30 research outputs found
Finite element LES and VMS methods on tetrahedral meshes
AbstractFinite element methods for problems given in complex domains are often based on tetrahedral meshes. This paper demonstrates that the so-called rational Large Eddy Simulation model and a projection-based Variational Multiscale method can be extended in a straightforward way to tetrahedral meshes. Numerical studies are performed with an inf–sup stable second order pair of finite elements with discontinuous pressure approximation
An analysis of a linearly extrapolated BDF2 subgrid artificial viscosity method for incompressible flows
This report extends the mathematical support of a subgrid artificial
viscosity (SAV) method to simulate the incompressible Navier-Stokes equations
to better performing a linearly extrapolated BDF2 (BDF2LE) time discretization.
The method considers the viscous term as a combination of the vorticity and the
grad-div stabilization term. SAV method introduces global stabilization by
adding a term, then antidiffuses through the extra mixed variables. We present
a detailed analysis of conservation laws, including both energy and helicity
balance of the method. We also show that the approximate solutions of the
method are unconditionally stable and optimally convergent. Several numerical
tests are presented for validating the support of the derived theoretical
results
Analysis of discretization errors in LES
All numerical simulations of turbulence (DNS or LES) involve some discretization errors. The integrity of such simulations therefore depend on our ability to quantify and control such errors. In the classical literature on analysis of errors in partial differential equations, one typically studies simple linear equations (such as the wave equation or Laplace's equation). The qualitative insight gained from studying such simple situations is then used to design numerical methods for more complex problems such as the Navier-Stokes equations. Though such an approach may seem reasonable as a first approximation, it should be recognized that strongly nonlinear problems, such as turbulence, have a feature that is absent in linear problems. This feature is the simultaneous presence of a continuum of space and time scales. Thus, in an analysis of errors in the one dimensional wave equation, one may, without loss of generality, rescale the equations so that the dependent variable is always of order unity. This is not possible in the turbulence problem since the amplitudes of the Fourier modes of the velocity field have a continuous distribution. The objective of the present research is to provide some quantitative measures of numerical errors in such situations. Though the focus of this work is LES, the methods introduced here can be just as easily applied to DNS. Errors due to discretization of the time-variable are neglected for the purpose of this analysis
Convergence Analysis of a Fully Discrete Family of Iterated Deconvolution Methods for Turbulence Modeling with Time Relaxation
We present a general theory for regularization models of the Navier-Stokes equations based on the Leray deconvolution model with a general deconvolution operator designed to fit a few important key properties. We provide examples of this type of operator, such as the (modified) Tikhonov-Lavrentiev and (modified) Iterated Tikhonov-Lavrentiev operators, and study their mathematical properties. An existence theory is derived for the family of models and a rigorous convergence theory is derived for the resulting algorithms. Our theoretical results are supported by numerical testing with the Taylor-Green vortex problem, presented for the special operator cases mentioned above
Symptom Status Predicts Patient Outcomes in Persons with HIV and Comorbid Liver Disease
Persons living with human immunodeficiency virus (HIV) are living longer; therefore, they are more likely to suffer significant morbidity due to potentially treatable liver diseases. Clinical evidence suggests that the growing number of individuals living with HIV and liver disease may have a poorer health-related quality of life (HRQOL) than persons living with HIV who do not have comorbid liver disease. Thus, this study examined the multiple components of HRQOL by testing Wilson and Cleary’s model in a sample of 532 individuals (305 persons with HIV and 227 persons living with HIV and liver disease) using structural equation modeling. The model components include biological/physiological factors (HIV viral load, CD4 counts), symptom status (Beck Depression Inventory II and the Medical Outcomes Study HIV Health Survey (MOS-HIV) mental function), functional status (missed appointments and MOS-HIV physical function), general health perceptions (perceived burden visual analogue scale and MOS-HIV health transition), and overall quality of life (QOL) (Satisfaction with Life Scale and MOS-HIV overall QOL). The Wilson and Cleary model was found to be useful in linking clinical indicators to patient-related outcomes. The findings provide the foundation for development and future testing of targeted biobehavioral nursing interventions to improve HRQOL in persons living with HIV and liver disease
Variable Time Step Method of DAHLQUIST, LINIGER and NEVANLINNA (DLN) for a Corrected Smagorinsky Model
Turbulent flows strain resources, both memory and CPU speed. The DLN method
has greater accuracy and allows larger time steps, requiring less memory and
fewer FLOPS. The DLN method can also be implemented adaptively. The classical
Smagorinsky model, as an effective way to approximate a (resolved) mean
velocity, has recently been corrected to represent a flow of energy from
unresolved fluctuations to the (resolved) mean velocity. In this paper, we
apply a family of second-order, G-stable time-stepping methods proposed by
Dahlquist, Liniger, and Nevanlinna (the DLN method) to one corrected
Smagorinsky model and provide the detailed numerical analysis of the stability
and consistency. We prove that the numerical solutions under any arbitrary time
step sequences are unconditionally stable in the long term and converge at
second order. We also provide error estimate under certain time step condition.
Numerical tests are given to confirm the rate of convergence and also to show
that the adaptive DLN algorithm helps to control numerical dissipation so that
backscatter is visible
Radially Symmetric Solutions of
We investigate solutions of and focus on the regime and . Our advance is to develop a technique to efficiently classify the behavior of solutions on , their maximal positive interval of existence. Our approach is to transform the nonautonomous equation into an autonomous ODE. This reduces the problem to analyzing the phase plane of the autonomous equation. We prove the existence of new families of solutions of the equation and describe their asymptotic behavior. In the subcritical case there is a well-known closed-form singular solution, , such that as and as . Our advance is to prove the existence of a family of solutions of the subcritical case which satisfies for infinitely many values . At the critical value there is a continuum of positive singular solutions, and a continuum of sign changing singular solutions. In the supercritical regime we prove the existence of a family of “super singular” sign changing singular solutions
Optimal analysis of finite element methods for the stochastic Stokes equations
Numerical analysis for the stochastic Stokes equations is still challenging
even though it has been well done for the corresponding deterministic
equations. In particular, the pre-existing error estimates of finite element
methods for the stochastic Stokes equations { in the norm} all suffer from the order reduction with respect to
the spatial discretizations. The best convergence result obtained for these
fully discrete schemes is only half-order in time and first-order in space,
which is not optimal in space in the traditional sense. The objective of this
article is to establish strong convergence of in the
norm for approximating the velocity, and
strong convergence of in the norm for approximating the time integral of pressure, where
and denote the temporal step size and spatial mesh size,
respectively. The error estimates are of optimal order for the spatial
discretization considered in this article (with MINI element), and consistent
with the numerical experiments. The analysis is based on the fully discrete
Stokes semigroup technique and the corresponding new estimates