Sixth-order Finite Volume Method for the 1d Biharmonic Operator: Application to the Intramedullary Nail Simulation

A new very high-order finite volume method to solve the harmonic and the biharmonic operators for one-dimensional geometries is proposed. The main ingredient is the 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 enables to design 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 the stress critical point determination for an intramedullary nail.