Pavel Blagoveston Bochev

A spectral mimetic least-squares method

We present a spectral mimetic least-squares method for a model diffusion–reaction problem, which preserves key conservation properties of the continuum problem. Casting the model problem into a first-order system for two scalar and two vector variables shifts material properties from the differential equations to a pair of constitutive relations. We also use this system to motivate a new least-squares functional involving all four fields and show that its minimizer satisfies the differential equations exactly. Discretization of the four-field least-squares functional by spectral spaces compatible with the differential operators leads to a least-squares method in which the differential equations are also satisfied exactly. Additionally, the latter are reduced to purely topological relationships for the degrees of freedom that can be satisfied without reference to basis functions. Furthermore, numerical experiments confirm the spectral accuracy of the method and its local conservation.

A Toolbox for a Class of Discontinuous Petrov-Galerkin Methods Using Trilinos

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 Sandias Trilinos library, for rapid development of solvers for DPG methods. We use this toolbox to develop solvers for the Poisson and Stokes problems.

Algebraically constrained extended edge element method (eXFEM-AC) for resolution of multi-material cells

Abstract Surface effects are critical to the predictive simulation of electromagnetics as current tends to concentrate near material interfaces. There are two principal difficulties in the accurate representation of these effects in discrete models. First, many applications of interest operate at large deformations, where body-fitted meshes are impractical. Second, physics-compatible discretizations of the governing equations require curl-conforming edge elements, for which no practical alternatives to body-fitted meshes exist. The main purpose of this paper is to develop such an alternative that avoids remeshing the problem. Our approach uses the existing edge element basis to dynamically construct an interface-conforming basis. We show that in the case of triangular grids in two dimensions, our approach generates a basis that spans the same space as edge elements on an interface-fitted mesh. We also demonstrate the efficacy of the approach computationally.

Reducing Uncertainty in High-Resolution Sea Ice Models

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 quantitativemorexa0» 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.«xa0less

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.

Formulation and computation of dynamic, interface-compatible Whitney complexes in three dimensions

Abstract A discrete De Rham complex enables compatible, structure-preserving discretizations for a broad range of partial differential equations problems. Such discretizations can correctly reproduce the physics of interface problems, provided the grid conforms to the interface. However, large deformations, complex geometries, and evolving interfaces makes generation of such grids difficult. We develop and demonstrate two formally equivalent approaches that, for a given background mesh, dynamically construct an interface-conforming discrete De Rham complex. Both approaches start by dividing cut elements into interface-conforming subelements but differ in how they build the finite element basis on these subelements. The first approach discards the existing non-conforming basis of the parent element and replaces it by a dynamic set of degrees of freedom of the same kind. The second approach defines the interface-conforming degrees of freedom on the subelements as superpositions of the basis functions of the parent element. These approaches generalize the Conformal Decomposition Finite Element Method (CDFEM) and the extended finite element method with algebraic constraints (XFEM-AC), respectively, across the De Rham complex.

Optimization-Based Coupling of Local and Nonlocal Models: Applications to Peridynamics

Nonlocal continuum theories such as peridynamics [?] and physics-based nonlocal elasticity [?] can capture strong nonlocal effects due to long-range forces at the mesoscale or microscale. For problems where these effects cannot be neglected, nonlocal models are more accurate than classical Partial Differential Equations (PDEs) that only consider interactions due to contact. However, the improved accuracy of nonlocal models comes at the price of a computational cost that is significantly higher than that of PDEs. The goal of Local-to-Nonlocal (LtN) coupling methods is to combine the computational efficiency of PDEs with the accuracy of nonlocal models. LtN couplings are imperative when the size of the computational domain or the extent of the nonlocal interactions are such that the nonlocal solution becomes prohibitively expensive to compute, yet the nonlocal model is required to accurately resolve small scale features (such as crack tips or dislocations that can affect the global material behavior). In this context, the main challenge of a coupling method is the stable and accurate merging of two fundamentally different mathematical descriptions of the same physical phenomena into a physically consistent coupled formulation.

An extended vector space model for information retrieval with generalized similarity measures : theory and applications.

We present an Extended Vector Space Model (EVSM) for information retrieval, endowed with a new set of similarity functions. Our model considers records as multisets of tokens. A token weight function maps records into a real vectors. Using this vector representation we define a p-norm of a record and pairwise conjunction and disjunction operations on records. These operations prompt consistent extensions of published set-based similarity functions and yield new `p distance-based similarities. We demonstrate that some well-known similarities form a subset of the new functions resulting from particular choices of token weights and p-values. In so doing,

Control Volume Finite Element Method with Multidimensional Edge Element Scharfetter-Gummel upwinding. Part 2. Computational Study

In [3] we proposed a new Control Volume Finite Element Method with multi-dimensional, edgebased 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.