Marsha J. Berger

Local adaptive mesh refinement for shock hydrodynamics

The aim of this work is the development of an automatic, adaptive mesh refinement strategy for solving hyperbolic conservation laws in two dimensions. There are two main difficulties in doing this. The first problem is due to the presence of discontinuities in the solution and the effect on them of discontinuities in the mesh. The second problem is how to organize the algorithm to minimize memory and CPU overhead. This is an important consideration and will continue to be important as more sophisticated algorithms that use data structures other than arrays are developed for use on vector and parallel computers.

Robust and Efficient Cartesian Mesh Generation for Component-Based Geometry

This work documents a new method for rapid and robust Cartesian mesh generation for component-based geometry. The new algorithm adopts a novel strategy that first intersects the components to extract the wetted surface before proceeding with volume mesh generation in a second phase. The intersection scheme is based on a robust geometry engine that uses adaptive precision arithmetic and automatically and consistently handles geometric degenerades with an algorithmic tie-breaking routine. The intersection procedure has worst-case computational complexity of O(N log N) and is demonstrated on test cases with up to 121 overlapping and intersecting components, including a variety of geometric degeneracies. The volume mesh generation takes the intersected surface triangulation as input and generates the mesh through cell division of an initially uniform coarse grid. In refining hexagonal cells to resolve the geometry, the new approach preserves the ability to directionally divide cells that are well aligned with local geometry. The mesh generation scheme has linear asymptotic complexity with memory requirements that total approximately 14-17 words/cell. The mesh generation speed is approximately 10 6 cells/minute on a typical engineering workstation

Three-dimensional adaptive mesh refinement for hyperbolic conservation laws

A local adaptive mesh refinement algorithm for solving hyperbolic systems of conservation laws in three space dimensions is described. The method is based on the use of local grid patches superimpo...

A Parallel Multilevel Method for Adaptively Refined Cartesian Grids with Embedded Boundaries

. Asrecently as five to ten years ago, mesh generation was fre-quently the most time consuming task in a typical CFD cycle.Adaptive Cartesian mesh generation methods are capable ofproducing millions of cells around complex geometries inminutes and have substantially removed this bottleneck.Why write yet another Euler solver? With robust mesh gener-ation largely in-hand, solution time resurfaces as the pacingitem in the CFD cycle. The current work attacks this issue bydesigning a scalable, accurate Cartesian solver with robustmultigrid convergence acceleration. Our primary motivationis to gain efficiency by capitalizing on the simplifications andspecialized data structures available on Cartesian grids. Sig-nificant savings in both CPU time and storage may be real-ized by taking advantage of the fact that cell faces arecoordinate aligned. In addition, higher-order methods withgood limiters are generally easier to design and perform morerobustly on uniform Cartesian meshes.Secondly, in any embedded-boundary Cartesian solver, thebody-intersectingcut-cellsdemand special attention. Thesecells can impose a substantial burden on the numerical dis-cretization since the arbitrary nature of geometric intersec-tion implies that a cut-cell may be orders of magnitudesmaller than its neighboring cells. This fact contrasts sharplywith the comparatively smooth meshes that are generallyfound on a good quality structured or unstructured mesh.Substantial research into these cut-cell issues have been stud-ied by references [9],[10],[6],[8], and [12] (among others)and we hope to take advantage of this investment.Thirdly, this work investigates a multigrid strategy that isspecialized for adaptively refined Cartesian meshes. In ourapproach, all grids in the multigrid hierarchy cover the entiredomain and include cells at many refinement levels. Thesmoother therefore iterates over the entire domain when it isinvoked on any grid in the hierarchy. In this respect, theapproach shares more with agglomeration or algebraic multi-grid techniques than with many other Cartesian or AMRmethods which iterate over only cells at the same level ofrefinement

An algorithm for point clustering and grid generation

A special-purpose point clustering algorithm is described, and its application to automatic grid generation, a technique used to solve partial differential equations, is considered. Extensions of techniques common in computer vision and pattern recognition literature are used to partition points into a set of enclosing rectangles. Examples from 2-D calculations are shown, but the algorithm generalizes readily to three dimensions. >

Adaptive Mesh Refinement Using Wave-Propagation Algorithms for Hyperbolic Systems

An adaptive mesh refinement algorithm developed for the Euler equations of gas dynamics has been extended to employ high-resolution wave-propagation algorithms in a more general framework. This allows its use on a variety of new problems, including hyperbolic equations not in conservation form, problems with source terms or capacity functions, and logically rectangular curvilinear grids. This framework requires a modified approach to maintaining consistency and conservation at grid interfaces, which is described in detail. The algorithm is implemented in the AMRCLAW package, which is freely available.

Automatic adaptive grid refinement for the Euler equations

A method of adaptive grid refinement for the solution of the steady Euler equations for transonic flow is presented. The algorithm automatically decides where the coarse grid accuracy is insufficient, and creates locally uniform refined grids in these regions. This typically occurs at the leading and trailing edges. The solution is then integrated to steady state using the same integrator (FLO52) in the interior of each grid. The boundary conditions needed on the fine grids are examined, and the importance of treating the fine/coarse grid interface conservatively is discussed. Numerical results indicate substantial computational savings for the same solution accuracy can be achieved.

On Conservation at Grid Interfaces

This paper considers the solution of hyperbolic systems of conservation laws on discontinuous grids. In particular, we consider what happens to conservation at grid interfaces. A procedure is presented to derive conservative difference approximations at the grid interfaces for two-dimensional grids which overlap in an arbitrary configuration. The same procedures are applied to compute interface formulas for grids which are refined in space and/or time, and for continuous grids where a switch in the scheme causes the discontinuity.

An Adaptive Cartesian Mesh Algorithm for the Euler Equations in Arbitrary Geometries

We present a Cartesian mesh algorithm with adaptive refinement to compute flows around arbitrary geometries. Cartesian meshes have been less popular than unstructured or body-fitted meshes because of several technical difficulties. We present an approach that resolves many of these problems. Cartesian meshes have the advantage of allowing the use of high resolution methods that are difficult to develop on unstructured grids. They also allow for efficient implementation on vector computers without using gather-scatter operations except at boundary cells. Some preliminary computational results using lower order boundary conditions are presented.

Tsunami modelling with adaptively refined finite volume methods

Numerical modelling of transoceanic tsunami propagation, together with the detailed modelling of inundation of small-scale coastal regions, poses a number of algorithmic challenges. The depth-averaged shallow water equations can be used to reduce this to a time-dependent problem in two space dimensions, but even so it is crucial to use adaptive mesh refinement in order to efficiently handle the vast differences in spatial scales. This must be done in a ‘wellbalanced’ manner that accurately captures very small perturbations to the steady state of the ocean at rest. Inundation can be modelled by allowing cells to dynamically change from dry to wet, but this must also be done carefully near refinement boundaries. We discuss these issues in the context of Riemann-solver-based finite volume methods for tsunami modelling. Several examples are presented using the GeoClaw software, and sample codes are available to accompany the paper. The techniques discussed also apply to a variety of other geophysical flows.