A finite element solver for ice sheet dynamics to be integrated with MPAS

Transcription

1 A finite element solver for ice sheet dynamics to be integrated with MPAS Mauro Perego in collaboration with FSU, ORNL, LANL, Sandia February 6, CESM LIWG Meeting, Boulder (CO), 202

2 Outline Introduction of finite element ice sheets component Integration of finite element ice sheet component with MPAS Results on ISMIP HOM and Greenland/Antarctica geometries Non linear solvers: features and issues

3 Ice Sheet Modeling Main components of an ice model: Ice flow equations (momentum and mass balance) r ¾ = ½g and r u = 0; with ¾ = pi = 2¹(") _ "_ pi, where ¹ viscosity, "_ shear rate Model for the evolution of the boundaries (thickness evolution = Hf lux Z u dz z ½cu rt + 2"¾ _

4 Ice Sheet Modeling Reference model: FULL STOKES Approximations based on their accuracy with respect to the ice sheet aspect ratio ± O(± 2 ) FO, Blatter Pattyn first order model2 (3D PDE, in horizontal velocities) O(±) Zeroth order, depth integrated models: SIA, Shallow Ice Approximation (slow sliding regimes), SSA Shallow Shelf Approximation (2D PDE) (fast sliding regimes) ' O(± 2 ) High order, depth integrated (2D) models: LL23, (LL)... Gagliardini and Zwinger, The Cryosphere. 2 Dukowicz, Price and Lipscomb, 200. J. Glaciol. 3 Schoof and Hindmarsh, 200. Q. J. Mech. Appl. Math.

5 Implementation Overview ice sheets FE dynamics component LifeV: Parallel, object oriented, C++ Finite Element Library: linear and quadratic finite elements assembling of finite element matrices handling of boundary conditions

6 Implementation Overview Trilinos: Parallel Data Structures (EPETRA) Parallel Linear Solvers (GMRES, CG...) Preconditioners (Multilevel, Multigrid, Incomplete LU) Nonlinear Solvers (NOX package: Newton, JFNK methods) ice sheets FE dynamics component LifeV: Parallel, object oriented, C++ Finite Element Library: linear and quadratic finite elements assembling of finite element matrices handling of boundary conditions Work in Progress MPAS (land ice component): Voronoi unstructured grids Evolution equation solvers (temperature and thickness equation)...

7 Interface MPAS LIFEV MPAS LIFEV 2D CVT mesh (Stereographic projection) thickness/elevation/layers bedrock sliding coefficient Solver options: model (FO, LL2, SSA, SIA) nonlinear solver (Newton, Picard, JFNK) Boundary condition (free-slip, no-slip, robin, coulomb) velocity heat dissipation viscosity ice-sheets component Land ice component temperature/ice flow factor

8 Interface Grids MPAS CVT Mesh (OK for Finite Volumes)

9 Interface Grids MPAS CVT Mesh (OK for Finite Volumes) velocity velocity Temperature Lifev Dual triangular Mesh (OK for Finite Elements)

10 Interface Grids MPAS CVT Mesh (OK for Finite Volumes) velocity velocity Temperature Based on 2D grid and thickness and layers build vertically structured 3D grid. Build prisms with triangular base and split them in tetrahedra. Lifev Dual triangular Mesh (OK for Finite Elements)

11 ISMIP HOM: Test C Surface velocity L = 5 km L = 40 km L = 0 km L = 80 km L = 20 km L = 60 km Perego, Gunzburger, Burkardt, Journal of Glaciology, 202.

12 Towards realistic simulations: Greenland, 5km resolution, sliding case Comparisons between different models SIA SSA LL2 FO

13 Towards realistic simulations: Greenland, 2km resolution, no slip SIA FO

14 Towards realistic simulations: Antarctica, 5km resolution, no slip SIA FO

15 Convergence of the nonlinear method (test C ISMIP HOM) Solve F (u) = 0: Newton method: J k (±u)k+ = F (uk ): (NOX library) The exact Jacobian matrix must be assembled at each nonlinear iteration. In order to increase robustness of Newton method, the Newton step is halved to achieve monotonic convergence. FO LL2 CPU time: 93s CPU time: 2s CPU time: 4s CPU time: 0.98s

16 Greenland, FO: convergence of the nonlinear method

17 First Order Equations FO is a nonlinear system of elliptic equations in the horizontal velocities: > > < r (2¹ "_ ) = > > : r (2¹ "_2 ) = n ( n ) ¹ = A "_e 2 q "_e = "_2xx + "_2yy + "_xx "_yy + "_2xy + "_2xz + "_2yz where s is the ice surface and, "_i;j = (@j ui uj ) ; 2 i; j 2 fx; y; zg; 2 6 "_ = 6 4 2"_xx + "_yy "_xy "_xz 3 Goldsby D. L. Kohlsted, JGR, 200. "_yx ; "_2 = 6 "_xx + 2"_yy "_yz Remark The nonlinear viscosity ¹ is singular when "_e = 0, however, ¹ "_ is not singular and the PDE is well de ned. ³p ( n ) ( ) n Viscosity regularization: "_e ¼ "_2e + ± 2 2

18 Greenland, FO: convergence of the nonlinear method Simpli ed Problem (similar kind of nonlinearity): x jxj k+ Picard method: x ( n ) solution: = CjCjn n ( ): = Cjx j k+ Newton method: x = C: k n ): ( = ( n)x + ncjx j k k µ x 2 Picard convergence: lim = = when n = 3 k k! x n 3 µ k+ x Newton convergence: lim = k 2 k! (x ) 2 n k+ regularization: x jxj ( n ³p ( n ) )¼x x2 + ± 2

19 Greenland, FO: convergence of the nonlinear method (comparison using different regularizations) Err: m/a Err: 7.2e 4 m/a Err: 0.34 m/a Viscosity regularization: ( n ) "_e ¼ ³p "_2e + ± 2 ( n )

20 Greenland, FO: convergence of the nonlinear method (comparison using different regularizations) ± variable : At each newton iteration ± is decreased from e-4 to e-9

21 Greenland, FO: convergence of the nonlinear method (LOCA continuation method) ±=e-9 ±=e-9 ±=e-4 ±=e-9 ±=.3e-7 ±=e-5 The parameter ± is decreased by LOCA from e-4 to e-9

22 Acknowledgment J. Burkardt, M. Gunzburger (FSU) B. Lipscomb, S. Price, X. Asay Davis, T. Ringler (LANL) K. Evans, J. Nichols (ORNL) A. Salinger (Sandia)