Spectral/hp penalty least-squares finite element formulation for the steady incompressible Navier–Stokes equations

Prabhakar, V.; Reddy, J. N.

doi:10.1016/j.jcp.2005.10.033

Cited by 40 publications

(23 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Consequently, most finite element models of the Navier-Stokes equations based on the weak-form Galerkin procedure do not guarantee the minimization of the error in the solution or in the differential equation. Least-squares finite element models offers an appealing alternative to the commonly used weak-form Galerkin procedure for fluids and have received substantial attention in the academic literature in recent years (see, for example, [21,22,24,31,33,28,27,36,37,35,30]). The least-squares formulation allows for the construction of finite element models for fluids that, when combined with high-order finite element technology [22,4,5,38,17,29,31,31,49] possess many of the attractive qualities associated with the well-known Ritz method [43] such as global minimization, best approximation with respect to a well-defined norm, and symmetric positive-definiteness of the resulting finite element coefficient matrix [9].…”

mentioning

confidence: 99%

RETRACTED: Higher-order Spectral/hp Finite Element Technology for Shells and Flows of Viscous Incompressible Fluids

Vallala

Reddy

2013

MCA

View full text Add to dashboard Cite

Abstract-This study deals with the use of high-order spectral/hp approximation functions in finite element models of various of nonlinear boundary-value and initial-value problems arising in the fields of structural mechanics and flows of viscous incompressible fluids. For many of these classes of problems, the high-order (typically, polynomial order 4  p ) spectral/hp finite element technology offers many computational advantages over traditional low-order (i.e., 3 < p ) finite elements. For instance, higher-order spectral/hp finite element procedures allow us to develop robust structural elements such as beams, plates, and shells in a purely displacement-based setting, which avoid all forms of numerical locking. For fluid flows, when combined with least-squares variational principles, the higher-order spectral/hp technology allows us to develop efficient finite element models that always yield a symmetric positive-definite (SPD) coefficient matrix and, hence, robust iterative solvers can be used. Also, the use of spectral/hp finite element technology results in a better conservation of physical quantities like dilatation, volume, and mass, and stable evolution of variables with time for transient flows. The present study considers the weak-form based displacement finite element models elastic shells and the least-squares finite element models of the Navier-Stokes equations governing flows of viscous incompressible fluids. Numerical solutions of several nontrivial benchmark problems are presented to illustrate the accuracy and robustness of the developed finite element technology.

show abstract

mentioning

confidence: 99%

RETRACTED: Higher-order Spectral/hp Finite Element Technology for Shells and Flows of Viscous Incompressible Fluids

Vallala

Reddy

2013

MCA

View full text Add to dashboard Cite

show abstract

“…The Type 4 Benchmark Example given in Section 3.1.1, the driven-cavity problem, demonstrates some of the difficulties encountered with solutions containing singularities. Prabhakar and Reddy [83]) eliminated the two singularities in the moving-lid corners by replacing the fixed speed of the moving lid with a speed that varies spatially near each of the corners. They clearly state that had they not removed the singularities, their numerical procedure would not have converged.…”

Section: Accuracy Assessment Issues For Type 4 Benchmark (Pde Numericmentioning

confidence: 99%

Verification and validation benchmarks

Oberkampf

Trucano²

2008

Nuclear Engineering and Design

180

View full text Add to dashboard Cite

Verification and validation (V&V) are the primary means to assess the accuracy and reliability of computational simulations. V&V methods and procedures have fundamentally improved the credibility of simulations in several high-consequence fields, such as nuclear reactor safety, underground nuclear waste storage, and nuclear weapon safety. Although the terminology is not uniform across engineering disciplines, code verification deals with assessing the reliability of the software coding, and solution verification deals with assessing the numerical accuracy of the solution to a computational model. Validation addresses the physics modeling accuracy of a computational simulation by comparing the computational results with experimental data. Code verification benchmarks and validation benchmarks have been constructed for a number of years in every field of computational simulation. However, no comprehensive guidelines have been proposed for the construction and use of V&V benchmarks. For example, the field of nuclear reactor safety has not focused on code verification benchmarks, but it has placed great emphasis on developing validation benchmarks. Many of these validation benchmarks are closely related to the operations of actual reactors at near-safety-critical conditions, as opposed to being more fundamental-physics benchmarks. This paper presents recommendations for the effective design and use of code verification benchmarks based on manufactured solutions, classical analytical solutions, and highly accurate numerical solutions. In addition, this paper presents recommendations for the design and use of validation benchmarks, highlighting the careful design of building-block experiments, the estimation of experimental measurement uncertainty for both inputs and outputs to the code, validation metrics, and the role of model calibration in validation. It is argued that the understanding of predictive capability of a computational model is built on the level of achievement in V&V activities, how closely related the V&V benchmarks are to the actual application of interest, and the quantification of uncertainties related to the application of interest.

show abstract

“…The seventh-order nodal expansion is used in each element, and there are a total of 39 204 degrees of freedom in the mesh. All internal degrees of freedom are condensed out using Schur complement method (see [6] for details), resulting in 10 980 interface degrees of freedom with a bandwidth of 788. The preconditioned conjugate gradient is used to solve for the interface degrees of freedom.…”

Section: Two-dimensional Lid-driven Cavity Flowmentioning

confidence: 99%

Spectral/hp penalty least‐squares finite element formulation for unsteady incompressible flows

Prabhakar

Reddy

2008

Numerical Methods in Fluids

Self Cite

View full text Add to dashboard Cite

SUMMARYIn this paper, we present spectral/hp penalty least-squares finite element formulation for the numerical solution of unsteady incompressible Navier-Stokes equations. Pressure is eliminated from Navier-Stokes equations using penalty method, and finite element model is developed in terms of velocity, vorticity and dilatation. High-order element expansions are used to construct discrete form. Unlike other penalty finite element formulations, equal-order Gauss integration is used for both viscous and penalty terms of the coefficient matrix. For time integration, space-time decoupled schemes are implemented. Secondorder accuracy of the time integration scheme is established using the method of manufactured solution. Numerical results are presented for impulsively started lid-driven cavity flow at Reynolds number of 5000 and transient flow over a backward-facing step. The effect of penalty parameter on the accuracy is investigated thoroughly in this paper and results are presented for a range of penalty parameter. Present formulation produces very accurate results for even very low penalty parameters (10-50).

show abstract

Spectral/hp penalty least-squares finite element formulation for the steady incompressible Navier–Stokes equations

Cited by 40 publications

References 25 publications

RETRACTED: Higher-order Spectral/hp Finite Element Technology for Shells and Flows of Viscous Incompressible Fluids

RETRACTED: Higher-order Spectral/hp Finite Element Technology for Shells and Flows of Viscous Incompressible Fluids

Verification and validation benchmarks

Spectral/hp penalty least‐squares finite element formulation for unsteady incompressible flows

Contact Info

Product

Resources

About