scholarly journals Recursion relation for instanton counting for SU(2) $$ \mathcal{N} $$ = 2 SYM in NS limit of Ω background

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Hasmik Poghosyan
Keyword(s):  
Numerical Method ◽  
Recursion Relation ◽  
Numerical Approach ◽  
Higher Order ◽  
Cycle Period ◽  
Recursive Method ◽  
Higher Order Terms ◽  
Pure Case ◽  
Analytic Expression ◽  
Standard Techniques

Abstract In this paper we investigate different ways of deriving the A-cycle period as a series in instanton counting parameter q for $$ \mathcal{N} $$ N = 2 SYM with up to four antifundamental hypermultiplets in NS limit of Ω background. We propose a new recursive method for calculating the period and demonstrate its efficiency by explicit calculations. The new way of doing instanton counting is more advantageous compared to known standard techniques and allows to reach substantially higher order terms with less effort. This approach is applied for the pure case as well as for the case with several hypermultiplets.In addition we suggest a numerical method for deriving the A-cycle period for arbitrary values of q. In the case when one has no hypermultiplets for the A-cycle an analytic expression for large q asymptotics is obtained using a conjecture by Alexei Zamolodchikov. We demonstrate that this expression is in convincing agreement with the numerical approach.

Using the Multi-Parameter Fracture Mechanics for more Accurate Description of Stress/Displacement Crack Tip Fields

2013 ◽  
Vol 586 ◽  
pp. 237-240 ◽  
Author(s):  
Lucie Šestáková
Keyword(s):  
Stress Distribution ◽  
Fracture Mechanics ◽  
Boundary Conditions ◽  
Displacement Field ◽  
Singular Term ◽  
Crack Tip ◽  
Higher Order ◽  
Precise Description ◽  
Accurate Knowledge ◽  
Higher Order Terms

Most of fracture analyses often require an accurate knowledge of the stress/displacement field over the investigated body. However, this can be sometimes problematic when only one (singular) term of the Williams expansion is considered. Therefore, also other terms should be taken into account. Such an approach, referred to as multi-parameter fracture mechanics is used and investigated in this paper. Its importance for short/long cracks and the influence of different boundary conditions are studied. It has been found out that higher-order terms of the Williams expansion can contribute to more precise description of the stress distribution near the crack tip especially for long cracks. Unfortunately, the dependences obtained from the analyses presented are not unambiguous and it cannot be strictly derived how many of the higher-order terms are sufficient.

The Skyrme model and higher order terms

1990 ◽  
Vol 235 (1-2) ◽  
pp. 141-146 ◽  
Author(s):  
Luc Marleau
Keyword(s):  
Higher Order ◽  
Skyrme Model ◽  
Higher Order Terms

scholarly journals Experimental determination and finite element analysis of coefficients of the multi-parameter Williams series expansion in the vicinity of the crack tip in linear elastic materials. Part I

2020 ◽  
pp. 237-249
Author(s):  
L. V Stepanova
Keyword(s):  
Series Expansion ◽  
Digital Image ◽  
Power Series Expansion ◽  
Crack Tip ◽  
Higher Order ◽  
Accurate Estimation ◽  
Principal Stresses ◽  
Isochromatic Fringe ◽  
Higher Order Terms ◽  
Fringe Patterns

This study aims at obtaining coefficients of the multi-parameter Williams series expansion for the stress field in the vicinity of the central crack in the rectangular plate and in the semi-circular notched disk under bending by the use of the digital photoelasticity method. The higher-order terms in the Williams asymptotic expansion are retained. It allows us to give a more accurate estimation of the near-crack-tip stress, strain and displacement fields and extend the domain of validity for the Williams power series expansion. The program is specially developed for the interpretation and processing of experimental data from the phototelasticity experiments. By means of the developed tool, the fringe patterns that contain the whole field stress information in terms of the difference in principal stresses (isochromatics) are captured as a digital image, which is processed for quantitative evaluations. The developed tool allows us to find points that belong to isochromatic fringes with the minimal light intensity. The digital image processing with the aid of the developed tool is performed. The points determined with the adopted tool are used further for the calculations of the stress intensity factor, T-stresses and coefficients of higher-order terms in the Williams series expansion. The iterative procedure of the over-deterministic method is utilized to find the higher order terms of the Williams series expansion. The procedure is based on the consistent correction of the coefficients of the Williams series expansion. The first fifteen coefficients are obtained. The experimentally obtained coefficients are used for the reconstruction of the isochromatic fringe pattern in the vicinity of the crack tip. The comparison of the theoretically reconstructed and experimental isochromatic fringe patterns shows that the coefficients of the Williams series expansion have a good match.

Convergence Study on Application of the Over-Deterministic Method for Determination of Near-Tip Fields in a Cracked Plate Loaded in Mixed-Mode

2012 ◽  
Vol 249-250 ◽  
pp. 76-81 ◽  
Author(s):  
Lucie Šestáková ◽  
Václav Veselý
Keyword(s):  
Mixed Mode ◽  
Brittle Materials ◽  
Higher Order ◽  
Displacement Fields ◽  
Deterministic Method ◽  
Crack Behavior ◽  
Stress And Displacement Fields ◽  
Higher Order Terms ◽  
Convergence Study

Multi-parameter description of crack behavior in quasi-brittle materials offers still enough space for investigations. Several studies have been carried out by the authors in this field [1-3]. One part of the publications by the authors (this work included) contain analyses of the accuracy, convergence and/or tuning of the over-deterministic method that enables determination of the coefficients of the higher-order terms in Williams expansion approximating the stress and displacement fields in a cracked body without any complicated FE formulations. These intermediate studies should bring together a list of recommendations how to use the ODM as effectively as possible and obtain reliable enough values of coefficients of the higher-order terms. Thus, the stress/displacement field can be determined precisely even in a larger distance from the crack tip, which is crucial for assessment of the fracture occurring in quasi-brittle materials.

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