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Lavrinenko N.M., DrSci, prof.

Donetsk State University of Economic and Trade

 

TIME-STEP ESTIMATES FOR SOME SIMPLE FINITE ELEMENTS

 

Sufficient conditions for stability in finite element analysis may be obtained from estimates of the maximum eigenvalues of individual elements. Consider some examples. If we assume a lumped mass matrix for two-node linear rod element, then

 

(1)

 

where  h is the element length and  is so-called bar-wave velocity, in which  is Young’s modulus and  is density. The critical time step for Newmark method with  (central-difference method) is

 

(2)

 

            which is the time required for a bar wave to traverse one element. If we assume a consistent mass matrix, then

 

(3)

 

            resulting in a reduced critical time step, viz.,

 

(4)

 

            This result is typical – Consistent-mass matrices tend to yield smaller critical time steps than lumped-mass matrices.

For three-node quadratic rod element, if we assume a lumped mass matrix based on a Simpson’s rule weighting (the ratio of the middle node mass to the end node masses is 4), we get

 

(5)

 

                       

(6)

 

Comparison of (6) with (2) reveals that the allowable time step is about 0.4082 that for linear elements with lumped mass. This is based upon equal element lengths. Perhaps a more equitable comparison is one based upon equal nodal spacing. In this case the ratio doubles to 0.8164, but still the advantage is with linear elements.

For linear beam element transverse displacement d face rotation are assumed to vary linearly over the element. One-point Gauss quadrature exactly integrates the bending stiffness and appropriately underintegrates the shear stiffness to avoid “locking”. We assume the trapezoidal rule is used to develop the lumped mass matrix. Solution of the eigenvalue problem results in

 

(7)

 

            where  is the bar-wave velocity,  , the beam shear-wave velocity,  is the cross-section area, is the shear area, I is the moment of inertia and   is the shear modulus.

To get a feeling for these quantities, we shall take a typical situation. Assume the cross section is rectangular with depth  and width 1. This results in  and   . We assume the ratio of wave speeds  . This corresponds to a Poisson’s ratio of ¼ and shear correction factor   , so it is a reasonable approximation for most metals. The time step incurred by bending mode will be critical when

 

(8)

 

This would be the case only for a very deep beam or an extremely fine mesh and is thus unlikely in practice. The more typical situation in structural analysis is when  (i.e. very thin beams or coarse meshing). In this case the critical time step is slightly less than   , the time for a bar wave to traverse the thickness. As this is an extremely small time step, the cost of explicit integration becomes prohibitive.

Flanagan and Belytshko [1] have performed a valuable analysis of the one-point quadrature  (four-node) quadrilateral and (eight-node) hexahedron, applicable to arbitrary geometric configurations of the elements. They obtain the following estimate of the maximum element frequency

 

(9)

                     

            where  , dilatational wave velocity,  and  are Lame parameters and  is a geometric parameter. The estimate (9) leads to a sufficient condition for stability. For the central difference method (9) results in

 

(10)

 

As an example of the restriction imposed by (10) consider a rectangular element with side lengths  and . In this case (10) becomes

 

(11)

 

             For higher-order elements, there appears that little of a precise nature has been done. Most results are of the form (10) where the geometric factor is approximated by trial and error. One would hope that  with the aid of the automatic symbolic manipulators, improved time-step estimates will become available in the ensuring years for more complex elements, material properties.

 

REFERENCES

1.D.P.Flaganan, T.Belytschko, A Uniform Strain Hexahedron and Quadrilateral with Orthogonal Hourglass Control. – International journal for numerical methods in engineering.- 1981, v.17.- p.679-706.