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Antiplane wave scattering from a cylindrical cavity in pre-stressed non-linear elastic media
Get PDFThe effect of a longitudinal stretch and a pressure-induced inhomogeneous radial deformation on the scattering of antiplane elastic waves from a cylindrical cavity is determined. Three popular non-linear strain energy functions are considered: the neo-Hookean, the Mooney-Rivlin and a two-term Arruda-Boyce model. A new method is developed to analyse and solve the governing wave equations. It exploits their properties to determine an asymptotic solution in the far-field, which is then used to derive a boundary condition to numerically evaluate the equations local to the cavity. This method could be applied to any linear ordinary differential equation whose inhomogeneous coefficients tend to a constant as its independent variable tends to infinity. The effect of the pre-stress is evaluated by considering the scattering cross-section. A longitudinal stretch is found to decrease the scattered power emanating from the cavity, whereas a compression increases it. The effect of the pressure difference depends on the strain energy function employed. For a Mooney-Rivlin material, a cavity inflation increases the scattered power and a deflation decreases it; for a neo-Hookean material, the scattering cross-section is unaffected by the radial deformation; and for a two-term Arruda-Boyce material, both inflation and deflation are found to decrease the scattered power
The point-line collinearity graph of the Fi'_{24} maximal 2-local geometry - the first three discs
Get PDFThe disc structure of the point-line collinearity graph for the maximal 2-local geometry associated with the largest simple Fischer group is investigated. For an arbitrary vertex of this graph the first three discs are determined. Additionally a fragment of the fourth disc is uncovered
A sparse linearization for Hermite interpolation matrix polynomials
Get PDFThe polynomial eigenvalue problem for Hermite interpolation matrix polynomials is
discussed. The standard approach to solve a polynomial eigenvalue problem is via
linearization. In this work we introduce a new linearization for Hermite interpolation
matrix polynomials expressed in the first barycentric form that is more sparse than
the ones known so far. In addition, we show that this linearization is a strong
linearization, and that eigenvectors of the polynomial and those of the linearization
are related in simple ways. Finally, the backward errors of computed eigenpairs of
the original and the linearized problem are compared as well as eigenvalue condition
numbers
Anderson Acceleration of the Alternating Projections Method for Computing the Nearest Correlation Matrix
Get PDFIn a wide range of applications it is required to compute the nearest correlation matrix in the Frobenius norm to a given symmetric but indefinite matrix. Of the available methods with guaranteed convergence to the unique solution of this problem the easiest to implement, and perhaps the most widely used, is the alternating projections method. However, the rate of convergence of this method is at best linear, and it can require a large number of iterations to converge to within a given tolerance. We show that Anderson acceleration, a technique for accelerating the convergence of fixed-point iterations, can be applied to the alternating projections method and that in practice it brings a significant reduction in both the number of iterations and the computation time. We also show that Anderson acceleration remains effective, and indeed can provide even greater improvements, when it is applied to the variants of the nearest correlation matrix problem in which specified elements are fixed or a lower bound is imposed on the smallest eigenvalue. Alternating projections is a general method for finding a point in the intersection of several sets and ours appears to be the first demonstration that this class of methods can benefit from Anderson acceleration
Constrained Approximation of Effective Generators for Multiscale Stochastic Reaction Networks and Application to Conditioned Path Sampling
Get PDFEfficient analysis and simulation of multiscale systems of chemical kinetics is an ongoing area for research, and is the source of many theoretical and compu- tational challenges. In this paper, we present a significant improvement to the constrained approach, which allows us to compute the effective generator of the slow variables, without the need for expensive stochastic simulations. This is done through finding the null space of the generator of the constrained system. For complex systems where this is not possible, the constrained approach can then be applied in turn to the constrained system in a nested manner, mean- ing that the problem can be broken down into solving many small eigenvalue problems. Moreover, this methodology does not rely on the quasi steady-state assumption, meaning that the effective dynamics that are approximated are highly accurate, and in the case of systems with only monomolecular reactions, are exact. We will demonstrate this with some numerics, and also use the effective generators to sample paths which are conditioned on their endpoints
Ranking the Importance of Nuclear Reactions for Activation and Transmutation Events
Get PDFPathways-reduced analysis is one of the techniques used by the Fispact-II nuclear activation and transmutation software to study the sensitivity of the computed inventories to uncertainties in reaction cross-sections. Although deciding which pathways are most important is very helpful in for example determining which nuclear data would benefit from further refinement, pathways-reduced analysis need not necessarily define the most critical reaction, since one reaction may contribute to several different path-
ways. This work examines three different techniques for ranking reactions in their order of importance in determining the final inventory, viz. a pathways based metric (PBM), the direct method and one based on the Pearson correlation coefficient. Reasons why the PBM is to be preferred are presented
Geometry of refractions and reflections through a biperiodic medium
Get PDFThe behaviour of light rays obeying Snell's Law in a medium made up of two materials with different refractive indices and which are arranged in a periodic chessboard pattern is described. The analysis is in some ways analogous to the study of rational billiards and uses a return map on one surface to prove, amongst other things, that the number of angles with which any individual ray intersects the lattice is bounded and that if the ratio of refractive indices is large enough then the dynamics can be described by interval exchange maps
Chebyshev rootfinding via computing eigenvalues of colleague matrices: when is it stable?
Get PDFComputing the roots of a scalar polynomial, or the eigenvalues of a matrix polynomial, expressed in the Chebyshev basis {T_k(x)} is a fundamental problem that arises in many applications.
In this work, we analyze the backward stability of the polynomial rootfinding problem solved with colleague matrices.
In other words, given a scalar polynomial p(x) or a matrix polynomial P(x) expressed in the Chebyshev basis, the question is to determine whether the whole set of computed eigenvalues of the colleague matrix, obtained with a backward stable algorithm, like the QR algorithm, are the set of roots of a nearby polynomial or not.
In order to do so, we derive a first order backward error analysis of the polynomial rootfinding algorithm using colleague matrices adapting the geometric arguments in [A. Edelman and H. Murakami, \emph{Polynomial roots for companion matrix eigenvalues}, Math. Comp. 210, 763--776, 1995] to the Chebyshev basis.
We show that, if the absolute value of the coefficients of p(x) (respectively, the norm of the coefficients of P(x)) are bounded by a moderate number, computing the roots of p(x) (respectively, the eigenvalues of P(x)) via the eigenvalues of its colleague matrix using a backward stable eigenvalue algorithm is backward stable.
This backward error analysis also expands on the very recent work [Y. Nakatsukasa and V. Noferini, \emph{On the stability of computing polynomial roots via confederate linearizations}, To appear in Math. Comp.] that already showed that this algorithm is not backward normwise stable if the coefficients of the polynomial p(x) do not have moderate norms
Efficient block preconditioning for a C1 finite element discretisation of the Dirichlet biharmonic problem
Get PDFWe present an efficient block preconditioner for the two-dimensional biharmonic Dirichlet problem discretised by C1 bicubic Hermite finite elements. In this formulation each node in the mesh has four different degrees of freedom (DOFs). Grouping DOFs of the same type together leads to a natural blocking of the Galerkin coefficient matrix. Based on this block structure, we develop two preconditioners: a 2x2 block diagonal preconditioner (BD) and a block bordered diagonal (BBD) preconditioner. We prove mesh independent bounds for the spectra of the BD-preconditioned Galerkin matrix under certain conditions. The eigenvalue analysis is based on the fact that the proposed preconditioner, like the coefficient matrix itself, is symmetric positive definite and is assembled from element matrices. We demonstrate the effectiveness of an inexact version of the BBD preconditioner, which exhibits near optimal scaling in terms of computational cost with respect to the discrete problem size. Finally, we study robustness of this preconditioner with respect to element stretching, domain distortion and non-convex domains
