scholarly journals T-systems with Boundaries from Network Solutions

10.37236/2645 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Philippe Di Francesco ◽  
Rinat Kedem
Keyword(s):  
Initial Data ◽  
Boundary Conditions ◽  
Cluster Algebra ◽  
Simple Explanation ◽  
Combinatorial Interpretation ◽  
Network Solution ◽  
Various Boundary Conditions ◽  
Periodicity Property ◽  
T System

In this paper, we use the network solution of the $A_r$ $T$-system to derive that of the unrestricted $A_\infty$ $T$-system, equivalent to the octahedron relation.  We then present a method for implementing various boundary conditions on this system, which consists of picking initial data with suitable symmetries. The corresponding restricted $T$-systems are solved exactly in terms of networks. This gives a simple explanation for phenomena such as the Zamolodchikov periodicity property for $T$-systems (corresponding to the case $A_\ell\times A_r$) and a combinatorial interpretation for the positive Laurent property for the variables of the associated cluster algebra. We also explain the relation between the $T$-system wrapped on a torus and the higher pentagram maps of Gekhtman et al.

scholarly journals The Solution of the $A_r$ T-System for Arbitrary Boundary

10.37236/361 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Philippe Di Francesco
Keyword(s):  
Partition Function ◽  
Initial Data ◽  
Boundary Conditions ◽  
Explicit Solution ◽  
Negative Integer ◽  
Laurent Polynomials ◽  
Arbitrary Boundary ◽  
Arbitrary Boundary Conditions ◽  
Integer Coefficients ◽  
T System

We present an explicit solution of the $A_r$ $T$-system for arbitrary boundary conditions. For each boundary, this is done by constructing a network, i.e. a graph with positively weighted edges, and the solution is expressed as the partition function for a family of non-intersecting paths on the network. This proves in particular the positive Laurent property, namely that the solutions are all Laurent polynomials of the initial data with non-negative integer coefficients.

Upon Impact Numerical Modeling of Foam Materials

2017 ◽  
Vol 54 (2) ◽  
pp. 195-202
Author(s):  
Vasile Nastasescu ◽  
Silvia Marzavan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Modeling ◽  
Numerical Analysis ◽  
Numerical Methods ◽  
Boundary Conditions ◽  
General Character ◽  
Foam Material ◽  
Various Boundary Conditions ◽  
Specific Behaviour

The paper presents some theoretical and practical issues, particularly useful to users of numerical methods, especially finite element method for the behaviour modelling of the foam materials. Given the characteristics of specific behaviour of the foam materials, the requirement which has to be taken into consideration is the compression, inclusive impact with bodies more rigid then a foam material, when this is used alone or in combination with other materials in the form of composite laminated with various boundary conditions. The results and conclusions presented in this paper are the results of our investigations in the field and relates to the use of LS-Dyna program, but many observations, findings and conclusions, have a general character, valid for use of any numerical analysis by FEM programs.

The Quasi-Static Analysis for Two-Dimensional Thin Plates

2011 ◽  
Vol 255-260 ◽  
pp. 166-169
Author(s):  
Li Chen ◽  
Yang Bai
Keyword(s):  
Boundary Conditions ◽  
Eigenfunction Expansion ◽  
Solution Space ◽  
Expansion Method ◽  
Fundamental Solutions ◽  
Thin Plates ◽  
Elastic Plates ◽  
Various Boundary Conditions ◽  
Eigenfunction Expansion Method ◽  
Concentration Phenomena

The eigenfunction expansion method is introduced into the numerical calculations of elastic plates. Based on the variational method, all the fundamental solutions of the governing equations are obtained directly. Using eigenfunction expansion method, various boundary conditions can be conveniently described by the combination of the eigenfunctions due to the completeness of the solution space. The coefficients of the combination are determined by the boundary conditions. In the numerical example, the stress concentration phenomena produced by the restriction of displacement conditions is discussed in detail.

Experimental and Theoretical Nonlinear Dynamic Response of Intact and Cracked Bone-Like Specimens With Various Boundary Conditions

2007 ◽  
Vol 129 (5) ◽  
pp. 541-549 ◽  
Author(s):  
Erick Ogam ◽  
Armand Wirgin ◽  
Z. E. A. Fellah ◽  
Yongzhi Xu
Keyword(s):  
Boundary Conditions ◽  
Duffing Oscillator ◽  
Contact Friction ◽  
Nonlinear Dynamic Response ◽  
Various Boundary Conditions ◽  
Vibration Data ◽  
Element Simulation ◽  
Resonance Frequencies ◽  
Vibration Experiment ◽  
Freely Suspended

The potentiality of employing nonlinear vibrations as a method for the detection of osteoporosis in human bones is assessed. We show that if the boundary conditions (BC), relative to the connection of the specimen to its surroundings, are not taken into account, the method is apparently unable to differentiate between defects (whose detection is the purpose of the method) and nonrelevant features (related to the boundary conditions). A simple nonlinear vibration experiment is described which employs piezoelectric transducers (PZT) and two idealized long bones in the form of nominally-identical drinking glasses, one intact, but in friction contact with a support, and the second cracked, but freely-suspended in air. The nonlinear dynamics of these specimens is described by the Duffing oscillator model. The nonlinear parameters recovered from vibration data coupled to the linear phenomena of mode splitting and shifting of resonance frequencies, show that, despite the similar soft spring behavior of the two dynamic systems, a crack is distinguishable from a contact friction BC. The frequency response of the intact glass with contact friction BC is modeled using a direct steady state finite element simulation with contact friction.

Static Analysis of Moderately Thick Functionally Graded Plates With Various Boundary Conditions

2010 ◽  
Author(s):  
Nastaran Shahmansouri ◽  
Mohammad Mohammadi Aghdam ◽  
Kasra Bigdeli
Keyword(s):  
Boundary Conditions ◽  
Volume Fraction ◽  
Closed Form Solution ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Rectangular Plates ◽  
Functionally Graded Plates ◽  
Various Boundary Conditions ◽  
Fg Plates ◽  
Ode Systems

The present study investigates static analyses of moderately thick FG plates. Using the First Order Shear Deformation Theory (FSDT), functionally graded plates subjected to transversely distributed loading with various boundary conditions are studied. Effective mechanical properties which vary from one surface of the plate to the other assumed to be defined by a power law form of distribution. Different ceramic-metal sets of materials are studied. Solution of the governing equations, including five equilibrium and eight constitutive equations, is obtained by the Extended Kantorovich Method (EKM). The system of thirteen Partial Differential Equations (PDEs) in terms of displacements, rotations, force and moment resultants are considered as multiplications of separable function of independent variables x and y. Then by successful utilization of the EKM these equations are converted to a double set of ODE systems in terms of x and y. The obtained ODE systems are then solved iteratively until final convergence is achieved. Closed form solution is presented for these ODE sets. It is shown that the method is very stable and provides fast convergence and highly accurate predictions for both thin and moderately thick plates. Comparison of the normal stresses at various points of rectangular plates and deflection of mid-point of the plate are presented and compared with available data in the literature. The effects of the volume fraction exponent n on the behavior of the normalized deflection, moment resultants and stresses of FG plates are also studied. To validate data for analysis fully clamped FG plates, another analysis was carried out using finite element code ANSYS. Close agreement is observed between predictions of the EKM and ANSYS.

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