On two-dimensional parametric variational problems

1999 ◽  
Vol 9 (3) ◽  
pp. 249-267 ◽  
Author(s):  
Stefan Hildebrandt ◽  
Heiko von der Mosel
Keyword(s):  
Variational Problems ◽  
Two Dimensional

VARIATIONAL PROBLEMS ON UNBOUNDED TWO-DIMENSIONAL DOMAINS

1992 ◽  
Vol 02 (02) ◽  
pp. 183-201
Author(s):  
ARIE LEIZAROWITZ
Keyword(s):  
Control Theory ◽  
Large Class ◽  
Optimality Criterion ◽  
Variational Problems ◽  
Two Dimensional ◽  
Minimal Energy ◽  
Chemical Systems ◽  
Overtaking Optimality ◽  
Special Case ◽  
Class Of Functions

We consider the functional IΩ(u) = ∫Ω [ψ (u(x,y)) + ½K (∇ u)]dxdy defined for real valued functions u on ℝ2 and study its minimization over a certain class of functions u(·, ·). We look for a minimizer u⋆ which is universal in the sense that IΩ(u⋆)≤IΩ(u) for every bounded domain (in a certain class) and for every u(·, ·) which satisfies u|∂Ω=u⋆|∂Ω. This optimality notion is an extension to a multivariable situation of the overtaking optimality criterion used in control theory, and the minimal-energy-configuration concept employed in the study of certain chemical systems. The existence of such universal minimizers is established for a large class of variational problems. In the special case were K(∇ u) = ½ |∇ u|2 these minimizers are characterized as the functions u⋆(x, y)=ϕ(ax+by+c) for some explicitly computable ϕ:ℝ1→ℝ1 and constants a, b and c.

Visual Topology and Variational Problems on Two-Dimensional Surfaces

2005 ◽  
pp. 99-123
Author(s):  
Anatoly T. Fomenko ◽  
Alexandr O. Ivanov ◽  
Alexey A. Tuzhilin
Keyword(s):  
Variational Problems ◽  
Two Dimensional

THE MESHLESS GALERKIN BOUNDARY NODE METHOD FOR TWO-DIMENSIONAL SOLIDS

2013 ◽  
Vol 10 (04) ◽  
pp. 1350013 ◽  
Author(s):  
XIAOLIN LI
Keyword(s):  
Solid Mechanics ◽  
Delta Function ◽  
Variational Problems ◽  
Positive Definiteness ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Governing Equations ◽  
Boundary Node ◽  
Numerical Tests ◽  
Variational Form

The Galerkin boundary node method (GBNM) is developed for two-dimensional solid mechanics problems. The GBNM is a boundary only meshless method that combines an equivalent variational form of boundary integral formulations for governing equations with the moving least-squares (MLS) approximations for construction of the trial and test functions. In this method, boundary conditions can be implemented directly and easily despite the MLS shape functions lack the delta function property, and the resulting formulation inherits the symmetry and positive definiteness of the variational problems. The optimal asymptotic error estimates of this approach for displacements and stresses are derived in detail in Sobolev spaces. Numerical tests are also given to demonstrate the developed algorithms.

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