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## RAPID analysis of variable stiffness beams and plates: Legendre polynomial triple-product formulation

Research output: Contribution to journalArticle

### Standard

In: International Journal for Numerical Methods in Engineering, Vol. 112, No. 1, 05.10.2017, p. 86-100.

Research output: Contribution to journalArticle

### Author

O'Donnell, Matt ; Weaver, Paul. / RAPID analysis of variable stiffness beams and plates : Legendre polynomial triple-product formulation. In: International Journal for Numerical Methods in Engineering. 2017 ; Vol. 112, No. 1. pp. 86-100.

### Bibtex

@article{bc150f3807e74a81b736e942b1c24ba9,
title = "RAPID analysis of variable stiffness beams and plates: Legendre polynomial triple-product formulation",
abstract = "Numerical integration techniques are commonly employed to formulate the system matrices encountered in the analysis of variable stiffness beams and plates using a Ritz based approach. Computing these integrals accurately is often computationally costly. Herein, a novel alternative is presented, the Recursive Analytical Polynomial Integral Definition (RAPID) formulation. The RAPID formulation offers a significant improvement in the speed of analysis, achieved by reducing the number of numerical integrations that are performed by an order of magnitude. A common Legendre Polynomial (LP) basis is employed for both trial functions and stiffness/load variations leading to a common form for the integrals encountered. The LP basis possesses algebraic recursion relations that allow these integrals to be reformulated as triple-products with known analytical solutions, defined compactly using the Wigner (3j) coefficient. The satisfaction of boundary conditions, calculation of derivatives, and transformation to other bases is achieved through combinations of matrix multiplication, with each matrix representing a unique boundary condition or physical effect, therefore permitting application of the RAPID approach to a variety of problems. Indicative performance studies demonstrate the advantage of the RAPID formulation when compared to direct analysis using MATLAB’s “integral” and “integral2”.",
keywords = "Structures, Composites, Integration",
author = "Matt O'Donnell and Paul Weaver",
year = "2017",
month = "10",
day = "5",
doi = "10.1002/nme.5528",
language = "English",
volume = "112",
pages = "86--100",
journal = "International Journal for Numerical Methods in Engineering",
issn = "0029-5981",
publisher = "John Wiley & Sons, Inc",
number = "1",

}

### RIS - suitable for import to EndNote

TY - JOUR

T1 - RAPID analysis of variable stiffness beams and plates

T2 - International Journal for Numerical Methods in Engineering

AU - O'Donnell, Matt

AU - Weaver, Paul

PY - 2017/10/5

Y1 - 2017/10/5

N2 - Numerical integration techniques are commonly employed to formulate the system matrices encountered in the analysis of variable stiffness beams and plates using a Ritz based approach. Computing these integrals accurately is often computationally costly. Herein, a novel alternative is presented, the Recursive Analytical Polynomial Integral Definition (RAPID) formulation. The RAPID formulation offers a significant improvement in the speed of analysis, achieved by reducing the number of numerical integrations that are performed by an order of magnitude. A common Legendre Polynomial (LP) basis is employed for both trial functions and stiffness/load variations leading to a common form for the integrals encountered. The LP basis possesses algebraic recursion relations that allow these integrals to be reformulated as triple-products with known analytical solutions, defined compactly using the Wigner (3j) coefficient. The satisfaction of boundary conditions, calculation of derivatives, and transformation to other bases is achieved through combinations of matrix multiplication, with each matrix representing a unique boundary condition or physical effect, therefore permitting application of the RAPID approach to a variety of problems. Indicative performance studies demonstrate the advantage of the RAPID formulation when compared to direct analysis using MATLAB’s “integral” and “integral2”.

AB - Numerical integration techniques are commonly employed to formulate the system matrices encountered in the analysis of variable stiffness beams and plates using a Ritz based approach. Computing these integrals accurately is often computationally costly. Herein, a novel alternative is presented, the Recursive Analytical Polynomial Integral Definition (RAPID) formulation. The RAPID formulation offers a significant improvement in the speed of analysis, achieved by reducing the number of numerical integrations that are performed by an order of magnitude. A common Legendre Polynomial (LP) basis is employed for both trial functions and stiffness/load variations leading to a common form for the integrals encountered. The LP basis possesses algebraic recursion relations that allow these integrals to be reformulated as triple-products with known analytical solutions, defined compactly using the Wigner (3j) coefficient. The satisfaction of boundary conditions, calculation of derivatives, and transformation to other bases is achieved through combinations of matrix multiplication, with each matrix representing a unique boundary condition or physical effect, therefore permitting application of the RAPID approach to a variety of problems. Indicative performance studies demonstrate the advantage of the RAPID formulation when compared to direct analysis using MATLAB’s “integral” and “integral2”.

KW - Structures

KW - Composites

KW - Integration

U2 - 10.1002/nme.5528

DO - 10.1002/nme.5528

M3 - Article

VL - 112

SP - 86

EP - 100

JO - International Journal for Numerical Methods in Engineering

JF - International Journal for Numerical Methods in Engineering

SN - 0029-5981

IS - 1

ER -