A Contribution to One-step Multiple-Value Methods for Computational Analysis of Problems in Non-Linear Dynamics

Kaunda, M.A.E.

doi:10.4203/ccp.108.109

Cited by 1 publication

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“…The simultaneous equations may be solved in conjunction with equations from balance laws of forces, momentum, and energy. The generalized Newmark scheme of Zienkiewicz et al is recognized as a case of the one‐step multiple‐value algorithms . Moreover, higher order differential equations may also be solved directly by the one‐step multiple‐value algorithms.…”

Section: One‐step Multiple‐value Algorithmsmentioning

confidence: 99%

“…Initial values for the one‐step multiple‐value methods were derived in Kaunda following important observations in Smith . The second‐order differential equation,

f (\dot{x}, x, t)

, is converted to a third‐order, fourth‐order, or fifth‐order (or even higher order) equivalent differential equation while holding the Jacobians ( J 11 , J 12 ) fixed over the interval of integration to match the Jacobian of the original equation, and maintaining the final value of the original differential equation.…”

Section: Equivalent Differential Equations and Higher Derivativesmentioning

confidence: 99%

“…The second‐order differential equation,

f (\dot{x}, x, t)

\begin{align} f_{1} false(\overset{x}{true}, x, t false) & \Leftrightarrow f_{2} false(f_{1}, \overset{x}{true}, J_{13} false) \Leftrightarrow f_{3} false(f_{1}, \overset{x}{true}, J_{13}, {\overset{J}{true}}_{13} false) \Leftrightarrow f_{4} false(f_{1}, \overset{x}{true}, J_{13}, {\overset{J}{true}}_{13}, {\overset{J}{true}}_{13} false) \Leftrightarrow f_{5} false(f_{1}, \overset{x}{true}, J_{13}, {\overset{J}{true}}_{13}, {\overset{J}{true}}_{13}, {\overset{J}{true}}_{13} false), \\ J_{11} & = \frac{\partial f_{1} false(\overset{x}{true}, x, t false)}{\partial \overset{x}{true}}, J_{12} = \frac{\partial f_{1} false(\overset{x}{true}, x, t false)}{\partial x}, J_{13} = \partial f_{1} false(\overset{x}{true} \end{align}

…”

Section: Equivalent Differential Equations and Higher Derivativesmentioning

confidence: 99%

“…Since late 1950s, the Newmark trapezoidal scheme, with limited accuracy and stability characteristics, was used for time integration of nonlinear finite‐element analysis of solids and structures until further solution procedures were developed to overcome the shortfalls of the Newmark scheme, such as in Hilber et al, Wood, Zienkiewicz et al, and Bathe et al Zienkiewicz et al developed a truncated Taylor series expansion for the displacement function and its derivatives, termed the “generalized Newmark” scheme. In Kaunda one‐step multiple‐value algorithms were developed using the “forward‐backward difference” schemes that included the generalized Newmark family of schemes of Zienkiewicz and Taylor as special cases. Standard performance criteria using spectral radius, algorithmic damping, period elongation, and amplitude decay were considered on a linear differential equation and Liapunov stability analysis was employed on a nonlinear equation.…”

Section: Introductionmentioning

confidence: 99%

“…Higher order accurate schemes have since been developed, for example, in Wood, Zienkiewicz et al, and Kaunda . In Reference , it was recommended in a rebuttal, to consider validation using the rival Runge‐Kutta method which is fourth‐order accurate, as well as quite stable.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Improved numerical solutions of nonlinear oscillatory systems

Kaunda

2019

Numerical Meth Engineering

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SUMMARY 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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Section: One‐step Multiple‐value Algorithmsmentioning

confidence: 99%

“…Initial values for the one‐step multiple‐value methods were derived in Kaunda following important observations in Smith . The second‐order differential equation,