Penalty method

The penalty method is based on the expression of pressure in terms of the incompressibility condition (i.e. the continuity equation) as... 

Elimination of the pressure term from the equation of motion does not automatically yield a robust scheme for incompressible flow and it is still necessary to satisfy the BB stability condition by a suitable technique in both forms of the penalty method. 

Equation (3.8) is the basic working equation of the continuous penalty method. 

The use of selectively reduced integration to obtain accurate non-trivial solutions for incompressible flow problems by the continuous penalty method is not robust and failure may occur. An alternative method called the discrete penalty technique was therefore developed. In this technique separate discretizations for the equation of motion and the penalty relationship (3.6) are first obtained and then the pressure in the equation of motion is substituted using these discretized forms. Finite elements used in conjunction with the discrete penalty scheme must provide appropriate interpolation orders for velocity and pressure to satisfy the BB condition. This is in contrast to the continuous penalty method in which the satisfaction of the stability condition is achieved indirectly through... 

Kheshgi, H. S. and Scriven, L. E., 1985. Variable penalty method for finite element analysis of incompressible flow. Int. J. Numer. Methods Fluids 5, 785-803. 

As described in the discrete penalty technique subsection in Chapter 3 in the discrete penalty method components of the equation of motion and the penalty relationship (i.e. the modified equation of continuity) are discretized separately and then used to eliminate the pressure term from the equation of motion. In order to illustrate this procedure we consider the following penalty relationship... 

After substitution of tressure in the equation of motion using Equation (4.80) and the application of Green s theorem to the second-order derivatives the working equations of the discrete penalty method are obtained as... 

Continuous penalty method - to discretize the continuity and (r, z) components of the equation of motion, Equations (5.22) and (5.24), for the calculation of r,. and v. Pressure is computed via the variational recovery procedure (Chapter 3, Section 4). 

As already mentioned, the present code corresponds to the solution of steady-state non-isothennal Navier-Stokes equations in two-dimensional Cartesian domains by the continuous penalty method. As an example, we consider modifications required to extend the program to the solution of creeping (Stokes) non-isothermal flow in axisymmetric domains ... 

We will prove this theorem in Section 1.3 by a penalty method. 

We prove the existence of solutions for the three-dimensional elastoplastic problem with Hencky s law and Neumann boundary conditions by elliptic regularization and the penalty method, both for the case of a smooth boundary and of an interior two-dimensional crack (see Brokate, Khludnev, 1998). It is shown in particular that the variational solution satisfies all boundary conditions. 

In this section the existence of a solution to the three-dimensional elastoplastic problem with the Prandtl-Reuss constitutive law and the Neumann boundary conditions is obtained. The proof is based on a suitable combination of the parabolic regularization of equations and the penalty method for the elastoplastic yield condition. The method is applied in the case of the domain with a smooth boundary as well as in the case of an interior two-dimensional crack. It is shown that the weak solutions to the elastoplastic problem satisfying the variational inequality meet all boundary conditions. The results of this section can be found in (Khludnev, Sokolowski, 1998a). 

We prove an existence theorem for elastoplastic plates having cracks. The presence of the cracks entails the domain to have a nonsmooth boundary. The proof of the theorem combines an elliptic regularization and the penalty method. We show that the solution satisfies all boundary conditions imposed at the external boundary and at the crack faces. The results of this section follow the paper (Khludnev, 1998). 

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. 

Kovtunenko V.A. (1993) An iterative methods for solving variational inequalities of the contact elastoplastic problem by the penalty method. Comp. Maths. Math. Phys. 33 (9), 1245-1249. 

Kovtunenko V.A. (1994b) An iteration penalty method for variational inequalities with strongly monotonous operators. Siberian Math. J. 35 (4), 735-738. 

Kovtunenko V.A. (1994c) Iteration penalty method for the contact elasto-plastic problem. Control and Cybernetics 23 (4), 803-808. 

Kovtunenko V.A. (1996b) An iterative penalty method for a problem with constraints on the inner boundary. Siberian Math. J. 37 (3), 508-512. 

Kovtunenko V.A. (1997b) Iterative penalty method for plate with a crack. Adv. Math. Sci. Appl. 7 (2), 667-674. 

The MINLP-problems were implemented in GAMS [7, 8] and solved by the outer approximation/equality relaxation/augmented penalty-method [9] as implemented in DICOPT. The algorithm generates a series of NLP and MILP subproblems, which were solved by the generalized reduced gradient method [10] as implemented in CONOPT and the integrality relaxation based branch and cut method as... 

The essential idea of a penalty method of nonlinear programming is to transform a constrained problem into a sequence of unconstrained problems. 

Because of the occurrence of ill-conditioning, pure penalty methods have been replaced by more efficient algorithms. In SLP and SQP, a merit function is used within the line search phase of these algorithms. 

Like penalty methods, barrier methods convert a constrained optimization problem into a series of unconstrained ones. The optimal solutions to these unconstrained subproblems are in the interior of the feasible region, and they converge to the constrained solution as a positive barrier parameter approaches zero. This approach contrasts with the behavior of penalty methods, whose unconstrained subproblem solutions converge from outside the feasible region. 

To illustrate, consider the example used at the start of Section 8.4 to illustrate penalty methods, but with the equality constraint changed to an inequality ... 

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. 

C. F. Shih and A. Needleman, Fully Plastic Crack Problems, Part I Solutions by a Penalty Method, J. Appl. Mech., 51[3], 48-56 (1984). 

Acar, R., and C. R. Vogel, 1994, Analysis of total variation penalty methods Inverse Problems, 10, 1217-1229. 

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