Compatible Finite Element Discretizations of Onno Bokhove - PowerPoint PPT Presentation

Compatible Schemes Compatible Finite Element Discretizations of Onno Bokhove Geometric Systems Introduction Non- Autonomous Systems Onno Bokhove NCP time flux Spatial NCP? School of Mathematics, University of Leeds with Elena Gagarina, Vijaya Ambati & Shavarsh Nurijanyan (Twente) Conclusion School of Mathematics, Leeds 2013

Compatible Schemes Onno Bokhove 1 Introduction Introduction Non- 2 Non-Autonomous Systems Autonomous Systems NCP time flux 3 NCP time flux Spatial NCP? Conclusion 4 Spatial NCP? 5 Conclusion

1. Introduction Compatible Schemes Onno Bokhove Numerical modelling of nonlinear waves and currents is often adequately done using conservative, Hamiltonian fluid Introduction dynamics, even in the presence of some forcing and damping. Non- Autonomous In the modelling of two laboratory experiments, Systems non-autonomous Hamiltonian/variational systems emerge: NCP time flux Spatial NCP? investigation of freak waves in wave tanks with Conclusion wave-makers [used for testing model offshore structures] wave-sloshing validations in a table-top Hele-Shaw cell with linear momentum damping. The question is how we can derive stable time integrators?

2. Non-Autonomous Hamiltonian Systems Compatible Schemes Onno Simulation (2D) of waves in MARIN’s Bokhove wave tank : Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion Workings of wave-maker

Hele-Shaw Wave Tank Compatible Schemes Onno Bokhove Simulation of damped, sloshing waves: initial conditions in Introduction model & experiment. Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

Maths of MARIN’s Wave Tank Compatible Mathematical formulation via Miles’ variational principle: Schemes Onno � T Bokhove 0 = δ L [ φ, h , t ] dt (1) Introduction 0 Non- � T � L φ s ∂ t h − 1 Autonomous 2 g ( h + b − H ) 2 = δ Systems 0 x w ( t ) NCP time flux � b + h � b + h 1 Spatial NCP? dx w 2 | ∇ φ | 2 dzdx − − dt φ w dzdt (2) Conclusion b b with potential φ = φ ( x , z , t ) such that velocity ( u , w ) T = ∇ φ = ( ∂ x φ, ∂ z φ ) T free-surface φ s ( x , t ) ≡ φ ( x , z = h + b , t ) at ∂ D s , depth h specified wave-maker piston x w ( t ) with φ w ≡ φ ( x w , z , t ). non-autonomous due to piston wave-maker.

FEM of MARIN’s Wave Tank FEM formulation of Miles’ variational principle. Compatible Schemes FEM test/basis functions ˜ ϕ j ( x , z , t ) , ˆ ϕ k ( x , t ) with i , j in Onno Bokhove D , k , l at free surface ∂ D s & m at wave maker. Introduction Substitute φ h ( x , z , t ) = φ j ( t ) ˜ ϕ j ( x , z , t ), Non- h h ( x , t ) = h k ( t ) ˆ ϕ k ( x , t ) in VP Autonomous Systems � T NCP time flux 0 = δ L [ φ j , h j , t ] dt (3) Spatial NCP? 0 � T Conclusion dh l dh l = δ φ k M kl dt − φ k D kl dt − . . . 0 − 1 2 g ( h k + b k − H ) M kl ( h l + b l − H ) − 1 2 φ i A ij φ j − w m ( t ) φ m dt . (4) M kl , D kl , A ij w m depend on { h k ( t ) , t } : mesh movement.

Maths of Hele Shaw Wave Tank Compatible Schemes Onno Substitution potential flow Ansatz (¯ u , ¯ w ) = ( ∂ x φ, ∂ z φ ) Bokhove into 2D Navier-Stokes eqns gives damped water waves: Introduction Non- �� L � T Autonomous φ s ∂ t h − 1 2 g ( h − H 0 ) 2 � � 0 = δ d x Systems 0 0 NCP time flux � � ˜ � L γ h e 3 ν t / l 2 d t 1 2 | ∇ φ | 2 d z d x Spatial NCP? − (5) 0 0 Conclusion Use experiment to validate linear momentum damping . Tilt tank till at rest: then drop it to create a linear tilt of the free surface“at rest”. Non-autonomous due to damping/integrating factor.

FEM of Hele Shaw Wave Tank Compatible Schemes Substitution potential flow Ansatz (¯ u , ¯ w ) = ( ∂ x φ, ∂ z φ ) Onno Bokhove into 2D Navier-Stokes eqns gives damped water waves: Introduction � T Non- 0 = δ L [ φ j , h j , t ] dt (6) Autonomous Systems 0 � T NCP time flux dh l � = δ φ k M kl Spatial NCP? dt 0 Conclusion − 1 2 g ( h k − H ) M kl ( h l − H ) − 1 e 3 ν t / l 2 dt . � 2 φ i A ij ( h k ) φ j (7) Explicit time dependence in e 3 ν t / l 2 due to damping.

Damped Water Waves: Model vs. Data Compatible Schemes Onno Bokhove Measure free surface & calculate potential energy P ( t ): Introduction Non- Autonomous Systems NCP time flux Spatial NCP? Conclusion

Non-Autonomous Variational/Hamiltonian System Compatible Schemes Onno Bokhove Both discretizations are succinctly summarized as: Introduction � T Non- dt − 1 p T M dq Autonomous 2 p T Ap � 0 = δ Systems 0 NCP time flux − 1 2 q T Mq − p T Dq − w ( t ) T p � f ( t ) dt (8) Spatial NCP? Conclusion MARIN’s tank: f ( t ) = 1 , A = A ( q , t ) , M = M ( q , t ), D = D ( q , t ) , w ( t ) � = 0. Hele-Shaw tank: w ( t ) = D = 0 , f ( t ) = exp (3 ν t / l 2 ).

Non-Autonomous Variational/Hamiltonian System Compatible Schemes Note: Newton’s equations for coupled linear oscillators in Onno Bokhove limit A ( q ) = S (constant) & f ( t ) = 1 , w ( t ) = 0: Introduction � T dt − 1 2 p T Sp − 1 p T M dq Non- � 2 q T Mq � 0 = dt Autonomous Systems 0 ⇒ M dq Sp = ∂ H NCP time flux ⇐ = ∂ p , Spatial NCP? dt Conclusion M dp − Mq = − ∂ H = (9) ∂ q dt for Hamiltonian H = 1 2( p T Sp + q T Mq ) . (10)

Damped nonlinear oscillator Compatible Schemes Toy example Onno Bokhove � T pdq e γ t dt � � 0 = δ dt − H ( p , q ) Introduction 0 Non- Autonomous dq p = ∂ H δ ( pe γ t ) : Systems = dt ∂ p NCP time flux − ( q + q 3 ) = − ∂ H dp Spatial NCP? δ q : dt + γ p = (11) ∂ q Conclusion 2 p 2 + 1 2 q 2 + 1 with energy/Hamiltonian H = H ( p , q ) = 1 4 q 4 . Note the integrating factor s.t.: d ( pe γ t ) = − ( q + q 3 ) e γ t . (12) dt

Damped nonlinear oscillator Compatible Dynamics becomes linear in long-time limit. The Schemes transformation Onno Bokhove q = Qe − γ t / 2 p = Pe − γ t / 2 (13) Introduction Non- shows from Autonomous Systems � T P dQ NCP time flux dt − ˜ 0 = δ Hdt Spatial NCP? 0 Conclusion that for t → ∞ d ˜ H dt = 0 with (14) 1 2 P 2 + 1 2 γ PQ + 1 2 Q 2 + 1 ˜ 4 Q 4 e − γ t = H (1 2 p 2 + 1 2 γ pq + 1 2 q 2 + 1 4 q 4 ) e γ t . =

Non-Conservative Products Compatible Schemes Goal: to derive stable variational time integrators with time Onno discontinuous FEM Bokhove Introduction Finite elements in time. Non- p h = p j ϕ j ( t ) and q h = q j ϕ j ( t ) expanded in piecewise Autonomous Systems continuous fashion, e.g.: NCP time flux Spatial NCP? Conclusion What to do with derivatives p T Mdq / dt at the jumps? No staggered C-grid in t : crux lies in choice numerical flux!

Non-Conservative Products Consider p dq dt � = dQ ( u ) Compatible dt or g ( u ) du with u = u ( p , q ). Schemes dt Dal Maso, LeFloch and Murat (1995) define Onno Bokhove ǫ → 0 g ( u ǫ ) du ǫ g ( u ) du dt = lim (15) Introduction dt Non- Autonomous Introduce a Lipschitz continuous path φ : [0 , 1] → ℜ with Systems φ (0) = u L and φ (1) = u R with limits u L , u R at t d : NCP time flux Spatial NCP? Conclusion Moreover, jump depends on path φ ( τ ): � 1 g ( u ǫ ) du ǫ d φ dt → C δ ( t − t d ) C = g ( φ ) τ d τ ( τ ) d τ with 0

Non-Conservative Products Compatible Schemes DLM assume a fixed family of paths with: Onno (i) φ (0; u L , u R ) = u L , φ (1; u L , u R ) = u R , Bokhove (ii) φ (0; u L , u L ) = u L , Introduction (iii) | d φ d τ ( τ ; u L , u R ) | ≤ K | u L − u R | Non- Autonomous Theorem by DLM: There is a unique real-valued bounded Systems Borel measure µ on ] a , b [ such that: if u is discontinuous NCP time flux at a position t d ∈ ] a , b [ then Spatial NCP? Conclusion � 1 g ( φ )( τ ; u l , u R ) d φ µ ( { t d } ) = d τ ( τ ; u L , u R ) d τ. (16) 0 µ is the nonconservative product of g ( u ) by du / dt . DGFEM in 3D & 4D: Rhebergen et al (2008ab, 2009) Idea is to explore NCP for p dq dt –term in VP.

Non-Conservative Products: VP Compatible Schemes Choice of path: open question. Onno Bokhove We generally chose a linear path: Introduction φ ( τ ; u L , u R ) = u L + τ ( u R − u L ) . (17) Non- Autonomous Systems Partition time in time slabs [ t n , t n +1 ] NCP time flux Spatial NCP? Using DLM-theorem, variational principle becomes Conclusion � t n +1 N − 1 dq h � e γ t dt � � 0 = δ p h dt − H ( p h , q h , t ) t n n =0 � 1 N φ p ( τ ; p L , p R ) d φ q � + d τ ( τ ; q L , q R ) d τ (18) 0 n = − 1

Non-Conservative Products: VP Compatible Schemes Using DLM-theorem, discrete variational principle Onno Bokhove � t n +1 N − 1 dq h � e γ t dt � � Introduction 0 = δ p h dt − H ( p h , q h , t ) t n Non- n =0 Autonomous � 1 Systems N φ p ( τ ; p L , p R ) d φ q � + d τ ( τ ; q L , q R ) d τ NCP time flux 0 Spatial NCP? n = − 1 Conclusion For quadratic & linear paths ( γ = 0): φ p = p L + 2 a 1 τ + 3 a 2 τ 2 & φ q = q L + τ ( q R − q L ) s.t.: � 1 φ p ( τ ; p L , p R ) d φ q d τ ( τ ; q L , q R ) d τ = ( α p L + β p R )( q R − q L ) 0 with 0 ≤ α, β ≤ 1, i.e., jump in q × weighted mean in p .