Markus Berndt

Superconvergence of the Velocity in Mimetic Finite Difference Methods on Quadrilaterals

Superconvergence of the velocity is established for mimetic finite difference approximations of second-order elliptic problems over

Two-step hybrid conservative remapping for multimaterial arbitrary Lagrangian-Eulerian methods

h^2

Convergence of mimetic finite difference discretizations of the diffusion equation

-uniform quadrilateral meshes. The superconvergence result holds for a full tensor coefficient. The analysis exploits the relation between mimetic finite differences and mixed finite element methods via a special quadrature rule for computing the scalar product in the velocity space. The theoretical results are confirmed by numerical experiments.

Analysis of First-Order System Least Squares (FOSLS) for Elliptic Problems with Discontinuous Coefficients: Part I

We present a new hybrid conservative remapping algorithm for multimaterial Arbitrary Lagrangian-Eulerian (ALE) methods. The hybrid remapping is performed in two steps. In the first step, only nodes of the grid that lie inside subdomains occupied by single materials are moved. At this stage, computationally cheap swept-region remapping is used. In the second step, nodes that are vertices of mixed cells (cells containing several materials) and vertices of some cells in a buffer zone around mixed cells are moved. At this stage, intersection-based remapping is used. The hybrid algorithm results in computational expense that lies between swept-region and intersection-based remapping We demonstrate the performance of our new method for both structured and unstructured polygonal grids in two dimensions, as well as for cell-centered and staggered discretizations.

Nonlinear Krylov acceleration applied to a discrete ordinates formulation of the k-eigenvalue problem

Abstract The main goal of this paper is to establish the convergence of mimetic discretizations of the first-order system that describes linear diffusion. Specifically, mimetic discretizations based on the support-operators methodology (SO) have been applied successfully in a number of application areas, including diffusion and electromagnetics. These discretizations have demonstrated excellent robustness, however, a rigorous convergence proof has been lacking. In this research, we prove convergence of the SO discretization for linear diffusion by first developing a connection of this mimetic discretization with Mixed Finite Element (MFE) methods. This connection facilitates the application of existing tools and error estimates from the finite element literature to establish convergence for the SO discretization. The convergence properties of the SO discretization are verified with numerical examples.

A mortar mimetic finite difference method on non-matching grids

First-order system least squares (FOSLS) is a recently developed methodology for solving partial differential equations. Among its advantages are that the finite element spaces are not restricted by the inf-sup condition imposed, for example, on mixed methods and that the least-squares functional itself serves as an appropriate error measure. This paper studies the FOSLS approach for scalar second-order elliptic boundary value problems with discontinuous coefficients, irregular boundaries, and mixed boundary conditions. A least-squares functional is defined, and ellipticity is established in a natural norm of an appropriately scaled least-squares bilinear form. For some geometries, this ellipticity is independent of the size of the jumps in the coefficients. The occurrence of singularities at interface corners, cross points, reentrant corners, and irregular boundary points is discussed, and a basis of singular functions with local support around singular points is established. A companion paper shows that the singular basis functions can be added at little extra cost and lead to optimal performance of standard finite element discretization and multilevel solver techniques, also independent of the size of coefficient jumps for some geometries.

Polyhedral Mesh Generation and Optimization for Non-manifold Domains

We compare a variant of Anderson Mixing with the Jacobian-Free Newton-Krylov and Broyden methods applied to an instance of the k-eigenvalue formulation of the linear Boltzmann transport equation. We present evidence that one variant of Anderson Mixing finds solutions in the fewest number of iterations. We examine and strengthen theoretical results of Anderson Mixing applied to linear problems.

Large-eddy simulation, fuel rod vibration and grid-to-rod fretting in pressurized water reactors

We consider mimetic finite difference approximations to second order elliptic problems on non-matching multiblock grids. Mortar finite elements are employed on the non-matching interfaces to impose weak flux continuity. Optimal convergence and, in certain cases, superconvergence is established for both the scalar variable and its flux. The theory is confirmed by computational results.

A hybrid incremental projection method for thermal-hydraulics applications

We present a preliminary method to generate polyhedral meshes of general non-manifold domains. The method is based on computing the dual of a general tetrahedral mesh. The resulting mesh respects the topology of the domain to the same extent as the input mesh. If the input tetrahedral mesh is Delaunay and well-centered, the resulting mesh is a Voronoi mesh with planar faces. For general tetrahedral meshes, the resulting mesh is a polyhedral mesh with straight edges but possibly curved faces. The initial mesh generation phase is followed by a mesh untangling and quality improvement technique.We demonstrate the technique on some simple to moderately complex domains.

Using the feasible set method for rezoning in ALE

Grid-to-rod fretting (GTRF) in pressurized water reactors is a flow-induced vibration phenomenon that results in wear and fretting of the cladding material on fuel rods. GTRF is responsible for over 70% of the fuel failures in pressurized water reactors in the United States. Predicting the GTRF wear and concomitant interval between failures is important because of the large costs associated with reactor shutdown and replacement of fuel rod assemblies. The GTRF-induced wear process involves turbulent flow, mechanical vibration, tribology, and time-varying irradiated material properties in complex fuel assembly geometries. This paper presents a new approach for predicting GTRF induced fuel rod wear that uses high-resolution implicit large-eddy simulation to drive nonlinear transient dynamics computations. The GTRF fluid-structure problem is separated into the simulation of the turbulent flow field in the complex-geometry fuel-rod bundles using implicit large-eddy simulation, the calculation of statistics of the resulting fluctuating structural forces, and the nonlinear transient dynamics analysis of the fuel rod. Ultimately, the methods developed here, can be used, in conjunction with operational management, to improve reactor core designs in which fuel rod failures are minimized or potentially eliminated. Robustness of the behavior of both the structural forces computed from the turbulent flow simulations and the results from the transient dynamics analyses highlight the progress made towards achieving a predictive simulation capability for the GTRF problem. A new approach for predicting grid-to-rod fretting wear in reactor fuel is presented.ILES validation is performed using data for a 5 × 5 fuel rod bundle.Richardson extrapolation of statistical fuel rod forces is used to bound wear work-rates.