the Jacobian matrix based on a local coordinate system on the surface,... Our proposed approach: the evolving manifold represented implicitly using a level set function. Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 6 / 33

17 Computing the backward FTLE σ T (x,t +T) for x M(t +T) With φ(x i,y j,t ) at t = t so that M = {φ = }, solve φ t +(u )φ = for up to t = t +T With Ψ (x i,y j,t ) = x i, Ψ (x i,y j,t ) = y j, solve the Liouville equations Ψ m t +(u )Ψ m =, m =, from t = t to t = t +T 3 Extend the map off the manifold by Ψ m τ +sgn(φ)(n )Ψ m =, m =, 4 Compute the deformation tensor = (DΨ) DΨ 5 Determine the backward FTLE σ T = lnλ max ( ) Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 7 / 33

18 Example: Double-Gyre flow The flow is modeled by the following stream-function ψ(x,y,t) = Asin[πf(x,t)]sin(πy), where f(x,t) = a(t)x +b(t)x, a(t) = ǫsin(ωt), b(t) = ǫsin(ωt) with A =., ω = π/ and ǫ =. Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 8 / 33

19 (a) (b) (c) (d) (a) The backward FTLE using x = y = /8 at t = 5, T = 5 and S =. (b) The backward FTLE using x = y = /5 at t = 5, T = 5 and S =. (c) The forward FTLE using x = y = /8 at t =, T = 5 and S =. (d) The forward FTLE using x = y = /5 at t =, T = 5 and S =. Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 9 / 33

20 (a) (b) (c) (d) (a) The backward FTLE with x = y = /5 at t = 5. using the Lagrangian approach (MATLAB function ode45) with T = 5. (b) The backward FTLE with x = y = /5 at t = 5. using the Lagrangian approach (RK4 with a fixed timestep) with T = 5. (c) The backward FTLE at t = 5 using the proposed Eulerian approach with x = y = /5 and T = 5 and S =. (d) The backward FTLE at t = 5 using the proposed Eulerian approach with x = y = /5 and T = 5. Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, / 33

21 (a) (b) (c) (d) (a) the forward FTLE with x = y = /5 at t =. using the Lagrangian approach (MATLAB function ode45) with T = 5. (b) The forward FTLE with x = y = /5 at t =. using the Lagrangian approach (RK4 with a fixed timestep) with T = 5. (d) The forward FTLE at t = using the proposed Eulerian approach with x = y = /5, T = 5 and S =. (c) The forward FTLE at t = using the proposed Eulerian approach with x = y = /5 and T = 5. Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, / 33

22 x Eul. Rate Lag. ( t = ) Rate Lag. ( t = x) Rate / / / / CPU times (in seconds) for the Eulerian approach with T = 5 and S =, and the Lagrangian approach with t = and t = x. Eulerian approach O( x 3 ) Lagrangian approach O( x 3 t ) Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, / 33

23 Example: Rayleigh-Bénard convection The velocity field is given by u = A ξsinkr cosz, k v = A y sinkr cosz, k w = Asinz (r coskr + k ) sinkr, where A =.4, k =., r = ξ +y, ξ = x g(t), and g(t) =.cos(πt). Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 3 / 33

24 (a) (b) (a) The backward FTLE using x = y = /3 at t = 5, T = and S = 38. (b) The forward FTLE using x = y = /3 at t =, T = and S = 38. Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 4 / 33

25 Example: FTLE on a Sphere (a) (b) The forward FTLE on a sphere under the rigid body rotation. The exact solution is zero everywhere on the sphere. The underlying mesh size (a) x = 3/3 and (b) x = 3/8. Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 5 / 33

26 (a) (b) The error in the forward FTLE on a sphere under the motion in the outward normal direction. The underlying mesh size (a) x = 3/3 and (b) x = 3/8. Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 6 / 33

27 (a) (b) The error in the forward FTLE on a sphere under the motion in the inward normal direction. The underlying mesh size (a) x = 3/3 and (b) x = 3/8. Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 7 / 33

28 Example: Four point vortices on a sphere The velocity of any marker particle satisfies the motion x = π 4 i= x i x ( x x i ), with the point vortices centered at (/ 3, / 3,/ 3), (/ 3, / 3, / 3), ( /3,/ 3,) and ( / 3, / 3,/ 3). Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 8 / 33

29 (a) (b) (a) The backward FTLE at t = with T =. (b) The forward FTLE at t =. with T =. Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 9 / 33

30 Example: Sphere under a vortex flow The sphere evolves under the vortex motion governed by u = sin (πx)sin(πy)sin(πz) v = sin (πy)sin(πx)sin(πz) w = sin (πz)sin(πy)sin(πx). (a) (b) (a) The backward FTLE at t =.5 with T =.5. (b) The forward FTLE at t =. with T =.5. Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 3 / 33

31 Conclusion Eulerian framework for approximating the FTLE based on the level set formulation Compute the related flow map by solving Liouville equations Determine the FTLE on an evolving manifold implicitly represented by a level set function Future work Extract the Lagrangian coherent structure from the FTLE Combine the current approaches with a CFD solver Incorporate our algorithms with flow fields from real life measurements Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 3 / 33

32 References Shingyu Leung () An Eulerian Approach for Computing the Finite Time Lyapunov Exponent. Journal of Computational Physics, Volume 3, Issue 9, May, Pages Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 3 / 33

33 Thank you Shingyu Leung (HKUST) An Eulerian Approach to the FTLE May, 33 / 33