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A unified set of single step algorithms part 3: The beta‐ m method, a generalization of the Newmark scheme
Author(s) -
Katona Michael C.,
Zienkiewicz O. C.
Publication year - 1985
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620210713
Subject(s) - generalization , algorithm , stability (learning theory) , newmark beta method , mathematics , scheme (mathematics) , simple (philosophy) , set (abstract data type) , series (stratigraphy) , numerical integration , computer science , mathematical analysis , finite element method , structural engineering , machine learning , engineering , biology , programming language , paleontology , philosophy , epistemology
Introduced herein is a generalization of Newmark's time marching integration scheme, called the β‐ m method. Like the SS pj method (introduced in Parts 1 and 2 of this series), the β‐ m method provides a gcneral single‐step scheme applicable to any set of initial value problems. The method is specialized by specifying the method order m along with rn integration parameters, β0, β1, …, β m −1 . For a particular choice of m , the integration parameters provide a subfamily of methods which control accuracy and stability, as well as offering options for explicit or implicit algorithms. For the most part, attention is focused on the application to structural dynamic equations. Most well‐known methods (e.g. Newmark, Wilson, Houbolt, etc.) are shown to be special cases within the β‐ m family. Hence, one computationally efficient and surprisingly simple algorithm unifies old and new methods. Stability and accuracy analyses are presented for method orders m = 2, 3 and 4 to determine optimal parameters for implicit and explicit schemes, along with numerical verification.