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

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Abstract

A new very high-order finite volume method to solve problems with harmonic and biharmonic operators for onedimensional 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.

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  • Audusse E. and Bristeau M.-O. (2007). Finite-volume solvers for a multilayer Saint-Venant system International Journal of Applied Mathematics and Computer Science 17(3): 311-320 DOI: 10.2478/v10006-007-0025-0.

  • Barreira L. Teixeira C. and Fonseca E. (2008). Avaliação da resistência do colo do fémur utilizando o modelo de elementos finitos Revista da Associação Portuguesa de Análise Experimental de Tensões 16: 19-24.

  • Branco C.M. (2011). Mecânica dos Materiais Fundação Calouste Gulbenkian Lisboa.

  • Clain S. Diot S. and Loubère R. (2011). A high-order polynomial finite volume method for hyperbolic system of conservation laws with multi-dimensional optimal order detection (MOOD) Journal of Computational Physics 230(10): 4028-4050.

  • Clain S. Machado G.J. Nóbrega J.M. and Pereira R.M.S. (2013). A sixth-order finite volume method for the convection-diffusion problem with discontinuous coefficients Computer Methods in Applied Mechanics and Engineering 267(1): 43-64.

  • Diot S. Clain S. and Loubère R. (2011). Multi-dimensional optimal order detection (mood)-a very high-order finite volume scheme for conservation laws on unstructured meshes 6th Finite Volume and Complex Application Prague Czech Republic pp. 263-271.

  • Dumbser M. and Munz C.-D. (2007). On source terms and boundary conditions using arbitrary high order discontinuous Galerkin schemes International Journal of Applied Mathematics and Computer Science 17(3): 297-310 DOI: 10.2478/v10006-007-0024-1.

  • Eymard R. Gallouët T. and Herbin R. (2000). The finite volume method in P. Ciarlet and J.L. Lions (Eds.) Handbook for Numerical Analysis North Holland Amsterdam pp. 715-1022.

  • Hern´andez J. (2002). High-order finite volume schemes for the advection-diffusion equation International Journal for Numerical Methods in Engineering 53(5): 1211-1234.

  • Kroner D. (1997). Numerical Schemes for Conservation Laws Wiley-Teubneur Publishers Chichester.

  • Leveque R.J. (2002). Finite Volume Methods for Hyperbolic Problems Cambridge Texts in Applied Mathematics Cambridge University Press Cambridge.

  • Ollivier-Gooch C. and Altena M.V. (2002). A high-order-accurate unstructured mesh finite-volume scheme for the advection-diffusion equation Journal of Computational Physics 181(2): 729-752.

  • Ramos A. and Simoes J.A. (2009). Caracterização de cavilhas de fixação intra-medular de estabilização de fracturas ósseas Revista da Associação Portuguesa de Análise Experimental de Tensões 17: 49-55.

  • Toro E. (2009). Riemann Solvers and Numerical Methods for Fluid Dynamics Springer Berlin/Heidelberg.

  • Toro E. and Hidalgo A. (2009). Ader finite volume schemes for nonlinear reaction-diffusion equations Applied Numerical Mathematics 59(1): 73-100.

  • Trangenstein J.A. (2009). Numerical Solution of Hyperbolic Partial Differential Equations Cambridge University Press Cambridge.

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