11,720 research outputs found
An unstructured parallel least-squares spectral element solver for incompressible flow problems
The parallelization of the least-squares spectral element formulation of the Stokes problem has recently been discussed for incompressible flow problems on structured grids. In the present work, the extension to unstructured grids is discussed. It will be shown that, to obtain an efficient and scalable method, two different kinds of distribution of data are required involving a rather complicated parallel conversion between the data. Once the data conversion has been performed, a large symmetric positive definite algebraic system has to be solved iteratively. It is well known that the Conjugate Gradient method is a good choice to solve such systems. To improve the convergence rate of the Conjugate Gradient process, both Jacobi and Additive Schwarz preconditioners are applied. The Additive Schwarz preconditioner is based on domain decomposition and can be implemented such that a preconditioning step corresponds to a parallel matrix-by-vector product. The new results reveal that the Additive Schwarz preconditioner is very suitable for the p-refinement version of the least-squares spectral element method. To obtain good portable programs which may run on distributed-memory multiprocessors, networks of workstations as well as shared-memory machines we use MPI (Message Passing Interface). Numerical simulations have been performed to validate the scalability of the different parts of the proposed method. The experiments entailed simulating several large scale incompressible flows on a Cray T3E and on an SGI Origin 3800 with the number of processors varying from one to more than one hundred. The results indicate that the present method has very good parallel scaling properties making it a powerful method for numerical simulations of incompressible flows
An Unstructured Parallel Least-Squares Spectral Element Solver for Incompressible Flow Problems
The parallelization of the least-squares spectral element formulation of the Stokes problem has recently been discussed for incompressible flow problems on structured grids. In the present work, the extension to unstructured grids is discussed. It will be shown that, to obtain an efficient and scalable method, two different kinds of distribution of data are required involving a rather complicated parallel conversion between the data. Once the data conversion has been performed, a large symmetric positive definite algebraic system has to be solved iteratively. It is well known that the Conjugate Gradient method is a good choice to solve such systems. To improve the convergence rate of the Conjugate Gradient process, both Jacobi and Additive Schwarz preconditioners are applied. The Additive Schwarz preconditioner is based on domain decomposition and can be implemented such that a preconditioning step corresponds to a parallel matrix-by-vector product. The new results reveal that the Additive Schwarz preconditioner is very suitable for the p-refinement version of the least-squares spectral element method. To obtain good portable programs which may run on distributed-memory multiprocessors, networks of workstations as well as shared-memory machines we use MPI (Message Passing Interface). Numerical simulations have been performed to validate the scalability of the different parts of the proposed method. The experiments entailed simulating several large scale incompressible flows on a Cray T3E and on an SGI Origin 3800 with the number of processors varying from one to more than one hundred. The results indicate that the present method has very good parallel scaling properties making it a powerful method for numerical simulations of incompressible flows
A p-version finite element method for steady incompressible fluid flow and convective heat transfer
A new p-version finite element formulation for steady, incompressible fluid flow and convective heat transfer problems is presented. The steady-state residual equations are obtained by considering a limiting case of the least-squares formulation for the transient problem. The method circumvents the Babuska-Brezzi condition, permitting the use of equal-order interpolation for velocity and pressure, without requiring the use of arbitrary parameters. Numerical results are presented to demonstrate the accuracy and generality of the method
Incorporating variable viscosity in vorticity-based formulations for Brinkman equations
In this brief note, we introduce a non-symmetric mixed finite element
formulation for Brinkman equations written in terms of velocity, vorticity and
pressure with non-constant viscosity. The analysis is performed by the
classical Babu\v{s}ka-Brezzi theory, and we state that any inf-sup stable
finite element pair for Stokes approximating velocity and pressure can be
coupled with a generic discrete space of arbitrary order for the vorticity. We
establish optimal a priori error estimates which are further confirmed through
computational example
Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. II: The incompressible Navier-Stokes equations
This paper presents the construction of a correct-energy stabilized finite
element method for the incompressible Navier-Stokes equations. The framework of
the methodology and the correct-energy concept have been developed in the
convective--diffusive context in the preceding paper [M.F.P. ten Eikelder, I.
Akkerman, Correct energy evolution of stabilized formulations: The relation
between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric
analysis. I: The convective--diffusive context, Comput. Methods Appl. Mech.
Engrg. 331 (2018) 259--280]. The current work extends ideas of the preceding
paper to build a stabilized method within the variational multiscale (VMS)
setting which displays correct-energy behavior. Similar to the
convection--diffusion case, a key ingredient is the proper dynamic and
orthogonal behavior of the small-scales. This is demanded for correct energy
behavior and links the VMS framework to the streamline-upwind Petrov-Galerkin
(SUPG) and the Galerkin/least-squares method (GLS).
The presented method is a Galerkin/least-squares formulation with dynamic
divergence-free small-scales (GLSDD). It is locally mass-conservative for both
the large- and small-scales separately. In addition, it locally conserves
linear and angular momentum. The computations require and employ NURBS-based
isogeometric analysis for the spatial discretization. The resulting formulation
numerically shows improved energy behavior for turbulent flows comparing with
the original VMS method.Comment: Update to postprint versio
Least squares deconvolution of the stellar intensity and polarization spectra
Least squares deconvolution (LSD) is a powerful method of extracting
high-precision average line profiles from the stellar intensity and
polarization spectra. Despite its common usage, the LSD method is poorly
documented and has never been tested using realistic synthetic spectra. In this
study we revisit the key assumptions of the LSD technique, clarify its
numerical implementation, discuss possible improvements and give
recommendations how to make LSD results understandable and reproducible. We
also address the problem of interpretation of the moments and shapes of the LSD
profiles in terms of physical parameters. We have developed an improved,
multiprofile version of LSD and have extended the deconvolution procedure to
linear polarization analysis taking into account anomalous Zeeman splitting of
spectral lines. This code is applied to the theoretical Stokes parameter
spectra. We test various methods of interpreting the mean profiles,
investigating how coarse approximations of the multiline technique translate
into errors of the derived parameters. We find that, generally, the Stokes
parameter LSD profiles do not behave as a real spectral line with respect to
the variation of magnetic field and elemental abundance. This problem is
especially prominent for the Stokes I variation with abundance and Stokes Q
variation with magnetic field. At the same time, the Stokes V LSD spectra
closely resemble profile of a properly chosen synthetic line for the magnetic
field strength up to 1 kG. We conclude that the usual method of interpreting
the LSD profiles by assuming that they are equivalent to a real spectral line
gives satisfactory results only in a limited parameter range and thus should be
applied with caution. A more trustworthy approach is to abandon the single-line
approximation of the average profiles and apply LSD consistently to
observations and synthetic spectra.Comment: Accepted for publication in Astronomy & Astrophysics; 15 pages, 12
figures; second version includes minor language correction
Least-squares solution of incompressible Navier-Stokes equations with the p-version of finite elements
A p-version of the least squares finite element method, based on the velocity-pressure-vorticity formulation, is developed for solving steady state incompressible viscous flow problems. The resulting system of symmetric and positive definite linear equations can be solved satisfactorily with the conjugate gradient method. In conjunction with the use of rapid operator application which avoids the formation of either element of global matrices, it is possible to achieve a highly compact and efficient solution scheme for the incompressible Navier-Stokes equations. Numerical results are presented for two-dimensional flow over a backward facing step. The effectiveness of simple outflow boundary conditions is also demonstrated
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