A new finite element formulation for computational fluid dynamics: IX. Fourier analysis of space-time Galerkin/least-squares algorithms

Shakib, Farzin; Hughes, Thomas J.R.

doi:10.1016/0045-7825(91)90145-v

Cited by 156 publications

(110 citation statements)

References 10 publications

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“…Certainly, when the time discretization is introduced the effective stabilization parameters have to be modified (as it is done for example in [2,26,27]), but when the steady-state is reached the subscaleũ that is obtained as solution to (26) satisfiesũ = τ 1P (R u ), so that the usual expression employed for stationary problems is recovered.…”

Section: Main Properties Of the Formulationmentioning

confidence: 99%

“…One can consider as numerical dissipation the one that affects the finite element component alone. If we write the subscales emanating from (26) and (28) as…”

Section: Numerical Dissipationmentioning

confidence: 99%

“…When obtaining energy balance statements is when the importance of orthogonal and dynamic subgrid scales is more evident. To this end, it is enlightening not only to take v h = u h , q h = p h and ψ h = ϑ h (for each t ∈ (0, T )), but also to test the equations for the subscales (26) and (28) (recall that we are assuming τ 2 = 0) byũ andθ, respectively. If this is done, we get:…”

Section: Conservation Of Kinetic Energy and Heat Energymentioning

confidence: 99%

“…• Suppose for example that the backward Euler scheme is used to integrate (26). From the point of view of the algebraic solver, the factor…”

Section: Main Properties Of the Formulationmentioning

confidence: 99%

See 3 more Smart Citations

Finite element approximation of turbulent thermally coupled incompressible flows with numerical sub-grid scale modeling

Codina¹,

Principe²,

Ávila³

2010

International Journal of Numerical Methods for Heat & Fluid

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Section: Main Properties Of the Formulationmentioning

confidence: 99%

“…One can consider as numerical dissipation the one that affects the finite element component alone. If we write the subscales emanating from (26) and (28) as…”

Section: Numerical Dissipationmentioning

confidence: 99%

Section: Conservation Of Kinetic Energy and Heat Energymentioning

confidence: 99%

“…• Suppose for example that the backward Euler scheme is used to integrate (26). From the point of view of the algebraic solver, the factor…”

Section: Main Properties Of the Formulationmentioning

confidence: 99%

See 2 more Smart Citations

Finite element approximation of turbulent thermally coupled incompressible flows with numerical sub-grid scale modeling

Codina¹,

Principe²,

Ávila³

2010

International Journal of Numerical Methods for Heat & Fluid

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“…Two widely used approaches for handling moving boundary problems numerically (e.g., FVM, FEM) are the (1) Arbitrary Lagrangian Eulerian (ALE) formulation (Hughes et al 1981;Dettmer and Peric 2006) and (2) Space-time formulation (Shakib, 1989;Shakib and Hughes, 1991;Tezduyar et al, 1992 a,b;Perrochet and Azerat, 1995;Guler et al, 1999). The ALE formulation takes into account the Lagrangian (mesh moves with fluid) as well as Eulerian (mesh fixed in space) methods simultaneously such that the mesh movement in ALE is independent of fluid motion.…”

Section: Space-time Finite-element Formulationmentioning

confidence: 99%

On the Response of a Freely Vibrating Thick Elliptic Cylinder of Low Mass Ratio

Sourav¹,

Sen²

2017

JAFM

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Two-dimensional space-time finite-element simulations are carried out to study the free vibrations of a rigid elliptic cylinder of aspect ratio 1.11 and low non-dimensional mass of unity. Undamped transverse-only as well as two-degrees-of-freedom oscillations are considered. The effect of damping is investigated on transverseonly motion. For all three cases, results for cylinder response are presented for 50≤Re≤180. In the absence of damping, transverse oscillations are mostly periodic except for a very narrow region near the end of lock-in. In contrast, a damping of 0.044 removes quasi-periodicity as well as secondary hysteresis from flow and body motion. For undamped motion, inclusion of in-line oscillations excites high amplitude oscillations, widens the range of synchronization and delays phase shift between lift and cross-stream response. In each case, synchronization between cylinder oscillations and vortex-shedding is 1:1. In addition, drag-lift phase plots are symmetric about mean lift (= 0) line. Thus, symmetrical shedding of two equally strong alternate vortices per oscillation cycle forms 2S, CNW(2S) or C(2S) modes. For each case, the lower branch initiates at Re = 65 where the oscillation or shedding frequency is found to be locally maximum.

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A SPACE-TIME COUPLED p-VERSION LEAST SQUARES FINITE ELEMENT FORMULATION FOR UNSTEADY TWO-DIMENSIONAL NAVIER-STOKES EQUATIONS

Bell

Surana

1996

Int. J. Numer. Meth. Engng.

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SUMMARYThis paper presents a p-version least squares finite element formulation for two-dimensional unsteady fluid flow described by Navier-Stokes equations where the effects of space and time are coupled. The dimensionless form of the Navier-Stokes equations are first cast into a set of first-order differential equations by introducing auxiliary variables. This permits the use of Co element approximation. The element properties are derived by utilizing the p-version approximation functions in both space and time and then minimizing the error functional given by the spac*time integral of the sum of squares of the errors resulting from the set of first-order differential equations. This results in a true space-time coupled least squares minimization procedure. The application of least squares minimization to the set of coupled first-order partial differential equations results in finding a solution vector {d} which makes gradient of error functional with respect to {a} a null vector. This is accomplished by using Newton's method with a line search.A time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step. Equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance. The space-time coupled p-version approximation functions provide the ability to control truncation error which, in turn, permits very large time steps. What literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space-time coupled approach. The generality, success and superiority of the present formulation procedure is demonstrated by presenting specific numerical examples for transient couette flow and transient lid driven cavity. The results are compared with the analytical solutions and those reported in the literature. The formulation presented here is ideally suited for space-time adaptive procedures. The element error functional values provide a mechanism for adaptive h, p or hp refinements.

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A new finite element formulation for computational fluid dynamics: IX. Fourier analysis of space-time Galerkin/least-squares algorithms

Cited by 156 publications

References 10 publications

Finite element approximation of turbulent thermally coupled incompressible flows with numerical sub-grid scale modeling

Finite element approximation of turbulent thermally coupled incompressible flows with numerical sub-grid scale modeling

On the Response of a Freely Vibrating Thick Elliptic Cylinder of Low Mass Ratio

A SPACE-TIME COUPLED p-VERSION LEAST SQUARES FINITE ELEMENT FORMULATION FOR UNSTEADY TWO-DIMENSIONAL NAVIER-STOKES EQUATIONS

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