Second Order Errors Related to Geometric Nonlinearity in Explicit Central Difference Operator

2008 ◽  
Author(s):  
Don R. Metzger ◽  
Young-Suk Kim
Keyword(s):  
Difference Method ◽  
Difference Operator ◽  
Time Integration ◽  
General Nature ◽  
Time Step ◽  
Central Difference ◽  
Central Difference Method ◽  
Dynamic Structures ◽  
The Stability ◽  
Stability And Accuracy

Numerical analysis of nonlinear dynamic structures frequently makes use of the central difference method to step the transient forward in time. The method is particularly robust, accommodating material and geometric nonlinearities as well as contact surfaces and constraints of a very general nature. The implementation of the method is most usually performed according to [1], where velocity terms (or more generally rate quantities) are taken half a time step from the displacement and acceleration terms. It was recognized that a proper check of energy balance, requires that velocity must also be interpolated to the integer steps [2]. The stability and accuracy of the central difference method is well established, and decades of experience including its use in numerous commercial finite element codes confirms why it is the method of choice for explicit time integration of transients.

Investigation Into Stability and Accuracy in Predicting Slender Bodies’ Hydroelasticity Using Loose Coupling Methods

2015 ◽  
Author(s):  
Leixin Ma ◽  
Shixiao Fu ◽  
Ke Hu ◽  
Qian Shi ◽  
Runpei Li
Keyword(s):  
Time Integration ◽  
Hydrodynamic Forces ◽  
Mass Force ◽  
Loose Coupling ◽  
Time Step ◽  
Time Saving ◽  
Coupling Methods ◽  
The Stability ◽  
Stability And Accuracy ◽  
Hydroelastic Response

Problems concerning fluid-structure-interaction are often encountered in aquaculture engineering. For a moving slender structure like fishing net or floater in currents and waves, modified Morison Equation is a widely employed formula to estimate its hydrodynamic loads. The hydrodynamic forces are closely dependent on the structures’ velocity and acceleration, and quadratic relative velocity in the equation even adds nonlinearity in the forces. To study the hydroelastic response, two time-saving loosely coupling methods, calculating the hydrodynamic forces based on the structure’s response in the previous time step without iteration, are proposed in this paper. The loose coupling methods were proved to affect the traditional stability criteria for time integration. Based on the two loose coupling methods, the stability and accuracy of a slender beam’s hydroelasticity undergoing large deformation were studied. The calculated responses were compared against strong coupling results. It was found that if loose coupling is assumed in added mass force, unconditional instability is likely to occur. On the other hand, the accuracy of numerical results can be improved with smaller time increments set if loose coupling is only assumed in the quadratic relative drag force.

A New Explicit Time Integration Scheme for Nonlinear Dynamic Analysis

2016 ◽  
Vol 16 (09) ◽  
pp. 1550054 ◽  
Author(s):  
Mohammad Rezaiee-Pajand ◽  
Mahdi Karimi-Rad
Keyword(s):  
Difference Method ◽  
Nonlinear Dynamic ◽  
Time Integration ◽  
Nonlinear Dynamic Analysis ◽  
Taylor Series Expansion ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Central Difference ◽  
Explicit Time Integration ◽  
Central Difference Method

An explicit time integration method is presented for the linear and nonlinear dynamic analyses of structures. Using two parameters and employing the Taylor series expansion, a family of second-order accurate methods for the solution of dynamic problems is derived. The proposed scheme includes the central difference method as a special case, while damping is shown to exert no effect on the solution accuracy. The proposed method is featured by the following facts: (i) the relative period error is almost zero for specific values of the parameters; (ii) the numerical dissipation contained can help filter out spurious high-frequency components; and (iii) the crucial lower modes are generally unaffected in the integration. Although the proposed method is conditionally stable, it has an appropriate region of stability, and is self-starting. The numerical tests indicate the improved performance of the proposed technique over the central difference method.

scholarly journals Frequency-Dependent FDTD Algorithm Using Newmark’s Method

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Bing Wei ◽  
Le Cao ◽  
Fei Wang ◽  
Qian Yang
Keyword(s):  
Differential Equation ◽  
Electric Field ◽  
Field Intensity ◽  
Time Domain ◽  
Difference Method ◽  
Electric Field Intensity ◽  
Polarization Vector ◽  
Solution Stability ◽  
Central Difference ◽  
Central Difference Method

According to the characteristics of the polarizability in frequency domain of three common models of dispersive media, the relation between the polarization vector and electric field intensity is converted into a time domain differential equation of second order with the polarization vector by using the conversion from frequency to time domain. Newmarkβγdifference method is employed to solve this equation. The electric field intensity to polarizability recursion is derived, and the electric flux to electric field intensity recursion is obtained by constitutive relation. Then FDTD iterative computation in time domain of electric and magnetic field components in dispersive medium is completed. By analyzing the solution stability of the above differential equation using central difference method, it is proved that this method has more advantages in the selection of time step. Theoretical analyses and numerical results demonstrate that this method is a general algorithm and it has advantages of higher accuracy and stability over the algorithms based on central difference method.

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