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The models created by UCSD (PressureDependMultiYield, PressureInDependMultiYield, PressureDependMultiYield02) have been implemented with an internal switch parameter that allows them to behave either as linear elastic, elasto-plastic, or elastic pressure-dependent. In order to switch this type of material to an elastic state for the elastic part of the gravity loading the following command is issued:
updateMaterialStage -material 1 -stage 0
The command above specifies that material 1 should behave according to stage flag 0, which is a linear elastic state.
Next, the analysis objects need to be defined:
set gamma 1.6
# create the SOE, ConstraintHandler, Integrator, Algorithm and Numberer
integrator Newmark $gamma [expr pow($gamma+0.5, 2)/4] 0.00 0.0 0.00 0.0
test EnergyIncr 1.0e-8 400 1;
constraints Transformation
algorithm Newton
numberer RCM
system ProfileSPD
analysis Transient
The above command creates a Newmark integrator with gamma=1.6. Beta b=(gamma+0.5)2/4 is chosen because this value provides unconditional stability of the integration algorithm. The Newmark integrator has been shown to produce spurious oscillations in the results, so it is advisable to use the HHT integrator, when possible. On the other hand, the choice of an integrator is also judged on a case to case basis based on its convergence and speed.
The test to judge when convergence has been achieved is based on increments of energy. This is a good criterion in an elasto-plastic problem where increments of displacement can be really large during yielding.
The constraint handler is set as the transformation handler, after recommendations from the creators of the quadUP elements when one is applying pore pressure boundary conditions.
The algorithm selected is a Newton algorithm. It is a robust and simple algorithm with asymptotically quadratic rate of convergence. The Newton-Raphson algorithms are computationally expensive and are known to suffer from residual flip flop due to sudden changes in the tangent stiffness matrix. Linear convergence can be achieved with ModifiedNewton algorithms. In principle, the fastest implemented algorithm should be the KrylovNewton, so it should be preferred, but there are situations that it might not converge to a solution.
The system used is a ProfileSPD. The Jacobian matrix of frictional materials is not symmetric, but in many situations ignoring the non-symmetric elements can help improve performance without significant differences in the results. A good alternative would be an unsymmetric system like UmfPack.
Finally the analysis object selected is Transient. This gives the ability to implement our own code to define the analysis sub-incrementation.