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International Journal for Numerical Methods in Engineering
Article . 2017 . Peer-reviewed
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zbMATH Open
Article . 2018
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On the equivalence of dynamic relaxation and the Newton‐Raphson method

On the equivalence of dynamic relaxation and the Newton-Raphson method
Authors: Jef Rombouts; Geert Lombaert; Lars De Laet; Mattias Schevenels;

On the equivalence of dynamic relaxation and the Newton‐Raphson method

Abstract

SummaryDynamic relaxation is an iterative method to solve nonlinear systems of equations, which is frequently used for form finding and analysis of structures that undergo large displacements. It is based on the solution of a fictitious dynamic problem where the vibrations of the structure are traced by means of a time integration scheme until a static equilibrium is reached. Fictitious values are used for the mass and damping parameters. Heuristic rules exist to determine these values in such a way that the time integration procedure converges rapidly without becoming unstable. Central to these heuristic rules is the assumption that the highest convergence rate is achieved when the ratio of the highest and lowest eigenfrequency of the structure is minimal. This short communication shows that all eigenfrequencies become identical when a fictitious mass matrix proportional to the stiffness matrix is used. If, in addition, specific values are used for the fictitious damping parameters and the time integration step, the dynamic relaxation method becomes completely equivalent to the Newton‐Raphson method. The Newton‐Raphson method can therefore be regarded as a specific form of dynamic relaxation. This insight may help to interpret and improve nonlinear solvers based on dynamic relaxation and/or the Newton‐Raphson method.

Country
Belgium
Keywords

Mathematics, Interdisciplinary Applications, Technology, Engineering, Multidisciplinary, COMPREHENSIVE EVALUATION, explicit time integration, Newton methods, 09 Engineering, implicit time integration, Engineering, finite element methods, Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs, FORMULATION, 40 Engineering, Science & Technology, Applied Mathematics, FINITE-ELEMENT EQUATIONS, stability, nonlinear solvers, numerical stability, Physical Sciences, Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs, Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs, Stability, Mathematics

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
13
Top 10%
Top 10%
Top 10%
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