Fourth-Order Function Synthesizers

2013 ◽  
pp. 157-193
Author(s):  
Cosmin Radu Popa
Keyword(s):  
Fourth Order ◽  
Order Function

An optimized constitutive model for reproducing flow stress–strain relationships of anisotropic materials

2018 ◽  
Vol 233 (4) ◽  
pp. 1357-1368 ◽  
Author(s):  
Yanli Lin ◽  
Guannan Chu ◽  
Caiyuan Lin ◽  
Yongda Yan
Keyword(s):  
Flow Stress ◽  
Fourth Order ◽  
Anisotropic Materials ◽  
Second Order ◽  
Least Square ◽  
Experimental Stress ◽  
Function Model ◽  
Stress Strain ◽  
Order Function ◽  
Ordinary Least Square

Due to the strong anisotropic property of the advanced metal materials used in automobile, aviation, and aerospace, experimental flow stress–strain relations including different stress states are necessary to provide the information of anisotropic hardening and plastic flow for constructing a constitutive model. Therefore, reasonably reproducing the experimental stress–strain relations is the most fundamental work to substitute adequate flow stress–strain curves into the constitutive equation at the same time. However, accurate and stable regression results are difficult to obtain through the current regression models such as power exponent, second-order function model, fourth-order function model, and so forth. In this paper, an optimized model named as a least square quadratic regression model (ordinary least square model) was proposed based on the most useful second-order function model. The significant difference is that all experimental points are used to reproduce the experimental stress–strain relations in ordinary least square model in place of only three experimental points adopted in second-order function model, which results in good regression accuracy. Through comparison, it is found that the regression results by power function are poor with regard to some experimental results, and the results reproduced by second-order function model or fourth-order function model are very sensitive to the experimental points selected to do the regression. The sum of squares for error (SSE) increases sharply when the selected points are unreasonable. In addition, for second-order function and fourth-order function models, only limited experimental points are adopted to do the regression, the best regression accuracy cannot be obtained even if the selected points are reasonable. In contrast, SSE of the regression curve by ordinary least square model reduces to less than 50% of the best regressed result by second-order function model, the yielding behavior and variable strain increment ratio of the anisotropic materials can be reflected more accurately. This is very important for accurately describing the plastic flow behaviors of anisotropic materials.

Application of a Sixth Order Generalized Stress Function for Determining Limit Loads for Plates with Triangular Penetration Patterns

2002 ◽  
Author(s):  
J. L. Gordon ◽  
D. P. Jones
Keyword(s):  
Finite Element ◽  
Fourth Order ◽  
Limit Load ◽  
Sixth Order ◽  
Numerical Representation ◽  
Order Function ◽  
Practical Applications ◽  
Perfectly Plastic ◽  
Unit Cells ◽  
Elastic Perfectly Plastic

The capability to obtain limit load solutions of plates with triangular penetration patterns using fourth order functions to represent the collapse surface has been presented in previous papers. These papers describe how equivalent solid plate elastic-perfectly plastic finite element capabilities are generated and demonstrate how such capabilities can be used to great advantage in the analysis of tubesheets in large heat exchanger applications. However, these papers have pointed out that although the fourth order functions can produce sufficient accuracy for many practical applications, there are situations where improvements in the accuracy of in-plane and transverse shear are desirable. This paper investigates the use of a sixth order function to represent the collapse surface for improved accuracy of the in-plane response. Explicit elastic-perfectly plastic finite element solutions are obtained for unit cells representing an infinite array of circular penetrations arranged in an equilateral triangular array. These cells are used to create a numerical representation of the complete collapse surfaces for a number of ligament efficiencies (h/P where h is the minimum ligament width and P is the distance between hole centers). Each collapse surface is then fit to a sixth order function that satisfies the periodicity of the hole pattern. Sixth-order collapse functions were developed for h/P values between .05 and .50. Accuracy of the sixth order and the fourth order functions are compared. It was found that the sixth order function is indeed more accurate, reducing the error from 12.2% for the fourth order function to less than 3% for the sixth order function.

Fourth order intermolecular correlation functions from spectroscopic data and molecular dynamics calculations

1981 ◽  
Vol 59 (10) ◽  
pp. 1532-1543 ◽  
Author(s):  
J. van der Elsken ◽  
D. M. Heyes
Keyword(s):  
Molecular Dynamics ◽  
Fourth Order ◽  
Correlation Functions ◽  
Spectroscopic Data ◽  
Noble Gas ◽  
Time Dependent ◽  
Order Function ◽  
Connected Part ◽  
Dipolar Molecules ◽  
Molecular Dynamics Calculations

The shape of collision induced absorption spectra of noble gas mixtures and the broadening of rotational lines of dipolar molecules dissolved in noble gas fluids are described in terms of time dependent perturbation theory up to fourth order. In combination with second and fourth order intermolecular correlation functions obtained from molecular dynamics calculations, conclusions are made about the importance of these functions for spectroscopic observations. The fourth order correlation functions are highly structured. The form taken by the connected part of the fourth order function depends markedly on the range of the probe interaction.

scholarly journals A Collapse Surface for a Perforated Plate With an Equilateral Triangular Array of Penetrations

2002 ◽  
Vol 124 (2) ◽  
pp. 201-206 ◽  
Author(s):  
J. L. Gordon ◽  
D. P. Jones ◽  
D. Banas ◽  
D. N. Hutula
Keyword(s):  
Fourth Order ◽  
Cell Model ◽  
Three Dimensional ◽  
Perforated Plate ◽  
Limit Load ◽  
Triangular Array ◽  
Unit Cell ◽  
Element Analysis ◽  
Order Function ◽  
Perfectly Plastic

A collapse surface is developed for use in limit-load analysis of plates containing a large number of small circular penetrations arranged in an equilateral triangular array of holes with a ligament efficiency of 0.31733. The collapse surface is obtained by calculating the limit load for a unit cell model of the penetration pattern using a three-dimensional elastic-perfectly plastic [EPP] finite element analysis [FEA] computer program. The EPP response from incipient yielding to plastic collapse for the unit cell is obtained for a sufficient number of load cases to define the complete collapse surface. The collapse surface is expressed analytically by using a fourth-order function that incorporates the periodicity dictated by the triangular hole pattern. The coefficients of the fourth-order function were obtained by statistically fitting the collapse surface generated by the EPP-FEA results. The resulting collapse surface was shown to be appropriate for development of an EPP-EQS theory for perforated plates. The analytic surface agrees to within 7 percent of the actual collapse surface obtained by EPP-FEA of the unit cell representing the penetration.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

Exploring the Fourth Order

2019 ◽  
Vol 2 (1) ◽  
Author(s):  
Barend KLITSIE ◽  
Rebecca PRICE ◽  
Christine DE LILLE
Keyword(s):  
Action Research ◽  
Fourth Order ◽  
Subject Matter ◽  
Action Research Project ◽  
Research Project ◽  
Building Trust ◽  
Current Business

Companies are organised to fulfil two distinctive functions: efficient and resilient exploitation of current business and parallel exploration of new possibilities. For the latter, companies require strong organisational infrastructure such as team compositions and functional structures to ensure exploration remains effective. This paper explores the potential for designing organisational infrastructure to be part of fourth order subject matter. In particular, it explores how organisational infrastructure could be designed in the context of an exploratory unit, operating in a large heritage airline. This paper leverages insights from a long-term action research project and finds that building trust and shared frames are crucial to designing infrastructure that affords the greater explorative agenda of an organisation.

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