We use tools from algebraic topology to show that a class of structural differential equations may be represented combinatorially and thus by a computer data structure. In particular, every differential k-form may be represented by a formal k-cochain over a cellular structure that we call a starplex, and exterior differentiation is equivalent to the coboundary operation on the corresponding k-cochain. Furthermore, there is a one to one correspondence between this model and the classical finite cellular model supported by the Generalized Stokes' Theorem, and translation between the two models can be completely automated.

A typical computer representation of a design includes geometric and physical information organized in a suitable combinatorial data structure. Queries and transformations of these design representations are used to formulate most algorithms in computational design, including analysis, optimization, evolution, generation, and synthesis. Formal properties, and in particular existence and validity of the computed solutions, must be assured and preserved by all such algorithms. Using tools

ions as a model for parallel computations[10]. These Phase Abstractions are identified by the XYZ levels of programming, where the X level corresponds to an individual sequential processor, the Y level represents a collection of processors working on a common task in a data parallel mode, and the Z level which represents the control structure "in the large" of an application. It is argued that this model maps well to most parallel architectures, and, as a result, applications built around this model will execute with high efficiency on a wide range of architectural variations. 2.1.4 Bulk-Synchronous Protocol The Bulk-Synchronous Protocol (BSP) has been proposed as a bridging model for parallel computations[11]. The term "bridging model" is used here to describe a model, much like the von Neumann model for sequential architectures, that facilitates development of computer 9 architectures such that certain quantitative guarantees regarding software performance are met. Given such a mod...

. Finite element meshes are large, richly structured sets whose internal relationships must be visible to different parts of a finite element program. This causes software engineerings problems that increase when adaptive mesh refinement and multilevel preconditioners are applied. Even more problems arise when finite element methods have to be implemented for parallel computers since the meshes have to be mapped onto the hardware topology so that their locality is preserved. We have designed a framework for parallel adaptive finite element methods that centers upon a problem-oriented index scheme as a new high level description method for finite element meshes. Within the index scheme, important mesh relations can be expressed by simple algebraic operations in Z n . We give an overview of the indexing methodology and outline the main parts of the framework. Special emphasis is on the reuse of several C++ template libraries---including standard container classes and the library for da...

Computational science and engineering are dominated by field problems. Traditionally, engineering practice involves repeated iterations of shape design (i.e., shaping and modeling of material properties), simulation of the physical field, evaluation of the result, and re-design. In this paper, we propose a specific interpretation of the algebraic topological formulation of field problems, which is conceptually simple, physically sound, computational effective and comprehensive. In the proposed approach, physical information is attached to an adaptive, full-dimensional decomposition of the domain of interest. Giving preeminence to the cells of highest dimension allows us to generate the geometry and to simulate the physics simultaneously. We will also demonstrate that our formulation removes artificial constraints on the shape of discrete elements and unifies commonly unrelated methods in a single computational framework. This framework, by using an efficient graph-representation of the domain of interest, unifies several geometric and physical finite formulations, and supports local progressive refinement (and coarsening) effected only where and when required.

In this report, a set of C++ classes is presented for representing unstructured triangular meshes of intrinsic dimension two; i.e. oriented 2-manifolds. Simple classes for the basic mesh objects, i.e. vertices, triangles, and line segments, are described. They define abstractions based on their incidence relations and a few geometric primitives for a mesh class, which is an intelligent container class of three lists of these simple mesh objects. The classes are intended to be components in an object oriented approach to software for meshing applications described in the report. This context differentiates the roles of the mesh class and the simple mesh object classes; these latter can be extended as the carriers of the applications data. The capability of the classes of this report to simultaneously simplify the coding of mesh methods and facilitate generalization of the code is discussed with examples. The report provides an overview of the class design and use, tutorial e...

The key to the success of mimetic discretization methods is that they discretize some description of continuum mechanics, e.g. vector calculus or differential forms.