A FAST AND ACCURATE STEP-BY-STEP SOLUTION PROCEDURE FOR DIRECT INTEGRATION

2011 ◽  
Vol 11 (03) ◽  
pp. 473-493 ◽  
Author(s):  
SHYH-RONG KUO ◽  
J. D. YAU
Keyword(s):  
Time Integration ◽  
Closed Form Solution ◽  
Solution Procedure ◽  
Small Time ◽  
Form Solution ◽  
Dynamic Problems ◽  
Sdof System ◽  
Linear Dynamic ◽  
The Stability ◽  
Stability And Accuracy

Very small time steps are usually needed in numerical computation as conventional time integration methods are used to compute the response of a structure subjected to a dynamic loading with rapid changes or load discontinuity. To overcome this drawback, this study proposed a fast, fourth-order accurate step-by-step time integration (FASSTI) algorithm that is unconditionally stable and allows larger time steps for linear dynamic problems. From the stability and accuracy analysis, it is shown that the FASSTI algorithm retains the features of unconditional stability, accuracy, and fast convergence than the Newmark method. As a first test, a closed-form solution of an excited single degree of freedom (SDOF) system is derived and used to verify the reliability of the present algorithm in solving linear dynamic problems. In the numerical examples, the accuracy and efficiency of the proposed method is demonstrated in the solution of the dynamic response of an SDOF system under a series of impulse-type forces.

scholarly journals Study of Dynamic Behavior of Multilayered Clamped Composite Skewed Hypar Shell Roofs under Impact Load

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Sanjoy Das Neogi ◽  
Amit Karmakar ◽  
Dipankar Chakravorty
Keyword(s):  
Structural Engineering ◽  
Impact Load ◽  
Time Integration ◽  
Closed Form Solution ◽  
Laminated Composite ◽  
Specific Weight ◽  
Form Solution ◽  
Hertzian Contact ◽  
Dead Weight ◽  
Industrial Sectors

With advancement in the field of structural engineering, hunt for smarter materials has channelised the research towards the application of composite material. It is the high specific weight and specific stiffness of this material that have drawn the interest of different industrial sectors. Civil engineers also picked up composites to use it as a roofing material. Laminated composite shells, which can cover large column-free area and reduces dead weight of structure, show vulnerability under sudden impact due to their low transverse shear resistances. This study utilises finite element tool to investigate the dynamic response of a multilayered laminated composite hypar shells for fully clamped boundary condition. This class of shells is unique in a sense that the curvature has only the radius of cross curvature and these shells do not admit easy closed form solution particularly when the boundary conditions are complicated. Contact behavior of impactor and impacted mass has been modeled by modified Hertzian contact law and time-dependent equations are solved using Newmark’s time integration technique. Basic aim is to analyse the shell for symmetrically placed multilayered angle and cross ply lamination under different impact velocities.

Stability and Imperfections in Quasistatic Viscoplastic Solutions

1990 ◽  
Vol 43 (5S) ◽  
pp. S251-S255 ◽  
Author(s):  
T. Belytschko ◽  
B. Moran ◽  
M. Kulkarni
Keyword(s):  
Strain Field ◽  
Strain Softening ◽  
Fourier Spectrum ◽  
Shear Bands ◽  
Closed Form Solution ◽  
Step Function ◽  
Form Solution ◽  
Viscoplastic Material ◽  
Strain Fields ◽  
The Stability

The effect of imperfections on the structure of shear bands in strain-softening viscoplasticity is studied via a closed form solution. The stability of various solutions is then examined by varying the data through imperfections. It is shown that a step-function imperfection, such as commonly used in finite element solutions, leads to a step-function shear strain field, which is an unstable solution. Arbitrary C0 and C1 imperfections lead to C0 and C1 strain fields, respectively. Fourier analyses show that the imperfection scales the response of the viscoplastic material: the Fourier spectrum of the strain field is strongly influenced by the Fourier spectrum of the imperfection.

Stability Analysis of Mechanical Seals With Two Flexibly Mounted Rotors

1998 ◽  
Vol 120 (2) ◽  
pp. 145-151 ◽  
Author(s):  
J. Wileman ◽  
I. Green
Keyword(s):  
Dynamic Properties ◽  
Equations Of Motion ◽  
Closed Form Solution ◽  
Form Solution ◽  
Fluid Film ◽  
Mechanical Seals ◽  
Stability Threshold ◽  
Seal Design ◽  
The Stability ◽  
Gyroscopic Effects

Dynamic stability is investigated for a mechanical seal configuration in which both seal elements are flexibly mounted to independently rotating shafts. The analysis is applicable to systems with both counterrotating and corotating shafts. The fluid film effects are modeled using rotor dynamic coefficients, and the equations of motion are presented including the dynamic properties of the flexible support. A closed-form solution for the stability criteria is presented for the simplifled case in which the support damping is neglected. A method is presented for obtaining the stability threshold of the general case, including support damping. This method allows instant determination of the stability threshold for a fully-defined seal design. A parametric study of an example seal is presented to illustrate the method and to examine the effects of various parameters in the seal design upon the stability threshold. The fluid film properties in the example seal are shown to affect stability much more than the support properties. Rotors having the form of short disks are shown to benefit from gyroscopic effects which give them a larger range of stable operating speeds than long rotors. For seals with one long rotor, counterrotating operation is shown to be superior because the increased fluid stiffness transfers restoring moments from the short rotor to the long.

An Investigation on Forced Convection in Clearance of Corotating Disks With Diverging Laminar Flow

2020 ◽  
Vol 143 (3) ◽  
Author(s):  
Achhaibar Singh ◽  
D. K. Singh
Keyword(s):  
Reynolds Number ◽  
Temperature Field ◽  
Closed Form ◽  
Forced Convection ◽  
Energy Equation ◽  
Closed Form Solution ◽  
Solution Procedure ◽  
Form Solution ◽  
Speed Ratio ◽  
Angular Velocities

Abstract In this work, forced convection in the laminar outflow between two disks is predicted. The disks can be corotating or contrarotating with equal or with different angular velocities. A published closed form solution is used to model the forced convection. The simplified energy equation is discretized using a finite difference method. The discretized equation is solved using the tridiagnal matrix algorithm (TDMA). Results are obtained for a heat flux as well as a specified temperature boundary condition and are presented for various flow parameters such as throughflow Reynolds number, rotational Reynolds number, gap ratio and speed ratio. The parameters except the speed ratio affect the temperature field strongly. The computed Nusselt number is in good agreement with the published experimental data. The coupling of closed form solution and energy equation predicts temperature field accurately. Additionally, the solution procedure is simplified considerably and the solution is obtained with lesser computational resources.

scholarly journals The RAW Filter: An Improvement to the Robert–Asselin Filter in Semi-Implicit Integrations

2011 ◽  
Vol 139 (6) ◽  
pp. 1996-2007 ◽  
Author(s):  
Paul D. Williams
Keyword(s):  
Small Time ◽  
Time Stepping ◽  
Phase Errors ◽  
Power Series Expansions ◽  
Analytic Expressions ◽  
Optimal Values ◽  
The Stability ◽  
Stability And Accuracy ◽  
Total Model ◽  
Elastic Pendulum

Abstract Errors caused by discrete time stepping may be an important component of total model error in contemporary atmospheric and oceanic simulations. To reduce time-stepping errors in leapfrog integrations, the Robert–Asselin–Williams (RAW) filter was proposed by the author as a simple improvement to the widely used Robert–Asselin (RA) filter. The present paper examines the behavior of the RAW filter in semi-implicit integrations. First, in a linear theoretical analysis, the stability and accuracy are interrogated by deriving analytic expressions for the amplitude errors and phase errors. Then, power-series expansions are used to interpret the leading-order errors for small time steps and hence to identify optimal values of the filter parameters. Finally, the RAW filter is tested in a realistic nonlinear setting, by applying it to semi-implicit integrations of the elastic pendulum equations. The results suggest that replacing the RA filter with the RAW filter could reduce time-stepping errors in semi-implicit integrations.

Initial Post-Buckling and Growth of a Circular Delamination Bridged by Nonlinear Fibers

2000 ◽  
Vol 67 (4) ◽  
pp. 777-784 ◽  
Author(s):  
S. Li ◽  
J. Nie ◽  
J. Qian ◽  
Y. Huang ◽  
Y. Hu
Keyword(s):  
Fracture Toughness ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Dynamic Effect ◽  
Form Solution ◽  
Delamination Growth ◽  
Nonlinear Fibers ◽  
Post Buckling ◽  
Bridging Model ◽  
The Stability

Axisymmetric buckling, initial post-buckling and growth of a circular delamination bridged by nonlinear fibers in three-dimensional composites are studied by a perturbation method. The through-thickness fibers are assumed to provide nonlinear restoring traction resisting the deflection of the delaminated layer. A closed-form solution for the central deflection of the delamination due to on applied compressive stress during initial post-buckling is obtained. In addition, some simple formulas for calculating the strain energy release rate and the mixed mode stress intensity ratio (i.e., Mode II versus Mode I) at the delamination crack tip are also established. Some interesting conclusions arising directly from the perturbation solutions are drawn. These include: (1) initial post-buckling behavior of a circular delamination is unstable for a softening bridging model; this may result in initial delamination growth for some materials with lower fracture toughness when the delamination buckles rather than post-buckles. However, stable growth is obtained for a hardening bridging model; (2) with an increase of the nonlinear fiber bridging parameter β¯, the residual stiffness of a three-dimensional composite structure with a circular delamination increases gradually; (3) bridging force changes the catastrophic nature of the delamination growth and increases the stability of the delamination. The range and the dynamic effect of the unstable delamination growth diminish or disappear as the bridging parameters increase; (4) for the bridged delamination, the higher the material fracture toughness, the higher the stability of the delamination growth, and the smaller the range and dynamic effect of its unstable growth. [S0021-8936(00)03203-7]

A Quantitative Evaluation of Two Classical Approximations Used to Predict the Extent of Vertical Hydraulic Fractures

1983 ◽  
Vol 105 (4) ◽  
pp. 512-527 ◽  
Author(s):  
M. B. Rubin
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Hydraulic Fracture ◽  
Closed Form Solution ◽  
Solution Procedure ◽  
Form Solution ◽  
Critical Parameters ◽  
Hydraulic Fractures ◽  
Fracture Width ◽  
Special Case

An integral equation was developed to predict the critical parameters (fracture width and length) associated with the propagation of a vertical hydraulic fracture and a numerical solution procedure was developed. The effects of the classical approximations of pressure and fracture width were investigated both separately and together. It was found that the effects associated with the pressure approximation were relatively insignificant, whereas those associated with the fracture width approximation were significant, particularly when the formation was only moderately permeable. Finally, an exact closed-form solution of the integral equation was developed for a special case. It was shown that when the formation is only moderately permeable, this solution provides a better approximation of the exact solution than the classical solution of Carter [2].

The effect of a nonlinear Mohr–Coulomb criterion on borehole stresses and damage-zone estimate

1994 ◽  
Vol 31 (1) ◽  
pp. 104-109 ◽  
Author(s):  
Yarlong Wang
Keyword(s):  
Oil Sands ◽  
Damage Zone ◽  
Closed Form Solution ◽  
Numerical Approach ◽  
Solution Procedure ◽  
Form Solution ◽  
Far Field ◽  
Circular Opening ◽  
Oil Sand ◽  
Coulomb Criterion

To investigate the possible error introduced by the assumption of a linear Mohr–Coulomb criterion on the stress distribution near a circular opening, a numerical approach is used to calculate both the damage-zone radius and the stresses near a circular opening. A general solution procedure for the stress calculation under a uniform far-field loading is presented and a closed-form solution for the near-opening stresses is given for a cohesionless medium. Based on the numerical results calculated, it is concluded that the linear assumption is acceptable for a strong rock with an unconfined compressive strength that is of similar magnitude to the far-field stress. However, the tangential stress calculated with this linear assumption may be overestimated, but the damage zone may be underestimated in weaker rock or soil such as an oil sand formation. Key words : Mohr–Coulomb criterion, circular opening, nonlinearity, oil sands, Gauss–Legendre method.

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