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Abstract
This paper aims at the exploitation of material forces to find an optimum mesh in the finite element method (FEM). The classical variational formulation provides the linear momentum equation in a Lagrangian description. A variational setting for the derivation of the canonical momentum equation in the Eulerian description is presented. The latter is based on an extremum principle for the total potential energy functional defined in terms of the inverse deformation function. This constitutes a theoretical framework which allows the formulation of the finite element method for the canonical momentum equation as well as the computation of the material forces arising from the discretization. Thus, apart from the finite element solution for the standard boundary value problem of elastostatics, a second one for the canonical momentum equation can be formulated and solved numerically. The former provides an optimum deformation by minimizing the standard total potential energy, namely solving the physical forces equilibrium equation. The latter provides an optimum discretization by minimizing the total potential energy in terms of the inverse deformation function, that is, solving the material force equilibrium equation. The latter provides an optimum discretization by minimizing the total potential energy in terms of the inverse deformation function, that is, solving the material forces equilibrium equation. The theoretical considerations are supported by providing a computational example.
Acknowledgements
The authors are grateful to Prof. G.A. Maugin for his remarks on an earlier version of this paper. Also the authors thank Prof. G.E. Stavroulakis for the valuable discussions they had with him during the preparation of this work. This research was funded by the program “Heraklitos” of the Operational Program for Education and Initial Vocational Training of the Hellenic Ministry of Education under the 3rd Community Support Framework and the European Social Fund.