Admissible and Inadmissible Simplifications of Variational Methods in Finite Element 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.…”

“…This result was supported by numerical tests. 25 In Appendix I we examine the e ects of employing our procedure on a simple model and show 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.…”

“…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.…”

A new penalty function element for thin shell analysis

“…While it is rather cumbersome to achieve a single C, continuous displacement field, it is much easier to introduce two independent C, continuous fields for the transverse displacement w and the rotation of the normal to the middle surface 8 using the so-called Mindlin plate theory.' This theory takes into account both the flexural and the shear stiffness of a plate.…”