Top-down metacomputing with algebraic dimensionality raising for automating theory-building to enable directly computable multiphysics models

Abstract

The need to address the fundamental issues associated with the computational model generation process in the context of multiphysics theories that observe conservation laws and relevant assumptions motivates the present work. Accordingly, the goal of this paper is to describe, discuss, and demonstrate the benefits and implications of using metacomputing (i.e., the utilization of computational technologies for computing or generating programs that, in turn, compute the desired results) for theory or model generation, further assisted by metacomputing for raising the semantic and syntactic dimensionality of physics models for continua. The issues associated with the model generation process are identified at the outset. Then, a top-down approach that contains three distinct metacomputing activities is described. First, a meta-theoretical description of generating continuum multiphysics models defined in terms of quantities instantiated from the field of real numbers is described, along with its computational implementation. Then, a computational framework for raising the semantic dimensionality from the reals to the hypercomplex numbers is presented for both 2D and 3D continuum multiphysics problems. The semantic dimensionality is expressed in terms of the order of the algebra used to define the tensorial quantities of interest and their associated operations, as they participate in representing physics models expressed initially as systems of partial differential equations. The utilization of hypercomplex algebras, such as those for complex numbers and quaternions, is demonstrated for representative applications of models. Subsequently, syntactic dimensionality is defined in terms of the basis of the space utilized to write equational systems. Finally, a prototype of a computational system for converting equational systems to directed graphs termed “Algebraic Solution Graphs” is described. Examples of how to utilize these technologies are also presented to convey the efficacy and efficiency of the proposed approach.