Application of an Ultra Weak Variational Formulation of Elliptic PDEs to the Two-Dimensional Helmholtz Problem

1998 ◽  
Vol 35 (1) ◽  
pp. 255-299 ◽  
Author(s):  
Olivier Cessenat ◽  
Bruno Despres
Keyword(s):  
Variational Formulation ◽  
Elliptic Pdes ◽  
Two Dimensional ◽  
Helmholtz Problem

Two-dimensional layered Ising models: Exact variational formulation and analysis

1994 ◽  
Vol 49 (1) ◽  
pp. 378-402 ◽  
Author(s):  
Lev V. Mikheev ◽  
Michael E. Fisher
Keyword(s):  
Variational Formulation ◽  
Ising Models ◽  
Two Dimensional

LUTTINGER-LIQUID BEHAVIOUR IN 2D: THE VARIATIONAL APPROACH

1993 ◽  
Vol 07 (03) ◽  
pp. 119-141 ◽  
Author(s):  
CLAUDIUS GROS ◽  
ROSER VALENTÍ
Keyword(s):  
Monte Carlo ◽  
Variational Formulation ◽  
Variational Approach ◽  
Order Parameter ◽  
Luttinger Liquid ◽  
Two Dimensions ◽  
Point Of View ◽  
Two Dimensional ◽  
Liquid Type ◽  
D Wave

We study a variational formulation of the Luttinger-liquid concept in two dimensions. We show that a Luttinger-liquid wavefunction with an algebraic singularity at the Fermiedge is given by a Jastrow-Gutzwiller type wavefunction, which we evaluate by variational Monte Carlo for lattices with up to 38 × 38 = 1444 sites. We therefore find that, from a variational point of view, the concept of a Luttinger liquid is well defined even in 2D. We also find that the Luttinger liquid state is energetically favoured by the projected kinetic energy in the context of the 2D t-J model. We study and find coexistence of d-wave superconductivity and Luttinger-liquid behaviour in two-dimensional projected wavefunctions. We then argue that generally, any two-dimensional d-wave superconductor should be unstable against Luttinger-liquid type correlations along the (quasi-1D) nodes of the d-wave order parameter, at temperatures small compared to the gap.

scholarly journals Two-Dimensional Tensor Product Variational Formulation

2001 ◽  
Vol 105 (3) ◽  
pp. 409-417 ◽  
Author(s):  
T. Nishino ◽  
Y. Hieida ◽  
K. Okunishi ◽  
N. Maeshima ◽  
Y. Akutsu ◽  
...  
Keyword(s):  
Tensor Product ◽  
Variational Formulation ◽  
Two Dimensional

Optimal Design for Two-Dimensional Structures Made of Composite Materials

1991 ◽  
Vol 113 (1) ◽  
pp. 88-92 ◽  
Author(s):  
G. Sacchi Landriani ◽  
M. Rovati
Keyword(s):  
Variational Formulation ◽  
Optimal Solution ◽  
The Body ◽  
Necessary Condition ◽  
Two Dimensional ◽  
Plane State ◽  
State Of Stress ◽  
Design Variables ◽  
Plane Stress Problem ◽  
Orthotropic Properties

The paper deals with the problem of finding the optimal orientation of orthotropic properties for an elastic body, subjected to a plane state of stress, in order to maximize the stiffness of the body itself. A variational formulation of the problem is presented and the computation of the necessary condition for an optimal solution is carried out. The physical meaning of such a condition is then outlined, and its generality pointed out with reference to the case of two-dimensional flexural systems. Finally, an extension of the plane stress problem is formulated, taking into account as design variables both the orientation of orthotropy axes and the mechanical properties of the material, according to a global constraint on the structural cost.

Mixed Fourier-Jacobi Spectral Method for Two-Dimensional Neumann Boundary Value Problems

2011 ◽  
Vol 1 (3) ◽  
pp. 284-296 ◽  
Author(s):  
Xu-Hong Yu ◽  
Zhong-Qing Wang
Keyword(s):  
Spectral Method ◽  
Stiffness Matrix ◽  
Neumann Boundary Condition ◽  
Variational Formulation ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Two Dimensional ◽  
Neumann Boundary Value Problem ◽  
Homogeneous Neumann Boundary Condition ◽  
Neumann Boundary Value Problems

AbstractIn this paper, we propose a mixed Fourier-Jacobi spectral method for two dimensional Neumann boundary value problem. This method differs from the classical spectral method. The homogeneous Neumann boundary condition is satisfied exactly. Moreover, a tridiagonal matrix is employed, instead of the full stiffness matrix encountered in the classical variational formulation. For analyzing the numerical error, we establish the mixed Fourier-Jacobi orthogonal approximation. The convergence of proposed scheme is proved. Numerical results demonstrate the efficiency of this approach.

scholarly journals ANALYSIS OF AN INVERSE PROBLEM ARISING IN PHOTOLITHOGRAPHY

2012 ◽  
Vol 22 (05) ◽  
pp. 1150026 ◽  
Author(s):  
LUCA RONDI ◽  
FADIL SANTOSA
Keyword(s):  
Inverse Problem ◽  
Variational Formulation ◽  
Diffractive Optics ◽  
Shape Design ◽  
Two Dimensional ◽  
Convergence Properties ◽  
Approximate Problem ◽  
The Relationship ◽  
Well Posed ◽  
Dimensional Domain

We consider the inverse problem of determining an optical mask that produces a desired circuit pattern in photolithography. We set the problem as a shape design problem in which the unknown is a two-dimensional domain. The relationship between the target shape and the unknown is modeled through diffractive optics. We develop a variational formulation that is well-posed and propose an approximation that can be shown to have convergence properties. The approximate problem can serve as a foundation to numerical methods, much like the Ambrosio–Tortorelli's approximation of the Mumford–Shah functional in image processing.

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