### Tech Briefs

The Noh-Bathe Explicit Time Integration Method

Explicit time integration is widely used in practice to solve wave propagation problems, and is also applied to solve structural vibration and even static problems. A common scheme used in explicit time integration is the central difference method. However, this method has two significant limitations.

Firstly, the predicted response obtained using the central difference method can show spurious oscillations in the high frequency modes. These are frequencies and modes that cannot be represented by the chosen mesh. Ideally, any response in these modes would be automatically suppressed — but without loss of accuracy in the frequencies and modes that can be represented by the mesh.

Secondly, to introduce physical damping when using the central difference method (in order to suppress spurious oscillations or obtain a static solution) only a diagonal damping matrix can be specified in order to keep the computational effectiveness [1]. This is also the case when using ADINA.

A new explicit time integration scheme [2] is available in ADINA 9.0, referred to as the Noh-Bathe method. The scheme automatically suppresses spurious high frequency response, without using any non-physical parameters, while accurately integrating those modes that can be spatially resolved. The scheme can also be used with a non-diagonal damping matrix. The computational cost of using the procedure for the same mesh, without a damping matrix, is only slightly larger than the cost of using the central difference method. The objective in this Brief is to illustrate the advantages of the Noh-Bathe explicit time integration scheme.

**The first example** gives the solution of a simple wave propagation problem (taken from ref. [2]). Figure 1 shows a schematic of the problem and Figure 2 shows some calculated response.

**Figure 1** Wave propagation in a pre-stressed membrane. A pulse is applied at the center. Due to symmetry only one quarter is modeled using a mesh of 88 x 88 MITC4 shell elements.

**Figure 2** Displacement contour plot at t = 9.250

As clearly shown in this comparison, the use of the Noh-Bathe method is significantly more effective in simulating the wave propagation response. Further details and response calculations for this example solution are given in ref. [2].

**In the second example**, we demonstrate the use of a non-diagonal damping matrix to obtain the static response in explicit time integration. This approach of solution for the static response can be valuable when very large finite element models need to be solved, and the factorization of the stiffness matrix is computationally very expensive or the matrix is quite ill-conditioned. In this simple illustrative solution, the reaction force on a rod is to be predicted. The rod, modeled with 20 truss elements, is fixed at one end and subjected to a compressive force at the other end. To obtain the static solution, Rayleigh damping is introduced

Using the Noh-Bathe method, the load can be applied rapidly and the static response is obtained in only 300 time steps. Figure 3 shows the results obtained for the reaction at the fixed end.

**Figure 3** Predicted response using the Noh-Bathe method

**In the third example**, we simulate an impact test crushing a tube. Such tests are commonly performed in the defense, automotive, and aerospace industries. Engineers are interested in the acceleration levels of the impactor during the event. Figure 4 shows the deformations at three different times, and Figure 5 shows the acceleration-time solution curves. Once again, we see that spurious oscillations are present in the central difference method solution, while the Noh-Bathe method solution does not show such oscillations.

**Figure 4** Tube-crush problem: Noh-Bathe method predicted deformations at t = 0.000, 0.010, and 0.015
s

**Figure 5** Impactor acceleration-time response for the tube

These example solutions and the references show the merits of using the Noh-Bathe method in explicit time integration. The method does not produce spurious high frequency oscillations in the solution, and non-diagonal damping terms (such as Rayleigh damping) can be included. This is an exciting development towards making ADINA even more effective for a wide range of analysis problems.

**References**

- K.J. Bathe,
*Finite Element Procedures*, Cambridge, MA: Klaus-Jürgen Bathe, 2006. - G. Noh and K.J. Bathe, "An Explicit Time Integration Scheme for the Analysis of Wave Propagations",
*Computers & Structures*, 129:178-193, 2013.

*Keywords:*

Explicit time integration, central difference method, spurious oscillations, chattering, Noh-Bathe method, Rayleigh damping, non-diagonal damping matrix