Dimensional analysis using toric ideals

TL;DR

This paper introduces a novel algebraic approach using toric ideals to automate and enhance dimensional analysis in engineering, enabling systematic identification of invariants.

Contribution

It applies algebraic geometry, specifically toric ideals, to develop an automatic method for dimensional analysis in engineering systems, expanding traditional techniques.

Findings

Method successfully identifies primitive invariants

Automates dimensional analysis process

Links algebraic geometry with engineering analysis

Abstract

Classical dimensional analysis is one of the cornerstones of qualitative physics and is also used in the analysis of engineering systems, for example in engineering design. The basic power product relationship in dimensional analysis is identical to one way of defining toric ideals in algebraic geometry, a large and growing field. This paper exploits the toric representation to provide a method for automatic dimensional analysis for engineering systems. In particular all "primitive", invariants for a particular problem, in a well defined sense, can be found using such methods.