Upper and lower bounds for natural frequencies: A property of the smoothed finite element methods

2010 ◽  
pp. n/a-n/a ◽  
Author(s):  
Zhi-Qian Zhang ◽  
G. R. Liu
Keyword(s):  
Finite Element ◽  
Lower Bounds ◽  
Finite Element Methods ◽  
Natural Frequencies ◽  
Upper And Lower Bounds

Pipe Whip—Bounding the Required Restraint Capacity

1985 ◽  
Vol 107 (4) ◽  
pp. 356-360
Author(s):  
R. Peek
Keyword(s):  
Finite Element ◽  
Energy Balance ◽  
Lower Bounds ◽  
Finite Element Methods ◽  
Plastic Hinge ◽  
Upper And Lower Bounds ◽  
Trial And Error ◽  
Rigid Plastic ◽  
Numerical Example ◽  
Plastic Pipe

Energy balance methods commonly in use for the design of pipe whip restraints are based on the solution for the motion of a rigid-plastic pipe before impact against the restraint, with the assumption that after impact, the whipping portion of the pipe continues to rotate about the plastic hinge location determined for conditions before impact. Such energy balance methods are not necessarily conservative because: 1) the plastic hinge which forms in the pipe moves after impact on the restraint; and 2) elastic pipe deformations are not considered. Here, upper and lower bounds to the required restraint capacity are derived. In contrast to finite element methods, which are very time-consuming, the upper and lower bounds can be evaluated by simple hand calculations. Another advantage is that the required restraint capacity is calculated directly. No trial and error design is required. A numerical example shows that for a typical pipe and restraint, the upper and lower bounds differ by as little as 20 percent.

The accuracy of Rayleigh’s method of calculating the natural frequencies of vibrating systems

1952 ◽  
Vol 211 (1105) ◽  
pp. 204-224 ◽  
Keyword(s):  
Lower Bounds ◽  
Natural Frequencies ◽  
Hermitian Operator ◽  
Normal Modes ◽  
Comparison Theorems ◽  
Upper And Lower Bounds ◽  
Subsequent Paper ◽  
Numerical Examples ◽  
Modes Of Vibration ◽  
Vibrating Systems

In this paper a theorem of Kato (1949) which provides upper and lower bounds for the eigenvalues of a Hermitian operator is modified and generalized so as to give upper and lower bounds for the normal frequencies of oscillation of a conservative dynamical system. The method given here is directly applicable to a system specified by generalized co-ordinates with both elastic and inertial couplings. It can be applied to any one of the normal modes of vibration of the system. The bounds obtained are much closer than those given by Rayleigh’s comparison theorems in which the inertia or elasticity of the system is changed, and they are in fact the ‘best possible’ bounds. The principles of the computation of upper and lower bounds is explained in this paper and will be illustrated by some numerical examples in a subsequent paper.

A Posteriori Error Estimator for a Weak Galerkin Finite Element Solution of the Stokes Problem

2017 ◽  
Vol 7 (3) ◽  
pp. 508-529 ◽  
Author(s):  
Xiaobo Zheng ◽  
Xiaoping Xie
Keyword(s):  
Finite Element ◽  
Lower Bounds ◽  
Stokes Problem ◽  
Three Dimensions ◽  
Upper And Lower Bounds ◽  
Error Estimator ◽  
A Posteriori ◽  
Weak Galerkin ◽  
Galerkin Finite Element ◽  
Velocity Approximation

AbstractA robust residual-based a posteriori error estimator is proposed for a weak Galerkin finite element method for the Stokes problem in two and three dimensions. The estimator consists of two terms, where the first term characterises the difference between the L2-projection of the velocity approximation on the element interfaces and the corresponding numerical trace, and the second is related to the jump of the velocity approximation between the adjacent elements. We show that the estimator is reliable and efficient through two estimates of global upper and global lower bounds, up to two data oscillation terms caused by the source term and the nonhomogeneous Dirichlet boundary condition. The estimator is also robust in the sense that the constant factors in the upper and lower bounds are independent of the viscosity coefficient. Numerical results are provided to verify the theoretical results.

Bounds for the Natural Frequencies of a Simply Supported beam Carrying a Uniformly Distributed Axial Load

1967 ◽  
Vol 9 (2) ◽  
pp. 149-156 ◽  
Author(s):  
G. Fauconneau ◽  
W. M. Laird
Keyword(s):  
Lower Bounds ◽  
Axial Load ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Simply Supported Beam ◽  
Simply Supported ◽  
Distributed Load ◽  
Made In

Upper and lower bounds for the eigenvalues of uniform simply supported beams carrying uniformly distributed axial load and constant end load are obtained. The upper bounds were calculated by the Rayleigh-Ritz method, and the lower bounds by a method due to Bazley and Fox. Some results are given in terms of two loading parameters. In most cases the gap between the bounds over their average is less than 1 per cent, except for values of the loading parameters corresponding to the beam near buckling. The results are compared with the eigenvalues of the same beam carrying half of the distributed load lumped at each end. The errors made in the lumping process are very large when the distributed load and the end load are of opposite signs. The results also indicate that the Rayleigh-Ritz upper bounds computed with the eigenfunctions of the unloaded beam as co-ordinate functions are quite accurate.

Numerical Methods for the Limit Analysis of Plates

1968 ◽  
Vol 35 (4) ◽  
pp. 796-802 ◽  
Author(s):  
P. G. Hodge ◽  
T. Belytschko
Keyword(s):  
Finite Element ◽  
Numerical Methods ◽  
Mathematical Programming ◽  
Rectangular Plate ◽  
Yield Point ◽  
Lower Bounds ◽  
Limit Analysis ◽  
Upper And Lower Bounds ◽  
Mathematical Programming Problems

The determination of upper and lower bounds on the yield-point loads of plates are formulated as mathematical programming problems by using finite element representations for the velocity and moment fields. Results are presented for a variety of square and rectangular plate problems and are compared to other available solutions.

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