Miriam Mehl

Multiphysics simulations: Challenges and opportunities

We consider multiphysics applications from algorithmic and architectural perspectives, where “algorithmic” includes both mathematical analysis and computational complexity, and “architectural” includes both software and hardware environments. Many diverse multiphysics applications can be reduced, en route to their computational simulation, to a common algebraic coupling paradigm. Mathematical analysis of multiphysics coupling in this form is not always practical for realistic applications, but model problems representative of applications discussed herein can provide insight. A variety of software frameworks for multiphysics applications have been constructed and refined within disciplinary communities and executed on leading-edge computer systems. We examine several of these, expose some commonalities among them, and attempt to extrapolate best practices to future systems. From our study, we summarize challenges and forecast opportunities.

A Cache-Aware Algorithm for PDEs on Hierarchical Data Structures Based on Space-Filling Curves

Competitive numerical algorithms for solving partial differential equations have to work with the most efficient numerical methods like multigrid and adaptive grid refinement and thus with hierarchical data structures. Unfortunately, in most implementations, hierarchical data—typically stored in trees—cause a nonnegligible overhead in data access. To overcome this quandary—numerical efficiency versus efficient implementation—our algorithm uses space-filling curves to build up data structures which are processed linearly. In fact, the only kind of data structure used in our implementation is stacks. Thus, data access becomes very fast—even faster than the common access to nonhierarchical data stored in matrices—and, in particular, cache misses are reduced considerably. Furthermore, the implementation of multigrid cycles and/or higher order discretizations as well as the parallelization of the whole algorithm become very easy and straightforward on these data structures.

Peano—A Traversal and Storage Scheme for Octree-Like Adaptive Cartesian Multiscale Grids

Almost all approaches to solving partial differential equations (PDEs) are based upon a spatial discretization of the computational domain—a grid. This paper presents an algorithm to generate, store, and traverse a hierarchy of

A cache‐oblivious self‐adaptive full multigrid method

d

A parallel adaptive cartesian PDE solver using space–filling curves

-dimensional Cartesian grids represented by a

Space-Filling Curves

(k=3)

Partitioned Simulation of Fluid-Structure Interaction on Cartesian Grids

-spacetree, a generalization of the well-known octree concept, and it also shows the correctness of the approach. These grids may change their adaptive structure throughout the traversal. The algorithm uses

A coupling environment for partitioned multiphysics simulations applied to fluid-structure interaction scenarios

2d+4

On the parallelization of a cache-optimal iterative solver for PDEs based on hierarchical data structures and space-filling curves

stacks as data structures for both cells and vertices, and the storage requirements for the pure grid reduce to one bit per vertex for both the complete grid connectivity structure and the multilevel grid relations. Since the traversal algorithm uses only stacks, the algorithms cache hit rate is continually higher than 99.9 percent, and the runtime per vertex remains almost constant; i.e., it does not depend on the overall number of vertices or the adaptivity pattern. We use the algorithmic approach as the fundamental concept for a mesh management for

Parallel coupling numerics for partitioned fluid–structure interaction simulations

d