# 2.5. FEM: Finite Element Method¶

The **FEM** tab will present users with a selection of FEM applications that will take a building model generated by the BIM application and the EVENT from the event application and perform a deterministic simulation of structural response. The default application is the OpenSees application.

Currently, there is one application, OpenSees. The current OpenSees implementation extends the standard OpenSees executable with a pre- and post-processor to take the BIM and EVENT files and use OpenSees to simulate the response and return it in an EDP file.

For the **OpenSees** application, the user must specify the
options to be used in the transient analysis. As shown in Fig. 2.5.1, this includes the choice of:

Solution Algorithm: The solution algorithm is the numerical algorithm used to solve the nonlinear equations at each time step/. The default is Newton Raphson; other options include ModifiedNewton, Newton -initial, BFGS, Linear.

System: The system defines how the matrix equation \(Ax = b\) is stored in memory and solved. The default is the unsymmetric sparse solver Umfpack. Other options available include: profileSPD, BandSPD, BandGEM, Diagonal, SuperLU.

Integration Scheme: The integration scheme specifies the time-stepping algorithm being employed to solve the transient problem (\(M \ddot{U}(t) + C \dot{U}(t) + Fs(U(t)) = P(t)\)). The integration scheme determined the coefficients in the \(A\) matrix, the meaning of the \(x\) and \(b\) vectors and how the nodal displacements, velocities, and accelerations should be updated given \(x\). The default is Newmark’s linear acceleration method (Newmark 0.5 0.25). Other Newmark methods can be employed by changing the \(\alpha\) and \(\beta\) terms, and other methods, HHT, CentralDifference, can be selected.

Convergence Test, the default is the norm of the unbalance force with a convergence tolerance of 1.0e-2 and a limit of 10 trial steps to achieve convergence. Other convergence criteria that can be chosen include NormDispIncr, EnergyIncrement, and the relative norms.

Damping Model: The pull-down menu defaults to

Rayleigh Damping. The other option isModal Damping. The theory behind the two is presented in Damping Options.

Rayleigh Damping. With Rayleigh damping, the user must provide the damping ratio, \(\zeta\) and specify two modes from which the damping coefficients \(a_0\) and \(a_1\) will be determined. Mass proportional damping can be obtained by setting the

Mode 2option to 0,stiffness proportional dampingby settingMode 1option to 0.Modal Damping: With modal damping, the user specifies the number of modes and the damping ratio \(\zeta\). In addition, the user has the option of selecting a

stiffness proportional damping ratio, which, as mentioned in the Damping Options is used to provide additional damping for the higher modes.

Analysis Script. This shall be left blank by default. Advanced users of OpenSees who have their preferred analysis script and wish to provide their own damping model can give it here. If this option is provided, the user in their script should use the variables

numStepanddt.

A default transient analysis script is run with these inputs. It is built for Version 3.0.0+ of OpenSees and uses a divide and conquer algorithm to overcome convergence issues. This new algorithm does not work for every nonlinear problem. The actual analysis command that is created based on the defaults is the following:

```
numberer RCM
system Umfpack
integrator Newmark 0.5 0.25
test NormUnbalance 0.01 20
algorithm Newton
analysis Transient -numSubLevels 2 -numSubSteps 10
analyze $numStep $dt
```

If the user specifies their own analysis script to run instead of the default, they can take advantage of the **numStep** and **dt** variables that are obtained from the EVENT and are automatically set by the program.