Use Of Penalty Method Research Articles

Enhanced functional evaluation for the finite element penalty method

We analyze the impact of using the penalty method on the estimation of a prescribed solution functional or ‘Quantity of Interest’ (QoI). Specifically, we consider the use of penalty methods to enforce Dirichlet boundary constraints, focusing our attention on boundary fluxes as QoIs. We propose an enhanced estimator of the boundary flux that includes a term involving the derivative of the flux with respect to the penalty parameter ϵ. We show that the new estimator reduces the error arising from the use of the penalty method from O(ϵ) to o(ϵ). A well-posed adjoint problem associated with the boundary flux is also proposed following an analysis of the penalty method. Errors in the enhanced flux estimator are then controlled using adjoint-based techniques. Several numerical experiments are presented to demonstrate that the enhanced estimator, in combination with adjoint error estimation, allows one to efficiently control both the discretization and penalty errors in target QoIs.

The finite element method scheme for a solution of an evolution variational inequality with a nonlocal space operator

We consider the parabolic inequality with monotone with respect to a gradient space operator, which is depended on integral with respect to space variables solution characteristic. We construct a two-layer differential scheme for this problem with use of penalty method, semidiscretization with respect to time variable method and the finite element method (FEM) with respect to space variables. We proved a convergence of constructed mothod.

Penalty function approaches for ship multidisciplinary design optimisation (MDO)

This paper focuses on the solution of difficult multidisciplinary optimisation formulations arising in ship design. The latter schemes are by nature the result of the interaction among several optimisation problems. Each optimisation problem summarises the issues related to a specific aspect (discipline) of the formulation, and it may be hardly solved by stand-alone methods which ignore the other disciplines. This usually yields very challenging numerical optimisation problems, due to the simultaneous solution of different schemes. In particular, in our ship design applications we stress the strong interaction between fluid-dynamics and optimisation, in order to get remarkable achievements. The ordinary stand-alone methods from mathematical programming prove to be often unsatisfactory on the latter multidisciplinary problems. This scenario requires a specific integration of both fluid-dynamics and optimisation, where constrained optimisation schemes frequently arise. We give evidence that the proper use of penalty methods, combined with global optimisation techniques, may both be a theoretically correct approach, and may yield a fruitful class of techniques for the solution of multidisciplinary problems. We provide numerical results with different penalty functions, over difficult multidisciplinary formulations from ship design. Here, the introduction of penalty methods proved to be a valuable tool since feasibility issues strongly affect the formulation.

Interior-Point Algorithms, Penalty Methods and Equilibrium Problems

In this paper we consider the question of solving equilibrium problems--formulated as complementarity problems and, more generally, mathematical programs with equilibrium constraints (MPECs)--as nonlinear programs, using an interior-point approach. These problems pose theoretical difficulties for nonlinear solvers, including interior-point methods. We examine the use of penalty methods to get around these difficulties and provide substantial numerical results. We go on to show that penalty methods can resolve some problems that interior-point algorithms encounter in general.

Stability of penalty finite‐element methods for nonconforming problems

AbstractPenalty methods have been proposed as a viable method for enforcing interelement continuity constraints on nonconforming elements. Particularly for fourth‐order problems in which C1‐continuity leads to elements of high degree or complex composite elements, the use of penalty methods to enforce the C1‐continuity constraint appears promising. In this study we demonstrate equivalence of the finite‐element penalty method to a hybrid method and provide a stability analysis which implies that the penalty method is stable only if reduced integration of a certain order is used. Numerical experiments confirm that the penalty method fails if this condition is not met.

Linear and Nonlinear Crack Analyses with the Use of Penalty Method

Linear and Nonlinear Crack Analyses with the Use of Penalty Method