On First-order Formulations of the Least-squares Finite Element Method for Incompressible Flows

Xu, Dong; Tsang, Tate T. H.

doi:10.1080/1061856031000123580

Cited by 11 publications

(14 citation statements)

References 20 publications

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“…Thus the Bayesian framework -with appropriate statistical modelling -leads us to the LSFEM+. Note that the connection between LSFEM+ and the statistical data-assimilation problem relies upon the particular choice of an additive discrepency δ in (22), with the specific covariance defined by (25). If an alternative covariance structure for δ is chosen, e.g.…”

Section: Calibrationmentioning

confidence: 99%

Data Assimilation for Navier-Stokes Using the Least-Squares Finite-Element Method

Schwarz

Dwight

2018

Int. J. UncertaintyQuantification

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We investigate theoretically and numerically the use of the Least-Squares Finite-element method (LSFEM) to approach data-assimilation problems for the steady-state, incompressible Navier-Stokes equations. Our LSFEM discretization is based on a stressvelocity-pressure (S-V-P) first-order formulation, using discrete counterparts of the Sobolev spaces H(div) × H 1 × L 2 for the variables respectively. In general S-V-P formulations are promising when the stresses are of special interest, e.g. for non-Newtonian, multiphase or turbulent flows. Resolution of the system is via minimization of a leastsquares functional representing the magnitude of the residual of the equations. A simple and immediate approach to extend this solver to data-assimilation is to add a datadiscrepancy term to the functional. Whereas most data-assimilation techniques require a large number of evaluations of the forward-simulation and are therefore very expensive, the approach proposed in this work uniquely has the same cost as a single forward run. However, the question arises: what is the statistical model implied by this choice? We answer this within the Bayesian framework, establishing the latent background covariance model and the likelihood. Further we demonstrate that -in the linear case -the method is equivalent to application of the Kalman filter, and derive the posterior covariance. We practically demonstrate the capabilities of our method on a backward-facing step case. Our LSFEM formulation (without data) is shown to have good approximation quality, even on relatively coarse meshes -in particular with respect to mass-conservation and reattachment location. Adding limited velocity measurements from experiment, we show that the method is able to correct for discretization error on very coarse meshes, as well as correct for the influence of unknown and uncertain boundary-conditions.

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Section: Calibrationmentioning

confidence: 99%

Data Assimilation for Navier-Stokes Using the Least-Squares Finite-Element Method

Schwarz

Dwight

2018

Int. J. UncertaintyQuantification

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“…A lid-driven cavity square cross section model was created, as shown in Figure 2(a). The three dimensions are 1 and three different numbers of cells with 30 3 , 60 3 and 120 3 were created to assess the acceleration ratio. The boundary condition is that, the upper lid given driving velocity of 1 m/s, the rest set as wall.…”

Section: Calculation Model and Computer Platformmentioning

confidence: 99%

“…In the past three decades, Jiang 1 and Bochev and Gunzburger 2 have developed the least squares finite element method (LSFEM) by combining the least squares method and the finite element method. LSFEM has been applied initially in the field of incompressible flow by Ding and Tsang 3 and Tang and Sun, 4 thermodynamics by Zhao et al 5 and Luo et al 6 and fluid-structure interaction by Kayser-Herold and Matthies. 7 LSFEM has the advantages of good convergence, good universality, good robustness and high accuracy.…”

Section: Introductionmentioning

confidence: 99%

“…Obviously, the less the time when dual GPUs are exchanging data, the higher the acceleration ratio. When the element number is small (less than 60 3 ), dual GPUs take more time than single GPU because of the time it takes to exchange data between GPUs. With the increase in element number, the proportion of time for data exchange decreases.…”

mentioning

confidence: 99%

“…Afterwards, lid-driven cavity flow cases with element number of 30 3 , 60 3 and 120 3 were calculated to assess the acceleration ratio. Companion between CPU and single GPU is shown in Table 2.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Massively parallel least squares finite element method with graphic processing unit

Zhong

Dai

et al. 2017

Advances in Mechanical Engineering

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For the reason of enormous computational expense, although the least squares finite element method has the advantages of high accuracy, robustness and strong versatility, the application of it is limited in computational fluid dynamics. The problems solved in this article include the rewriting of branching statements and the kernel function, variable distribution and data transfer between graphic processing units, and library functions rewriting. To the best knowledge of the authors, this article is the first time to develop the parallel computing codes for single and multiple graphic processing units based on the least squares finite element method. The computational results of single and multiple graphic processing units are verified by lid-driven cavity flow. Compared with a single central processing unit on the condition of 120 3 grids, the acceleration ratios of single and dual graphic processing units are up to 70.5 times and 95.2 times, respectively, which is much higher than the previous value of 7.7. With the increase in the grid number, the acceleration ratio of single and multiple graphic processing units is expected to increase, which can greatly enhance the computational efficiency of the least squares finite element method. Therefore, it is possible to solve the massive turbulence computing by the least squares finite element method with higher efficiency.

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Stress-Velocity Mixed Least-Squares FEMs for the Time-Dependent Incompressible Navier-Stokes Equations

Schwarz

Nisters

Averweg

et al. 2018

Large-Scale Scientific Computing

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On First-order Formulations of the Least-squares Finite Element Method for Incompressible Flows

Cited by 11 publications

References 20 publications

Data Assimilation for Navier-Stokes Using the Least-Squares Finite-Element Method

Data Assimilation for Navier-Stokes Using the Least-Squares Finite-Element Method

Massively parallel least squares finite element method with graphic processing unit

Stress-Velocity Mixed Least-Squares FEMs for the Time-Dependent Incompressible Navier-Stokes Equations

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