Penalty formulation

Two Dimensional Penalty Formulation for Creeping Flow Problems... 

This results in a system of three equations with three variables ux, uy and p. It is very common to eliminate the pressure from the set of equations by introducing a slight compressibility to the fluid. This approach is often referred to as the penalty formulation. The compressibility factor 7 is introduced in the continuity equation as... 

Next, we apply Galerkin s weighted residual method and reduce the order of integration of the various terms in the above equations using the Green-Gauss Theorem (9.1.2) for each element. For a simpler presentation we will deal with each term in the above equations separately. The terms of the x-component (eqn. (9.95)) of the penalty formulation momentum balance become... 

In this chapter, we have derived the two-dimensional finite element penalty formulation for creeping flows where the pressure was eliminated by assuming a compressible flow. Here, we will use a mixed formulation, where the pressure is included among the unknown variables. In the mixed formulation, we use different order of approximation for the pressure as we will for the velocity. For instance, if tetrahedral elements are used, we can use a quadratic representation for the velocity (10 nodes) and a linear representation for the pressure (4 nodes). Hence, we must use different shape functions for the velocity and pressure. For such a formulation we can write... 

There are several methods for imposing the loop closure conditions. The Lagrange multipliers method uses a vector of unknown reaction forces that act at the cut joints. The dynamic equilibrium equations together with the constraint equations form a set of DAE s that must be integrated. In order to reduce the size of the final system of equations, Avello et al.(1993) presented an approach based on the penalty formulation developed by Bayo et al. (1988). In this paper, the reduction of the original system of DAE s to a set of ODE s is achieved through a second velocity transformation. The numerical efficiency of the last two approaches in terms of CPU time per function evaluation is very similar, as it is shown by the practical examples solved in section 4. 

Cut joint constraints imposed through a penalty formulation Proposed method (constraints imposed through a second velocity transformation)... 

It is clear that the situation in the penalty method is very similar to the one based on the Lagrange multiplier method. The technique to reduce the number of constraints in the penalty formulation is to split the constitutive equations into the deviatoric and the dilatational parts, and then use a lower order integration scheme for the dilatational part (thus reducing the number of the incompressibility constraints) while keeping the normal order of numerical integration for the deviatoric part. 

The results given here are on a laboratory basis. During formulation in the refinery, the cold characteristics are much less satisfactory with a penalty of around for cases where it is desired to keep the same yield from the crude oil. 

The described application of Green s theorem which results in the derivation of the weak statements is an essential step in the formulation of robu.st U-V-P and penalty schemes for non-Newtonian flow problems. 

In this section we deal with the simplified nonpenetration condition of the crack faces considered in the previous section. We formulate the model of a plate with a crack accounting for only horizontal displacements and construct approximate equations using penalty and iterative methods. The convergence of these solutions is proved and its application to the onedimensional problem is discussed. Analytical solutions for the model of a bar with a cut are obtained. The results of this section can be found in (Kovtunenko, 1996c, 1996d). 

To improve the convergence of the gradient-type method, Tannor et al. [81, 93] suggested employing the Krotov iteration method [102]. In formulating their method, they utilize a penalty function of the form /[e(f)] = pe (f). In Tannor s Krotov method, the fcth iteration step of the solution process is given by... 

Implicit estimation offers the opportunity to avoid the computationally demanding state estimation by formulating a suitable optimality criterion. The penalty one pays is that additional distributional assumptions must be made. Implicit formulation is based on residuals that are implicit functions of the state variables as opposed to the explicit estimation where the residuals are the errors in the state variables. The assumptions that are made are the following ... 

The Complex algorithm was modified slightly to include the two extra summation constraints as penalty functions. The response of leachability was minimized at a value of -3.53 with the following formulation ... 

Thus, the corresponding mean-absolute deviation (MAD) of the expected penalty costs is formulated as ... 

The caramelized liquid is sent by railroad tank cars to the cola syrup formulators, who then test the solution for quality. If it is too light in color—penalty if it has too many carbon particles per unit volume, then the whole tank car is rejected. There is thus a delicate balance between underreacting and overreacting. 

Remark 3 Kocis and Grossmann (1989a) suggested another alternative formulation of the feasibility problem, in which a penalty-type contribution is added to the objective function that is,... 

The basic idea in GOA is similar to the one in OA, with the key differences being the (i) treatment of infeasibilities, (ii) new formulation of the master problem that considers the infeasibilities explicitly, and (iii) unified treatment of exact penalty functions. 

In GOA/EP, we do not distinguish between feasible and infeasible primal problem, but instead formulate the following primal problem suggested by Fletcher and Leyffer (1994) which is based on an exact penalty function ... 

Section 6.6 discusses the Outer Approximation with Equality Relaxation and Augmented Penalty OA/ER/AP approach. In Sections 6.6.1 and 6.6.2 the formulation and basic idea are presented, while in section 6.6.3 the master problem is derived. Section 6.6.4 presents the OA/ER/AP algorithm and illustrates it with a nonconvex example problem. The reader is referred to the suggested references in sections 6.4,6.5 and 6.6 for further reading in the outer approximation based algorithms. 

Section 6.7 presents the Generalized Outer Approximation GOA approach. After a brief discussion on the problem formulation, the primal and master subproblem formulations are developed, and the GOA algorithm is stated in section 6.7.4. In Section 6.7.5, the worst case analysis of GOA is discussed, while in section 6.7.6 the Generalized Outer Approximation with exact Penalty GOA/EP and its finite convergence are discussed. 

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