scholarly journals On the Energy Scaling Behaviour of a Singularly Perturbed Tartar Square

2021 ◽  
Author(s):  
Angkana Rüland ◽  
Antonio Tribuzio
Keyword(s):  
Sobolev Space ◽  
Upper Bound ◽  
Scaling Law ◽  
Approximate Solutions ◽  
Bootstrap Argument ◽  
Perturbation Problem ◽  
Energy Scaling ◽  
A Chain ◽  
Negative Sobolev Space ◽  
Numer Math

AbstractIn this article we derive an (almost) optimal scaling law for a singular perturbation problem associated with the Tartar square. As in Winter (Eur J Appl Math 8(2):185–207, 1997), Chipot (Numer Math 83(3):325–352, 1999), our upper bound quantifies the well-known construction which is used in the literature to prove the flexibility of the Tartar square in the sense of the flexibility of approximate solutions to the differential inclusion. The main novelty of our article is the derivation of an (up to logarithmic powers matching) ansatz free lower bound which relies on a bootstrap argument in Fourier space and is related to a quantification of the interaction of a nonlinearity and a negative Sobolev space in the form of “a chain rule in a negative Sobolev space”. Both the lower and the upper bound arguments give evidence of the involved “infinite order of lamination”.

scholarly journals On the explosion of chain-thermal reactions

1982 ◽  
Vol 24 (2) ◽  
pp. 194-202 ◽  
Author(s):  
R. O. Ayeni
Keyword(s):  
Activation Energy ◽  
Asymptotic Analysis ◽  
Upper Bound ◽  
Homogeneous Model ◽  
Chain Reaction ◽  
Isothermal Reaction ◽  
Thermal Reactions ◽  
Spatially Homogeneous ◽  
Branched Chain ◽  
A Chain

AbstractA chain reaction of oxygen (reactant) and hydrogen (active intermediary) with mtrosyl chloride (sensitizer) as a catalyst may be modelled mathematically as a non-isothermal reaction. In this paper we present an asymptotic analysis of a spatially homogeneous model of a non-isothermal branched-chain reaction. Of particular interest is the so-called explosion time and we provide an upper bound for it as a function of the activation energy which can vary over all positive values. We also establish a bound on the temperature when the activation energy is finite.

A new edge-based energy scaling law for W7-AS and its implications for boundary-island divertor operation

2007 ◽  
Vol 363-365 ◽  
pp. 389-394 ◽  
Author(s):  
K. McCormick ◽  
P. Grigull ◽  
H. Ehmler ◽  
E. Pasch
Keyword(s):  
Scaling Law ◽  
Energy Scaling ◽  
Edge Based

scholarly journals TOP EVALUATION FOR THE RA TION FOR THE RATE OF FUNC TE OF FUNCTIONAL OF ERROR AL OF ERROR WEIGHT CUBATURE FORMUL TURE FORMULA IN SP A IN SPACE

2019 ◽  
Vol 3 (4) ◽  
pp. 32-37
Author(s):  
Ozodjon Isomidinovich Jalolov ◽  
◽  
Khurshidzhon Usmanovich Khayatov
Keyword(s):  
Sobolev Space ◽  
Upper Bound ◽  
Cubature Formula ◽  
Integration Formulas ◽  
Error Functional ◽  
The Error Functional ◽  
Error Weight ◽  
Norm Of The Error ◽  
Approximate Integration

An upper bound is obtained for the norm of the error functional of the weight cubature formula in the Sobolev space . The modern formulation of the problem of optimization of approximate integration formulas is to minimize the norm of the error functional of the formula on the selected normalized spaces. In these works, the problem of optimality with respect to some definite space is investigated. Most of the problems are considered in the Sobolev space

AN IMPROVED REDUCTION METHOD FOR THE ROBUST OPTIMIZATION OF THE ASSIGNMENT PROBLEM

2012 ◽  
Vol 29 (05) ◽  
pp. 1250032 ◽  
Author(s):  
BYUNGJUN YOU ◽  
DAISUKE YOKOYA ◽  
TAKEO YAMADA
Keyword(s):  
Lower Bound ◽  
Exact Solutions ◽  
Upper Bound ◽  
Assignment Problem ◽  
Numerical Experiments ◽  
Reduction Method ◽  
Computation Time ◽  
Approximate Solutions ◽  
Cpu Time ◽  
Surrogate Relaxation

We are concerned with a variation of the assignment problem, where the assignment costs differ under different scenarios. We give a surrogate relaxation approach to derive a lower bound and an upper bound quickly, and show that the pegging test known for zero–one programming problems is also applicable to this problem. Next, we discuss how the computation time for pegging can be shortened by taking the special structure of the assignment problem into account. Finally, through numerical experiments we show that the developed method finds exact solutions for instances with small number of scenarios in relatively small CPU time, and good approximate solutions in case of many scenarios.

The Mechanics of the Coining Process

1962 ◽  
Vol 84 (4) ◽  
pp. 491-501 ◽  
Author(s):  
Y. Bocharov ◽  
S. Kobayashi ◽  
E. G. Thomsen
Keyword(s):  
Upper Bound ◽  
Slip Line ◽  
Local Stress ◽  
Approximate Solutions ◽  
Pure Lead ◽  
Strip Method ◽  
State Of Stress ◽  
Methods Of Analysis ◽  
Method Of Solution ◽  
The Coefficient Of Friction

The mechanism of coining is analyzed and several approximate solutions are given. The solutions were based on the strip, slip-line, and upper-bound methods of analysis. Comparison of the solutions for the local stress distribution could only be made for the strip and slip-line methods of analysis and it was found that the particular solutions obtained agreed reasonably well with each other in predicting coining pressures as functions of degree of coining. A comparison of the strip and upper-bound method of solution revealed that the predicted average coining pressures for a single central square groove are nearly identical for blanks having thicknesses of h0 = 1/16 in. for all values of b/b0 (degree of coining) except when b/b0 approaches unity. At b/b0 = 1 (deg of coining = 100 per cent), the modified strip method of analysis, using a cylindrical state of stress in the square corner of the groove, gave the highest average coining pressures. The upper-bound solution seems to overestimate the forging pressures for h0 > 1/16 in. for all values of b/b0, except when b/b0 is unity. A comparison of experimental average pressures, for coining 1-in-diameter blanks of commercially pure lead and aluminum, with the theoretical solutions, revealed that the modified strip method appears to be the best method for predicting the pressures when the ratio of b/b0 approaches unity. However, the exact pressure at b/b0 = 1 is indeterminable by use of this method. The analyses presented in the paper and comparisons of the solutions with experimental data reveal that a lowering of the coefficient of friction has a profound influence on decreasing the required coining pressures. The analyses further show that for high friction the pressure required for getting exact definition rises rapidly as the ratio b/b0 approaches unity.

Extension of the Upper Bound Method to Include Estimation of Stresses

1991 ◽  
Vol 58 (2) ◽  
pp. 493-498 ◽  
Author(s):  
A. Azarkhin ◽  
O. Richmond
Keyword(s):  
Plastic Deformation ◽  
Stress Field ◽  
Upper Bound ◽  
Plastic Material ◽  
Approximate Solutions ◽  
Incompressible Material ◽  
Bulk Deformation ◽  
Upper Bound Method ◽  
Convenient Tool ◽  
Perfectly Plastic

The upper bound method is a convenient tool for evaluating the rate of work in processes involving predominantly plastic deformation of rigid/perfectly plastic material. Since the rate of work for an incompressible material depends only on the deviator portion of the stress, the hydrostatic portion does not enter the formulation and the stress field is not determined. Here we show that this limitation can be overcome by adding a relatively simple postprocessing procedure. We then apply this technique to examples of rigid asperities penetrating a plastic material undergoing subsurface bulk deformation and compare our results with previous approximate solutions.

scholarly journals Laser Energy Scaling Law for the Yield of Neutrons Generated by Intense Femtosecond Laser-Cluster Interactions

2009 ◽  
Vol 4 ◽  
pp. 041-041 ◽  
Author(s):  
Shuji SAKABE ◽  
Masaki HASHIDA ◽  
Shigeki TOKITA ◽  
Kazuto OTANI
Keyword(s):  
Femtosecond Laser ◽  
Laser Energy ◽  
Scaling Law ◽  
Energy Scaling
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