25,475 research outputs found
Spectral/hp element methods: recent developments, applications, and perspectives
The spectral/hp element method combines the geometric flexibility of the
classical h-type finite element technique with the desirable numerical
properties of spectral methods, employing high-degree piecewise polynomial
basis functions on coarse finite element-type meshes. The spatial approximation
is based upon orthogonal polynomials, such as Legendre or Chebychev
polynomials, modified to accommodate C0-continuous expansions. Computationally
and theoretically, by increasing the polynomial order p, high-precision
solutions and fast convergence can be obtained and, in particular, under
certain regularity assumptions an exponential reduction in approximation error
between numerical and exact solutions can be achieved. This method has now been
applied in many simulation studies of both fundamental and practical
engineering flows. This paper briefly describes the formulation of the
spectral/hp element method and provides an overview of its application to
computational fluid dynamics. In particular, it focuses on the use the
spectral/hp element method in transitional flows and ocean engineering.
Finally, some of the major challenges to be overcome in order to use the
spectral/hp element method in more complex science and engineering applications
are discussed
Derivation of asymptotic two-dimensional time-dependent equations for ocean wave propagation
A general method for the derivation of asymptotic nonlinear shallow water and
deep water models is presented. Starting from a general dimensionless version
of the water-wave equations, we reduce the problem to a system of two equations
on the surface elevation and the velocity potential at the free surface. These
equations involve a Dirichlet-Neumann operator and we show that all the
asymptotic models can be recovered by a simple asymptotic expansion of this
operator, in function of the shallowness parameter (shallow water limit) or the
steepness parameter (deep water limit). Based on this method, a new
two-dimensional fully dispersive model for small wave steepness is also
derived, which extends to uneven bottom the approach developed by Matsuno
\cite{matsuno3} and Choi \cite{choi}. This model is still valid in shallow
water but with less precision than what can be achieved with Green-Naghdi
model, when fully nonlinear waves are considered. The combination, or the
coupling, of the new fully dispersive equations with the fully nonlinear
shallow water Green-Naghdi equations represents a relevant model for describing
ocean wave propagation from deep to shallow waters
A well-balanced meshless tsunami propagation and inundation model
We present a novel meshless tsunami propagation and inundation model. We
discretize the nonlinear shallow-water equations using a well-balanced scheme
relying on radial basis function based finite differences. The inundation model
relies on radial basis function generated extrapolation from the wet points
closest to the wet-dry interface into the dry region. Numerical results against
standard one- and two-dimensional benchmarks are presented.Comment: 20 pages, 13 figure
A Comprehensive Three-Dimensional Model of the Cochlea
The human cochlea is a remarkable device, able to discern extremely small
amplitude sound pressure waves, and discriminate between very close
frequencies. Simulation of the cochlea is computationally challenging due to
its complex geometry, intricate construction and small physical size. We have
developed, and are continuing to refine, a detailed three-dimensional
computational model based on an accurate cochlear geometry obtained from
physical measurements. In the model, the immersed boundary method is used to
calculate the fluid-structure interactions produced in response to incoming
sound waves. The model includes a detailed and realistic description of the
various elastic structures present.
In this paper, we describe the computational model and its performance on the
latest generation of shared memory servers from Hewlett Packard. Using compiler
generated threads and OpenMP directives, we have achieved a high degree of
parallelism in the executable, which has made possible several large scale
numerical simulation experiments that study the interesting features of the
cochlear system. We show several results from these simulations, reproducing
some of the basic known characteristics of cochlear mechanics.Comment: 22 pages, 5 figure
Implementation of the LANS-alpha turbulence model in a primitive equation ocean model
This paper presents the first numerical implementation and tests of the
Lagrangian-averaged Navier-Stokes-alpha (LANS-alpha) turbulence model in a
primitive equation ocean model. The ocean model in which we work is the Los
Alamos Parallel Ocean Program (POP); we refer to POP and our implementation of
LANS-alpha as POP-alpha. Two versions of POP-alpha are presented: the full
POP-alpha algorithm is derived from the LANS-alpha primitive equations, but
requires a nested iteration that makes it too slow for practical simulations; a
reduced POP-alpha algorithm is proposed, which lacks the nested iteration and
is two to three times faster than the full algorithm. The reduced algorithm
does not follow from a formal derivation of the LANS-alpha model equations.
Despite this, simulations of the reduced algorithm are nearly identical to the
full algorithm, as judged by globally averaged temperature and kinetic energy,
and snapshots of temperature and velocity fields. Both POP-alpha algorithms can
run stably with longer timesteps than standard POP.
Comparison of implementations of full and reduced POP-alpha algorithms are
made within an idealized test problem that captures some aspects of the
Antarctic Circumpolar Current, a problem in which baroclinic instability is
prominent. Both POP-alpha algorithms produce statistics that resemble
higher-resolution simulations of standard POP.
A linear stability analysis shows that both the full and reduced POP-alpha
algorithms benefit from the way the LANS-alpha equations take into account the
effects of the small scales on the large. Both algorithms (1) are stable; (2)
make the Rossby Radius effectively larger; and (3) slow down Rossby and gravity
waves.Comment: Submitted to J. Computational Physics March 21, 200
A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces
In this paper we present a high-order kernel method for numerically solving
diffusion and reaction-diffusion partial differential equations (PDEs) on
smooth, closed surfaces embedded in . For two-dimensional
surfaces embedded in , these types of problems have received
growing interest in biology, chemistry, and computer graphics to model such
things as diffusion of chemicals on biological cells or membranes, pattern
formations in biology, nonlinear chemical oscillators in excitable media, and
texture mappings. Our kernel method is based on radial basis functions (RBFs)
and uses a semi-discrete approach (or the method-of-lines) in which the surface
derivative operators that appear in the PDEs are approximated using
collocation. The method only requires nodes at "scattered" locations on the
surface and the corresponding normal vectors to the surface. Additionally, it
does not rely on any surface-based metrics and avoids any intrinsic coordinate
systems, and thus does not suffer from any coordinate distortions or
singularities. We provide error estimates for the kernel-based approximate
surface derivative operators and numerically study the accuracy and stability
of the method. Applications to different non-linear systems of PDEs that arise
in biology and chemistry are also presented
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