A Symbolic Transformation Language and its Application to a Multiscale Method

TL;DR

This paper introduces a symbolic transformation language implemented as a Maple package to facilitate the automatic derivation of multiscale models for micro- and nanosystems, enhancing the development of MEMSALab.

Contribution

It presents a novel rule-based transformation language integrated with Maple to simplify multiscale model derivations in complex geometries.

Findings

Successfully encoded two multiscale derivations

Demonstrated practical application in heat equation modeling

Enhanced model derivation efficiency

Abstract

The context of this work is the design of a software, called MEMSALab, dedicated to the automatic derivation of multiscale models of arrays of micro- and nanosystems. In this domain a model is a partial differential equation. Multiscale methods approximate it by another partial differential equation which can be numerically simulated in a reasonable time. The challenge consists in taking into account a wide range of geometries combining thin and periodic structures with the possibility of multiple nested scales. In this paper we present a transformation language that will make the development of MEMSALab more feasible. It is proposed as a Maple package for rule-based programming, rewriting strategies and their combination with standard Maple code. We illustrate the practical interest of this language by using it to encode two examples of multiscale derivations, namely the two-scale limit of the derivative operator and the two-scale model of the stationary heat equation.