Three new modules are integrated in OpenSees Model and Exampel Download: OS_Explicit_Example.zip We have added three new modules into OpenSees based on the 2.4.3 version: (1) A three node triangular shell element DKGT, and its geometric nonlinear formula NLDKGT. (2) Leapfrog method, an explicit integration method. (3) A new DiagonalSOE with modal damping for leapfrog method. The source codes of these modules are submitted to the official website of OpenSees. 1. Triangular shell element DKGT/NLDKGT DKGT is a threenode triangular thin shell element, and its extended formula NLDKGT can be used for geometric nonlinear problems. They are established by combining the membrane element GT9 and plate bending element DKT. More details can be seen in the codes, and the following are some validations. 
Case 1 ScordelisLo roof problem The ScordelisLo roof problem is shown in Figure 1. The cylindrical panel is loaded vertically with a uniform selfweight of g = 90. Both edges are fixed. Due to the symmetry, only one quarter of the panel is modeled. Three meshes are used in this analysis, as shown in Table 1. The vertical deflection of Point A is recorded. The exact solution of 0.3024 provided by MacNeal and Harder is used as a reference. The results are shown in Table 1. Figure 1 ScordelisLo roof problem Table 1 Results of ScordelisLo roof problem

Case 2 Twisted beam problem The twisted beam problem is shown in Figure 2. A concentrated load is applied at the free end in the inplane (P) and outofplane (Q) directions, respectively. A mesh of 2 × 12 is adopted for this problem. Two load cases are analyzed: (1) P = 1, Q = 0; and (2) Q = 1, P = 0. The displacement along the loading direction at the free end is recorded. The exact solutions provided by MacNeal and Harder are used as a reference. The comparison is shown in Table 2. Figure 2 Twisted beam problem Table 2 Results comparison for the twisted beam problem

2. Leapfrog method Because the central difference method cannot ensure the equations decoupled, which limits the computational speed, the leapfrog method is integrated into OpenSees. The leapfrog method has a simple formula, and can ensure the equations decoupled. However, it should be noted that there must be no zero element among the diagonal elements of the mass matrix when using the leapfrog method. However, some elements in OpenSees have zero inertia for the rotation degreeoffreedoms. Such element should not be used together with the leapfrog method. The fundamental equations of the leapfrog method are shown as follows: The critical time step size of the leapfrog method is as follow: Note that when the leapfrog method is used with Rayleigh damping, the damping ratio of the higher vibration modes is overestimated. Consequently, the critical time step size should be much smaller than that for the implicit method. 3. Modal damping The critical time step size of the leapfrog method is controlled by damping ratio. Because the Rayleigh damping model is not suitable for the leapfrog method, we added Modal damping into OpenSees. We made the modal damping as a subclass of LinearSOE based on the diagonal class because diagonalSOE is suitable for the leapfrog method. More details can be seen in the codes. In addition, in order to get the pointers and data from other modules, many codes are added in some other classes. The command to use modal damping is shown as follows:
$Dratio is the target damping ratio; $startfreq and $endfreq specify the frequency range to be controlled; $alphaM is the mass proportional coefficient of Rayleigh damping. $alphaM (a_{1}) is introduced here because we often use modal damping and mass damping together as follow: $Dratio is the total damping ratio; Thus, the damping ratio of modal damping _{ } can be calculate by $Dratio and $alphaM. Case 3 Earthquakeinduced collapse simulation of a frame structure Figure 3 shows an eightstory frame structure, with a total height of 31.5 m. There are 3 spans in the x direction, 2 spans in the y direction and each span is 5 m. ElCentro ground motion is used as input in the x direction. More details of the modes can be seen in the Tcl files. First, when PGA = 510 cm/s^{2}, the time history of the top displacement is shown in Figures 4  6. Then the PGA is increased to 3500 cm/s^{2} to perform the collapse simulation. The top displacement results are shown in Figure 7. It should be noted that in order to make the leapfrog method work, we added some codes in dispcolumnbeam.cpp to set the moment of inertia equal to the translational mass. The vibration periods of the frame with the moment of inertia have little difference with the original ones. Figures 4  6 illustrate that the mass & modal damping has better accuracy than using the mass damping or modal damping only. Figure 7 shows that the Newmark method (implicit) encounters numerical problem in convergence, but the leapfrog method can simulate the entire collapse process. Figure 3 A eightstory frame structure Figure 4 Linear time history responses of the frame structure (PGA=510cm/s^{2}) Figure 5 Nonlinear time history response of the frame structure (PGA=510cm/s^{2}) Figure 6 Comparison of the modal damping and the mass & modal damping Figure 7 Collapse simulation of the frame structure (PGA=3500cm/s^{2}) 