## Convergence problem in solving a nonlinear viscoelastic material user

## Convergence problem in solving a nonlinear viscoelastic material user

(OP)

I am having trouble in solving a nonlinear viscoelastic material problem and would appreciate it if someone can suggest any solution or has advice.

I have written a linear viscoelastic user material routine for ABAQUS (Implicit) Version 5.8 which was verified with the build-in ABAQUS linear viscoelastic material model. My problem is to implement a nonlinear viscoelastic material model, based on the modified Swanson theory, for solving the non-linear viscoelastic response of propellants. The main principle is that the linear viscoelastic response is modified by a correction function that is dependant on the strain rate, temperature and the actual strain at a specific integration point. In this case the temperature is kept constant thus not forming part of the equation.

The correction factor is implemented by obtaining the principle strains and their direction, use these strain values and the time increment to determine the strain rates, interpolate between two experimental curves performed at two different strain rates, obtain the corresponding correction factors for each principle directions (which differ for each direction), rotate the correction factors to the global directions and then modify the Jacobian stiffness matrix and stress tensor of the linear viscoelastic response with the globally rotated correction factors.

For a one dimensional problem, using brick elements were the one end is constraint and the other end displaced a specific distance during a specific time period, the solution has no problem in converging. Printing the correction factors and the stress tensor, it is clear that only the principle stress is modified, the rest are very near to zero. The model is a 10X10X100 mm bar sample.

However, in a three-dimensional case, convergence it not achieved. In this case the model is a 120X10X30 mm sample with aluminium tabs on its ends. By displacing the tab-ends a three dimensional stress is induced. Printing the correction factors in the global directions it is clear that all stress components are modified.

In the first increment the solution converges since the correction factor is unity. However, from the second interval the strain increase suggested by ABAQUS is so great that it becomes ridiculous, typical jumping from 0.5% strain to 500% in the same iteration of an increment. Printing the suggested strain increase, it is clear that the solution will never be found.

I have tried everything possible that I could think of to improve the solution rate, from using ridiculous small time increments to using the quasi Newton and Risks methods, even extensive mesh refinements. Modifying the conversion criteria was also ineffective.

My idea is that the complexity of the solution is too severe. The solution must try to find convergence not on a fixed curve, as is the normal case, but inside a surface comprising out of curves at various strain rates. This discrepancy will never lead to convergence of the internal forces.

The question now is ….. can I change the suggested strain increment and how?…. Is the problem too complex to solve? ….. Can I use other convergence criteria, e.g. strain energy, to influence the convergence rate and suggested strain increment? … Is there a more suitable solution method than the quasi-Newton or Risks method? … Has anybody written such a routine for a non-linear viscoelastic material model for propellants? … Can somebody maybe point out the pitfalls of implementing the UMAT routine….. Any additional information that is not in the User Manuals?……….. Any other ABAQUS or MARC implemented nonlinear viscoelastic material models available for propellants?……….. Any ideas ……?

Any information, help, suggestions, ideas, proposals, etc will be highly appreciated.

Kind regards

Detlef

I have written a linear viscoelastic user material routine for ABAQUS (Implicit) Version 5.8 which was verified with the build-in ABAQUS linear viscoelastic material model. My problem is to implement a nonlinear viscoelastic material model, based on the modified Swanson theory, for solving the non-linear viscoelastic response of propellants. The main principle is that the linear viscoelastic response is modified by a correction function that is dependant on the strain rate, temperature and the actual strain at a specific integration point. In this case the temperature is kept constant thus not forming part of the equation.

The correction factor is implemented by obtaining the principle strains and their direction, use these strain values and the time increment to determine the strain rates, interpolate between two experimental curves performed at two different strain rates, obtain the corresponding correction factors for each principle directions (which differ for each direction), rotate the correction factors to the global directions and then modify the Jacobian stiffness matrix and stress tensor of the linear viscoelastic response with the globally rotated correction factors.

For a one dimensional problem, using brick elements were the one end is constraint and the other end displaced a specific distance during a specific time period, the solution has no problem in converging. Printing the correction factors and the stress tensor, it is clear that only the principle stress is modified, the rest are very near to zero. The model is a 10X10X100 mm bar sample.

However, in a three-dimensional case, convergence it not achieved. In this case the model is a 120X10X30 mm sample with aluminium tabs on its ends. By displacing the tab-ends a three dimensional stress is induced. Printing the correction factors in the global directions it is clear that all stress components are modified.

In the first increment the solution converges since the correction factor is unity. However, from the second interval the strain increase suggested by ABAQUS is so great that it becomes ridiculous, typical jumping from 0.5% strain to 500% in the same iteration of an increment. Printing the suggested strain increase, it is clear that the solution will never be found.

I have tried everything possible that I could think of to improve the solution rate, from using ridiculous small time increments to using the quasi Newton and Risks methods, even extensive mesh refinements. Modifying the conversion criteria was also ineffective.

My idea is that the complexity of the solution is too severe. The solution must try to find convergence not on a fixed curve, as is the normal case, but inside a surface comprising out of curves at various strain rates. This discrepancy will never lead to convergence of the internal forces.

The question now is ….. can I change the suggested strain increment and how?…. Is the problem too complex to solve? ….. Can I use other convergence criteria, e.g. strain energy, to influence the convergence rate and suggested strain increment? … Is there a more suitable solution method than the quasi-Newton or Risks method? … Has anybody written such a routine for a non-linear viscoelastic material model for propellants? … Can somebody maybe point out the pitfalls of implementing the UMAT routine….. Any additional information that is not in the User Manuals?……….. Any other ABAQUS or MARC implemented nonlinear viscoelastic material models available for propellants?……….. Any ideas ……?

Any information, help, suggestions, ideas, proposals, etc will be highly appreciated.

Kind regards

Detlef

## RE: Convergence problem in solving a nonlinear viscoelastic material user

Sounds like an interesting problem. Fundamentally I don't see this problem being one in which a RIKS solution method is appropriate; it seems like it should be able to be solved using the quasi-Newton. I recognize that what you have described is essentially a "surface", but it if it is smooth, I would still expect reasonable convergence.

Two suggestions: Is this surface indeed smooth? If it is not, that may be the source of your problems in obtaining convergence.

Second, I presume that you have not obtained help from your Abaqus vendor due to this being a User Subroutine; however you may be able to get help from the Yahoo e-Groups Abaqus user's group. Go to http://groups.yahoo.com/group.abaqus

This is a site with many academics and commercial Abaqus users where these types of things get brought up for discusssion. I guarantee that you will get several thoughtful responses from this question.

Brad