A nonlinear analysis is performed in two load steps. In the first step the spring is compressed, and in the second step the spring is slowly relaxed. Geometric nonlinearity is included, and automatic time increments are used. Initial, minimum, and maximum substeps are the same for each analysis method.

During any iteration, contact elements between the spring and the blocks may go from a closed status to an open status causing rigid body motion and hence convergence difficulty. To overcome this instability, five different methods are used.

Static Analysis Using Weak Springs

The Weak Springs option under the Solver Controls menu in the Ansys Mechanical application is used to create weak springs on the corner nodes of the bounding box of each part. The nodes of the bounding box are attached to ground using COMBIN14 elements, and a program-controlled stiffness is chosen. Adding stiffness to the stiffness matrix via springs can help avoid the near-zero pivoting issue and suppress the rigid body modes discussed in the introduction.

In this example, five types of weak springs (one for each part) are used with spring stiffnesses (N/mm) as shown by the following R commands:

 r,15,1.276e-003
 r,16,1.276e-003
 r,17,5.805e-004
 r,18,6.746e-004
 r,19,6.746e-004

Static Analysis Using Nonlinear Stabilization

Nonlinear stabilization adds artificial damping to all nodes of the structure. This damping matrix stabilizes the system of equations in a static solution that would have been near-singular due to loss of constraints. The Ansys Mechanical application default stabilization setting (constant energy method) is used, as shown by the following STABILIZE command:

stabilize,constant,energy,1e-04,,0.2

Static Analysis Using Contact Stabilization

Contact stabilization damping is another technique that adds damping in the static solution to stabilize the structure. This damping is more local than the damping introduced by the STABILIZE command as the damping is only added to the contact nodes, thereby helping to keep the artificial damping energy local and smaller. In this analysis, contact stabilization damping is controlled by KEYOPT(15) and real constant FDMT (tangential stabilization damping factor) of the CONTA174 element.

Contact between the spring and the blocks is stabilized by adding contact damping with default damping coefficients as shown by the following KEYOPT and RMODIF commands:

keyopt,6,15,3    
keyopt,8,15,3   
rmodif,6,32,1   
rmodif,8,32,1   

Full Transient Analysis Using the Quasi-Static Setting

Another approach is to run a full transient analysis. Because of the mass matrix in a full transient analysis, the system of equations becomes stable and can easily simulate rigid body motion. In a quasi-static solution, the backward Euler time integration is used to solve the transient problem. The high numerical dissipation in the algorithm helps to keep the response slow to static.

One advantage of this method over the other methods is that you do not need to adjust any parameters.

The quasi-static solver is invoked by issuing the following ANTYPE and TINTP commands:

antype,trans
tintp,quas

Full Transient Analysis Using the Low Speed Setting

Another option for the full transient analysis is to use the second order time integration (HHT) method with the low speed option. This option automatically sets the integration constants and the time incrementation scheme that is best suited for the slow speed application.

This approach is activated by the following ANTYPE, TRNOPT, and TINTP commands:

antype,trans
trnopt,full,,,,,hht
tintp,losp