Publications

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Surface effects are critical to the accurate simulation of electromagnetics (EM) as current tends to concentrate near material surfaces. Sandia EM applications, which include exploding bridge wires for detonator design, electromagnetic launch of flyer plates for material testing and gun design, lightning blast-through for weapon safety, electromagnetic armor, and magnetic flux compression generators, all require accurate resolution of surface effects. These applications operate in a large deformation regime, where body-fitted meshes are impractical and multimaterial elements are the only feasible option. State-of-the-art methods use various mixture models to approximate the multi-physics of these elements. The empirical nature of these models can significantly compromise the accuracy of the simulation in this very important surface region. We propose to substantially improve the predictive capability of electromagnetic simulations by removing the need for empirical mixture models at material surfaces. We do this by developing an eXtended Finite Element Method (XFEM) and an associated Conformal Decomposition Finite Element Method (CDFEM) which satisfy the physically required compatibility conditions at material interfaces. We demonstrate the effectiveness of these methods for diffusion and diffusion-like problems on node, edge and face elements in 2D and 3D. We also present preliminary work on h -hierarchical elements and remap algorithms.

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We present a new optimization-based method for atomistic-to-continuum (AtC) coupling. The main idea is to cast the latter as a constrained optimization problem with virtual Dirichlet controls on the interfaces between the atomistic and continuum subdomains. The optimization objective is to minimize the error between the atomistic and continuum solutions on the overlap between the two subdomains, while the atomistic and continuum force balance equations provide the constraints. Separation, rather then blending of the atomistic and continuum problems, and their subsequent use as constraints in the optimization problem distinguishes our approach from the existing AtC formulations. We present and analyze the method in the context of a one-dimensional chain of atoms modeled using a linearized two-body potential with next-nearest neighbor interactions.

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We present a new extended finite element method with algebraic constraints (XFEM-AC) for recovering weakly discontinuous solutions across internal element interfaces. If necessary, cut elements are further partitioned by a local secondary cut into body-fitting subelements. Each resulting subelement contributes an enrichment of the parent element. The enriched solutions are then tied using algebraic constraints, which enforce C0 continuity across both cuts. These constraints impose equivalence of the enriched and body-fitted finite element solutions, and are the key differentiating feature of the XFEM-AC. In so doing, a stable mixed formulation is obtained without having to explicitly construct a compatible Lagrange multiplier space and prove a formal inf-sup condition. Likewise, convergence of the XFEM-AC solution follows from its equivalence to the interface-fitted finite element solution. This relationship is further exploited to improve the numerical solution of the resulting XFEM-AC linear system. Examples are shown demonstrating the new approach for both steady-state and transient diffusion problems. © 2013 Elsevier B.V.

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Arctic sea ice is an important component of the global climate system, reflecting a significant amount of solar radiation, insulating the ocean from the atmosphere and influencing ocean circulation by modifying the salinity of the upper ocean. The thickness and extent of Arctic sea ice have shown a significant decline in recent decades with implications for global climate as well as regional geopolitics. Increasing interest in exploration as well as climate feedback effects make predictive mathematical modeling of sea ice a task of tremendous practical import. Satellite data obtained over the last few decades have provided a wealth of information on sea ice motion and deformation. The data clearly show that ice deformation is focused along narrow linear features and this type of deformation is not well-represented in existing models. To improve sea ice dynamics we have incorporated an anisotropic rheology into the Los Alamos National Laboratory global sea ice model, CICE. Sensitivity analyses were performed using the Design Analysis Kit for Optimization and Terascale Applications (DAKOTA) to determine the impact of material parameters on sea ice response functions. Two material strength parameters that exhibited the most significant impact on responses were further analyzed to evaluate their influence on quantitative comparisons between model output and data. The sensitivity analysis along with ten year model runs indicate that while the anisotropic rheology provides some benefit in velocity predictions, additional improvements are required to make this material model a viable alternative for global sea ice simulations.

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Conventional least-squares finite element methods (LSFEMs) for incompressible flows conserve mass only approximately. For some problems, mass loss levels are large and result in unphysical solutions. In this paper we formulate a new, locally conservative LSFEM for the Stokes equations wherein a discrete velocity field is computed that is point-wise divergence free on each element. The central idea is to allow discontinuous velocity approximations and then to define the velocity field on each element using a local stream-function. The effect of the new LSFEM approach on improved local and global mass conservation is compared with a conventional LSFEM for the Stokes equations employing standard C 0 Lagrangian elements. © 2011 John Wiley & Sons, Ltd.

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The class of discontinuous Petrov-Galerkin finite element methods (DPG) proposed by L. Demkowicz and J. Gopalakrishnan guarantees the optimality of the solution in an energy norm and produces a symmetric positive definite stiffness matrix, among other desirable properties. In this paper, we describe a toolbox, implemented atop Sandia's Trilinos library, for rapid development of solvers for DPG methods. We use this toolbox to develop solvers for the Poisson and Stokes problems.

In [3] we proposed a new Control Volume Finite Element Method with multi-dimensional, edge- based Scharfetter-Gummel upwinding (CVFEM-MDEU). This report follows up with a detailed computational study of the method. The study compares the CVFEM-MDEU method with other CVFEM and FEM formulations for a set of standard scalar advection-diffusion test problems in two dimensions. The first two CVFEM formulations are derived from the CVFEM-MDEU by simplifying the computation of the flux integrals on the sides of the control volumes, the third is the nodal CVFEM [2] without upwinding, and the fourth is the streamline upwind version of CVFEM [10]. The finite elements in our study are the standard Galerkin, SUPG and artificial diffusion methods. All studies employ logically Cartesian partitions of the unit square into quadrilateral elements. Both uniform and non-uniform grids are considered. Our results demonstrate that CVFEM-MDEU and its simplified versions perform equally well on rectangular or nearly rectangular grids. However, performance of the simplified versions significantly degrades on non-affine grids, whereas the CVFEM-MDEU remains stable and accurate over a wide range of mesh Peclet numbers and non-affine grids. Compared to FEM formulations the CVFEM-MDEU appears to be slightly more dissipative than the SUPG, but has much less local overshoots and undershoots.

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We develop a new formulation of the Control Volume Finite Element Method (CVFEM) with a multidimensional Scharfetter-Gummel (SG) upwinding for the drift-diffusion equations. The formulation uses standard nodal elements for the concentrations and expands the flux in terms of the lowest-order Nedelec H(curl; {Omega})-compatible finite element basis. The SG formula is applied to the edges of the elements to express the Nedelec element degree of freedom on this edge in terms of the nodal degrees of freedom associated with the endpoints of the edge. The resulting upwind flux incorporates the upwind effects from all edges and is defined at the interior of the element. This allows for accurate evaluation of integrals on the boundaries of the control volumes for arbitrary quadrilateral elements. The new formulation admits efficient implementation through a standard loop over the elements in the mesh followed by loops over the element nodes (associated with control volume fractions in the element) and element edges (associated with flux degrees of freedom). The quantities required for the SG formula can be precomputed and stored for each edge in the mesh for additional efficiency gains. For clarity the details are presented for two-dimensional quadrilateral grids. Extension to other element shapes and three dimensions is straightforward.

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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 state to internal model parameters. A new sea ice model that holds some promise for improving 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 this MPM sea ice code and compare it with the Los Alamos National Laboratory CICE 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.

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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.

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Arctic sea ice plays an important role in global climate by reflecting solar radiation and insulating the ocean from the atmosphere. Due to feedback effects, the Arctic sea ice cover is changing rapidly. To accurately model this change, high-resolution calculations must incorporate: (1) annual cycle of growth and melt due to radiative forcing; (2) mechanical deformation due to surface winds, ocean currents and Coriolis forces; and (3) localized effects of leads and ridges. We have demonstrated a new mathematical algorithm for solving the sea ice governing equations using the material-point method with an elastic-decohesive constitutive model. An initial comparison with the LANL CICE code indicates that the ice edge is sharper using Materials-Point Method (MPM), but that many of the overall features are similar.