Filippini, Andrea Gilberto

Kazolea, Maria

Ricchiuto, Mario

Tsunamis en Atlantique et MaNche : Définition des Effets par Modélisatio

FILIPPINI, Andrea Gilberto

KAZOLEA, Maria

RICCHIUTO, Mario

Oskar Bordeaux

Hybrid finite-volume/finite-element simulations of fully-nonlinear/weakly dispersive wave propagation, breaking, and runup on unstructured grids

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HAL Id: hal-01625308https://hal.archives-ouvertes.fr/hal-01625308Submitted on 20 Jul 2018HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Hybrid finite-volume/finite-element simulations offully-nonlinear/weakly dispersive wave propagation,breaking, and runup on unstructured gridsAndrea Gilberto Filippini, Maria Kazolea, Mario RicchiutoTo cite this version:Andrea Gilberto Filippini, Maria Kazolea, Mario Ricchiuto. Hybrid finite-volume/finite-element sim-ulations of fully-nonlinear/weakly dispersive wave propagation, breaking, and runup on unstructuredgrids. SIAM Conference on Mathematical and Computational Issues in the Geosciences, Sep 2017,Erlangen, Germany. ￿hal-01625308￿Hybrid finite-volume/finite-element simulations offully-nonlinear/weakly dispersive wave propagation,breaking, and runup on unstructured gridsA.G. Filippini, M. KazoleaMario RicchiutoTeam CARDAMOMINRIA Bordeaux - Sud-Ouest, FranceSIAM-GS17Erlangen, September 14th 2017Thanks to ...I R. Chassagne (Inria CARDAMOM), P. Bonneton (EPOC Bordeaux, France):tidal bore dispersive propagationI D. Lannes (IMB Bordeaux, France), F. Marche (U. Montpellier, France):Boussinesq modellingI R. Pedreros & S. LeRoy (BRGM, France), R. Ata (EDF Chatou, France):Tohoku tsunami dataSome referencesSource material1. MR and A.G. Filippini, J.Comput.Phys. 271, 20142. A.G. Filippini, M. Kazolea and MR, J.Comput.Phys. 310, 20163. A.G. Filippini, M. Kazolea and MR, ISOPE proc.s , 2017see also the PhD of A.G. Filippini (December 2016, available online on hal)Additional references/related work1. J.T. Kirby and G. Wei, J. Waterway, Port, Coastal, and Ocean Eng. 1995;2. F. Shi, et al. Ocean Modelling, 20123. M. Kazolea et al., Coast.Eng. 2012 and J.Comput.Phys, 20144. N. Aissiouene et al, Networks and Heterogeneous media, 2016Tohoku exampleTohoku exampleWave arrival in Sendai BayTohoku exampleBoussinesq Shallow WaterModelling approach 1/6Near shore: wave transformation due tointeraction with complex bathymetriesI strong nonlinearity and dispersion ;I refraction and diffraction ;I shoaling;I breaking;I induced currents;I run-up and inundation; 2004 Sumatra tsunami reachingthe coast of ThailandSea waves diffracting around a peninsula Rip current, Park beach(Coffs harbour, NSW Australia)Modelling approach 2/6Near shore hydrodynamics: modelling standpoint(Ribbed channel clip).Propagation: large scales,dispersion, shoaling, etcPOTENTIAL FLOWWave vreaking:dissipation, vorticityCLOSURE MODELRunup/flooding :hydrostatic shallow waterSHALLOW WATERModelling approach 4/6Near shore hydrodynamics: modelling standpointPropagation vs wave breaking closure1. Potential/dispersive PDE for propagation + 3D Navier-Stokes (or SPH)for breaking/impact ;2. Dispersive PDE for propagation + eddy viscosity to model dissipation insurf zone/breakers ;3. Coupling dispersive PDEs with shallow water/hydrostatic limit:I Kirby, Grilli, et al (FUNWAVE-TVD)I Lynett et al USC (COULWave)I Delis, Kazolea, Synolakis (TUCWave) Coast.Eng. 2011, JCP 2014I Smit, Zijlema et al DELFT (SWASH)I See also:Tonelli, Petti Coast.Eng. 2009, Bonneton et al. JCP 2011, Coast.Eng. 2012Modelling approach 5/6Breaking closure: shallow water dissipationClosure model1. Detect breaking regions2. Remove dispersive terms3. → shallow water shock4. Total energy E = gh2/2 + hu2/25. Dissipation :Db = [FE − σE] ≈ γb[H]3aModelling approach 5/6Breaking closure: shallow water dissipationClosure model1. Detect breaking regions2. Remove dispersive terms3. → shallow water shock4. Total energy E = gh2/2 + hu2/25. Dissipation :Db = [FE − σE] ≈ γb[H]3Modelling approach 6/6Breaking closure: shallow water dissipationaClosure model1. Detect breaking regions2. Remove dispersive terms3. → shallow water shock4. Total energy E = gh2/2 + hu2/25. Dissipation :Db = [FE − σE] ≈ γb[H]3aNeed to handle shallow water:I “Upwinding”/numerical dissipationI Shock capturing in breakingregionsI etcAll the std. artillery....Principle of the talkDispersive models for propagationWhich dispersion dominates ?1. Continuous dispersive models have a range of validity related tothe model dispersion error2. Discrete dispersive models have a range of validity related tothe scheme dispersion errorHow can we exploit this knowledge to constructefficient low dispersion schemes on unstructured grids ?Dispersive surface waves: Boussinesq approach (1/8)Undular bore (Garonne river,Bonneton et al J.Geophys.Res. 2015)x/Lb0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8η/h0-0.45-0.4-0.35-0.3-0.25-0.2-0.15GNNLSWTidal wave distorsion in a converging estuaryx/Lb1.2 1.21 1.22 1.23 1.24 1.25η/h0-0.45-0.4-0.35-0.3-0.25-0.2GNNLSWDispersive surface waves: Boussinesq approach (2/8)Linear travelling waves: incompressible Euler equationsTheory due to (G.B. Airy, Encyclopædia Metropolitana, 1841)η = A cos(κx− ωt) travelling waveΦ = B(z)A sin(κx− ωt) potentialκ wave numberC = ω/κ phase velocityxΦzηz=  (t,x)d0B(z) =gωcosh(κ(d0 + z))cosh(κ(d0)C2= C20tanh(κ d0)κ d0Phase relation and phase dispersionC20 = g d0 shallow water (hyperbolic) celerity. C = C(κ)!!Dispersive surface waves: Boussinesq approach (3/8)a wave amplitudeλ = 2π/κ wave lengthT wave periodλ = C(κ)T = C(2π/λ)T phase relationaλDimensionless parametersI µ =d0λ=κd02πdispersionI ε =ad0nonlinearitymmPhysical hypothesesI Long waves: smallness of µ2ad0I Weak-nonlinearity: smallness of ε = O(µ2)ad0I Full-nonlinearity: ε = O(1)ad0Dispersive surface waves: Boussinesq approach (4/8)Modelling principles1Starting from the 3D nonlinear wave equations:1. dimensionless form ;2. asymptotic development w.r.t. µ2: ∇Φ = ∇Φ0 + µ2∇Φ2 + µ4∇Φ4 + ...3. depth averaging: 3D → 2D4. retain appropriate order terms ....1J. Boussinesq, J.Math.Pures Appl., 1872 – M.W. Dingemans, World Scientific, 1997Dispersive surface waves: Boussinesq approach (5/8)Example: shallow water equations (zero-th order model)h̃t+q̃x = 0q̃t+ε(ũq̃)x + h̃η̃x = 0 +O(µ2)With the notationI dimensionless depth : h̃ = d̃+ εη̃I dimensionless volume flux : q = h̃ũI dimensionless depth averaged velocity: ũI Red : hyperbolic shallow water equationsI Blue terms are responsible for dispersionNonlinear – non dispersiveDispersive surface waves: Boussinesq approach (6/8)Example: Peregrine’s equations2h̃t+q̃x = 0q̃t+ε(ũq̃)x + h̃η̃x = µ2h̃(d̃23utxx +d̃d̃x3ũtx)+O(µ4, εµ2)With the notationI dimensionless depth : h̃ = d̃+ εη̃I dimensionless volume flux : q = h̃ũI dimensionless depth averaged velocity: ũI Red : hyperbolic shallow water equationsI Blue terms are responsible for dispersionWeakly nonlinear – weakly dispersive2D.H. Peregnine. J.Fluid.Mech, 1967Dispersive surface waves: Boussinesq approach (7/8)Example: Madsen & Sorensen’s enhanced equations3h̃t+q̃x = 0q̃t+ε(ũq̃)x + h̃η̃x = µ2(βd̃2q̃txx +d̃d̃x3q̃tx +Bd̃3η̃xxx + 2Bd̃2d̃xη̃xx)+O(µ4, εµ2)With the notationI dimensionless depth : h̃ = d̃+ εη̃I dimensionless volume flux : q = h̃ũI dimensionless depth averaged velocity: ũI Red : hyperbolic shallow water equationsI Blue terms are responsible for dispersionWeakly nonlinear – weakly dispersivePhase enhancement vie the tunable coeff. B and β = B + 1/3 (cf. later)3P.A. Madsen and O.R. Sorensen Coast.Eng., 1992Dispersive surface waves: Boussinesq approach (8/8)Example: enhanced Serre-Green-Naghdi equations4h̃t+q̃x = 0q̃t+ε(ũq̃)x + h̃η̃x = µ2 h̃ψ̃ +O(µ4)ψ̃ =α[∂x(h̃2∂x(ũt + ũũx))]+ (α− 1)[∂x(h̃2∂xxη̃)]+Qψ(ũ, h̃, d̃; ũx, h̃x, d̃x)With the notationI dimensionless depth : h̃ = d̃+ εη̃I dimensionless volume flux : q = h̃ũI dimensionless depth averaged velocity: ũI Red : hyperbolic shallow water equationsI Blue terms are responsible for dispersionFully nonlinear – weakly dispersivePhase enhancement vie the tunable coeff. α (cf. later)4A.E. Green J.Fluid Mech., 1976 – F. Chazel et al. J.Sci.Comp., 2011Modelling error (1/3)What are these models good for: nonlinear behaviorShoaling test, weakly nonlinear models5x/h00 5 10 15 20 25 30 35z[m]00.10.20.30.40.50.620 21 22 23 24 25 26 270.40.60.811.21.41.6x/h0ηmax/h0  Exp. dataPBNMSPNABNAMSNAMany variations for a given asymptotic accuracy (same linear limit), e.g.I µ2d̃ ≈ µ2h̃ as εµ2η̃ is negligible (weakly nonlinear)I µ2(d̃ũ)xxt ≈ µ2q̃xxt as εµ2(η̃ũ)xx is negligible (weakly nonlinear)I etc.5S.T. Grilli et al J.Waterw.Port.C.-ASCE, 1994 – A.G. Filippini et al. Coast.Eng., 2015Modelling error (2/3)What are these models good for: nonlinear behaviorShoaling test, fully nonlinear modelsx/h00 5 10 15 20 25 30 35z[m]00.10.20.30.40.50.6Modelling error (3/3)What are these models good for: phase behaviorkd00 1 2 3 4 5 6ω/ωAiry0.950.960.970.980.9911.011.021.031.041.05Shallow WaterPeregrineMadsen-S ørensenenhanced GNDeep Waterkd0≈ 0.56kd0≈ 2.3kd0≈ 3.23CAVEAT: MS and eGN are equivalent. The ”classical” optimized MS obtained changing αContinous and discreteContinuous: error w.r.t. Euler eq.sI Range of validity in terms of reduced wave number κd0 = 2πd0/λI Near shore: (κd0)max ≈ π, (d0/λ)max ≈ 2Discrete: error w.r.t. continuous modelI Range of validity in terms of 1/N = ∆x/λ : pts per wavelengthOverall error w.r.t. Euler eq.s ???Truncation error heuristicsExample: linearized shallow water∂tW +A∂xW = 0with (in dimensionless form)W =(ηu), A =(0 11 0), |A| = Id2where we recall that (all .̃ are removed for simplicity, and c20 = gd0)η =ηdimεd0, u =udimεc0, x =xdimλ, t =tdimλ/c0Truncation error heuristicsExample: linearized shallow water1st order upwind method∂tW +AWi+1 −Wi−12∆x=∆x2Wi+1 − 2Wi +Wi−1∆x22nd order centered differencing∂tW +AWi+1 −Wi−12∆x= 0Truncation error heuristicsExample: linearized shallow water1st order upwind and 2nd order centered differencing: modified equation/TE∂tWsmooth +A∂xWsmooth =∆x2∂xxWsmooth−A∆x26∂xxxWsmooth +O(∆x3)∂tWsmooth +A∂xWsmooth = −A∆x26∂xxxWsmooth +O(∆x4)It looks like “if we could see the dispersive effects of the 1st order scheme, theywould be the same as those of the second order”..but the numerical viscosity is too high...Truncation error heuristicsExample: linearized shallow waterTurn down the viscosity: 3rd upwind scheme∂tW+(AWRi+1/2+WLi+1/22∆x−WRi+1/2−WLi+1/22∆x)−(AWRi−1/2 +WLi−1/22∆x−WRi−1/2 −WLi−1/22∆x)= 0With “quadratically reconstructed ” left and right values at xi ±∆x/2WLi+1/2 =Wi +Wi −Wi−16+Wi+1 −Wi3WRi+1/2 =Wi+1 −Wi −Wi−16−Wi+2 −Wi+13Truncation error heuristicsExample: linearized shallow water3rd order upwind vs 4th order central differencing: modified eq./TE∂tWsmooth +A∂xWsmooth =−∆x312∂xxxxWsmooth−A∆x430∂xxxxxWsmooth +O(∆x5)∂tWsmooth +A∂xWsmooth = −A∆x430∂xxxxxWsmooth +O(∆x6)I Low dissipation: O(∆x3) viscosity (ok for hyperbolic + explicit time stepping)I Same dispersion of the fourth order FD !Truncation error heuristics: dispersive modelsExample: enhanced linearized Madsen-Sorensen∂tη + ∂xu = 0∂tu− µ2(13+ β)∂xxtu+ ∂xη − µ2β∂xxxη = 0Truncation error heuristics: dispersive modelsExample: enhanced linearized Madsen-Sorensen∂tη + ∂xu = 0∂tw + ∂xζ = 0u− µ2(13+ β)∂xxu = wη − µ2β∂xxη = ζFirst order (“hyperbolic”) systemOverhead w.r.t. hyperbolic systemHow accurate must the discretization of these red terms be ?Dispersion error heuristicsLinearized Madsen-Sorensen (with GN αopt)Dispersion error heuristicsLinearized Madsen-Sorensen (with GN αopt)Dispersion error heuristicsLinearized Madsen-Sorensen (with GN αopt)Dispersion error heuristicsLinearized Madsen-Sorensen (with GN αopt)Dispersion error heuristicsLinearized Madsen-Sorensen (with GN αopt)Lesson learnedHyperbolic operatorWe MUST use at least a third order schemeElliptic operatorWaste of effort using more than second order for near shore modelsNext stepUnstructured grid generalization for the eGN modelDiscretization of the (breaking) eGN on unstructured gridsThe eGN equations (in 1D)h̃t+q̃x = 0q̃t+ε(ũq̃)x + h̃η̃x = µ2 h̃ψ̃ψ̃ =α[∂x(h̃2∂x(ũt + ũũx))]+ (α− 1)[∂x(h̃2∂xxη̃)]+Qψ(ũ, h̃, d̃; ũx, h̃x, d̃x)With the notationI dimensionless depth : h̃ = d̃+ εη̃I dimensionless volume flux : q = h̃ũ withI ũ the (dimensionless) depth averaged velocityI Red terms provide the hyperbolic shallow water equationsI Blue terms are responsible for dispersionFully nonlinear – weakly dispersivePhase enhancement vie the tunable coeff. α (cf. later)Discretization of the (breaking) eGN on unstructured gridsThe eGN equations (in 2D)∂th+∇ · q = 0 ;(I + αTh)(∂tq +∇ ·(q⊗ qh)+ gh∇η)− Th(gh∇η)+ hQ(u) = 0 .With the notationI depth : h = d+ ηI (vector) volume flux : q = h̃u withI q the (dimensionless) depth averaged velocity vectorI Th the dispersive elliptic operator given byTh(·) = hT (·h)where T is a self adjoint operator6:T (·) = S∗(S(·)) , S(·) =h√3∇ · (·)−√3− 12∇b · (·)6Alvarez-Samaniego and Lannes, Indiana Univ. Math J., 2008Discretization of the (breaking) eGN on unstructured gridsThe eGN equationsWe recast the system in two independent steps:∂th+∇ · q = 0 ;∂tq +∇ ·(q⊗ qh)+ gh∇η = hψ ;(I + αT)(ψ )− T (g∇η) +Q(u) = 0 .a→ hyperbolic step→ elliptic stepThis reformulation aims at exploiting the self-adjoint character of TDiscretization of the (breaking) eGN on unstructured gridsSolution algorithmAt each time-step n:1. elliptic step is solved : ψ =(I + αT)−1(RHS) using (hn,qn) ;2. shallow water solver + non-hydrostatic term ψ → (hn+1,qn+1) .Discretization of the (breaking) eGN on unstructured gridsElliptic stepI Continuous P 1 Galerkin FE exploiting the self-adjoint character of T 7∫Ων ·ψh + α∫ΩS(ν) S(ψh) = RHS(ηh, hh,uh, bh) ;I Linear system :(MGH + αT) Ψ = T δh − Q with δh the L2 projection of g∇ηI Coercivity of I + αT −→ inversibility of (MGH + αT)(block SPD + diagonally dominant) ;η hϕi1i7Alvarez-Samaniego and Lannes Indiana Univ. Math J., 2008Discretization of the (breaking) eGN on unstructured gridsElliptic stepI Propagation: add hψ to the rhsI Wave breaking:1. Flag nodes2. Agglomerate elements and enlargebreaking region in wave direction3. Set ψ to zero:breakers as shallow water shocks9I Wave breaking detection10I either |ηt| > γ√gH with γ ∈ [0.4, 0.6]I or ‖∇η‖ > tgθ with θ ∈ [15, 30]◦I and Fr > Frcr9P. Bonneton Ocean Eng., 200710Kazolea, Delis and Synolakis, JCP 2014Discretization of the (breaking) eGN on unstructured gridsHyperbolic stepI Node - centered FV schemeI∂Ui∂t+1|Ci|∫∂Ω(Fn̂x + Gn̂y)=1|Ci|∫Ω(Sb + hΨ);I Roe’s Riemann solver + Harten-Hyman entropy fixI High order reconstruction: weighted least squares11 (quadratic or cubic) ;I Well-balanced treatment of topography, wet/dry fronts, etc1211Ollivier-Gooch et al. AIAA J. 2009; Wang et al JCP 201710Bermudez&Vazquez, CAF 1994; Hubbard&Garcia-Navarro, JCP 2000; Brufau et al. IJNMF 2002;Castro, Math.&Computer Mod. 2005; etc.etc.Discretization of the (breaking) eGN on unstructured gridsDispersion analysis of the schemeBenchmarking: propagation test 1Test Description: a0 = 0.2 [m], h0 = 1 [m]log10(hK)-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6log10||err||2-10-9-8-7-6-5-4-3-2FV3slope 2.5slope 30 50 100 150 200 250 300 3500.20ζ [m]x [m]250 300 350−0.0500.050.10.150.20.25ζ [m]x [m]  NumericalExactBenchmarking: propagation test 2Shoaling on a shelfBenchmarking: propagation test 2Shoaling on a shelfFinal solutionBenchmarking: diffraction on an elliptic shoalI Range : a/h0 = 0.0515, T = 1 [s] ;I Energy transfer to higher harmonics;I Experiments: (Berkhoff et al., 1982)I Adapted mesh 88760 nodes ;Incomingperiodic waveSec. 1Sec. 8Sec. 4Sec. 5Sec. 2Sec. 3Sec. 6 Sec. 7Benchmarking: diffraction on an elliptic shoalSection 2 and Section 4x [m]-5 0 5Waveheight[m]00.511.522.5Exp.Num.Incomingperiodic waveSec. 1Sec. 8Sec. 4Sec. 5Sec. 2Sec. 3Sec. 6 Sec. 7Section 7 and Section 8x [m]0 5 10Waveheight[m]00.511.522.5Exp.Num.x [m]0 5 10Waveheight[m]00.511.522.5Exp.Num.Incomingperiodic waveSec. 1Sec. 8Sec. 4Sec. 5Sec. 2Sec. 3Sec. 6 Sec. 7Benchmarking: overtopping on a three dimensional reefBenchmarking: overtopping on a three dimensional reeft√(g/h)0 10 20 30 40 50 60 70η/h-0.100.10.20.30.4WG7Experim.FrictionNo Frictiont√(g/h)0 20 40 60η/h-0.0500.050.10.150.2WG3Experim.FrictionNo Frictiont(s)0 10 20 30 40 50 60 70u(m/s)-1-0.500.511.522.5ADV 3Experim.FrictionNo Frictiont(s)0 10 20 30 40 50 60 70v(m/s)-0.4-0.200.20.40.60.8ADV 3Experim.FrictionNo FrictionApplication: study of undular bores and Treske experimentsUndular bore (Garonne river,Bonneton et al J.Geophys.Res. 2015)Application: study of undular bores and Treske experimentsComparison with experiments by TreskeThe short of it ...I Interaction modelling/discretization error for near shore BoussinesqI Frst (hyperbolic) order system vs second order elliptic operator :• High order on hyperbolic : third order (at least) for good dispersion• Elliptic component: can be treated with a second order method• Unstructured grid eGN: high(er) order FV + P1 FEM for elliptic partI Wave breaking: revert to SW + shock capturing2 6 10 12h0=2.5 [m ]slope=1÷12h0=0.14 [m ]a0=0.75[m ]Perspectives and open issuesA few ongoing/foreseen extensionsI Systematic orders/CPU time investigationI Other methods, in particular DG and RDI Implicit time integration + energy conserving in space and time ?I Wave breaking via PDE based eddy viscosityI Study of deep water/non-hydrostatic (multi-layer) modelsI Moving meshes and adaptation(cf. parallel session on rupture based tsunami simulation)

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Hybrid finite-volume/finite-element simulations of fully-nonlinear/weakly dispersive wave propagation, breaking, and runup on unstructured grids

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