8. Modal Transient

Standard Sierra/SD has a fine set of modal based solutions, including a modal transient integrator. However, Sierra/SD is designed to focus on massively parallel solutions. It is not uncommon for an analyst to generate a small modal solution, and to use the modal solution as part of a small transient run. Since in modal space, the solution is diagonal, this completely uncouples the modes and allows for an independent solution of each modal amplitude, \(q_i\).

Sierra/SD uses these solutions, but it assumes that the full solution on all output degrees of freedom is required. In other words, the quantity \(q_i(t)\) is easily computed, but to transform back to physical space, a fair amount of calculation must be performed, and it is performed on the full system model. For transient dynamics, Sierra/SD performs the following operations.

1. Compute \(q_i(t)\) for all modes, \(i\), at time \(t\).

2. Expand to physical space. \(x(t)=\phi q(t)\).

   This requires participation of all processors that were involved in the calculation of the modes.

3. Contract to a reduced physical space, if history output is requested.

   This requires communication between processors.

In cases where the analyst requires only a subset of the data, this process can be streamlined by performing the integration outside of Sierra/SD. The calculation is fast, and can be performed in serial.

8.1. Process for serial integration

8.1.1. Compute modes of the system model

Modes are extracted in the usual way, i.e. perform a standard eigen extraction on the full system model. Output a reduced order model by extracting a small portion of the eigenvectors to the history file. Element variables of stress and strain may also be output.

HISTORY
  nodeset 1
  block 12
  displacement
  stress
END

8.1.2. Extract Modal force

The modal force, \(\tilde{F}(t)\), can be written by specifying ‘mfile’ in the OUTPUT section of the Sierra/SD input. The file is named “ModalFv.m”. The file contains a matrix of size \(N_{vect}\) x \(N_{modes}\), where \(N_{modes}\) is the number of normal modes computed, and \(N_{vect}\) is the number of spatial load vectors.

Recall that Sierra/SD defines time dependent loads as a sum of products of spatial and temporal functions. For example, consider this example loads section.

LOADS
  nodeset 111
    force 1 0 0
    function 111
  sideset 22
    pressure 1.0
    function 2
END

This example time dependent force could be written as follows.

\[ F(x,t) = N_{111}(x)F_{111}(t) + N_{22}(x)F_2(t) \]

where the \(N(x)\) represents a function of space only, and \(F(t)\) is a function of time only. In this example, there are two spatially varying functions, and \(N_{vect}=2\).

We assume that the analyst has access to the time varying functions, \(F(t)\), since they are part of the input. Each of the spatial terms is multiplied by the eigenvectors to arrive it the modal contribution.

\[ ModalFv=( \Phi^T N_j )^T \]

The total generalized force is then,

\[ \tilde{f}_j(t) = \sum_i ModalFv_{ji} F_i(t) \]

8.1.3. Perform Time Integration of Modal Space

Time integration can be performed in MATLAB or other suitable integrator. The file, “modal_int.m” provides an example time integrator using the standard trapezoidal rule (the Newmark-Beta integrator with \(\beta=1/4\), and \(\gamma=1/2\)).

We can think of the integration as the solution of three equations in three unknowns.

\[\begin{eqnarray*} \tilde{k}q_{n+1} + \tilde{c}\dot{q}_{n+1} + \ddot{q}_{n+1} & = & \tilde{f}(t) \\ q_{n+1} &=& q_n + \frac{1}{2}(\dot{q}_n + \dot{q}_{n+1}) \\ \dot{q}_{n+1} &=& \dot{q}_n + \frac{1}{2}(\ddot{q}_n + \ddot{q}_{n+1}) \\ \end{eqnarray*}\]

The latter two equations are used to eliminate the \(\dot{q}_{n+1}\) and \(\ddot{q}_{n+1}\) terms, resulting in the algebraic equation for \(q_{n+1}\).

\[ \left[ \tilde{k} + \frac{2}{\Delta t} \tilde{c} + \frac{4}{\Delta t^2} \right] q_{n+1} = \tilde{f} + \tilde{c}(\dot{q}_n + \frac{2}{\Delta t} q_n) + \left( \ddot{q}_n + \frac{4}{\Delta t}\dot{q}_n + \frac{4}{\Delta t^2} q_n \right) \]

The result is \(q_j(t_i)\), for each mode \(j\) in the system, and for each time value \(t_i\).

8.1.4. Expand to Physical Space

The integrated time values can be represented as a matrix \(Q\), where each row of \(Q\) corresponds to a normal mode coordinate, and each column represents a time value. The physical space is represented by the product, \(\tilde{\phi} Q\), where \(\tilde{\phi}\) is the eigenvector in the reduced space.

Using exo2mat the eigenvectors are put into six variables. They can be reshaped into \(\phi\) as follows.

phi = [ nvar01 nvar02 nvar03 nvar04 nvar05 nvar06];
phi = reshape(something)

The transformation to physical space is,

XXX = phi * Q;
XX  = reshape(XXX,n,6);
x = X(:,1);
y = X(:,2);
z = X(:,3);

Determining the element variables is not much different. A set of element results “eigenvectors” is obtained using evarXX in place of nvarXX. The result is the product \(\psi Q\).

8.2. How to Use Results

The results from this calculation cannot be easily visualized as an animated structure because there is typically insufficient data to reconstruct the model. However, time histories of nodal and element data can be examined and plotted.

8.2.1. Related Calculations

Similar calculations are possible with other modal based solutions. For example, a modal frequency response calculation is performed in the same way except that the modal amplitude is given by the following.

\[ q_i(\omega) = \frac{\tilde{f}_i(\omega)} {\omega^2 -\omega_i^2 + 2\gamma_i\omega\omega_i} \]

where \(\tilde{f}_i(\omega)=\phi F(\omega)\) is given by \(Modalfv\) as before. The modal amplitude in this problem is complex of course.

8.3. Limitations

The entire modal must fit in memory. Since this is a linear superposition model, only linear results can be used. Further, while natural stresses can be computed, von Mises and other principal stresses cannot be directly computed, as they are not linear functions of displacement.

The modal superposition method has significant limitations, independent of the particular solution methodology. In particular, the method may be slow to converge spatially if the loading is not well represented by a low frequency mode. Other methods such as the Craig-Bampton reduction can be much better in these cases, though they suffer from having a coupled system of equations.

8.4. Verification

The simplest verification is to run a portion of the time history through the standard Sierra/SD modal transient, and compare the results with the results from the reduced order model.