Solvers Overview: Nonlinear Transient Dynamic

Description

The Nonlinear Transient Dynamic solver calculates the time history of the dynamic response of a structure subjected to arbitrary forcing functions (loading) and/or initial conditions. The solver produces results in the time domain and considers the effects of inertia, damping and nonlinearity including material, geometric and contact nonlinearity

Forcing Function

Two types of loading conditions can be applied: dynamic loads and base acceleration excitation.

Loads applied to the model in the various load cases can be factored by Factor vs Time tables to represent the time history of the loads. All load types can be included. Multiple load cases can act simultaneously; they are combined according to their load factors to form the loading condition at each time step.

Multiple freedom cases can also be included and factored in the same way as the loads in the load cases. This provides support for time dependent enforced displacements.
Base excitations can be expressed as acceleration, velocity or displacement time histories. The base of the model refers to all restrained nodes. The time history of the excitation is specified via a direction vector and associated Factor vs Time tables. Up to three excitations may act simultaneously, each in one of the global axis directions.

Initial Conditions

Four types of initial conditions can be specified:

A static solution that specifies the initial displacements of a structure under certain static loads. This can be a Nonlinear Static or a Quasi-static result file.
A time step in a previously run Nonlinear Transient Dynamic analysis, which specifies the dynamic response of the structure at a time instance. The solution will start from any selected time step available in the file. To enable the possibility of this type of restart, the node velocity and acceleration results must be available in the previously run analysis file.
Initial velocity and acceleration of all free nodes.
Initial velocity (via the Node Velocity attribute) and zero acceleration, as specified independently at each node in one or more load cases.

Procedure

The Nonlinear Transient Dynamic solver executes the following steps:

Initialises the nodal displacement, velocity and acceleration vectors according to the specified initial conditions. For a restart, all quantities required for describing the current element deformation and/or stress status are recovered from the initial solution file and its associated restart file.
Starts a new time step and calculates the constants required for the time integration if the time stepping is different from the previous one.
Calculates and assembles element stiffness, mass and damping matrices, equivalent element force vectors and external nodal force vectors. In the stiffness calculation, material temperature dependency is considered (see Special Topics: Temperature Dependence). Damping can be included (see Special Topics: Damping). Constraints and links are also assembled in this process, and the constant terms in enforced displacements, multi-point links and shrink links are combined and applied according to the respective factors in the relevant freedom cases. At the end of this assembly procedure, the three global matrices in the equation of dynamic equilibrium are formed:

where

= global mass matrix,

= global damping matrix,

= global stiffness matrix,

= unknown nodal displacement, velocity and acceleration vectors, respectively, and

= applied load vector (which may be time dependent) .
Uses the Newmark-beta method to calculate the displacement, velocity and acceleration vectors and updates the current total displacement vector.
Checks convergence:

Displacement norm , and

Residual force norm ,

where and are convergence tolerances on displacement and residual force, respectively, and are norms of iterative and total displacement vectors, respectively, is the norm of the currently applied force vector, and is the norm of the residual force vector in the current iteration.
If either of the criteria is not satisfied, continues the iteration by returning to Step 3. If both of the convergence criteria are satisfied, returns to Step 2 to start the next time step, or stops if at the last time step.

Notes

The results of a transient dynamic analysis are provided as a series of solutions for discrete points in time throughout the time period of interest based on the time stepping and saving settings. The results include the nodal displacements, velocity and acceleration, as well as element stresses, strains and other quantities at each time step.
The choice of time step is important to ensure that the complete response of the structure is captured by the solution and that the solution is stable and free from divergence when nonlinear solutions are executed.

In general, the smaller the time step the more accurate the solution will be. However, there are practical limits on how small the time step can be; more solution steps will be required for a smaller time step and the run time will increase accordingly.

If the time step is too large, much of the higher frequency response of the structure will be missed.