Kara J. Peterson

Using the material‐point method to model sea ice dynamics

[1] The material-point method (MPM) is a numerical method for continuum mechanics that combines the best aspects of Lagrangian and Eulerian discretizations. The material points provide a Lagrangian description of the ice that models convection naturally. Thus properties such as ice thickness and compactness are computed in a Lagrangian frame and do not suffer from errors associated with Eulerian advection schemes, such as artificial diffusion, dispersion, or oscillations near discontinuities. This desirable property is illustrated by solving transport of ice in uniform, rotational and convergent velocity fields. Moreover, the ice geometry is represented by unconnected material points rather than a grid. This representation facilitates modeling the large deformations observed in the Arctic, as well as localized deformation along leads, and admits a sharp representation of the ice edge. MPM also easily allows the use of any ice constitutive model. The versatility of MPM is demonstrated by using two constitutive models for simulations of wind-driven ice. The first model is a standard viscous-plastic model with two thickness categories. The MPM solution to the viscous-plastic model agrees with previously published results using finite elements. The second model is a new elastic-decohesive model that explicitly represents leads. The model includes a mechanism to initiate leads, and to predict their orientation and width. The elastic-decohesion model can provide similar overall deformation as the viscous-plastic model; however, explicit regions of opening and shear are predicted. Furthermore, the efficiency of MPM with the elastic-decohesive model is competitive with the current best methods for sea ice dynamics.

Solving PDEs with Intrepid

Intrepid is a Trilinos package for advanced discretizations of Partial Differential Equations PDEs. The package provides a comprehensive set of tools for local, cell-based construction of a wide range of numerical methods for PDEs. This paper describes the mathematical ideas and software design principles incorporated in the package. We also provide representative examples showcasing the use of Intrepid both in the context of numerical PDEs and the more general context of data analysis.

Optimization-based remap and transport: A divide and conquer strategy for feature-preserving discretizations

This paper examines the application of optimization and control ideas to the formulation of feature-preserving numerical methods, with particular emphasis on the conservative and bound-preserving remap (constrained interpolation) and transport (advection) of a single scalar quantity. We present a general optimization framework for the preservation of physical properties and specialize it to a generic optimization-based remap (OBR) of mass density. The latter casts remap as a quadratic program whose optimal solution minimizes the distance to a suitable target quantity, subject to a system of linear inequality constraints. The approximation of an exact mass update operator defines the target quantity, which provides the best possible accuracy of the new masses without regard to any physical constraints such as conservation of mass or local bounds. The latter are enforced by the system of linear inequalities. In so doing, the generic OBR formulation separates accuracy considerations from the enforcement of physical properties. We proceed to show how the generic OBR formulation yields the recently introduced flux-variable flux-target (FVFT) [1] and mass-variable mass-target (MVMT) [2] formulations of remap and then follow with a formal examination of their relationship. Using an intermediate flux-variable mass-target (FVMT) formulation we show the equivalence of FVFT and MVMT optimal solutions. To underscore the scope and the versatility of the generic OBR formulation we introduce the notion of adaptable targets, i.e., target quantities that reflect local solution properties, extend FVFT and MVMT to remap on the sphere, and use OBR to formulate adaptable, conservative and bound-preserving optimization-based transport algorithms for Cartesian and latitude/longitude coordinate systems. A selection of representative numerical examples on two-dimensional grids demonstrates the computational properties of our approach.

A parameter-free stabilized finite element method for scalar advection-diffusion problems

We formulate and study numerically a new, parameter-free stabilized finite element method for advection-diffusion problems. Using properties of compatible finite element spaces we establish connection between nodal diffusive fluxes and one-dimensional diffusion equations on the edges of the mesh. To define the stabilized method we extend this relationship to the advection-diffusion case by solving simplified one-dimensional versions of the governing equations on the edges. Then we use H(curl)-conforming edge elements to expand the resulting edge fluxes into an exponentially fitted flux field inside each element. Substitution of the nodal flux by this new flux completes the formulation of the method. Utilization of edge elements to define the numerical flux and the lack of stabilization parameters differentiate our approach from other stabilized methods. Numerical studies with representative advection-diffusion test problems confirm the excellent stability and robustness of the new method. In particular, the results show minimal overshoots and undershoots for both internal and boundary layers on uniform and non-uniform grids.

Analysis and Computation of Compatible Least-Squares Methods for div-curl Equations

We develop and analyze least-squares finite element methods for two complementary div-curl elliptic boundary value problems. The first one prescribes the tangential component of the vector field on the boundary and is solved using curl-conforming elements. The second problem specifies the normal component of the vector field and is handled by div-conforming elements. We prove that both least-squares formulations are norm-equivalent with respect to suitable discrete norms, yield optimal asymptotic error estimates, and give rise to algebraic systems that can be solved by efficient algebraic multigrid methods. Numerical results that illustrate scalability of iterative solvers and optimal rates of convergence are also included.

Optimization-Based modeling with applications to transport: part 3. computational studies

This paper is the final of three related articles that develop and demonstrate a new optimization-based framework for computational modeling. The framework uses optimization and control ideas to assemble and decompose multiphysics operators and to preserve their fundamental physical properties in the discretization process. One application of the framework is in the formulation of robust algorithms for optimization-based transport (OBT). Based on the theoretical foundations established in Part 1 and the optimization algorithm for the solution of the remap subproblem, derived in Part 2, this paper focuses on the application of OBT to a set of benchmark transport problems. Numerical comparisons with two other transport schemes based on incremental remapping, featuring flux-corrected remap and the linear reconstruction with van Leer limiting, respectively, demonstrate that OBT is a competitive transport algorithm.

Formulation and Analysis of a Parameter-Free Stabilized Finite Element Method

We present and study a parameter-free stabilized finite element method for the scalar advection-diffusion equation. Edge element lifting of diffusive edge fluxes defines the stabilization. The amount of edge diffusion varies with the edge Peclet number and adapts to solution features without mesh-dependent parameters. We prove that the method is first-order accurate in the advective limit. Numerical studies confirm the theoretical estimates.

ParNCL and ParGAL: Data-parallel Tools for Postprocessing of Large-scale Earth Science Data☆

Abstract Earth science high-performance applications often require extensive analysis of their output in order to complete the scien- tific goals or produce a visual image or animation. Often this analysis cannot be done in situ because it requires calculating time-series statistics from state sampled over the entire length of the run or analyzing the relationship between similar time series from previous simulations or observations. Many of the tools used for this postprocessing are not themselves high- performance applications, but the new Parallel Gridded Analysis Library (ParGAL) provides high-performance data-parallel versions of several common analysis algorithms for data from a structured or unstructured grid simulation. The library builds on several scalable systems, including the Mesh Oriented DataBase (MOAB), a library for representing mesh data that sup- ports structured, unstructured finite element, and polyhedral grids; the Parallel-NetCDF (PNetCDF) library; and Intrepid, an extensible library for computing operators (such as gradient, curl, and divergence) acting on discretized fields. We have used ParGAL to implement a parallel version of the NCAR Command Language (NCL) a scripting language widely used in the climate community for analysis and visualization. The data-parallel algorithms in ParGAL/ParNCL are both higher performing and more flexible than their serial counterparts.

Development, sensitivity analysis, and uncertainty quantification of high-fidelity arctic sea ice models.

Arctic sea ice is an important component of the global climate system and due to feedback effects the Arctic ice cover is changing rapidly. Predictive mathematical models are of paramount importance for accurate estimates of the future ice trajectory. However, the sea ice components of Global Climate Models (GCMs) vary significantly in their prediction of the future state of Arctic sea ice and have generally underestimated the rate of decline in minimum sea ice extent seen over the past thirty years. One of the contributing factors to this variability is the sensitivity of the sea ice to model physical parameters. A new sea ice model that has the potential to improve sea ice predictions incorporates an anisotropic elastic-decohesive rheology and dynamics solved using the material-point method (MPM), which combines Lagrangian particles for advection with a background grid for gradient computations. We evaluate the variability of the Los Alamos National Laboratory CICE code and the MPM sea ice code for a single year simulation of the Arctic basin using consistent ocean and atmospheric forcing. Sensitivities of ice volume, ice area, ice extent, root mean square (RMS) ice speed, central Arctic ice thickness, and central Arctic ice speed with respect to ten different dynamic and thermodynamic parameters are evaluated both individually and in combination using the Design Analysis Kit for Optimization and Terascale Applications (DAKOTA). We find similar responses for the two codes and some interesting seasonal variability in the strength of the parameters on the solution.

A Virtual Control Coupling Approach for Problems with Non-coincident Discrete Interfaces

Independent meshing of subdomains separated by an interface can lead to spatially non-coincident discrete interfaces. We present an optimization-based coupling method for such problems, which does not require a common mesh refinement of the interface, has optimal \(H^1\) convergence rates, and passes a patch test. The method minimizes the mismatch of the state and normal stress extensions on discrete interfaces subject to the subdomain equations, while interface “fluxes” provide virtual Neumann controls.