Penalty parameter

Level of enforcement of the incompressibility condition depends on the magnitude of the penalty parameter. If this parameter is chosen to be excessively large then the working equations of the scheme will be dominated by the incompressibility constraint and may become singular. On the other hand, if the selected penalty parameter is too small then the mass conservation will not be assured. In non-Newtonian flow problems, where shear-dependent viscosity varies locally, to enforce the continuity at the right level it is necessary to maintain a balance between the viscosity and the penalty parameter. To achieve this the penalty parameter should be related to the viscosity as A = Xorj (Nakazawa et al, 1982) where Ao is a large dimensionless parameter and tj is the local viscosity. The recommended value for Ao in typical polymer flow problems is about 10. ... 

In the continuous penalty scheme used here the penalty parameter is defined as... 

GETMAT Reads and echo prints physical and rheological parameters and the penalty parameter used in the simulation. 

C RVISC MEU NOUGHT CONSISTENCY COEFFICIENT C RPLAM PENALTY PARAMETER... 

We prove an existence of solutions for the Prandtl-Reuss model of elastoplastic plates with cracks. The proof is based on a special combination of a parabolic regularization and the penalty method. With the appropriate a priori estimates, uniform with respect to the regularization and penalty parameters, a passage to the limit along the parameters is fulfilled. Both the smooth and nonsmooth domains are considered in the present section. The results obtained provide a fulfilment of all original boundary conditions. 

The above subproblem can be solved very efficiently for fixed values of the multipliers X and v and penalty parameter p. Here a gradient projection trust region method is applied. Once subproblem (3-104) is solved, the multipliers and penalty parameter are updated in an outer loop and the cycle repeats until the KKT conditions for (3-85) are satisfied. LANCELOT works best when exact second derivatives are available. This promotes a fast convergence rate in solving each... 

Intuitively, Px is exact and the squared penalty function P2 is not because squaring a small infeasibility makes it much smaller, that is, (10-4)2 = 10-8. Hence the penalty parameter r in P2 must increase faster as the infeasibilities get small, and it can never be large enough to make all infeasibilities vanish. 

Step 1. For a given set of Lagrange multipliers and penalty parameter minimize the Lagrangian function L R) to obtain an improved estimate of the factorized 2-RDM at the energy minimum. 

Step 3. If Step 2 is not executed, then the penalty parameter is decreased to better enforce the constraints... 

Steps 1-3 are repeated until the maximum absolute error in the constraints falls below a target threshhold. Before the first iteration the Lagrange multipliers may be initialized to zero and the penalty parameter set to 0.1. The constraints are not fully enforced until convergence, and the energy in the primal program approaches the optimal value from below. 

The two contaminant plumes are represented in the first stage of the optimization formulation with a set of 110 control points along the plume boundaries. These same control point locations are used as starting points for particles when forward tracking is used in the second stage of the solution process. For the unconfined simulation, additional constraints are included to require a minimum saturated thickness of 1.5 m at each well cell. Both confined and unconfined assumptions are simulated under two sets of penalty parameters. Recall that the solution algorithm uses the penalty method for the plume capture constraints, in which each constraint violation is multiplied by a penalty parameter and added to the objective function. 

Very recently Sargent 88 has shown that, although the Lagrangian function is not an exact penalty function, it can indeed be used in a descent test to force global convergence. Moreover the test is satisfied without step-reduction in the final stages, so the convergence is superlinear. Thus the penalty parameter and its associated problems are eliminated. 

Volume fraction of components x and j Strain penalty parameter Time constant, s Oscillating strain or deformation Shear rate, s ... 

Figure 4.2 (a) Optimal control field forthe system shown in Fig. 4.1 for a penalty parameter k = 400 (see Eq. (4.4)). The pulse triggers quantum tunneling between the two lowest states whose populations are shown in (b) for details see also Ref. [50]. 

It has been demonstrated that for certain finite values of its penalty parameters, the minimization of the function 4>i solves the original constrained problem exactly. 

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