Advances in Studies and Applications of Centroidal Voronoi Tessellations

Centroidal Voronoi tessellations(CVTs) have become a useful tool in many applications ranging from geometric modeling,image and data analysis,and numerical partial differential equations,to problems in physics,astrophysics,chemistry,and biology. In this paper,we briefly review the CVT concept and a few of its generalizations and well-known properties.We then present an overview of recent advances in both mathematical and computational studies and in practical applications of CVTs.Whenever possible,we point out some outstanding issues that still need investigating.

A Second-Order Implicit-Explicit Scheme for the Baroclinic-Barotropic Split System of Primitive Equations

The baroclinic-barotropic mode splitting technique is commonly employed in numerical solutions of the primitive equations for ocean modeling to deal with the multiple time scales of ocean dynamics.In this paper,a second-order implicit-explicit(IMEX)scheme is proposed to advance the baroclinic-barotropic split system.Specifically,the baroclinic mode and the layer thickness of fluid are evolved explicitly via the second-order strong stability preserving Runge-Kutta scheme,while the barotropic mode is advanced implicitly using the linearized Crank-Nicolson scheme.At each time step,the baroclinic velocity is first computed using an intermediate barotropic velocity.The barotropic velocity is then corrected by re-advancing the barotropic mode with an improved barotropic forcing.Finally,the layer thickness is updated by coupling the baroclinic and barotropic velocities together.In addition,numerical inconsistencies on the discretized sea surface height caused by the mode splitting are alleviated via a reconciliation process with carefully calculated flux deficits.Temporal truncation error is also analyzed to validate the second-order accuracy of the scheme.Finally,two benchmark tests from the MPAS-Ocean platform are conducted to numerically demonstrate the performance of the proposed IMEX scheme.

A SPARSE-GRID METHOD FOR MULTI-DIMENSIONAL BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

A sparse-grid method for solving multi-dimensional backward stochastic differential equations （BSDEs） based on a multi-step time discretization scheme [31] is presented. In the multi-dimensional spatial domain, i.e. the Brownian space, the conditional mathe- matical expectations derived from the original equation are approximated using sparse-grid Gauss-Hermite quadrature rule and （adaptive） hierarchical sparse-grid interpolation. Error estimates are proved for the proposed fully-discrete scheme for multi-dimensional BSDEs with certain types of simplified generator functions. Finally, several numerical examples are provided to illustrate the accuracy and efficiency of our scheme.

Coupled Models and Parallel Simulations for Three-Dimensional Full-Stokes Ice Sheet Modeling

A three-dimensional full-Stokes computational model is considered for determining the dynamics,temperature,and thickness of ice sheets.The governing thermomechanical equations consist of the three-dimensional full-Stokes system with nonlinear rheology for the momentum,an advective-diffusion energy equation for temperature evolution,and a mass conservation equation for ice-thickness changes.Here,we discuss the variable resolution meshes,the finite element discretizations,and the parallel algorithms employed by the model components.The solvers are integrated through a well-designed coupler for the exchange of parametric data between components.The discretization utilizes high-quality,variable-resolution centroidal Voronoi Delaunay triangulation meshing and existing parallel solvers.We demonstrate the gridding technology,discretization schemes,and the efficiency and scalability of the parallel solvers through computational experiments using both simplified geometries arising from benchmark test problems and a realistic Greenland ice sheet geometry.

A Parallel Computational Model for Three-Dimensional,Thermo-Mechanical Stokes Flow Simulations of Glaciers and Ice Sheets

This paper focuses on the development of an efficient,three-dimensional,thermo-mechanical,nonlinear-Stokes flow computational model for ice sheet simulation.The model is based on the parallel finite element model developed in[14]which features high-order accurate finite element discretizations on variable resolution grids.Here,we add an improved iterative solution method for treating the nonlinearity of the Stokes problem,a new high-order accurate finite element solver for the temperature equation,and a new conservative finite volume solver for handling mass conservation.The result is an accurate and efficient numerical model for thermo-mechanical glacier and ice-sheet simulations.We demonstrate the improved efficiency of the Stokes solver using the ISMIP-HOM Benchmark experiments and a realistic test case for the Greenland ice-sheet.We also apply our model to the EISMINT-II benchmark experiments and demonstrate stable thermo-mechanical ice sheet evolution on both structured and unstructured meshes.Notably,we find no evidence for the“cold spoke”instabilities observed for these same experiments when using finite difference,shallow-ice approximation models on structured grids.

Auxiliary Equations Approach for the Stochastic Unsteady Navier-Stokes Equations with Additive Random Noise

This paper presents a Martingale regularization method for the stochas-tic Navier–Stokes equations with additive noise.The original system is split into two equivalent parts,the linear stochastic Stokes equations with Martingale solution and the stochastic modified Navier–Stokes equations with relatively-higher regular-ities.Meanwhile,a fractional Laplace operator is introduced to regularize the noise term.The stability and convergence of numerical scheme for the pathwise modified Navier–Stokes equations are proved.The comparisons of non-regularized and reg-ularized noises for the Navier–Stokes system are numerically presented to further demonstrate the efficiency of our numerical scheme.

Partial and Spectral-Viscosity Models for Geophysical Flows

Two models based on the hydrostatic primitive equations are proposed.The first model is the primitive equations with partial viscosity only,and is oriented towards large-scale wave structures in the ocean and atmosphere.The second model is the viscous primitive equations with spectral eddy viscosity,and is oriented towards turbulent geophysical flows.For both models,the existence and uniqueness of global strong solutions are established.For the second model,the convergence of the solutions to the solutions of the classical primitive equations as eddy viscosity parameters tend to zero is also established.

Peridynamic State-Based Models and the Embedded-Atom Model

We investigate connections between nonlocal continuum models and molecular dynamics.A continuous upscaling of molecular dynamics models of the form of the embedded-atom model is presented,providing means for simulating molecular dynamics systems at greatly reduced cost.Results are presented for structured and structureless material models,supported by computational experiments.The nonlocal continuum models are shown to be instances of the state-based peridynamics theory.Connections relating multibody peridynamic models and upscaled nonlocal continuum models are derived.

Computational Investigation of the Effects of Sample Geometry on the Superconducting-Normal Phase Boundary and Vortex-Antivortex States in Mesoscopic Superconductors

A computational study of superconducting states near the superconductingnormal phase boundary in mesoscopic finite cylinders is presented.The computational approach uses a finite element method to find numerical solutions of the linearized Ginzburg-Landau equation for samples with various sizes,aspect ratios,and crosssectional shapes,i.e.,squares,triangles,circles,pentagons,and four star shapes.The vector potential is determined using a finite element method with two penalty terms to enforce the gauge conditions that the vector potential is solenoidal and its normal component vanishes at the surface(s)of the sample.The eigenvalue problem for the linearized Ginzburg-Landau equations with homogeneous Neumann boundary conditions is solved and used to construct the superconducting-normal phase boundary for each sample.Vortex-antivortex(V-AV)configurations for each sample that accurately reflect the discrete symmetry of each sample boundary were found through the computational approach.These V-AV configurations are realized just within the phase boundary in the magnetic field-temperature phase diagram.Comparisons are made between the results obtained for the different sample shapes.

Bridging Methods for Atomistic-to-Continuum Coupling and Their Implementation

Several issues connected with bridging methods for atomistic-to-continuum(AtC)coupling are examined.Different coupling approaches using various energy blending models are studied as well as the influence that model parameters,blending functions,and grids have on simulation results.We use the Lagrange multiplier method for enforcing constraints on the atomistic and continuum displacements in the bridge region.We also show that continuum models are not appropriate for dealing with problems with singular loads,whereas AtC bridging methods yield correct results,thus justifying the need for a multiscale method.We investigate models that involve multiple-neighbor interactions in the atomistic region,particularly focusing on a comparison of several approaches for dealing with Dirichlet boundary conditions.