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A sixth-order finite volume method for the 1D biharmonic operator: Application to intramedullary nail simulation

Costa R., Machado G. J., Clain S.

International Journal of Applied Mathematics and Computer Science

2015

Vol. 25, no. 3

529--537

A new very high-order finite volume method to solve problems with harmonic and biharmonic operators for one-dimensional geometries is proposed. The main ingredient is polynomial reconstruction based on local interpolations of mean values providing accurate approximations of the solution up to the sixth-order accuracy. First developed with the harmonic operator, an extension for the biharmonic operator is obtained, which allows designing a very high-order finite volume scheme where the solution is obtained by solving a matrix-free problem. An application in elasticity coupling the two operators is presented. We consider a beam subject to a combination of tensile and bending loads, where the main goal is the stress critical point determination for an intramedullary nail.

Chemical attack in free boundary domains

Clain S.

Journal of Applied Analysis

1999

Vol. 5, nr 1

35-58

This paper presents a mathematical model for a chemical process used to machine cristal as glass or silica. A short physical description is presented from which we draw the mathematical model. We obtain a coupled parabolic equations system on a free boundary domain with a non-linear condition on the boundary. The existence and the uniqueness is proved in the one-dimensional case.