The solution of nonlinear direct and inverse problems for beam by means of the Trefftz functions

Mathematical modelling makes it possible to analyse physical phenomena which are frequently described by non-linear differential equations. Those equations can be solved by a number of numerical methods, but not all are efficient. The paper presents an approximate solution of the nonlinear problems...

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Veröffentlicht in:European journal of mechanics, A, Solids Jg. 92; S. 104476
Hauptverfasser: Maciag, Artur, Pawinska, Anna
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin Elsevier Masson SAS 01.03.2022
Elsevier BV
Schlagworte:
Beam vibration
Inhomogeneity
Inverse problems
Iterative methods
Linear operators
Mathematical analysis
Nonlinear differential equations
Nonlinear partial differential equation
Nonlinear vibrations
Nonlinearity
Numerical methods
Operators (mathematics)
Picard’s iterations
Trefftz functions
Picard’s iterations
Nonlinear partial differential equation
Nonlinear vibrations
Beam vibration
Trefftz functions
ISSN:0997-7538, 1873-7285
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Zusammenfassung:Mathematical modelling makes it possible to analyse physical phenomena which are frequently described by non-linear differential equations. Those equations can be solved by a number of numerical methods, but not all are efficient. The paper presents an approximate solution of the nonlinear problems for beam vibrations. The main aim of this work is to develop an efficient method for solving inverse nonlinear and time-dependent problems. The main idea of this method is a decomposition of a nonlinear operator into a linear and nonlinear part. Next, the Picard’s iterations are used. In each iteration a linear combination of the Trefftz functions for the linear operator is used. The nonlinearity is treated as an inhomogeneity for the linear operator. Hence, in each iteration a linear inhomogeneous equation is solved. The presented numerical examples confirm efficiency of the method in finding a stable solution of inverse nonlinear problems, even with noisy data. •An approximate solution of the nonlinear problems for beam vibrations was found.•Picard’s iterations with the Trefftz functions for solving the nonlinear direct and inverse problems were used.•The sensitivity of the solutions to the disturbances of input data was tested and proved to be low.•This is the first report describing Trefftz approach for non-linear non-stationary problems.
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ISSN:0997-7538
1873-7285
DOI:10.1016/j.euromechsol.2021.104476