A finite element solver for ice sheet dynamics to be ......A finite element solver for ice sheet dynamics to be integrated with MPAS February 16, CESM LIWG Meeting, Boulder (CO), 2012

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)


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


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


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


Interface Grids

MPAS CVT Mesh(OK for Finite Volumes)


Interface Grids

Lifev Dual triangular Mesh(OK for Finite Elements)


Temperaturevelocity

velocity


Interface Grids

Lifev Dual triangular Mesh(OK for Finite Elements)


Temperaturevelocity

velocity


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


Perego, Gunzburger, Burkardt, Journal of Glaciology, 2012.

Comparisons between different models

SIA SSA L1L2 FO

Towards realistic simulations: Greenland, 5km resolution, sliding case


FOSIA

Towards realistic simulations: Greenland, 2km resolution, noslip


SIA FO


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


Solve F (u) = 0:Newton method: Jk (±u)k+1 = ¡F (uk):

Greenland, FO: convergence of the nonlinear method


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;


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


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)


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)


± variable : At each newton iteration ± is decreased from 1e-4 to 1e-9

Greenland, FO: convergence of the nonlinear method(LOCA continuation method)


±=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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Documents

A finite element solver for ice sheet dynamics to be ......A finite element solver for ice sheet dynamics to be integrated with MPAS February 16, CESM LIWG Meeting, Boulder (CO), 2012