Large Scale Eigenvalue Calculations for Computing the Stability of Buoyancy Driven Flows

Burroughs, Elizabeth A.; Romero, Louis A.; Lehoucq, Richard B.; Salinger, Andrew G.

doi:10.2172/782594

Cited by 17 publications

(18 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This capability has been verified and validated for numerous fluid flow applications and has demonstrated parallel scaling to millions of unknowns [12,13], and is briefly described in Section 2.3. The LOCA (Library of Continuation Algorithms) library [14] has also been interfaced with the MPSalsa code for directly calculating bifurcations [15].…”

Section: 0604x10 5 =mentioning

confidence: 99%

See 1 more Smart Citation

Computational bifurcation and stability studies of the 8: 1 thermal cavity problem

Salinger

Lehoucq

Pawlowski

et al. 2002

Numerical Methods in Fluids

View full text Add to dashboard Cite

Summary:Stability analysis algorithms coupled with a robust Newton-Krylov steady state iterative solver are used to understand the behavior of the 2D model problem of thermal convection in a 8:1 differentially heated cavity. Parameter continuation methods along with bifurcation and linear stability analysis are used to study transition from steady to transient flow as a function of Rayleigh number. To carry out this study the steady state form of the governing PDEs is discretized using a Galerkin/Least Squares Finite Element formulation, and solved on parallel computers using a fully coupled Newton method and preconditioned Krylov iterative linear solvers. Linear stability analysis employing a large scale eigenvalue capability is used to determine the stability of the steady solutions. The boundary between steady and time dependent flows is determined by a Hopf bifurcation tracking capability that is used to directly track the instability with respect to the aspect ratio of the system and with respect to mesh resolution. The effect of upwinding stabilization terms in the finite element formulation on the computed value of critical Rayleigh number is investigated. The Hopf bifurcation signaling the onset of flow is determined to occur at a critical Rayleigh number of .

show abstract

Section: 0604x10 5 =mentioning

confidence: 99%

“…Details relating to the methods and parallel implementation of the linear stability analysis algorithms can be found in [24,12,13,25]. The analysis begins at a given steady state solution point.…”

Section: Linear Stability Analysis Algorithmsmentioning

confidence: 99%

Computational bifurcation and stability studies of the 8: 1 thermal cavity problem

Salinger

Lehoucq

Pawlowski

et al. 2002

Numerical Methods in Fluids

View full text Add to dashboard Cite

show abstract

“…Although similar techniques that compute the eigenvalues of the Jacobian matrix originated from the finite element discretization have been employed in other fields (e.g. stability of fluid dynamics [23] or thermal capillarity instability [24]), to the authors' knowledge, this methodology has never been employed to study local instabilities of shear bands.…”

Section: Background On Shear Bandsmentioning

confidence: 99%

Onset of shear band localization by a local generalized eigenvalue analysis

Arriaga

McAuliffe

Waisman

2015

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

Shear bands are material instabilities associated with highly localized intense plastic deformation zones which can form in materials undergoing high strain rates. Determining the onset of shear band localization is a difficult task and past work reported in the literature attempt to detect this instability by computing the eigenvalues of the acoustic tensor or by studying the linear stability of the perturbed governing equations. However, both methods have their limitations and are not suited for general rate dependent materials in multidimensions.In this work we propose a novel approach to determine the onset of shear band localization and alleviate the limitations of the above mentioned methods.Owing to the implicit mixed finite elements discretization employed in this work, we propose to cast the instability analysis as a generalized eigenvalue problem by employing a particular decomposition of the element Jacobian matrix. We show that this approach is attractive, as it is applicable to general rate dependent multidimensional cases where no special simplifying assumptions ought to be made.To verify the accuracy of the proposed eigenvalue analysis, we first extend an analytical criterion by applying linear perturbation techniques to the continuous PDE model, considering an elastoplastic material with thermal diffusion and a nonlinear Taylor-Quinney coefficient. While this extension is novel on its own, it requires strenuous derivations and is not easily extended to general multidimension applications. Hence, herein it is only used for verification purposes in 1D.Numerical results on one-dimensional problems show that the eigenvalue analysis exactly recovers the instability point predicted by the analytical criterion with non-linear Taylor-Quinney coefficient. In addition, the proposed generalized eigenvalue analysis is applied on two-dimensional problems where propagation of the instability can be easily determined.

show abstract

“…We have used the P A W A C K package [1] [2] to indentify a few select eigenvalues and eigenmodes. With this release of LOCA, we are including routines that drive P-ARPACK to perform the Cayley transformation, which we have found to work well for large scale problems [3] [4].…”

Section: Phase-transition-continuationmentioning

confidence: 99%

LOCA 1.0 Library of Continuation Algorithms: Theory and Implementation Manual

Salinger¹,

Bou-Rabee²,

Burroughs³

et al. 2002

Self Cite

103

View full text Add to dashboard Cite

LOCA, the Library of Continuation Algorithms, is a software library for performing stability analysis of large-scale applications. LOC A enables the tracking of solution branches as a function of a system parameter, the direct tracking of bifurcation points, and, when linked with the ARPACK library, a linear stability analysis capability. It is designed to be easy to implement around codes that already use Newton's method to converge to steady-state solutions. The algorithms are chosen to work for large problems, such as those that arise from discretizations of partial differential equations, and to run on distributed memory parallel machines. This manual presents LOCA's continuation and bifurcation analysis algorithms, and instructions on how to implement LOCA with an application code. The LOCA code is being made publicly available at www . cs . sandia. gov/loca. 3

show abstract

Large Scale Eigenvalue Calculations for Computing the Stability of Buoyancy Driven Flows

Cited by 17 publications

References 41 publications

Computational bifurcation and stability studies of the 8: 1 thermal cavity problem

Computational bifurcation and stability studies of the 8: 1 thermal cavity problem

Onset of shear band localization by a local generalized eigenvalue analysis

LOCA 1.0 Library of Continuation Algorithms: Theory and Implementation Manual

Contact Info

Product

Resources

About