Algorithms

A basic menu of the algorithms and problem types available in Zebulon are as follows:

• Newton/Quasi-Newton The best convergence for non-linear problems is achieved with the Newton method when a high quality consistent tangent matrix is supplied for each element integration. Because of the cost of inverting the global stiffness, the initial iteration can be used repeatedly for subsequent iterations giving slower convergence but perhaps faster CPU time.
• BFGS is a quasi-Newton algorithm initially developed for optimization problems, which updates an already inverted matrix (e.g. the initial problem matrix, or the 1st iteration inverse). The updates are made using the residual vector and the displacement increment is quite efficient.
• Riks is an algorithm used to bypass instability points, or situations where "snap-through" can occur without going to a dynamic solution.
• 2D/3D Contact Contact is available using an associated stiffness method which employs iterations to solve a sub-problem (much like a super element). The iterations generally are quickly achieved relative to the global inversion, so contact problems are not very expensive. The contact can be enforced exactly, or use "soft" contact allowing contact to interact with boundary conditions at the cost of very small penetrations.
• Implicit dynamic with contact is implemented using a trapezoid rule time integration.
• Explicit dynamic with contact (under development). The next version of Zebulon will include a full featured explicit solver.
• Eigen solutions is available with post processing of the deformed mode shapes. A stiffness matrix can be loaded from a deformed configuration.
• Thermal stationary/transient solutions are available. The model kinematics can be imposed also from previously calculated mechanical solutions (i.e. during coupled solution).
• Multi-problem "stepped" coupling Currently only stepped coupling is available. This can include any number and order of solutions and the import/export of model data is quite general.

• Automatic time stepping, init-ddof are features allowing a fine tuned control over the increments in a solution. These options can ensure the quality of quality and maximize efficiency of a solution. An example would be to ensure that the change in Mises stress over the increment is below 10MPa.
• Arbitrary external parameters (fields) are essentially loading for the problem, affecting parametric strains (thermal dilatation) and coefficient values. Any number of field variables are possible, with different "locations," and any material coefficient can depend on any number of parameter.
• Debonding with contact activation Debonding interfaces are available with automatically activated contact surfaces. A number of different "debond material" models are available to predict different types of damage which occur such as the degradation of interfaces between fiber and matrix in composites. The interface models are user-definable of course.
• Post step computations coupled to BCs Algorithms have the ability for post-step computations such as strain energy, J, delta J, etc, which can be used to control subsequent boundary conditions.
• Cycle skipping extrapolates the internal state variables on a cyclic basis.