9,911 research outputs found
A note on the penalty parameter in Nitsche's method for unfitted boundary value problems
Nitsche's method is a popular approach to implement Dirichlet-type boundary
conditions in situations where a strong imposition is either inconvenient or
simply not feasible. The method is widely applied in the context of unfitted
finite element methods. From the classical (symmetric) Nitsche's method it is
well-known that the stabilization parameter in the method has to be chosen
sufficiently large to obtain unique solvability of discrete systems. In this
short note we discuss an often used strategy to set the stabilization parameter
and describe a possible problem that can arise from this. We show that in
specific situations error bounds can deteriorate and give examples of
computations where Nitsche's method yields large and even diverging
discretization errors
Exploring Periodic Orbit Expansions and Renormalisation with the Quantum Triangular Billiard
A study of the quantum triangular billiard requires consideration of a
boundary value problem for the Green's function of the Laplacian on a trianglar
domain. Our main result is a reformulation of this problem in terms of coupled
non--singular integral equations. A non--singular formulation, via Fredholm's
theory, guarantees uniqueness and provides a mathematically firm foundation for
both numerical and analytic studies. We compare and contrast our reformulation,
based on the exact solution for the wedge, with the standard singular integral
equations using numerical discretisation techniques. We consider in detail the
(integrable) equilateral triangle and the Pythagorean 3-4-5 triangle. Our
non--singular formulation produces results which are well behaved
mathematically. In contrast, while resolving the eigenvalues very well, the
standard approach displays various behaviours demonstrating the need for some
sort of ``renormalisation''. The non-singular formulation provides a
mathematically firm basis for the generation and analysis of periodic orbit
expansions. We discuss their convergence paying particular emphasis to the
computational effort required in comparision with Einstein--Brillouin--Keller
quantisation and the standard discretisation, which is analogous to the method
of Bogomolny. We also discuss the generalisation of our technique to smooth,
chaotic billiards.Comment: 50 pages LaTeX2e. Uses graphicx, amsmath, amsfonts, psfrag and
subfigure. 17 figures. To appear Annals of Physics, southern sprin
The statistical mechanics of multi-index matching problems with site disorder
We study the statistical mechanics of multi-index matching problems where the
quenched disorder is a geometric site disorder rather than a link disorder. A
recently developed functional formalism is exploited which yields exact results
in the finite temperature thermodynamic limit. Particular attention is paid to
the zero temperature limit of maximal matching problems where the method allows
us to obtain the average value of the optimal match and also sheds light on the
algorithmic heuristics leading to that optimal matchComment: 11 pages 11 figures, RevTe
Conformally invariant scaling limits in planar critical percolation
This is an introductory account of the emergence of conformal invariance in
the scaling limit of planar critical percolation. We give an exposition of
Smirnov's theorem (2001) on the conformal invariance of crossing probabilities
in site percolation on the triangular lattice. We also give an introductory
account of Schramm-Loewner evolutions (SLE(k)), a one-parameter family of
conformally invariant random curves discovered by Schramm (2000). The article
is organized around the aim of proving the result, due to Smirnov (2001) and to
Camia and Newman (2007), that the percolation exploration path converges in the
scaling limit to chordal SLE(6). No prior knowledge is assumed beyond some
general complex analysis and probability theory.Comment: 55 pages, 10 figure
The finite element method in low speed aerodynamics
The finite element procedure is shown to be of significant impact in design of the 'computational wind tunnel' for low speed aerodynamics. The uniformity of the mathematical differential equation description, for viscous and/or inviscid, multi-dimensional subsonic flows about practical aerodynamic system configurations, is utilized to establish the general form of the finite element algorithm. Numerical results for inviscid flow analysis, as well as viscous boundary layer, parabolic, and full Navier Stokes flow descriptions verify the capabilities and overall versatility of the fundamental algorithm for aerodynamics. The proven mathematical basis, coupled with the distinct user-orientation features of the computer program embodiment, indicate near-term evolution of a highly useful analytical design tool to support computational configuration studies in low speed aerodynamics
Approximation and geometric modeling with simplex B-splines associated with irregular triangles
Bivariate quadratic simplical B-splines defined by their corresponding set of knots derived from a (suboptimal) constrained Delaunay triangulation of the domain are employed to obtain a C1-smooth surface. The generation of triangle vertices is adjusted to the areal distribution of the data in the domain. We emphasize here that the vertices of the triangles initially define the knots of the B-splines and do generally not coincide with the abscissae of the data. Thus, this approach is well suited to process scattered data.\ud
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With each vertex of a given triangle we associate two additional points which give rise to six configurations of five knots defining six linearly independent bivariate quadratic B-splines supported on the convex hull of the corresponding five knots.\ud
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If we consider the vertices of the triangulation as threefold knots, the bivariate quadratic B-splines turn into the well known bivariate quadratic Bernstein-BĂ©zier-form polynomials on triangles. Thus we might be led to think of B-splines as of smoothed versions of Bernstein-BĂ©zier polynomials with respect to the entire domain. From the degenerate Bernstein-BĂ©zier situation we deduce rules how to locate the additional points associated with each vertex to establish knot configurations that allow the modeling of discontinuities of the function itself or any of its directional derivatives. We find that four collinear knots out of the set of five defining an individual quadratic B-spline generate a discontinuity in the surface along the line they constitute, and that analogously three collinear knots generate a discontinuity in a first derivative.\ud
Finally, the coefficients of the linear combinations of normalized simplicial B-splines are visualized as geometric control points satisfying the convex hull property.\ud
Thus, bivariate quadratic B-splines associated with irregular triangles provide a great flexibility to approximate and model fast changing or even functions with any given discontinuities from scattered data.\ud
An example for least squares approximation with simplex splines is presented
Grid generation for the solution of partial differential equations
A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given
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