Fundamental examples of the polynomial method

This paper evaluates eight different analytical procedures used in determining elastic stress intensity factors for gradient or nonlinear stress fields. From a fracture viewpoint, the main interest in this problem comes from the nuclear industry where the safety of the nuclear system is of concern. A fracture mechanics analysis is then required to demonstrate the vessel integrity under these postulated accident conditions. The geometry chosen for his study is that of a 10-in. thick flawed plate with nonuniform stress distribution through the thickness. Two loading conditions are evaluated, both nonlinear and both defined by polynomials. The assumed cracks are infinitely long surface defects. Eight methods are used to find the stress intensity factor: 1–maximum stress, 2–linear envelope, 3–linearization over the crack length from ASME Code, Section XI, 4–equivalent linear moment from ASME Code, Section III, Appendix G for thermal loadings, 5–integration method from WRC 175, Appendix 4 for thermal loadings, 6–8-node singularity (quarter-point) isoparametric element in conjunction with the displacement method, 7–polynomial method, and 8–semi-infinite edge crack linear distribution over crack. Comparisons are made between all eight procedures with the finding that the methods can be ranked in order of decreasing conservatism and ease of application as follows: 1–maximum stress, 2–linear envelope, 3–linearization over the crack length, 4–polynomial method, and 5–singularity element method. Good agreement is found between the last three of these methods. The remaining three methods produce nonconservative results.

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Optimization of coupled Lamb wave parameters for defect detection in anisotropic composite three-layer with Kelvin-Voigt viscoelasticity using Legendre polynomial method

Composite Structures ◽

10.1016/j.compstruct.2021.114158 ◽

2021 ◽

pp. 114158

Author(s):

Souhail Dahmen ◽

Christ Glorieux

Keyword(s):

Defect Detection ◽

Legendre Polynomial ◽

Lamb Wave ◽

Polynomial Method ◽

Wave Parameters ◽

Anisotropic Composite

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Importance of coefficient diagram in polynomial method

42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475) ◽

10.1109/cdc.2003.1271687 ◽

2004 ◽

Cited By ~ 58

Author(s):

S. Manabe

Keyword(s):

Polynomial Method

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General Polynomial Method to Design Discrete Regulators

Control Engineering: MATLAB Exercises - Advanced Textbooks in Control and Signal Processing ◽

10.1007/978-981-10-8321-1_15 ◽

2018 ◽

pp. 261-264

Author(s):

László Keviczky ◽

Ruth Bars ◽

Jenő Hetthéssy ◽

Csilla Bányász

Keyword(s):

Polynomial Method ◽

General Polynomial

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Reconfiguring Smart Structures Using Approximate Heteroclinic Connections in a Spring-Mass Model

Volume 1: Development and Characterization of Multifunctional Materials; Mechanics and Behavior of Active Materials; Modeling, Simulation and Control of Adaptive Systems ◽

10.1115/smasis2015-8819 ◽

2015 ◽

Cited By ~ 1

Author(s):

Jiaying Zhang ◽

Colin R. McInnes

Keyword(s):

Optimal Control ◽

Control Method ◽

Smart Structures ◽

Polynomial Method ◽

Reference Trajectory ◽

Sensors And Actuators ◽

Embedded Computing ◽

Equal Energy ◽

Optimal Control Method ◽

Heteroclinic Connection

Several new methods are proposed to reconfigure smart structures with embedded computing, sensors and actuators. These methods are based on heteroclinic connections between equal-energy unstable equilibria in an idealised spring-mass smart structure model. Transitions between equal-energy unstable (but actively controlled) equilibria are considered since in an ideal model zero net energy input is required, compared to transitions between stable equilibria across a potential barrier. Dynamical system theory is used firstly to identify sets of equal-energy unstable configurations in the model, and then to connect them through heteroclinic connection in the phase space numerically. However, it is difficult to obtain such heteroclinic connections numerically in complex dynamical systems, so an optimal control method is investigated to seek transitions between unstable equilibria, which approximate the ideal heteroclinic connection. The optimal control method is verified to be effective through comparison with the results of the exact heteroclinic connection. In addition, we explore the use of polynomials of varying order to approximate the heteroclinic connection, and then develop an inverse method to control the dynamics of the system to track the polynomial reference trajectory. It is found that high order polynomials can provide a good approximation to true heteroclinic connections and provide an efficient means of generating such trajectories. The polynomial method is envisaged as being computationally efficient to form the basis for real-time reconfiguration of real, complex smart structures with embedded computing, sensors and actuators.

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