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 Bathe method w/o physical damping Newmark method w/o damping

Accurate Implicit Time Integration in Nonlinear Dynamic Analysis

The accurate solution of the ordinary differential equations resulting from the finite element discretization of dynamic problems has been the subject of extensive research for the last few decades. While the issue of stability and accuracy of the solution of dynamic problems has been amply addressed for linear problems [1], there are still open questions regarding some classes of nonlinear dynamic problems. In this News, we consider one such problem and compare the results of different time marching methods to assess the accuracy and suitability of these methods.

It has been shown that the trapezoidal rule when applied to nonlinear dynamic problems can become unstable especially for large deformation, long duration dynamic problems [2]. In this study we consider another important class of problems where the instability manifests itself due to the presence of contact, which results in contact chatter and furthermore affects the coupling between the fluid and structure in fluid structure interaction analyses.

A test finite element analysis was set up to perform studies and to obtain some insights, see Figure 1.

The model consists of an elastic shell fully clamped at its base and a fluid surrounding it which is contained by an exterior rigid wall. The MITC4 shell elements and subsonic potential based fluid elements are used to represent the media. The shell structure consists of two pieces with frictional contact conditions between them. The model is subjected to a sudden fluid flux representing a pipe break. The resulting shock waves cause the internal parts of the model that are in contact to rapidly change status. For the implicit dynamic analysis of such problems usually the Newmark time integration is used. However, when contact conditions are included between internal parts, the contact surfaces repeatedly stick and slip, which results in rapid pressure pulses in the fluid. Due to the coupling to the structure, high frequency vibrations of the walls are also observed (see the top right movie). These high frequency oscillations are spurious in the Newmark method solution and grow with time. After a while the solution becomes obviously very erroneous and may even diverge. The results using the Newmark method without damping are shown in Figure 2. Note the highly oscillatory response of the flange, the non-smooth contact status between the internal parts and also the parasitic pressure distributions in both the solid and fluid phases.

To overcome this problem different techniques can be used:

• Adding physical damping to the model (e.g. Rayleigh damping). In this case the damping will only be applied to the structure and the question is how much damping to introduce when physically it is negligible.

• Adding numerical damping. This reduces the numerical oscillations, but also reduces the physical response which should be solved for, and the question is how much numerical damping to introduce in order to obtain acceptable results.

• Using the Bathe time integration, available in ADINA. The method is based on two substeps per time step. The first substep uses the standard trapezoidal rule while the second substep uses the backward difference method. The method is second-order accurate and provides a small numerical damping (with no parameters to adjust and only dependent on the size of the time step) that effectively damps out the higher frequency modes [2].

Figure 1  Schematic of the problem

Figures 3 to 5 depict the response of the system using the different time integration techniques mentioned above and always the same time step size:

Figure 2  Newmark method, no damping (δ = 0.5, α = 0.25)

Figure 3  Newmark method with Rayleigh damping, with C = 0.001K

Figure 4  Newmark method with numerical damping, (δ = 0.6, α = 0.3025)

Figure 5  Bathe method, no physical damping

Considering the above case studies, it is observed that while the presence of physical damping or numerical damping improves the results using the Newmark method, to suppress all oscillations, the damping must be increased to high levels, which is not desirable. However, when the Bathe method was used, a significant improvement was found. Note that in this case no numerical parameter had to be adjusted and no artificial physical damping was introduced in the model.

Hence, the Bathe method, which has been described as suitable for long duration problems with large deformations, is also very attractive in these nonlinear FSI problems where the contact chatter may cause large errors when using the standard Newmark method or other methods that use parameters to introduce numerical damping, see ref. [2].

Although the above analysis focused on a simple problem, the above phenomenon is rather universal and also occurs in large-scale practical problems. Due to the proprietary nature of the data, here we only show some parts of the actual nuclear reactor, provided by Onsala Ingenjörsbyrå, where this phenomenon was observed. As a result Onsala Ingenjörsbyrå conducted the study that we described above and graciously provided the data for this Tech Brief.

Figure 6  Finite element model of the actual nuclear reactor and its internal components

Finally, while the Bathe method uses around 50% more solution time, if the same time step is used as in the Newmark method, clearly the stability and accuracy obtained outweigh this added solution cost. Also, in many analyses significantly larger time steps can be used in the Bathe method.

For similar applications of ADINA, see our page on the Nuclear Industry.

References

1. K. J. Bathe, Finite Element Procedures, Prentice Hall, 1996.

2. K. J. Bathe, "Conserving energy and momentum in nonlinear dynamics: A simple implicit time integration scheme", Computers and Structures 85:437-445, 2007.

Keywords:
Implicit, Newmark, Bathe time integration, step by step time integration, accuracy, stability, second order accurate, contact chatter, fluid structure interaction, subsonic potential based fluid, nonlinear dynamics, trapezoidal rule, numerical damping

Courtesy Onsala Ingenjörsbyrå