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A finite element solver for iceA finite element solver for icesheet dynamics to be integrated sheet dynamics to be integrated
with MPASwith MPAS
February 16, CESM LIWG Meeting, Boulder (CO), 2012February 16, CESM LIWG Meeting, Boulder (CO), 2012
Mauro PeregoMauro Perego
in collaboration with FSU, ORNL, LANL, Sandiain collaboration with FSU, ORNL, LANL, Sandia
Outline
● Introduction of finite element icesheets component
● Integration of finite element icesheet component with MPAS
● Results on ISMIPHOM and Greenland/Antarctica geometries
● Non linear solvers: features and issues
Ice Sheet Modeling
Ice flow equations (momentum and mass balance)
Model for the evolution of the boundaries (thickness evolution equation)
Temperature equation
Main components of an ice model:
M.Perego, [email protected]
¡r ¢ ¾ = ½g and r ¢ u = 0;
@H
@t= Hflux ¡r ¢
Z
z
u dz
½c@T
@t=@
@z
µk@T
@z
¶¡ ½cu ¢ rT + 2 _"¾
with ¾ = ¿ ¡ pI = 2¹( _") _"¡ pI,where ¹ viscosity, _" shear rate
Ice Sheet Modeling
±
FO, BlatterPattyn first order model2 (3D PDE, in horizontal velocities)
Zeroth order, depth integrated models:SIA, Shallow Ice Approximation (slow sliding regimes) ,SSA Shallow Shelf Approximation (2D PDE) (fast sliding regimes)
High order, depth integrated (2D) models: L1L23, (L1L1)...
Approximations based on their accuracy with respect to the icesheet aspect ratio
“Reference” model: FULL STOKES1
2Dukowicz, Price and Lipscomb, 2010. J. Glaciol.
1Gagliardini and Zwinger, 2008. The Cryosphere.
3Schoof and Hindmarsh, 2010. Q. J. Mech. Appl. Math.
O(±2)
O(±)
' O(±2)
M.Perego, [email protected]
icesheets FE dynamics componentLifeV: Parallel, object oriented, C++ Finite Element Library:● linear and quadratic finite elements● assembling of finite element matrices● handling of boundary conditions
Implementation Overview
M.Perego, [email protected]
Trilinos:● Parallel Data Structures (EPETRA) ● Parallel Linear Solvers (GMRES, CG...) ● Preconditioners (Multilevel, Multigrid, Incomplete LU)● Nonlinear Solvers (NOX package: Newton, JFNK methods)
MPAS (land ice component):● Voronoi unstructured grids ● Evolution equation solvers (temperature and thickness equation)
● ...
Implementation Overview
Work in Progress
M.Perego, [email protected]
icesheets FE dynamics componentLifeV: Parallel, object oriented, C++ Finite Element Library:● linear and quadratic finite elements● assembling of finite element matrices● handling of boundary conditions
ice-
shee
ts c
ompo
nent
Land
ice
com
pone
ntMPAS LIFEV
2D CVT mesh(Stereographic projection)
temperature/ice flow factorbedrock sliding coefficient
velocityheat dissipation
viscosity
thickness/elevation/layers
Solver options:model (FO, L1L2, SSA, SIA)nonlinear solver (Newton, Picard, JFNK)Boundary condition (free-slip, no-slip, robin, coulomb)
Interface MPASLIFEV
M.Perego, [email protected]
Interface Grids
MPAS CVT Mesh(OK for Finite Volumes)
M.Perego, [email protected]
Interface Grids
Lifev Dual triangular Mesh(OK for Finite Elements)
MPAS CVT Mesh(OK for Finite Volumes)
Temperaturevelocity
velocity
M.Perego, [email protected]
Interface Grids
Lifev Dual triangular Mesh(OK for Finite Elements)
MPAS CVT Mesh(OK for Finite Volumes)
Temperaturevelocity
velocity
M.Perego, [email protected]
Based on 2D grid and thickness and layers build vertically structured 3D grid.
Build prisms with triangular base and split them in tetrahedra.
L = 5 km L = 10 km
L = 40 km L = 80 kmL = 80 km L = 160 km
L = 20 km
ISMIPHOM: Test C
Surface velocity
M.Perego, [email protected]
Perego, Gunzburger, Burkardt, Journal of Glaciology, 2012.
Comparisons between different models
SIA SSA L1L2 FO
Towards realistic simulations: Greenland, 5km resolution, sliding case
M.Perego, [email protected]
FOSIA
Towards realistic simulations: Greenland, 2km resolution, noslip
M.Perego, [email protected]
SIA FO
M.Perego, [email protected]
Towards realistic simulations: Antarctica, 5km resolution, noslip
Convergence of the nonlinear method (test C ISMIPHOM)
CPU time: 21s
CPU time: 93s CPU time: 4s
CPU time: 0.98s
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.
(NOX library)
FO L1L2
M.Perego, [email protected]
Solve F (u) = 0:Newton method: Jk (±u)k+1 = ¡F (uk):
Greenland, FO: convergence of the nonlinear method
M.Perego, [email protected]
First Order Equations
_"e =q_"2xx + _"
2yy + _"xx _"yy + _"
2xy + _"
2xz + _"
2yz
¹ =1
2A¡
1n _"( 1n¡1)e
8>><>>:
¡r ¢ (2¹ _"1) = ¡½g@s
@x
¡r ¢ (2¹ _"2) = ¡½g@s
@y;
M.Perego, [email protected]
where s is the ice surface and,
_"2 =
2664
_"yx
_"xx + 2 _"yy
_"yz
3775_"1 =
2664
2 _"xx + _"yy
_"xy
_"xz
3775 ;
FO is a nonlinear system of elliptic equations in the horizontal velocities:
_"i;j =1
2(@jui + @iuj) ; i; j 2 fx; y; zg;
Remark The nonlinear viscosity ¹ is singular when _"e = 0,however, ¹ _"1 is not singular and the PDE is well de¯ned.
1Goldsby D. L. Kohlsted, JGR, 2001.
Viscosity1 regularization: _"¡(1¡ 1n)e ¼
³p_"2e + ±
2´¡(1¡ 1n )
Greenland, FO: convergence of the nonlinear method
M.Perego, [email protected]
Picard convergence: limk!1
xk+1 ¡ ®xk ¡ ® = 1¡
1
n
µ=2
3when n = 3
¶
Newton convergence: limk!1
xk+1 ¡ ®(xk ¡ ®)2 =
1
2®
µ1
n¡ 1¶
regularization: x jxj¡(1¡ 1n ) ¼ x³p
x2 + ±2´¡(1¡ 1n)
Simpli¯ed Problem (similar kind of nonlinearity):
x jxj¡(1¡ 1n ) = C: solution: ® = CjCjn¡1
Picard method: xk+1 = Cjxk j(1¡ 1n ):
Newton method: xk+1 = (1¡ n)xk + nCjxk j(1¡ 1n):
Greenland, FO: convergence of the nonlinear method(comparison using different regularizations)
M.Perego, [email protected]
Err: 0.34 m/a
Err: 0.026 m/a Err: 7.2e4 m/a
Viscosity regularization: _"¡(1¡ 1n)e ¼
³p_"2e + ±
2´¡(1¡ 1n )
Greenland, FO: convergence of the nonlinear method(comparison using different regularizations)
M.Perego, [email protected]
± variable : At each newton iteration ± is decreased from 1e-4 to 1e-9
Greenland, FO: convergence of the nonlinear method(LOCA continuation method)
M.Perego, [email protected]
±=1e-4
±=1e-5±=1.3e-7
±=1e-9
±=1e-9
The parameter ± is decreased by LOCA from 1e-4 to 1e-9
±=1e-9
Acknowledgment
J. Burkardt, M. Gunzburger (FSU)
B. Lipscomb, S. Price, X. AsayDavis, T. Ringler (LANL)
K. Evans, J. Nichols (ORNL)
A. Salinger (Sandia)
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