# Implicit Example

A shallow cylindrical roof upon which an imposed velocity is applied at its mid-point. Analysis uses an implicit approach.

The purpose of this example is to study a snap-thru problem with a single instability. Thus, a structure that will bend when under a load will be used. The results are compared to a reference solution. 1 An implicit strategy using an arc-length method is illustrated.

## Options and Keywords Used

• Implicit solver, time step control by arc-length method
• Static nonlinear analysis
• Stability, snap-thru, and limit load
• T3 Shell
• Boundary conditions (/BCS)
• Implicit options (Implicit Solution)
• Imposed velocity (/IMPVEL)
• Rigid body (/RBODY)

The limit point causes major nonlinearities. Therefore, a static nonlinear analysis is performed using the arc-length displacement strategy. The time step is determined by a displacement norm control. In order to exceed the limit point characterized by a null tangent on the load displacement curve and to describe the increasing and decreasing parts of the nonlinear path, a small time step is required, which is ensured by setting a maximum value.

A solver method is required to resolve Ax=b in each iteration of a nonlinear cycle. It is defined in /IMPL/SOLVER.
Linear Implicit Options
Linear solver
Direct solver MUMPS
Precondition methods
Factored approximate Inverse
Maximum iterations number
System dimension (NDOF)
Stop criteria
Relative residual in force
Tolerance for stop criteria
Machine precision
The input implicit options set in *_001.rad are:
/IMPL/PRINT/NONL/-1
Printout frequency for nonlinear iteration
/IMPL/SOLVER/2 5 0 0 0.0
Solver method (solve Ax=b)
/IMPL/NONLIN 3 1 0.20e-3
Static nonlinear computation
/IMPL/DTINI 10
/IMPL/DT/STOP 0.5 10
Min Max values for time step
/IMPL/DT/2 6.0 20 0.8 1.05
Time step control method 2 - Arc-length+Line-search will be used with this method to accelerate and control convergence

## Input Files

Refer to Access the Model Files to download the required model file(s).

The model files used in this example are available in:

## Model Descripton

A shallow cylindrical roof, pinned along its straight edges, upon which an imposed velocity is applied at its mid-point.

Units: mm, ms, g, N, MPa

Geometrical data are indicated in Figure 1, with the following dimensions:
l
254 mm
R
2540 mm
Shell thickness
t = 12.7 mm
$\theta$
The materials used follow a linear elastic law with the following characteristics:
Material Properties
Initial density
7.85x10-3 $\left[\frac{g}{m{m}^{3}}\right]$
Young's modulus
3102.75 $\left[\mathrm{MPa}\right]$
Poisson ratio
0.3

### Model Method

The modeling problem described in the explicit study remains unchanged.

The implicit computation requires specific implicit parameters that must be defined in the Engine file *_001.rad using the options beginning with /IMPL.
The imposed velocity is considered using the implicit method. Thus, the constant input curve is converted into an imposed displacement according to the computation time.

## Results

### Curves and Animations

Only a quarter of the total load is applied due to the symmetry. Thus, force Fz of the rigid body as indicated in the Time History must be multiplied by 4 in order to obtain force, P.

Figure 4 represents the characteristic load displacement curve for a snap-thru. This diagram plots the reaction at point C of the shell as the function of its vertical displacement. The implicit results are compared with the experimental data.
For a time step equal to or less than 10 ms (maximum value set in the implicit /IMPL/DT/STOP option), agreement with Radioss is achieved, with good results obtained using the reference. Accuracy is improved by decreasing the maximum time step, even though the CPU time is increased.

### Implicit and Explicit Comparison Results

The load displacement curves achieved through implicit computations (time step limit set to 10 ms) and explicit computations are very close. A maximum time step of 100 ms does not allow the nonlinear path of the load displacement curve to be described accurately. However, the final static solution is correct.
Comparison of the computation time between the explicit and implicit (maximum time step set to 10 ms) approaches is shown in Table 1:
Table 1. Computation Time
Implicit Solver Explicit Solver
Normalized CPU 1 2.45
Cycles (normalized) 1 237

In comparison with the implicit computation, which uses a maximum time step of 10 ms, the saved CPU time using a maximum time step fixed at 100 ms, approximately corresponds to factor 4.

### References

1 Finite Element Instability Analysis of Free Formed Shells. Report 77-2, 1977, Norwegian Institute Of Technology, Trondheim, HORRIGMOE G.