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Research Collection Research Collection Doctoral Thesis Computational methods for ice flow simulation Author(s): Kallen-Brown, Jedediah Aspen Publication Date: 2011 Permanent Link: https://doi.org/10.3929/ethz-a-007316245 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library Diss. ETH No. 19948 Computational methods for ice flow simulation A dissertation submitted to ETH ZURICH¨ for the degree of Doctor of Sciences presented by JEDEDIAH ASPEN KALLEN-BROWN Master of Science (Mathematics) University of Alaska Fairbanks Bachelor of Science (Physics) University of Alaska Fairbanks Bachelor of Science (Mathematics) University of Alaska Fairbanks born 4 June, 1983 citizen of United States of America accepted on the recommendation of Prof. Dr. M. Funk, examiner Prof. Dr. W. Kinzelbach, examiner Dr. M. Luthi,¨ examiner Prof. Dr. E. L. Bueler, co-examiner Dr. B. F. Smith, co-examiner 2011 Contents Abstract . ix Zusammenfassung . xi 1 Introduction 1 2 Efficient nonlinear solvers for nodal high-order finite element methods in 3D 4 2.1 Introduction . .4 2.2 Newton-Krylov . .6 2.2.1 The Newton iteration . .6 2.2.2 Krylov methods . .6 2.2.3 Performance . .7 2.3 High order finite element methods . .8 2.3.1 Discretization . .8 2.3.2 Residual evaluation . .8 2.3.3 Jacobian representation . .9 2.3.4 Preconditioning . 10 2.3.5 Performance of matrix operations . 10 2.4 Numerical examples . 11 2.4.1 p-Poisson . 11 2.4.2 Stokes . 14 2.5 Discussion . 17 3 Textbook multigrid efficiency for hydrostatic ice sheet flow 19 3.1 Introduction . 19 3.2 Equations and Discretization . 20 3.3 Solver and Implementation . 21 3.3.1 Anisotropic Meshes . 21 3.3.2 Dirichlet Boundary Conditions . 23 3.3.3 Matrices . 23 3.4 Numerical Examples . 24 3.4.1 Algorithmic Scalability . 24 3.4.2 Parallel Scalability on Blue Gene/P . 26 3.4.3 Algebraic Methods . 27 3.5 Conclusion . 30 3.6 Addendum: Implicit free surface and erosion . 32 4 Software 35 4.1 Multiphysics coupling . 36 4.1.1 Field-split preconditioning . 37 4.1.2 Linear algebraic interfaces to facilitate field-split preconditioning . 39 4.2 High throughput on cache-based architectures . 41 4.2.1 Sparse matrix kernels . 41 4.2.2 Small dense tensor product kernels . 45 4.3 Generic finite elements . 46 4.3.1 Dual order finite elements . 47 4.3.2 Input/output and visualization for high-order mixed spaces . 51 4.4 Verification . 52 4.5 Time integration . 57 5 Discretization issues 59 5.1 Regularity and approximation . 59 5.1.1 Singularities in the continuum formulation . 60 5.1.2 Approximation spaces . 61 5.2 Implementation of boundary conditions . 63 5.2.1 Dirichlet boundary conditions . 63 5.2.2 Slip . 64 6 Steady-state viscous heat transport at Jakobshavn Isbræ 67 6.1 Formulation and methods . 68 6.1.1 Problem description . 68 6.1.2 Numerical solution . 73 6.1.3 Verification . 77 6.1.4 A simple problem . 79 6.2 Jakobshavn Isbræ . 82 6.2.1 Scaling . 82 6.2.2 Meshing . 82 6.2.3 Solutions . 84 7 Conclusion and Outlook 89 7.1 Future directions . 89 References . 93 List of Figures 2.1 The left panel shows truncation error as a function of total degrees of freedom for a 3D Poisson problem with smooth but rapidly varying solution. The thick horizontal line represents an acceptable accuracy and is only modestly within the asymptotic range for all approximation orders. The right panel shows the total number of nonzeros in the Jacobian which is an optimistic lower bound for the flops required for the solve. .5 2.2 A patch of four Q5 elements with one Q1 subelement shaded. 10 −8 2.3 Linear solve time for 3D Poisson with relative tolerance of 10 using assembled Q2 elements and unassembled Q3, Q5, and Q7 elements preconditioned by an assembled Q1 operator. 12 2.4 Two all-hexahedral meshes, twist and random, containing nearly degenerate elements (element condition number shown). 13 −6 2.5 Linear solve time for 3D Stokes with relative tolerance of 10 . For Q2 −Q1 and Q3 −Q2 elements, convergence is significantly slower than with Qk −Qk−2, but apparently scalable despite being somewhat erratic. 16 3.1 A cutout colored by velocity magnitude for flow over the bumpy bed of test X at L = 10km with m = 1=10 nearly plastic basal yield model. The cut on the left shows along-flow velocity as the ice hits the sticky region, the cut on the right shows across-flow shear structure. 25 3.2 Grid-sequenced Newton-Krylov solution of test X. The solid lines denote nonlinear iterations, and the dotted lines with × denote linear residuals. 26 3.3 Grid sequenced Newton-Krylov convergence for test Y................... 26 3.4 Average number of Krylov iterations per nonlinear iteration. Each nonlinear system was solved to a relative tolerance of 10−2............................ 27 3.5 Strong scaling on Shaheen for different size coarse levels problems and different coarse level solvers (see text for details). The straight lines on the strong scaling plot have slope −1 which is optimal. Grid sequencing is used, but only the nonlinear solve on the finest level is shown since strong scalability is most important when many time steps are needed. 28 3.6 Weak scaling on Shaheen with a breakdown of time spent in different phases of the solution process. Times are for the full grid-sequenced problem instead of just the finest level solve. 28 3.7 A steady-state solution for ISMIP-HOM test C (Pattyn et al., 2008) at 10km computed in 19 iterations. The elevated surface is exaggerated surface height and the color in the solid domain is velocity. 33 3.8 Bed profile eroded from a flat bed after 300ka with test C slipperiness perturbation. Time steps are 30ka at this point in the simulation. 33 4.1 Memory and floating point requirements for matrix-free tensor-product application of an operator versus representation as an assembled matrix stored in BAIJ(b) format. The same operation y Ax is applied in both cases, the storage is just different. A “result” is a single scalar entry in y, regardless of the block size b................... 47 4.2 Solution of the large-deformation nonlinear elasticity problem with manufactured solution on a Q5 mesh. 56 6.1 The numerical approximation to the manufactured solution (6.1.15) as computed using a Q3 − Q2 − Q3 finite element discretization on a 12 × 12 × 12 mesh. The converging streamlines are symmetric from below. Energy isosurfaces are shown, with the phase transition occuring at approximately E = 0. The numerical solution is accurate to 4 digits for momentum and energy and 2 digits for pressure, evaluated using the maximum norm and a higher order quadrature rule. The gradients are accurate to 3 digits for momentum and energy, with 3% error in the pressure gradient. 78 6.2 Convergence rates for Q3 − Q2 − Q3 under h-refinement. 79 6.3 Energy isosurfaces and velocity streamlines for the block on an inclined plate. The energy scale has been shifted so that the phase transition occurs at approximately E = 0. The warmest region (inside the red isosurface) occurs where the singular viscous heat production (see Figure 6.4) balances advection and combined thermal and moisture diffusion. 80 6.4 Isosurfaces of the viscous heat production rate hDui :Dui and velocity streamlines for the block on an inclined plate. Viscous heat production has a 1=r singularity at both the upstream and downstream corners, clearly stronger at the downstream corner. 81 6.5 Regions of interest for Jakobshavn Isbræ. The outer black line marks the area on which bed and surface measurements were accurate enough to perform decimation. The inner (thick) black line marks the meshed region. 83 6.6 The volume to be meshed resting on the geometric model for the bed. 84 6.7 Computed momentum density streamlines for the ice stream region at Jakobshavn Isbræ. The highest velocity occurs in a deep, narrow region of the ice stream. 85 6.8 Computed energy density (top) and momentum density (bottom) with flow streamlines for the ice stream region at Jakobshavn Isbræ. ..
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