A Hybridizable Discontinuous Galerkin formulation for Fluid-Structure Interaction

Transcription

1 A Hybridizable Discontinuous Galerkin formulation for Fluid-Structure Interaction Jason P. Sheldon Sco/ T. Miller Jonathan S. Pi/ Distribu8on Statement A Approved for public release DistribuMon unlimited Funding: Naval Sea Systems Command, Advanced Submarine Systems Development Office (SEA 073R) Fi#h deal.ii Users and Developers Workshop Texas A&M University, College Station, TX, USA; August 3-7,

2 Motivation Fluid- Structure Interac8on (FSI) The interacmon of a deformable structure with a surrounding or internal fluid Source

3 Problem statement Problem statement High- order accurate FSI calculamons are o#en too computamonally expensive for real- world problems Solu8on requirements High- order accurate computamons Reasonable computamonal expenses Proposed solu8on Develop the first FSI model that uses the state of the art hybridizable disconmnuous Galerkin (HDG) method 3

4 HDG best of both worlds ConMnuous Galerkin (CG) DisconMnuous Galerkin (DG) ComputaMonally efficient for low orders Hybridizable DisconMnuous Galerkin (HDG) UMlizes opmmized solvers such as stamc condensamon Scales well for high order Preserves local conservamon of mass 4

5 HDG discretized domain CG DG HDG - Global DOFs - Local DOFs Local solution Global solution trace Global boundary Example: 2D 1x2 quadratic elements Variables: u, v, e - 8 global DOFS per ( ) CG: 120 global DOFs DG: 144 global DOFs Variables: u, v, e - 8 local DOFS per (, ) μ - 2 global DOFS per ( ) HDG: 42 global DOFs + 72 local DOFs per cell 5

6 Fluid-structure interaction modeling Domain informamon and boundary condimons Fluid model Solid model Mesh model Couple Solve and post process Fluid Domain Solid Domain Mesh Domain Fluid-Solid Interface Fluid Boundary Solid Boundary 6

7 Strong form Linear elastostatics: HDG discretization Div (C [e]) = b e Sym(Grad u) = 0 (Mesh MoMon) - μ - u, e 1. Weight by weighmng funcmons ( eu, e) 2. Integrate over cell () 3. Integrate by parts > boundary terms (Γ) Weak form e : e d Grad eu : C (e)d eu [C (e)n d = Sym(e) : Grad u d + Sym(e)n (u µ) d =0 Flux continuity eµ [C (e)n d =0 eu b d Stabilization [C (e) :=C (e) S (u µ) n 7

8 ARL Weak form e : e d (Mesh MoMon) Linear elastostatics: HDG discretization Grad eu : C (e)d eu [C (e)n d = Sym(e) : Grad u d + Sym(e)n (u µ) d =0 [C (e) :=C (e) S (u µ) n eµ [C (e)n d =0 eu b d local local local global global local global global sollocal sol global = RHSlocal RHS global Using the Schur complement, any equation of the form!! A B U = F!, C D G can be rearranged into the following system of equations: D CA 1 B = G CA 1 F, AU = F B. 8

9 Linear elastostatics: verification h- refinement (Q3) p- refinement L 2 error norm u 0 e 00 e 01 L 2 error norm u 0 Q1 u 0 Q2 u 0 Q log(1/h) log(1/h) All errors converge at the expected rate of h p+1 Manufactured solumon funcmon of sines and cosines Pure Dirichlet boundary condimons 9

10 Preliminary results: Sandia geometry Upper (fixed) end of test specimen Source The 2014 Sandia Fracture Challenge (SFC2) (hole ahead of the notch) (no hole ahead of the notch) Lower (actuated) end of test specimen 10

11 Preliminary results: Sandia geometry Undeformed mesh 7777 cells Deformed solumon Showing stress Element Order Local # of DOFs Increase from Q1 Global # of DOFs Increase from Q1 Q Q % % Q % % Q % % Size of global system increases minimally, even for high orders! 11

12 Nonlinear elastodynamics: governing equations Variables: u, v, E μ - local - global Restricted to hyperelastic materials (Saint Venant-irchhoff) Weak form ee:e d Sym( e E):Grad u d + Grad eu:f C (E) d eu bh d = 1 e 2E:(Grad u) T Grad u d + Sym( E)n e (u µ) d d ev v d =0 eµ bh d =0 eu b d where b h := [F C (E)] n S (u µ)) F = I + Grad u 12

13 Preliminary results: Turek and Hron CSM3 A 0 Tip displacement at point A F g x displacement (m) Turek data HDG Q2 data unit interval (s) y displacement (m) Source Proposal for numerical benchmarking of fluid- structure interacmon between an elasmc object and laminar incompressible flow, Turek Turek data HDG Q2 data unit interval (s) 13

14 Source: HDG Navier- Stokes formulamon from MarMn ronbichler s unpublished work Navier-Stokes: governing equations Variables: v, p, L - local υ, ψ - global Weak d grad ev : v v L d + ev grad p k d + ev bh d = el : L d el : grad v d + eln (v )d =0 grad ep k v d + ep k n d =0 Flux stabilization bh := [ L + pi] n + S (v ) bh := [ L] n + S (v ) e e bh d n d =0 =0 ev f d Introduce ψ to close the system := 1 p d Remove the kernel of the gradient p d =0 14

15 Arbitrary Lagrangian-Eulerian (ALE) transformation HDG ALE transform L = grad v = Grad v f F 1 m = L f [v m ], Auxiliary HDG variables simplify ALE transformamon IntegraMon domain ALE transform f (x)dx = f f (X) J m dx, F (t) F (0) 15

16 German research association (DFG) Benchmark 2D channel flow around a cylinder Reynolds number Re=100 Source 16

17 Preliminary results: flow over cylinder DFG benchmark - > Matching HDG Navier- Stokes with DFG benchmark ALE HDG Navier- Stokes proof of concept: oscilla8ng cylinder 17

18 Fluid-Structure interaction modeling Fluid model Domain informamon and boundary condimons Solid model Mesh model Solid System Matrix Fluid Solid Coupling Mesh Solid Coupling Monolithic Coupling Solid Fluid Coupling Fluid System Matrix Mesh Fluid Coupling Solid Mesh Coupling Fluid Mesh Coupling Mesh System Matrix Solid Unkns Fluid Unkns Mesh Unkns 3 2 = Solid RHS Fluid RHS Mesh RHS Γ FS Couple Ω S µ s µ m Ω F υ s υ f Solve and post process T s n f T f n s 18

19 Proposed HDG FSI formulation Problem 1 (Fluid sub-problem) Find {L h f, vh f,ph f, h f, f h}2lh f Vf h Q h f Vh f f h such h f ev f, f J m + ev f, f J m L h f v f v h m + ev f,j m Fm T Grad p k f f f D E + Grad ev f,µ f J m L h f F T elf,j m L h f f De f, b T [n] f E elf,j m Grad v h f F 1 h f \ FSI + m f + D + elf Fm T [n f ],J m vf h f Fm T Grad ep k f,jm vf h + F T f De f, b T [n] f + b T [n] s E 8{ L e f, ev f, ep f, e f, e f}2l h f Vf h Q h f V e f f h, where p 0 f + p k f = p h f bt [n] f := FSI µ fj m L h f Fm T [n f ]+S f vf h bt [n] f := J m µf L h f + p k f + f h I F T S f := µf l f + f v h f L2 I, ev f, T b [n] f E h =(ev f,j m f f ) f, =0, m [n f ] ep k h f,j m f + h h e f, S f = he f s f,j m g Nf i FSI N, D E ef,j m Fm T h [n f ] f h h f, p 0 f,j m p 0 f m [n f ]+S f v h f Problem 3 (Mesh sub-problem) Find {u h m, F h m, µ h m}2um h F h m Mm h such that Grad eu m, C m F h m Deu m, T b E [n] m = (Grad eu m, C m (I)) heu m, C m (I) n m +(eu m, D E efm, Fm h efm, Grad um h + efm n m, u h m µ h m = efm, Deµ m, T b E [n] m + eµ m, µ h\ m µ h s = heµ m, C m (I) n m i FSI h m \ + heµ m, g FSI N N, 8{eu m, e F, eµ} 2U h m F h m f M m, where bt [n] m := C m F h m n m S m u h m µ h m, S m := µ l I. T h f =0, h f, Problem 2 (Solid sub-problem) Find {u h s, vs h, E h s, µ h h s, s }2Us h Vs h E h s Ms h V h s, such that ev s + Grad ev s, F s C s E h s + Dev b E T [n] s s =(ev s, b s ) s ees, E h s Sym( e E) s, Grad u h 1 e s 2E s, Grad u h T s Grad u h s s s s D + Sym( E) e E s n, u h s µ h s eu s eu s, vs h s s eµs, u h s eµs, µ h s De s, b T [n] s h s \ FSI + De s, b T [n] s + b T [n] f E 8{eu s, ev s, E e s, eµ s, e s }2U h Vs h E h s M f s V h s, where Solid System Matrix Fluid Solid Coupling Mesh Solid Coupling F s = I + Grad u h s bt [n] s := F s C s E h s Solid Fluid Coupling Fluid System Matrix Mesh Fluid Coupling S s := µ s l s I. Solid Mesh Coupling Fluid Mesh Coupling Mesh System + h h e s, S s s f FSI ns + S s Monolithic Coupling Solid Unkns Fluid Unkns Mesh Unkns v h s 3 h s, 2 = =0, FSI Solid RHS Fluid RHS Mesh RHS = he s, g Ns i Ns,

20 Actually implementing this in deal Mul8ple Triangula8ons (for each material) and sets of dokandlers, fe- sytems, etc Using blockmatrix system Each diagonal block gets a different physics model Fluid- solid interface (off- diagonal blocks) We have two sets of global (trace) degrees of freedom unlike everywhere else in the domain Face/point tracking between abunng trias Use weak form of coupling condimons with single point quadrature from the adjacent tria Need global- global coupling! Don t want to couple the local and global solumons 20

21 Turek-Hron FSI benchmark parameter symbol value [m] property value Channel length L 2.5 s [ kg m ] Channel height H 0.41 s 0.4 Cylinder center C (0.2, 0.2) E s [e6 kg ms ] Cylinder radius r 0.05 f [ kg m ] Flag length l 0.35 µ f [ kg ms ] 1 Flag height h 0.02 Ū [ m s ] 1 Reference point A (0.6, 0.2) Reference point B (0.15, 0.2) v f in [ m y(h y) s ] 1.5 Ū (H/2) 2 Mesh: 2900 Cells 21

22 Tip displacement plots (at A) Turek- Hron FSI benchmark 8p displacement Ploqed against CG and HDG results, both using backward Euler Mme- stepping on same mesh x displacement (m) y displacement (m) Turek data CG Q1 data CG Q2 data HDG Q1 data unit interval (s) Turek data CG Q1 data CG Q2 data HDG Q1 data unit interval (s) Q1 HDG results match oscillamon period for Q1 CG results Q1 HDG results are closer in amplitude to Q2 CG results than Q1 22

23 Summary Goal: Develop new computamonal tool, a HDG FSI model HDG allows for arbitrarily high order polynomials and their inherently parallel nature leads to significant increase in computamonal efficiency Progress: Models implemented, tested, and coupled Implemented and verified HDG elasmcity and Navier- Stokes models Considered preliminary test cases Implemented HDG FSI formulamon HDG FSI: Benchmarked against Turek and Hron (backward Euler in Mme) Q1 HDG results match oscillamon period for Q1 CG results Q1 HDG results are closer in amplitude to Q2 CG results than Q1 23

24 Future research tasks / questions Implement higher order Mme- stepping, such as Diagonally implicit Runge- uqa Perform verificamon with the method of manufactured solumons for the fully- coupled FSI problem. Implement superconvergent postprocess capabilimes for HDG FSI as done for individual components. Are the various HDG formulamons for the different physics models algebraically equivalent? Are there opmmal formulamons in terms of accuracy, computamonal efficiency, and/or stability? Can we quanmfy the effects on convergence and dissipamon of varying the stability parameter S? Plonng error vs. wall Mme, at what order element does using the HDG method for FSI become more computamonally efficient than using other methods, such as our CG FSI model? 24

25 Questions Thank you QuesMons? 25