Recent developments in parametrized variational principles for mechanics

Abstract

This paper gives an introduction to the formulation of parametrized variational principles (PVPs) in mechanics. This is complemented by more advanced material describing selected recent developments in hybrid and nonlinear variational principles. A PVP is a variational principle containing free parameters that have no effect on the Euler-Lagrange equations and natural boundary conditions. The theory of single-field PVPs, based on gauge functions, is a subset of the Inverse Problem of Variational Calculus that has limited value. On the other hand, multified PVPs are more interesting from both theoretical and practical standpoints. The two-dimensional Poisson equation is used to present, in a tutorial fashion, the formulation of parametrized mixed functionals. This treatment is then extended to internal interfaces, which are useful in treatment of discontinuities, subdomain linkage and construction of parametrized hybrid functionals. This is followed by a similar but more compact treatment of three-dimensional classical elasticity, and a parametrization of nonlinear hyperelasticity.

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