International Journal for Numerical Methods in Engineering, 2019
We consider numerical solutions of nonlinear oscillatory systems where closed-form solutions do n... more We consider numerical solutions of nonlinear oscillatory systems where closed-form solutions do not exist. Such systems occur in buckling of columns, electrical oscillations of circuits containing inductance with an iron core, and vibration of mechanical systems with nonlinear restoring forces. We have improved the accuracy, stabilty and speed of the generalised Newmark scheme of Zienkiewicz and Taylor by using one-step multiple-value algorithms. Previously, a Laplace transform-based method was developed to determine initial conditions of higher order derivatives of predictor algorithms together with the corrector algorithms cast in form of higherorder Newton-Raphson schemes, which are considered as consistent tangent operators that preserve quadratic rate of asymptotic convergence characteristics. For convenience, algorithms are applied to the solution of the van der Pol and Duffing's equations in order to show the concepts, but the procedures can also be applied to other systems such as the Bouc's, Coulomb's and Mathieu equations. Comparisons are carried out regarding speed, accuracy and stability, of results from the Runge-Kutta and one-step multiple-value methods with remainder (truncation error) terms, previously not considered. Results compare favourably. A system of two-degrees of freedom is also covered to illustrate how to extend the methods to deal with multiple-degree of freedom systems.
One-step multiple-value methods are developed which involve an accurate predictor method with hig... more One-step multiple-value methods are developed which involve an accurate predictor method with higher derivatives, followed by a corrector method cast in form of an enhanced Newton-Raphson scheme. The generalized Newmark (GNpj) method may be recovered as a special case. The algorithms serve to match the accuracy of the fourth-order Runge-Kutta-Fehlberg method. Challenges to solve more reliably, accurately and efficiently non-linear differential equations are highlighted as stemming from amplitude and phase shift errors introduced by discretization in space and time-a continuous-discrete transformation. The classical stability tool of spectral radius is performed on linear systems whereas Liapunov method on nonlinear systems.
We consider numerical solutions of nonlinear oscillatory systems where closed-form solutions do n... more We consider numerical solutions of nonlinear oscillatory systems where closed-form solutions do not exist. Such systems occur in buckling of columns, electrical oscillations of circuits containing inductance with an iron core, and vibration of mechanical systems with nonlinear restoring forces. We have improved the accuracy, stability, and speed of the generalized Newmark scheme of Zienkiewicz and Taylor using one-step multiple-value algorithms. Previously, a Laplace transform-based method was developed to determine initial conditions of higher order derivatives of predictor algorithms together with the corrector algorithms cast in form of higher order Newton-Raphson schemes, which are considered as consistent tangent operators that preserve quadratic rate of asymptotic convergence characteristics. For convenience, algorithms are applied to the solution of the van der Pol and Duffing's equations to show the concepts, but the procedures can also be applied to other systems such as the Bouc's, Coulomb's, and Mathieu equations. Comparisons are carried out regarding speed, accuracy, and stability, of results from the Runge-Kutta and one-step multiple-value methods with remainder (truncation error) terms, previously not considered. Results compare favorably. A system of two-degrees of freedom is also covered to illustrate how to extend the methods to deal with multiple-degree of freedom systems.
International Journal for Numerical Methods in Engineering, 2019
We consider numerical solutions of nonlinear oscillatory systems where closed-form solutions do n... more We consider numerical solutions of nonlinear oscillatory systems where closed-form solutions do not exist. Such systems occur in buckling of columns, electrical oscillations of circuits containing inductance with an iron core, and vibration of mechanical systems with nonlinear restoring forces. We have improved the accuracy, stabilty and speed of the generalised Newmark scheme of Zienkiewicz and Taylor by using one-step multiple-value algorithms. Previously, a Laplace transform-based method was developed to determine initial conditions of higher order derivatives of predictor algorithms together with the corrector algorithms cast in form of higherorder Newton-Raphson schemes, which are considered as consistent tangent operators that preserve quadratic rate of asymptotic convergence characteristics. For convenience, algorithms are applied to the solution of the van der Pol and Duffing's equations in order to show the concepts, but the procedures can also be applied to other systems such as the Bouc's, Coulomb's and Mathieu equations. Comparisons are carried out regarding speed, accuracy and stability, of results from the Runge-Kutta and one-step multiple-value methods with remainder (truncation error) terms, previously not considered. Results compare favourably. A system of two-degrees of freedom is also covered to illustrate how to extend the methods to deal with multiple-degree of freedom systems.
One-step multiple-value methods are developed which involve an accurate predictor method with hig... more One-step multiple-value methods are developed which involve an accurate predictor method with higher derivatives, followed by a corrector method cast in form of an enhanced Newton-Raphson scheme. The generalized Newmark (GNpj) method may be recovered as a special case. The algorithms serve to match the accuracy of the fourth-order Runge-Kutta-Fehlberg method. Challenges to solve more reliably, accurately and efficiently non-linear differential equations are highlighted as stemming from amplitude and phase shift errors introduced by discretization in space and time-a continuous-discrete transformation. The classical stability tool of spectral radius is performed on linear systems whereas Liapunov method on nonlinear systems.
We consider numerical solutions of nonlinear oscillatory systems where closed-form solutions do n... more We consider numerical solutions of nonlinear oscillatory systems where closed-form solutions do not exist. Such systems occur in buckling of columns, electrical oscillations of circuits containing inductance with an iron core, and vibration of mechanical systems with nonlinear restoring forces. We have improved the accuracy, stability, and speed of the generalized Newmark scheme of Zienkiewicz and Taylor using one-step multiple-value algorithms. Previously, a Laplace transform-based method was developed to determine initial conditions of higher order derivatives of predictor algorithms together with the corrector algorithms cast in form of higher order Newton-Raphson schemes, which are considered as consistent tangent operators that preserve quadratic rate of asymptotic convergence characteristics. For convenience, algorithms are applied to the solution of the van der Pol and Duffing's equations to show the concepts, but the procedures can also be applied to other systems such as the Bouc's, Coulomb's, and Mathieu equations. Comparisons are carried out regarding speed, accuracy, and stability, of results from the Runge-Kutta and one-step multiple-value methods with remainder (truncation error) terms, previously not considered. Results compare favorably. A system of two-degrees of freedom is also covered to illustrate how to extend the methods to deal with multiple-degree of freedom systems.
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