Appending boundary conditions by Lagrange multipliers: Analysis of the LBB condition

2001 ◽  
Vol 88 (1) ◽  
pp. 9-42 ◽  
Author(s):  
Wolfgang Dahmen ◽  
Angela Kunoth
Keyword(s):  
Boundary Conditions ◽  
Lagrange Multipliers ◽  
Lbb Condition

scholarly journals The Ritz Method for Boundary Problems with Essential Conditions as Constraints

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Vojin Jovanovic ◽  
Sergiy Koshkin
Keyword(s):  
Boundary Conditions ◽  
Saddle Point ◽  
Lagrange Multipliers ◽  
Ritz Method ◽  
Higher Dimensions ◽  
Boundary Problems ◽  
Elementary Derivation ◽  
Convergence Proof

We give an elementary derivation of an extension of the Ritz method to trial functions that do not satisfy essential boundary conditions. As in the Babuška-Brezzi approach boundary conditions are treated as variational constraints and Lagrange multipliers are used to remove them. However, we avoid the saddle point reformulation of the problem and therefore do not have to deal with the Babuška-Brezzi inf-sup condition. In higher dimensions boundary weights are used to approximate the boundary conditions, and the assumptions in our convergence proof are stated in terms of completeness of the trial functions and of the boundary weights. These assumptions are much more straightforward to verify than the Babuška-Brezzi condition. We also discuss limitations of the method and implementation issues that follow from our analysis and examine a number of examples, both analytic and numerical.

Generalized Hellinger-Reissner Principle

1998 ◽  
Vol 67 (2) ◽  
pp. 326-331 ◽  
Author(s):  
J.-H. He
Keyword(s):  
Boundary Conditions ◽  
Lagrange Multipliers ◽  
Arbitrary Constant ◽  
Inverse Method ◽  
Variational Principles ◽  
Present Theory ◽  
Governing Equations

By the semi-inverse method of establishing variational principles, the Hellinger-Reissner principle can be obtained straightforwardly from energy trial-functionals without using Lagrange multipliers, and a family of generalized Hellinger-Reissner principles with an arbitrary constant are also obtained, some of which are unknown to us at the present time. The present theory provides a straightforward tool to search for various variational principles directly from governing equations and boundary conditions. [S0021-8936(00)00702-9]

Periodic and Anti-Periodic Boundary Conditions With the Lagrange Multipliers in the FEM

2010 ◽  
Vol 46 (8) ◽  
pp. 3417-3420 ◽  
Author(s):  
M. Aubertin ◽  
Thomas Henneron ◽  
F. Piriou ◽  
P. Guerin ◽  
J.-C. Mipo
Keyword(s):  
Boundary Conditions ◽  
Lagrange Multipliers ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions

scholarly journals Free In-Plane Vibration of Super-Elliptical Plates

2011 ◽  
Vol 18 (3) ◽  
pp. 471-484 ◽  
Author(s):  
Murat Altekin
Keyword(s):  
Boundary Conditions ◽  
Lagrange Multipliers ◽  
Ritz Method ◽  
Rectangular Plates ◽  
Uniform Thickness ◽  
Plane Vibration ◽  
Good Agreement ◽  
Geometrical Boundary

Free in-plane vibration of super-elliptical plates of uniform thickness was investigated by the Ritz method. A large variety of plate shapes ranging from an ellipse to a rectangle were examined. Two cases were considered: (1) a completely free, and (2) a point-supported plate. The geometrical boundary conditions were satisfied by the Lagrange multipliers. The results were compared with those of rectangular plates. Basically good agreement was obtained. Matching results were reported, and the discrepancies were highlighted.

scholarly journals AN ALGORITHM FOR MESH REFINEMENT AND UN-REFINEMENT IN FAST TRANSIENT DYNAMICS

2013 ◽  
Vol 10 (04) ◽  
pp. 1350018 ◽  
Author(s):  
FOLCO CASADEI ◽  
PEDRO DÍEZ ◽  
FRANCESC VERDUGO
Keyword(s):  
Wave Propagation ◽  
Data Structure ◽  
Boundary Conditions ◽  
Real Number ◽  
Lagrange Multipliers ◽  
Computational Grid ◽  
Transient Dynamics ◽  
Nonconforming Element ◽  
Nodal Coordinates ◽  
Refinement Strategy

A procedure to locally refine and un-refine an unstructured computational grid of four-node quadrilaterals (in 2D) or of eight-node hexahedra (in 3D) is presented. The chosen refinement strategy generates only elements of the same type as their parents, but also produces so-called hanging nodes along nonconforming element-to-element interfaces. Continuity of the solution across such interfaces is enforced strongly by Lagrange multipliers. The element split and un-split algorithm is entirely integer-based. It relies only upon element connectivity and makes no use of nodal coordinates or other real-number quantities. The chosen data structure and the continuous tracking of the nature of each node facilitate the treatment of natural and essential boundary conditions in adaptivity. A generalization of the concept of neighbor elements allows transport calculations in adaptive fluid calculations. The proposed procedure is tested in structure and fluid wave propagation problems in explicit transient dynamics.

scholarly journals General rule for boundary conditions from the action principle

2016 ◽  
Vol 31 (08) ◽  
pp. 1650032
Author(s):  
Roee Steiner
Keyword(s):  
Boundary Conditions ◽  
Lagrange Multipliers ◽  
General Rule ◽  
Action Principle ◽  
Physical Models ◽  
The Past ◽  
Initial And Boundary Conditions

We construct models where initial and boundary conditions can be found from the fundamental rules of physics, without the need to assume them, they will be derived from the action principle. Those constraints are established from physical viewpoint, and it is not in the form of Lagrange multipliers. We show some examples from the past and some new examples that can be useful, where constraint can be obtained from the action principle. Those actions represent physical models. We show that it is possible to use our rule to get those constraints directly.

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