Reduced Integration Penalty Research Articles

A simple staggered approach for comprehensive analysis of forced convection heat transfer using the improved element-free Galerkin–reduced integration penalty method to solve the fluid dynamics problem

Forced convection heat transfer problems are successfully solved in this work via a simple staggered approach based on the improved element-free Galerkin (IEFG) formulation, which involves a fluid-dynamic analysis conducted in the framework of Navier–Stokes equations written in terms of velocity via a reduced integration penalty method (RIPM). Subsequently, the velocity field achieved from the solution of the Navier–Stokes equations is used in the IEFG-based solution of the internal energy balance formulated in terms of temperature. The proposed approach is tested in the solution and analysis of forced convection heat transfer problems concerning (i) the simultaneously (thermally and fluid-dynamically) developing flow between parallel plates and, (ii) the confined recirculating flow in non-isothermal lid-driven square cavities (also including viscous dissipation effects). A comprehensive parametric analysis developed in terms of characteristic dimensionless numbers is conducted for both problems, providing reliable benchmark results that will be useful to assess the performance of numerical techniques to model forced convection heat transfer phenomena. The versatility and reliability of the proposed approach are also demonstrated in the solution of a more complex problem with curved geometry, such as the non-isothermal lid-driven semicircular cavity with a circular obstacle. The accuracy and feasibility of solving forced convection heat transfer problems via the mesh-less procedure proposed in this communication are proven by comparison with results achieved via mesh-based techniques, and also with analytical solutions available in the literature. The outcomes demonstrate that the appropriate implementation of such numerical technique allows the achievement of accurate and stable results in a straightforward and remarkably simple manner, even under markedly advection-dominated flow conditions in both thermal and fluid dynamics problems.

Solving steady-state lid-driven square cavity flows at high Reynolds numbers via a coupled improved element-free Galerkin–reduced integration penalty method

Steady-state two-dimensional lid-driven square cavity flows at high Reynolds numbers are solved in this communication using a velocity-based formulation developed in the context of the improved element-free Galerkin–reduced integration penalty method (IEFG–RIPM). The analyses based on the IEFG–RIPM are performed under a standard Galerkin weak-formulation, i.e. without the need of introducing streamline-upwind or pressure stabilizing terms in order to suppress the appearance of non-physical oscillations. Solutions in a wide range of high Reynolds numbers are successfully achieved, and the results obtained have exhibited an excellent agreement with mesh-based solutions reported in the literature. The numerical performance of the proposed IEFG–RIPM in the solution of such benchmark problem has been analysed in terms of velocity distribution, streamline patterns, vorticity and pressure contours, and also in terms of the properties of primary and secondary vortices. The outcomes of this study demonstrate the potential of the IEFG–RIPM as a feasible and reliable numerical technique for the analysis of lid-driven square cavity flows at high Reynolds numbers, allowing the fulfilment of accuracy and stability numerical requirements demanded in the solution of this benchmark problem in a remarkably simple manner.

A plausible extension of standard penalty, streamline upwind and immersed boundary techniques to the improved element-free Galerkin-based solution of incompressible Navier–Stokes equations

The present work has been conducted in order to propose the extension of standard penalty and stabilization techniques to the improved element-free Galerkin (IEFG) method, for the numerical solution of incompressible fluid flow problems. In principle, the numerical procedures to be implemented in this communication have been conceived for finite element method (FEM)-based solutions, and these include the reduced integration penalty method (RIPM), the streamline upwind Petrov–Galerkin (SUPG) scheme, and a penalty-based immersed boundary method (PBIBM) for the imposition of essential boundary conditions along internal fluid–solid interfaces. The linear momentum balance and mass conservation equations have been coupled via the RIPM, in order to obtain a global weak formulation where the IEFG model is entirely developed in terms of improved moving least squares (IMLS) approximations of the velocity field. A detailed explanation concerning the appropriate extension of both the RIPM and SUPG procedures to the context of IEFG formulations, has also been provided. The resulting formulation has been applied to the solution of two well-known benchmark problems: i) Lid-driven square cavity flow, and ii) Flow past a fixed cylinder. Regarding the flow past a fixed cylinder benchmark problem, the fluid–solid interaction has been imposed as an internal immersed boundary condition via the PBIBM. The feasibility and reliability of implementing the RIPM, SUPG and PBIBM procedures in the IEFG formulation, have been proven by comparison with experimental and mesh-based numerical results reported in the literature. The results obtained in this study have revealed that a proper extension of the aforementioned penalty and stabilization techniques to the IEFG formulation, allows the achievement of accurate and stable numerical results during the solution of incompressible fluid-dynamics problems.

Implementation of standard penalty procedures for the solution of incompressible Navier–Stokes equations, employing the element-free Galerkin method

The present work introduces an alternative implementation of the element-free Galerkin method (EFGM) for the steady incompressible Navier–Stokes equations. The linear momentum balance and mass conservation equations have been developed on the basis of this global weak formulation. The weak forms of both equations have been coupled by means of two standard penalty procedures, which have been previously formulated and successfully employed in mesh-based numerical techniques such as the finite element method (FEM). These include the consistent penalty method (CPM) and reduced integration penalty method (RIPM). A detailed explanation concerning the characteristics inherent to the implementation of both penalty procedures in the EFGM based solutions for the incompressible Navier–Stokes equations, has also been provided. The resulting systems of equations have been adapted to the solution of different well-known incompressible flow benchmark problems. The feasibility and reliability of extending the implementation of these penalty procedures to the EFGM based solutions has been verified by comparison with the numerical techniques proposed and the results reported by other researchers. Results have revealed that this technique could be successfully used in the solution of Newtonian incompressible flow problems under the aforementioned penalty approaches.

Alternative least‐squares finite element models of Navier‐Stokes equations for power‐law fluids

PurposeMost studies of power‐law fluids are carried out using stress‐based system of Navier‐Stokes equations; and least‐squares finite element models for vorticity‐based equations of power‐law fluids have not been explored yet. Also, there has been no study of the weak‐form Galerkin formulation using the reduced integration penalty method (RIP) for power‐law fluids. Based on these observations, the purpose of this paper is to fulfill the two‐fold objective of formulating the least‐squares finite element model for power‐law fluids, and the weak‐form RIP Galerkin model of power‐law fluids, and compare it with the least‐squares finite element model.Design/methodology/approachFor least‐squares finite element model, the original governing partial differential equations are transformed into an equivalent first‐order system by introducing additional independent variables, and then formulating the least‐squares model based on the lower‐order system. For RIP Galerkin model, the penalty function method is used to reformulate the original problem as a variational problem subjected to a constraint that is satisfied in a least‐squares (i.e. approximate) sense. The advantage of the constrained problem is that the pressure variable does not appear in the formulation.FindingsThe non‐Newtonian fluids require higher‐order polynomial approximation functions and higher‐order Gaussian quadrature compared to Newtonian fluids. There is some tangible effect of linearization before and after minimization on the accuracy of the solution, which is more pronounced for lower power‐law indices compared to higher power‐law indices. The case of linearization before minimization converges at a faster rate compared to the case of linearization after minimization. There is slight locking that causes the matrices to be ill‐conditioned especially for lower values of power‐law indices. Also, the results obtained with RIP penalty model are equally good at higher values of penalty parameters.Originality/valueVorticity‐based least‐squares finite element models are developed for power‐law fluids and effects of linearizations are explored. Also, the weak‐form RIP Galerkin model is developed.

Analysis of some mixed elements for the Stokes problem

In this paper we discuss some mixed finite element methods related to the reduced integration penalty method for solving the Stokes problem. We prove optimal order error estimates for bilinear-constant and biquadratic-bilinear velocity-pressure finite element solutions. The result for the biquadratic-bilinear element is new, while that for the bilinear-constant element improves the convergence analysis of Johnson and Pitkäranta (1982). In the degenerate case when the penalty parameter is set to be zero, our results reduce to some related known results proved in by Brezzi and Fortin (1991) for the bilinear-constant element, and Bercovier and Pironneau (1979) for the biquadratic-bilinear element. Our theoretical results are consistent with the numerical results reported by Carey and Krishnan (1982) and Oden et al. (1982).

Alternative methods for the solution of hyperelastic problems with incompressibility

A new approach for analysing incompressible hyperelastic and rubber-like material behaviour is discussed. The approach is based on extending the linear analysis bubble function concept into nonlinear analysis. To eliminate external or residual forces on the bubble function or nodeless degrees of freedom, two iterative methods are discussed. The new approach is compared to other approaches available in the literature including multified principle, reduced integration penalty and average constraint procedures. A unified numerical treatment for handling various constitutive relations of hyperelastic and rubber-like materials is presented. Numerical examples and comparisons include thick-walled cylinder subjected to internal pressure, inflation of a circular rubber plate and inflation of a hollow sphere.

On the computed pressures for Navier‐Stokes problems at increasing Reynolds numbers using the penalty finite element method

DISCUSSION The reduced integration penalty finite element method has been frequently used to compute approximate velocity solutions to Navier- Stokes problems.’-4 The velocity approximation can be post-processed to obtain an approximate pressure solution. Numerical studies for Stokes have shown that for problems where the data (boundary conditions or the forcing function) are ‘rough’, these computed pressures may exhibit oscillations for certain elements and reduced integration schemes, whereas smoother pressures are obtained for other (‘stable’) elements. An example of such a problem is the familiar ‘driven cavity’ problem. For this problem, both in the case of Stokes flow and low Reynolds number Navier-Stokes flow, when the 4-node bilinear element with 1-point Gauss integration of the penalty term is used, the pressures computed from the velocity field are found to be oscillatory, whereas when the 9-node biquadratic element with 1point integration of the penalty term is used, smooth pressure profiles are obtained. Recently, we have been investigating the use of continuation techniques and iterative methods for computing approximate solutions at higher Reynolds numbers. On examining the computed pressures for the driven cavity we observed that as the Reynolds number increased, the local pressure oscillations for the bilinear element diminish and by Re = 2000 the pressure profiles at representative sections appear smooth. As an example, results are now given for the cavity problem and bilinear I-point element for calculations on a uniform mesh of size h = 1/32 and with penalty parameter E = The pressure profile at Re = 10 along the section y = 17/64 is given in Figure 1 (marked + ) and is seen to contain local pressure oscillations. The projected pressure obtained by averaging the four element pressures adjacent to a node is also given and is smooth (marked x ). Analogous results are also observed for Stokes flow (Re = 0). However, the computed pressure profile at J’ = 17/64 for Re = 2000 appears smooth (Figure 2).+ Similar behaviour is observed at other representative sections and is not shown here. Results with penalty parameter E = and E = Approximate solutions were also computed using the mixed method with Co biquadratic velocity and Co bilinear pressure on uniform meshes with h = 1/16 and h = 1/25. This element is known to be stable. The pressure solution is smooth as anticipated and qualitatively agrees with that obtained with the penalty method. However, quantitatively the pressure profiles agree were essentially the same.

Fully Plastic Crack Problems, Part 1: Solutions by a Penalty Method

We formulate a finite-element reduced integration penalty method applicable to plane-strain problems with incompressible material behavior. This numerical method is employed to generate crack solutions for pure power-hardening solids. For two configurations of interest in applications, an edge cracked panel subject to remote tension and an edge-cracked panel subject to remote bending, we obtain solutions for a wide range of crack lengths and strain-hardening behaviors.

Two comments on: Consistent vs reduced integration penalty methods for incompressible media using several old and new elements1

Two comments on: Consistent vs reduced integration penalty methods for incompressible media using several old and new elements¹

A smoothing technique for reduced integration penalty methods in contact problems

AbstractThe paper presents a smoothing technique to the oscillated contact pressure obtained by penalty methods for a class of unilateral contact problems in linear elasticity. The main result is to show that the smoothed contact pressure satisfies the so‐called Babuska–Brezzi condition which dominates the convergence of the penalty method. One numerical example is described.

Consistent vs. reduced integration penalty methods for incompressible media using several old and new elements

AbstractThe frequently used reduced integration method for solving incompressible flow problems ‘a la penalty’ is critically examined vis‐a‐vis the consistent penalty method. For the limited number of quadrilateral and hexahedral elements studied, it is shown that the former method is only equivalent to the latter in certain special cases. In the general case, the consistent penalty method is shown to be more accurate. Finally, we demonstrate significant advantages of a new element, employing biquadratic (2‐D) or triquadratic (3‐D) velocity and linear pressure over that using the same velocity but employing bilinear (2‐D) or trilinear (3‐D) pressure approximation.