Penalty Technique Research Articles

Numerical analysis of fluid squeezed between two parallel plates by meshless method

This paper presents the steady-state and transient analysis of the fluid squeezed between two long parallel plates. The governing coupled partial differential equations have been discretized by element free Galerkin method and implemented using variational approach. Penalty and Lagrange multiplier techniques have been utilized to enforce the essential boundary conditions. Four point Gauss quadrature has been used to evaluate the viscous terms in the coefficient matrix whereas reduced integration scheme (i.e. one point Gauss quadrature) has been used to evaluate the penalty terms over two-dimensional domain ( Ω). Cubicspline, exponential and rational weight functions have been used in the present work. The results obtained by EFG method are compared with those obtained by finite element and analytical methods. The effect of scaling and penalty parameters on EFG results has been discussed in detail.

Using a modified genetic algorithm to minimize the production costs for slabs of precast prestressed concrete joists

Genetic algorithms (GAs) are search methods that have been successfully applied to a variety of tasks. This paper describes the use of a modified GA as an optimization method in structural engineering for minimizing the production costs of slabs using precast prestressed concrete joists. The work initially identifies and describes the multiple costs involved in the production of these slabs and then combines them into a function subjected to 28 equality and inequality constraints. The experiments conducted address the minimization of this function using GA, where constraints are treated using a penalty technique. In addition, results obtained with a conventional optimization method are presented, for comparison.

Rank‐based ant colony algorithms for truss weight minimization with discrete variables

AbstractOne variant of the ant colony optimization (ACO) metaheuristic, known as rank‐based ant system (RBAS), is proposed for weight minimization of structures involving discrete design variables. Stress and displacements constraints are handled using a penalty technique, and the structural analysis is performed by the finite element method.Results obtained for several discrete weight minimization problems show that the RBAS algorithm implemented is effective and competitive when compared with results found by genetic algorithms and another RBAS variant. Copyright © 2006 John Wiley & Sons, Ltd.

An interior penalty Galerkin method for the MHD equations in heterogeneous domains

The Maxwell equations in the magnetohydrodynamic (MHD) limit in heterogeneous domains composed of conducting and non-conducting regions are solved by using Lagrange finite elements and by enforcing continuities across interfaces using an Interior Penalty technique à la Baker [Finite element methods for elliptic equations using non-conforming elements, Math. Comp. 31 (137) (1977) 45–59]. The method is shown to be stable and convergent and is validated by convergence tests. It is used to compute Ohmic decay in various compact conducting domains and to simulate the kinematic dynamo action in two different geometries.

Penalty guided genetic search for redundancy optimization in multi-state series-parallel power system

This paper presents a genetic algorithm (GA) for parallel redundancy optimization in series-parallel power systems exhibiting multi-state behavior, optimizing the reliability subject to constraints. The components are binary and chosen from a list of products available in the market, and are being characterized by their feeding capacity, reliability, cost and weight. System reliability is defined as the ability to satisfy consumer demand and is presented as a piecewise cumulative load curve. In GA, to handle infeasible solutions penalty strategies are used. Penalty technique keep a certain amount of infeasible solutions in each generation so as to enforce genetic search towards an optimal solution from sides of, both, feasible and infeasible regions. We here present a dynamic adaptive penalty function which helps the algorithm to search efficiently for optimal/near optimal solution. To evaluate system reliability, a fast procedure, based on universal generating function, is used. An example considering a multi-state series-parallel power system is solved considering both homogeneous and heterogeneous types of redundancy. Also an example considering price discounts is solved. The effectiveness of the penalty function and the proposed algorithm is studied and shown graphically.

Element-free Galerkin method for static and dynamic analysis of spatial shell structures

The implementation of the element free Galerkin method (EFG) for static and free vibration analysis of general shell structures is presented in this paper. The formulation of the discrete system equations is derived from the governing equations of stress resultant geometrically exact shell theory based on the Cosserat surface. A discrete singularity-free mapping between the five- and the six-degrees-of-freedom formulation is constructed by exploiting the geometry connection between the orthogonal group and the unit sphere. Moving least-squares approximation is used in both the construction of shape functions based on arbitrarily distributed nodes as well as in the surface approximation of general spatial shell geometry. Discrete system equations are obtained by incorporating these interpolations into the Galerkin weak form. The formulation is verified through numerical examples of static stress analysis and frequency analysis of spatial shell structures. The phenomena of shear locking and membrane locking are illustrated by showing the membrane and shear energy as fractions of the total energy. Essential boundary conditions are efficiently imposed through a penalty technique for both static analysis and frequency analysis. The formulation is tested on several benchmark problems and the results compare favorably with closed-form solutions and that of finite element analyses.

Heat transfer analysis of composite slabs using meshless element Free Galerkin method

This paper deals with three dimensional heat transfer analysis of composite slabs using meshless element free Galerkin method. The element free Galerkin method (EFG) method utilizes moving least square (MLS) approximants to approximate the unknown function of temperature Tx). These approximants are constructed by using a weight function, a basis function and a set of coefficients that depends on position. Penalty and Lagrange multiplier techniques have been used to enforce the essential boundary conditions. MATLAB codes have been developed to obtain the EFG results. Two new basis functions namely trigonometric and polynomial have been proposed. A comparison has been made among the results obtained using existing (linear) and proposed (trigonometric and polynomial) basis functions for three dimensional heat transfer in composite slabs. The effect of penalty parameter on EFG results has also been discussed. The results obtained by EFG method are compared with those obtained by finite element method

Improvement of the flow formulation by recovery of the hydrostatic pressure using the finite point method

AbstractMost of the existing numerical methods based on penalty technique, lead to pressure fields, which are not satisfactory especially when crude meshes are used. The present work for the first time, using a simple and effective finite point approach approximates the total hydrostatic stresses. By this new technique, the nodal hydrostatic pressure values are calculated so that the gradient of hydrostatic pressure satisfies the equilibrium equation, with regard to the current distribution of deviatoric stresses. This approach is verified using a cantilever beam with known analytical stress field. It is also applied to a typical 2D axisymmetric bulk metal forming process. The calculated pressure field is then compared with available solutions. The results of the present approach are very encouraging. Copyright © 2005 John Wiley & Sons, Ltd.

Application of the element-free Galerkin meshless method to 3-D fracture mechanics problems

The present paper deals with the development of a simple meshless method, known as element-free Galerkin method (EFG), and its numerical implementation and application for the solution of 3-D elastic fracture mechanics problems. Meshless methods are rather new computational techniques that do not require the use of any connectivity concept, such as those used in the finite element method (FEM); since only a cloud of nodes is required, the EFG method is particularly suitable for problems involving internal boundaries, geometry changes, and so on. In the present paper the development of the EFG method and its numerical implementation to 3-D linear elasticity is presented with emphasis to the solution of problems with geometric discontinuities such as cracks. The description of the 3-D body is performed by simply employing triangles in space to describe edges (external or internal) and by generating a grid of internal points; if required, a local cloud of nodes concentrated around high stressed zones can be added to the model. The “visibility criterion” is used to detect internal or external boundaries and a Gauss-type weight function, together with a penalty technique, is employed to enforce the boundary conditions. With the developed EFG method some fundamental 3-D fracture mechanics problems are solved in order to verify the computational capability and accuracy of the method.

Adaptive Wavelet Method for Incompressible Flows in Complex Domains

Abstract An adaptive wavelet-based method provides an alternative means to refine grids according to local demands of the physical solution. One of the prominent challenges of such a method is the application to problems defined on complex domains. In the case of incompressible flow, the application to problems with complicated domains is made possible by the use of the Navier-Stokes–Brinkman equations. These equations take into account solid obstacles by adding a penalized velocity term in the momentum equation. In this study, an adaptive wavelet collocation method, based on interpolating wavelets, is first applied to a benchmark problem defined on a simple domain to demonstrate the accuracy and efficiency of the method. Then the penalty technique is used to simulate flows over obstacles. The numerical results are compared to those obtained by other computational approaches as well as to experiments.

A self-adaptive penalty approach for nonlinear programming

For penalty approaches, it is extremely difficult to choose penalty parameter and penalty term to avoid ill-conditions. In this work, penalty techniques fornonlinear programming problems are revisited. A new penalty approach, which is called self-adaptive penalty method, is proposed. When penalty parameter and penalty term are considered, we adjust them according to the feedback information at each step (and the previous steps) in the new method so that the second information of the subproblem may be a semidefinite positive matrix whose conditioned number is not too large.

Optimal control of salinity boundary condition in a tidal model using a variational inverse method

A variational inverse data assimilation scheme is developed to estimate the salinity boundary conditions in a three-dimensional tidal hydrodynamic and salinity transport model. In this paper, the maximum incoming salinity value and the recovery time from the outflow salinity to the maximum incoming salinity at model open boundaries are treated as poorly known model control variables, and estimated using a variational inverse data assimilation scheme. The variational inverse model is tested in an idealized estuary using identical twin experiments, in which observed data are generated from the same model. Model tests with different initial guesses of the model control variables are conducted to evaluate the capability of the inverse model. A penalty technique is used to eliminate oscillations in the solution during the minimization process. The effects of preconditioning and penalty terms on the convergence rate are investigated. Model results demonstrate that the variational inverse model can be used to efficiently determine the optimal salinity open boundary conditions and improve the model state when there are no observed data available to specify the proper salinity open boundary conditions in a tidal model.

A spectral multidomain penalty method model for the simulation of high Reynolds number localized incompressible stratified turbulence

A spectral multidomain penalty method model has been developed for the simulation of high Reynolds number localized stratified turbulence. This is the first time that a penalty method, with a particular focus on subdomain interface treatment, has been used with a multidomain scheme to simulate incompressible flows. The temporal discretization ensures maximum temporal accuracy by combining third order stiffly stable and backward differentiation schemes with a high-order boundary condition for the pressure. In the non-periodic vertical direction, a spectral multidomain discretization is used and its stability for under-resolved simulations at high Reynolds numbers is ensured through use of penalty techniques, spectral filtering and strong adaptive interfacial averaging. The penalty method is implemented in different formulations for both the explicit non-linear term advancement and the implicit treatment of the viscous terms. The multidomain model is validated by comparing results of simulations of the mid-to-late time stratified turbulent wake with non-zero net momentum to the corresponding laboratory data for a towed sphere. The model replicates correctly the characteristic vorticity and internal wave structure of the stratified wake and exhibits robust agreement with experiments in terms of the temporal power laws in the evolution of mean profile characteristic velocity and lengthscales.

A POSTERIORI ANALYSIS OF A PENALTY METHOD AND APPLICATION TO THE STOKES PROBLEM

We derive an a posteriori error estimate for an abstract saddle-point problem when a penalty term is added to stabilize the variational formulation, the aim being to optimize the choice of the penalty parameter. As an application, we consider a discretization of the Stokes problem obtained by combining the penalty technique and the finite element method, we perform its a posteriori analysis in a detailed way and present some numerical experiments on adaptive meshes which are in good agreement with the results of the analysis and confirm its interest.

Availability allocation to repairable systems with genetic algorithms: a multi-objective formulation

Abstract This paper describes a methodology based on genetic algorithms (GA) and experiments plan to optimize the availability and the cost of reparable parallel-series systems. It is a NP-hard problem of multi-objective combinatorial optimization, modeled with continuous and discrete variables. By using the weighting technique, the problem is transformed into a single-objective optimization problem whose constraints are then relaxed by the exterior penalty technique. We then propose a search of solution through GA, whose parameters are adjusted using experiments plan technique. A numerical example is used to assess the method.

Comparison between the radial point interpolation and the Kriging interpolation used in meshfree methods

The radial point interpolation method (RPIM) based on local supported radial basis function (RBF) and the Galerkin weak form has been developed and successfully applied to many engineering problems. Recently, a new meshfree method was proposed based on the universal moving Kriging interpolation. This paper studies the difference between the meshfree shape functions created based on the point interpolation and the Kriging interpolation. It is found that both the two methods yield the same shape function as long as the same radial basis function or semivariogram is adopted for interpolation. Although the two methods lead to the same shape function, the theorem in Kriging formulation may provide an alternative theoretical support for the RPIM. Some common semivariograms used in Kriging may also be incorporated in the RPIM. In addition, in order to satisfy the conformability requirements, a penalty technique is introduced in this paper to form a conforming Kriging, which can pass the standard patch test exactly.

A mesh-free method for static and free vibration analysis of shear deformable laminated composite plates

A mesh-free method is presented to analyze the static deflection and natural frequencies of thin and thick laminated composite plates using high order shear deformation theory. In the present method, the problem domain is represented by a set of properly scattered nodes and no element conformability is required. Moving least-squares method is applied to construct the shape functions. Variational principle is used to derive the discrete system equations based on the third order shear deformation theory (TSDT) of Reddy. Essential boundary conditions are efficiently implemented by a penalty technique for both the static deflection and natural frequency analysis. Several examples are solved to demonstrate the convergence, accuracy and validity of the proposed method. The present solutions are verified with those available values by analytical as well as finite element method. The results from classical plate theory and first order shear deformation theory are also computed and compared with those of TSDT. The effects of the material coefficients, side-to-thickness ratio, nodal distribution and shear correction factor are discussed.

Analysis of contact of elastic rods subject to large displacements

We present a mathematical model describing the motion of two elastic rods in contact. The model allows for large displacements and is essentially based on Cosserat's modelling of rods. Using a penalty technique, we prove existence of a static solution in the case of a unique rod. The contact modelling involves unilateral constraints on the central lines of the rods. Existence is also proved for this contact problem. 2003 Elsevier Science Ltd. All rights reserved.

Bending analysis of mindlin-reissner plates by the element free galerkin method with penalty technique

In this work, a new penalty formulation is proposed for the analysis of Mindlin-Reissner plates by using the element-free Galerkin method. A penalized weak form for the Mindlin-Reissner Plates is constructed through the exterior penalty method to enforce the essential boundary conditions of rotations as well as transverse displacements. In the numerical examples, some typical problems of Mindlin-Reissner plates are analyzed, and parametric studies on the order of integration and the size of influence domain are also carried out. The effect of the types of background cells on the accuracy of numerical solutions is observed and a proper type of background cell for obtaining optimal accuracy is suggested. Further, optimal order of integration and basis order of Moving Least Squares approximation are suggested to efficiently handle the irregularly distributed nodes through the triangular type of background cells. From the numerical tests, it is identified that unlike the finite element method, the proposed element-free Galerkin method with penalty technique gives highly accurate solution without shear locking in dealing with Mindlin-Reissner plates.

Alternative Quasi-Newton Methods for Capacitated User Equilibrium Assignment

Two quasi-Newton methods are proposed to deal with traffic assignment in a capacitated network. The methods combine Newton formula, column generation, and penalty techniques. The first method uses the gradient of the objective function to obtain an improving feasible direction scaled by the second-order derivatives. The second one uses a Rosen gradient to obtain an improving direction scaled by the corresponding origin–destination demand. Both methods make a line search to obtain an optimal step size to guarantee the feasibility of either path or link flow. The proposed methods are of fast convergence and high accuracy at the expense of saving path information. Numerical examples verify their efficiency and stability. The quasi-Newton method with a straight gradient demonstrates more stability than the Rosen gradient for capacitated traffic assignment.