66 research outputs found
A Symbolic Transformation Language and its Application to a Multiscale Method
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.Comment: 36 page
Computer-Aided Derivation of Multi-scale Models: A Rewriting Framework
We introduce a framework for computer-aided derivation of multi-scale models.
It relies on a combination of an asymptotic method used in the field of partial
differential equations with term rewriting techniques coming from computer
science.
In our approach, a multi-scale model derivation is characterized by the
features taken into account in the asymptotic analysis. Its formulation
consists in a derivation of a reference model associated to an elementary
nominal model, and in a set of transformations to apply to this proof until it
takes into account the wanted features. In addition to the reference model
proof, the framework includes first order rewriting principles designed for
asymptotic model derivations, and second order rewriting principles dedicated
to transformations of model derivations. We apply the method to generate a
family of homogenized models for second order elliptic equations with periodic
coefficients that could be posed in multi-dimensional domains, with possibly
multi-domains and/or thin domains.Comment: 26 page
Closed combination of context-embedding iterative strategies
This work is motivated by the challenging problem of the computer-aided generation of approximations (viewed as a series of transformations) of partial derivative equations. In this framework, the approximations posed over complex settings are incrementally constructed by extending an approximation posed on a simple setting and combining these extensions. In order to formalize these extensions and their combination, we introduce a class of rewriting strategies, called context-embedding iterative strategies (CE-strategies, for short). Roughly speaking, the class of CE-strategies is constructed by means of adding contexts and an iteration operator allowing the definition of recursive strategies. We show that the class of CE-strategies is closed under combination with respect to a correctness-completeness criterion. It turns out that the class CE-strategies enjoy nice algebraic properties, namely, the combination is associative, has a neutral element, and all the elements are idempotents
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