On an Improper Modification of a Variational Principle for Finite Element Plate Analysis

Boundary infinite elements for the Helmholtz equation in exterior domains

“…A similar procedure of enforcing boundary conditions by Lagrange multipliers which are subsequently removed was described in the context of elasticity problems by Key. 22 Key's formulation, sometimes referred to as a "Simpliÿed Variational Principle", was explored by Haugeneder et al [23][24][25][26] in various studies. A criticism based on analysis of single line elements indicates that this approach may reduce accuracy on the boundary.…”

“…A piecewise linear ÿnite element solution u h that is based on Galerkin approximation of this formulation is nodally exact since the corresponding Green's function is piecewise linear (see, e.g., Reference 44, pp. [24][25][26][27]. Thus, for a mesh of n+1 nodes such that x 1 = 0, x A ¡x A+1 ; A = 1; : : : ; n, and x n+1 = 1 u h (x A ) = u(x A ); A= 1; : : : ; n (96) and u h (1) = g is an admissibility requirement of the approximate trial solution.…”

“…22 Analysis of single line elements indicates that this approach may reduce accuracy on the boundary. [23][24][25][26] In Appendix I we show for a model that is simple, yet more general than a single element, that the reduction in pointwise accuracy of the approximate solution on the boundary is a result of increased accuracy of its derivative. Consequently, for our purposes (namely, enforcing ux continuity across the boundary) this procedure can lead to an e ective and accurate numerical method.…”

Finite element formulations for exterior problems: application to hybrid methods, non-reflecting boundary conditions, and infinite elements

“…19 Key's formulation, sometimes referred to as a 'Simpliÿed Variational Principle,' was explored with mixed results by Mang, Gallagher and Haugeneder in various studies. [20][21][22][23] Our formulation di ers signiÿcantly from Key's, however, in at least two respects. First, we integrate by parts in the unbounded domain and therefore require our basis functions there to exactly satisfy the di erential equation.…”

Numerical and spectral investigations of Trefftz infinite elements

“…This formulation is exact and free of constraint degrees of freedom, and can be used to recover many well-known formulations developed previously [11; 16], including the DtN boundary conditions. Criticism [23][24][25][26][27][28] of a similar approach [29] (also related to Reference 30) is examined in Reference 10, leading to the conclusion that for our purposes (namely, enforcing ux continuity across the boundary) this procedure can lead to an e ective and accurate numerical method. Conservation properties and the well-posedness of the formulation in terms of uniqueness of its solutions are established.…”