27,110 research outputs found
Boundary knot method: A meshless, exponential convergence, integration-free, and boundary-only RBF technique
Based on the radial basis function (RBF), non-singular general solution and
dual reciprocity principle (DRM), this paper presents an inheretnly meshless,
exponential convergence, integration-free, boundary-only collocation techniques
for numerical solution of general partial differential equation systems. The
basic ideas behind this methodology are very mathematically simple and
generally effective. The RBFs are used in this study to approximate the
inhomogeneous terms of system equations in terms of the DRM, while non-singular
general solution leads to a boundary-only RBF formulation. The present method
is named as the boundary knot method (BKM) to differentiate it from the other
numerical techniques. In particular, due to the use of non-singular general
solutions rather than singular fundamental solutions, the BKM is different from
the method of fundamental solution in that the former does no need to introduce
the artificial boundary and results in the symmetric system equations under
certain conditions. It is also found that the BKM can solve nonlinear partial
differential equations one-step without iteration if only boundary knots are
used. The efficiency and utility of this new technique are validated through
some typical numerical examples. Some promising developments of the BKM are
also discussed.Comment: 36 pages, 2 figures, Welcome to contact me on this paper: Email:
[email protected] or [email protected]
Fast finite difference solvers for singular solutions of the elliptic Monge-Amp\`ere equation
The elliptic Monge-Ampere equation is a fully nonlinear Partial Differential
Equation which originated in geometric surface theory, and has been applied in
dynamic meteorology, elasticity, geometric optics, image processing and image
registration. Solutions can be singular, in which case standard numerical
approaches fail. In this article we build a finite difference solver for the
Monge-Ampere equation, which converges even for singular solutions. Regularity
results are used to select a priori between a stable, provably convergent
monotone discretization and an accurate finite difference discretization in
different regions of the computational domain. This allows singular solutions
to be computed using a stable method, and regular solutions to be computed more
accurately. The resulting nonlinear equations are then solved by Newton's
method. Computational results in two and three dimensions validate the claims
of accuracy and solution speed. A computational example is presented which
demonstrates the necessity of the use of the monotone scheme near
singularities.Comment: 23 pages, 4 figures, 4 tables; added arxiv links to references, added
coment
Multiparameter spectral analysis for aeroelastic instability problems
This paper presents a novel application of multiparameter spectral theory to
the study of structural stability, with particular emphasis on aeroelastic
flutter. Methods of multiparameter analysis allow the development of new
solution algorithms for aeroelastic flutter problems; most significantly, a
direct solver for polynomial problems of arbitrary order and size, something
which has not before been achieved. Two major variants of this direct solver
are presented, and their computational characteristics are compared. Both are
effective for smaller problems arising in reduced-order modelling and
preliminary design optimization. Extensions and improvements to this new
conceptual framework and solution method are then discussed.Comment: 20 pages, 8 figure
Refraction-corrected ray-based inversion for three-dimensional ultrasound tomography of the breast
Ultrasound Tomography has seen a revival of interest in the past decade,
especially for breast imaging, due to improvements in both ultrasound and
computing hardware. In particular, three-dimensional ultrasound tomography, a
fully tomographic method in which the medium to be imaged is surrounded by
ultrasound transducers, has become feasible. In this paper, a comprehensive
derivation and study of a robust framework for large-scale bent-ray ultrasound
tomography in 3D for a hemispherical detector array is presented. Two
ray-tracing approaches are derived and compared. More significantly, the
problem of linking the rays between emitters and receivers, which is
challenging in 3D due to the high number of degrees of freedom for the
trajectory of rays, is analysed both as a minimisation and as a root-finding
problem. The ray-linking problem is parameterised for a convex detection
surface and three robust, accurate, and efficient ray-linking algorithms are
formulated and demonstrated. To stabilise these methods, novel
adaptive-smoothing approaches are proposed that control the conditioning of the
update matrices to ensure accurate linking. The nonlinear UST problem of
estimating the sound speed was recast as a series of linearised subproblems,
each solved using the above algorithms and within a steepest descent scheme.
The whole imaging algorithm was demonstrated to be robust and accurate on
realistic data simulated using a full-wave acoustic model and an anatomical
breast phantom, and incorporating the errors due to time-of-flight picking that
would be present with measured data. This method can used to provide a
low-artefact, quantitatively accurate, 3D sound speed maps. In addition to
being useful in their own right, such 3D sound speed maps can be used to
initialise full-wave inversion methods, or as an input to photoacoustic
tomography reconstructions
An efficient parallel immersed boundary algorithm using a pseudo-compressible fluid solver
We propose an efficient algorithm for the immersed boundary method on
distributed-memory architectures, with the computational complexity of a
completely explicit method and excellent parallel scaling. The algorithm
utilizes the pseudo-compressibility method recently proposed by Guermond and
Minev [Comptes Rendus Mathematique, 348:581-585, 2010] that uses a directional
splitting strategy to discretize the incompressible Navier-Stokes equations,
thereby reducing the linear systems to a series of one-dimensional tridiagonal
systems. We perform numerical simulations of several fluid-structure
interaction problems in two and three dimensions and study the accuracy and
convergence rates of the proposed algorithm. For these problems, we compare the
proposed algorithm against other second-order projection-based fluid solvers.
Lastly, the strong and weak scaling properties of the proposed algorithm are
investigated
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