Nonconforming Case Research Articles

A BDDC Method for Mortar Discretizations Using a Transformation of Basis

A BDDC (balancing domain decomposition by constraints) method is developed for elliptic equations, with discontinuous coefficients, discretized by mortar finite element methods for geometrically nonconforming partitions in both two and three space dimensions. The coarse component of the preconditioner is defined in terms of one mortar constraint for each edge/face, which is the intersection of the boundaries of a pair of subdomains. A condition number bound of the form $C\max_i\{(1+\log(H_i/h_i))^2\}$ is established under certain assumptions on the geometrically nonconforming subdomain partition in the three-dimensional case. Here $H_i$ and $h_i$ are the subdomain diameters and the mesh sizes, respectively. In the geometrically conforming case and the geometrically nonconforming cases in two dimensions, no assumptions on the subdomain partition are required. This BDDC preconditioner is also shown to be closely related to the Neumann-Dirichlet version of the FETI-DP algorithm. The results are illustrated by numerical experiments which confirm the theoretical results.

“Finding what doesn’t fit”: Adjusting the focus on education in country towns as a limiting case for the “learning or earning” years

Limiting, or nonconforming cases are good tests in research for extending a concept’s reach and broadening generalisations, but the method is under utilised in the formation and assessment of educational policy. In this paper a rural town is described as a case, among others, not fitting policy regulations for “learning or earning” in Years 11 and 12 in Queensland. It is argued, through a description of the production and reproduction of the town’s service economy, that neither the curriculum nor the workforce can be placed within a theory of social and economic networking underpinning policies on retention and pathways upon which the Queensland legislation rests. Cases such as these are difficult to uncover because factors that best describe their true economic and social conditions tend to be compressed within statistical reporting models for educational regions. While the focus is on a rural town, the purpose is to advance policy design and implementation that seeks to sustain educational change in diverse conditions.

Estimation of interpolation error constants for the P0 and P1 triangular finite elements

We give some fundamental results on the error constants for the piecewise constant interpolation function and the piecewise linear one over triangles. For the piecewise linear one, we mainly analyze the conforming case, but some results are also given for the non-conforming case. We obtain explicit relations for the dependence of such error constants on the geometric parameters of triangles. In particular, we explicitly determine the Babuška–Aziz constant, which plays an essential role in the interpolation error estimation of the linear triangular finite element. The equation for determination is the transcendental equation λ + tan λ = 0 , so that the solution can be numerically obtained with desired accuracy and verification. Such highly accurate approximate values for the constant as well as estimates for other constants can be widely used for a priori and a posteriori error estimations in adaptive computation and numerical verification of finite element solutions.

Subadditive pressure for triangular maps

We investigate properties of the zero of the subadditive pressure used by Falconer, Barreira and Zhang to estimate the box and Hausdorff dimension of a non-conformal repeller. In the conformal case, and in Falconer's 1-bunched non-conformal case, the contraction rates satisfy bounded distortion and so this zero is insensitive to where on each cylinder the contraction is evaluated. We study some nonlinear two-dimensional examples which do not satisfy bounded distortion but do exhibit the same insensitivity. Here the contraction rate fails to specify ellipses that can be used to cover cylinders.

Online Mutual Aid, “RAQ”: A Study of Anonymous and Traceable Interactions

Our study proposes mutual aid platforms for the resolution of uncommon or non-conforming cases in both administrative and commercial procedures. On reviewing inter-customer interaction, we came to rate RAQ over FAQ. We also reviewed former works on categorizing anonymity and related areas to support our research objectives. Conducting a study of knowledge search service, we were able to identify the differences between anonymous and traceable interaction in three non-conforming administrative cases. From a quantitative standpoint, we noted a tendency for anonymous communications to involve more participants. The accuracy of answers appeared not significantly variable, which was unexpected. From a qualitative view, more personal data were disclosed in anonymous interaction than in traceable interaction.

An adaptive least squares mixed finite element method for the stress‐displacement formulation of linear elasticity

AbstractA least‐squares mixed finite element method for linear elasticity, based on a stress‐displacement formulation, is investigated in terms of computational efficiency. For the stress approximation quadratic Raviart‐Thomas elements are used and these are coupled with the quadratic nonconforming finite element spaces of Fortin and Soulie for approximating the displacement. The local evaluation of the least‐squares functional serves as an a posteriori error estimator to be used in an adaptive refinement algorithm. We present computational results for a benchmark test problem of planar elasticity including nearly incompressible material parameters in order to verify the effectiveness of our adaptive strategy. For comparison, conforming quadratic finite elements are also used for the displacement approximation showing convergence orders similar to the nonconforming case, which are, however, not independent of the Lamé parameters. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

Minimal surfaces and reggeization in the AdS/CFT correspondence

Abstract We address the problem of computing scattering amplitudes related to the correlation function of two Wilson lines and/or loops elongated along light-cone directions in strongly coupled gauge theories. Using the AdS/CFT correspondence in the classical approximation, the amplitudes are shown to be related to minimal surfaces generalizing the helicoid in various AdS 5 backgrounds. Infra-red divergences appearing for Wilson lines can be factorized out or can be cured by considering the IR finite case of correlation functions of two Wilson loops. In non-conformal cases related to confining theories, reggeized amplitudes with linear trajectories and unit intercept are obtained and shown to come from the approximately flat metrics near the horizon, which sets the scale for the Regge slope. In the conformal case the absence of confinement leads to a different solution. A transition between both regimes appears, in a confining theory, when varying impact parameter.

A new tool for dose conformity evaluation of radiosurgery treatment plans

Purpose: A novel method for dose conformity evaluation of treatment plans produced by the stereotactic radiosurgery treatment planning system is postulated. Methods and Materials: By consolidating the information contained in the integral dose-volume histogram and the treatment volume ratio, a plot of treatment volume ratio versus percentage dose may be considered as a useful tool for plan evaluation. To validate the suggested argument, two simple experiments simulating the conformal and nonconformal cases were conducted on the geometric phantom that is commercially available from Radionics. An actual patient treatment plan is also included to explore the effectiveness of the proposed parameters. It is an attempt to establish the baseline of a conformal plan. Results: A plot showed the ability to give the user an idea whether the size of the collimator was adequate to cover the delineated lesion when the user-defined criteria had been in place. Two parameters, namely take-off dose (TOD) and take-off volume (TOV) were defined. The former was defined as the maximum dose level found on the surface of the target volume. The TOD is also the maximum possible dose to be received by the adjacent normal tissue. The latter was defined as the percentage of the target volume that received the TOD. Another parameter, irradiated percentage volume (IPV), was defined here as the percentage of the target volume receiving at least the prescribed dose. When the prescribed dose is also the TOD, the IPV becomes the TOV. They were proved to be effective in evaluating the dose conformity. Another term known as equivalent fall-off distance (EFOD) was defined as the equivalent radial distance calculated between two isodose lines. In fact, the dose fall-off rate can also act as a measuring index for plan comparison, because a fast dose fall-off rate is often a requirement for radiosurgery in order to minimize the risk of radiation damage to the surrounding structures. The two phantom studies showed consistent results with the theoretical predictions. The ability of the plot was further explored in the patient treatment plan studies. It was demonstrated that the plot had a remarkable ability to check whether the hot spot is in the vicinity of the lesion. A baseline of a conformal plan was also established; for example, a plan is said to be conformal if its IPV has attained a value of not less than 95% and its associated TVR is not greater than 2. Conclusion: The proposed method has demonstrated the effectiveness in dose conformity evaluation. It supplements the integral dose-volume histogram to provide a complete information of a treatment plan in terms of dose uniformity and conformity.

Nonconformal fluctuations in radiation dominated anisotropic cosmology

Using simple path integral, Feynman propagator method and the relation between conformal timeη and scale factor Τ, we investigate the non-conformal quantum fluctuations (of expansion and shear) and axisymmetric singularity case in radiation dominated anisotropic cosmology. We show that near the classical singularity the quantum fluctuations tend to diverge.

Nonconforming finite element methods without numerical locking

We analyze the numerical approximation of a class of elliptic problems which depend on a small parameter $\varepsilon$ . We give a generalization to the nonconforming case of a recent result established by Chenais and Paumier for a conforming discretization. For both the situations where numerical integration is used or not, a uniform convergence in $\varepsilon$ and h is proved, numerical locking being thus avoided. Important tools in the proof of such a result are compactness properties for nonconforming spaces as well as the passage to the limit problem.

HAMILTONIAN DIAGONALIZATION FOR A NONCONFORMAL SCALAR FIELD IN AN ISOTROPIC GRAVITATIONAL BACKGROUND

The metrical Hamiltonian is investigated for a nonconformal scalar field with arbitrary coupling in a nonstationary spatially homogeneous isotropic gravitational field. It is shown that this Hamiltonian may be diagonalized by the canonical Bogoliubov transformations of creation-annihilation operators only for the values of coupling coefficient 0 ≤ ξ ≤ 1/6. The quasienergy of quasiparticles diagonalizing the Hamiltonian is shown to differ for the nonconformal case from the effective oscillator frequency of the wave equation. Different definitions of the initial vacuum state for the nonconformal scalar field are discussed.

Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems

We analyze nonconforming finite element approximations of streamline-diffusion type for solving convection-diffusion problems. Both the theoretical and numerical investigations show that additional jump terms have to be added in the nonconforming case in order to get the same \(O(h^{k+1/2})\) order of convergence in L\(^2\) as in the conforming case for convection dominated problems. A rigorous error analysis supported by numerical experiments is given.

Archbishop Abbot and the problem of ‘puritanism’

ABSTRACTThis article is a study of Archbishop Abbot's attitude to puritan nonconformity. It contends that the accusations of laxness levelled against him by Laudian apologists such as Heylin and Clarendon can be discounted as the inevitable consequence of the very different standpoint from which they judged him. It was not laziness that led the primate to turn a blind eye to certain types of nonconformity, but rather a detached attitude – not shared by his anti-Calvinist opponents – which allowed him to differentiate between mild cases of conscience-induced ceremonial nonconformity which posed no danger to the status quo, and the factious high-profile stance of those who undermined the authority of the established church. By taking a very hard line against separatists and their ilk, Abbot felt he could justify the degree of tolerance he granted to those peaceable and orthodox preaching pastors with scruples over certain adiaphora, whose role was so crucial in the unremitting battle against popery to which he dedicated his life.

On different finite element methods for approximating the gradient of the solution to the helmholtz equation

We consider the numerical solution of the Helmholtz equation by different finite element methods. In particular, we are interested in finding an efficient method for approximating the gradient of the solution. We first approximate the gradient by the standard Ritz-Galerkin method. As a second method a two-stage method due to Aziz and Werschulz (1980) is presented. It is shown that this method gives the same accuracy in the computed gradient and in the computed solution also in the nonconforming case. Finally, a direct method with asymptotic error estimates is given. It turns out that the presented direct method is of lowest computational complexity. Test examples are presented to illustrate the accuracy of the methods.