Forward–backward-difference time-integrating schemes with higher order derivatives for non-linear finite element analysis of solids and structures

Kaunda, M.A.E.

doi:10.1016/j.compstruc.2015.02.026

Cited by 3 publications

(18 citation statements)

References 9 publications

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“…The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function,

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

, and of all of its derivatives are known at a single point, and then the function is unique. We approximate recurrence equations of displacement and velocity (as well as acceleration, and higher order derivatives), in a similar way as in Zienkiewicz et al using a truncated Taylor series expansion, repeated here from Kaunda, for convenience.

s_{n + 1} + \sum_{k = 1}^{k = p} \frac{{(- 1)}^{k}}{k!} {[γ_{1 k} Δ t \frac{d}{d t}]}^{k} s_{n + 1} = s^{*} = s_{n} + \sum_{k = 1}^{k = p} \frac{1}{k!} {[β_{1 k} Δ t \frac{d}{d t}]}^{k} s_{n},

v_{n + 1} + \sum_{k = 1}^{k = p prefix- 1} \frac{{(- 1)}^{k}}{k!} {[γ_{2 k} Δ t \frac{d}{d t}]}^{k} v_{n + 1} = v^{*} = v_{n} + \sum_{k = 1}^{k = p prefix- 1} \frac{1}{k!}

…”

Section: One‐step Multiple‐value Algorithmsmentioning

confidence: 99%

“…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%

“…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. However, comparisons with the Runge‐Kutta methods are yet to be done.…”

Section: Introductionmentioning

confidence: 99%

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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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“…The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function,

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

s_{n + 1} + \sum_{k = 1}^{k = p} \frac{{(- 1)}^{k}}{k!} {[γ_{1 k} Δ t \frac{d}{d t}]}^{k} s_{n + 1} = s^{*} = s_{n} + \sum_{k = 1}^{k = p} \frac{1}{k!} {[β_{1 k} Δ t \frac{d}{d t}]}^{k} s_{n},

v_{n + 1} + \sum_{k = 1}^{k = p prefix- 1} \frac{{(- 1)}^{k}}{k!} {[γ_{2 k} Δ t \frac{d}{d t}]}^{k} v_{n + 1} = v^{*} = v_{n} + \sum_{k = 1}^{k = p prefix- 1} \frac{1}{k!}

…”

Section: One‐step Multiple‐value Algorithmsmentioning

confidence: 99%

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)

Section: Equivalent Differential Equations and Higher Derivativesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Improved numerical solutions of nonlinear oscillatory systems

Kaunda

2019

Numerical Meth Engineering

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Semi-closed-form solutions of the van der Pol oscillator system

Kaunda

2024

E3S Web Conf.

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Second order vector-valued nonlinear differential equations occurring in science and engineering have been considered which generally do not have closed-form solutions. Explicit incremental semi-analytical numerical solution procedures for nonlinear multiple-degree-of-freedom systems have been developed. Higher order equivalent differential equations were formulated and then subsequent values of vectors were updated using explicit Taylor series expansions. As the time-step tends to zero, the values of displacement and velocity are exact in the Taylor series expansions involving as many higher order derivatives as necessary. A typical second order differential equation considered was, the van der Pol oscillator. Further developments consisted of closed-form solutions of the van der Pol equation. What remains to be determined is the closed-form solution of displacement, which is being addressed. Further applications of the semi-analytical procedures to time-dependent systems should also include, time-independent equations that are differentiable in terms of other independent variables, such as partial differential equations that have many independent variables.

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The Cross-Scale Life Prediction for the High-Speed Train Gearbox Shell Based on the Three-Interval Method

et al. 2022

Scientific Programming

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The gearbox shell is a key component of class A of high-speed trains. In engineering applications, the fatigue life prediction of the gearbox shell is a critical issue to be addressed. It is not feasible to obtain fatigue life results for the gearbox shell experimentally because of its long design life, lack of actual failure data, complex structure, high test cost, and material dispersion. Therefore, the cross-scale method was introduced to accurately predict the fatigue life of the gearbox shell. In this study, the entire gearbox shell is divided into two scales of “material structure.” Firstly, the S-N curve is plotted within the material layer, based on the data from the rotating bending fatigue test. Secondly, the finite element model of the gearbox shell is established within the structural layer via the simulation platform. The characteristics and random vibration of the established model are analyzed and presented. Additionally, the first ten-order frequency of modal analysis and power spectral density responses of the gearbox are obtained. The fatigue life of the gearbox shell and the safe running distance of the train are calculated by using the three-interval method and the linear cumulative damage rule, respectively, by combining the fatigue analysis from the material layer with the simulation analysis from the structure layer. Finally, to illustrate the application of the proposed method, a group of small-scale test examples is provided. The proposed method can be used in fatigue life prediction more effectively than the single finite element simulation method.

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Forward–backward-difference time-integrating schemes with higher order derivatives for non-linear finite element analysis of solids and structures

Cited by 3 publications

References 9 publications

Improved numerical solutions of nonlinear oscillatory systems

Improved numerical solutions of nonlinear oscillatory systems

Semi-closed-form solutions of the van der Pol oscillator system

The Cross-Scale Life Prediction for the High-Speed Train Gearbox Shell Based on the Three-Interval Method

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