Piecewise Smooth Boundary Research Articles

Acoustically invisible gateways

In recent years considerable interest has been directed at devising obstacles which appear 'invisible' to various types of wave propagation. That is; suppose for example we can construct an obstacle coated with an appropriate material such that when illuminated by an incident field (e.g plane wave) the wave scattered by the obstacle has zero cross section(equivalently radiation pattern)for all incident directions and frequencies then the obstacle appears to have no effect on the illuminating wave and the obstacle can be considered invisible. Such an electromagnetic cloaking device has been constructed by Schurig et. al [18].Motivated by recent work [1] concerning the problem of a parallel flow ofparticles falling on a body with piecewise smooth boundary and leaving no trace the analogous problem of acoustic scattering of a plane wave illuminating the same body, a gateway, is considered. It is shown that at high frequencies, with use ofthe Kirchoff approximation and the geometrical theory of diffraction, the scattered far field in a range of observeddirections, e. g the back scattered direction, is zero for a discrete set of wave numbers. That is the gatewayis acoustically 'invisible' in that direction.

Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains

We consider a generalized Stochastic Cahn-Hilliard equation with multiplicative white noise posed on bounded convex domains in $R^d$, $d=1,2,3$, with piece-wise smooth boundary, and introduce an additive time dependent white noise term in the chemical potential. Since the Green's function of the problem is induced by a convolution semigroup, we present the equation in a weak stochastic integral formulation and prove existence of solution when $d\leq 2$ for general domains, and for $d=3$ for domains with minimum eigenfunction growth, without making use of any explicit expression of the spectrum and the eigenfunctions. The analysis is based on stochastic integral calculus, Galerkin approximations and the asymptotic spectral properties of the Neumann Laplacian operator. Existence is also derived for some non-convex cases when the boundary is smooth.

The dirichlet problem for harmonic functions from variable exponent smirnov classes in domains with piecewise smooth boundary

The Dirichlet problem is solved for harmonic functions from variable exponent Smirnov classes in domains with piecewise smooth boundaries. The solvability conditions are established. Depending on the boundary geometry and value of the space exponent at angular points, the Dirichlet problem may turn out to be unsolvable, solvable uniquely and non-uniquely. In the unsolvable case, for boundary functions the necessary and sufficient conditions are found, which govern the solvability. In all solvability cases, solutions are constructed in explicit form. Bibliography: 19 titles.

Manifold-based surfaces with boundaries

We present a manifold-based surface construction extending the C ∞ construction of Ying and Zorin (2004). Our surfaces allow for piecewise-smooth boundaries and have an improved derivative and visual behavior. Their degree of smoothness can be controlled by the user. Two-flexibility of our surface construction is confirmed numerically for a range of local mesh configurations.

Regularized method of fundamental solutions for boundary identification in two-dimensional isotropic linear elasticity

We investigate the stable numerical reconstruction of an unknown portion of the boundary of a two-dimensional domain occupied by an isotropic linear elastic material from a prescribed boundary condition on this part of the boundary and additional displacement and traction measurements (i.e. Cauchy data) on the remaining known portion of the boundary. This inverse geometric problem is approached by combining the method of fundamental solutions (MFS) and the Tikhonov regularization method, whilst the optimal value of the regularization parameter is chosen according to the discrepancy principle. Various geometries are considered, i.e. convex and non-convex domains with a smooth or piecewise smooth boundary, in order to show the numerical stability, convergence, consistency and computational efficiency of the proposed method.

Essentially different distributions of current in the inverse problem for the Grad-Shafranov equation

The paper answers a question debated by physicists for many years. It is proved that, for almost equal gradients of the magnetic flux u at its zero-level curve ∂ω, which is the piecewise smooth boundary of a simply-connected domain ω ⋐ ℝ2, the inverse problem for the Grad-Shafranov equation of plasma equilibrium in a tokamak (in the cylindrical approximation) admits essentially different profiles of distributions fu: ω ∋ (x, y) ↦ f(u(x, y)) = uxx(x, y) + uyy(x, y) ⩾ 0 in the class of third-order polynomials f(u) = Σm=03amum.

An over-determined boundary problem for the Helmholtz equation in a semiinfinite domain with a curvilinear boundary

In this paper we consider an over-determined Cauchy problem for the Helmholtz equation in a semiinfinite domain with a piecewise smooth curvilinear boundary. Applying the Fourier transform method in the space of distributions of slow growth, we establish the necessary and sufficient solvability conditions which connect the boundary functions. We construct integral representations of a solution.

A Gauss-Bonnet-type formula on Riemann-Finsler surfaces with nonconstant indicatrix volume

We prove a Gauss-Bonnet type formula for Riemann-Finsler surfaces of non-constant indicatrix volume and with regular piecewise smooth boundary. We give a Hadamard type theorem for N-parallels of a Landsberg surface.

Invariant measures associated to degenerate elliptic operators

This paper is devoted to the study of the existence and uniqueness of the invariant measure associated to the transition semigroup of a diffusion process in a bounded open subset of ℝ n . For this purpose, we investigate first the invariance of a bounded open domain with piecewise smooth boundary showing that such a property holds true under the same conditions that insure the invariance of the closure of the domain. A uniqueness result for the invariant measure is obtained in the class of all probability measures that are absolutely continuous with respect to Lebesgue's measure. A sufficient condition for the existence of such a measure is also provided.

The Riemann–Hilbert Problem in a Domain with Piecewise Smooth Boundaries in Weight Classes of Cauchy Type Integrals with a Density from Variable Exponent Lebesgue Spaces

Abstract The Riemann–Hilbert problem for an analytic function is solved in weighted classes of Cauchy type integrals in a simply connected domain not containing 𝑧 = ∞ and having a density from variable exponent Lebesgue spaces. It is assumed that the domain boundary is a piecewise smooth curve. The solvability conditions are established and solutions are constructed. The solution is found to essentially depend on the coefficients from the boundary condition, the weight, space exponent values at the angular points of the boundary curve and also on the angle values. The non-Fredholmian case is investigated. An application of the obtained results to the Neumann problem is given.

Shrinkage estimation in elliptically contoured distribution with restricted parameter space

Abstract This paper deals with the problem of estimating the mean vector of an elliptically contoured distribution with unknown scale matrix, where the mean vector is restricted to an unbounded and closed convex set with a smooth or a piecewise smooth boundary. A shrinkage-type estimator is shown to be better than the maximum likelihood estimator subject to the restriction relative to a quadratic loss.

New asymptotic effects for the spectrum of problems on concentrated masses near the boundary

The Dirichlet and Neumann spectral problems for the Laplace operator in a bounded domain Ω ⊂ R 2 are considered. We assume that Ω has a piecewise smooth boundary ∂ Ω and the density function is equal to 1 + ε − m χ ε in Ω, where ε > 0 is a small parameter, m ∈ R and χ ε is the characteristic function of the union ω ε 0 ∪ ⋯ ∪ ω ε J − 1 of small sets (the concentrated masses) distributed periodically near a straight segment Γ ⊂ ∂ Ω . We describe asymptotics for the eigenelements of both problems as ε → 0 . To cite this article: S.A. Nazarov, E. Pérez, C. R. Mecanique 337 (2009).

Piecewise oblique boundary treatment for the elastic–plastic wave equation on a cartesian grid

Numerical schemes for hyperbolic conservation laws in 2-D on a Cartesian grid usually have the advantage of being easy to implement and showing good computational performances, without allowing the simulation of “real-world” problems on arbitrarily shaped domains. In this paper a numerical treatment of boundary conditions for the elastic–plastic wave equation is developed, which allows the simulation of problems on an arbitrarily shaped physical domain surrounded by a piece-wise smooth boundary curve, but using a PDE solver on a rectangular Cartesian grid with the afore-mentioned advantages.

The Cauchy problem in Sobolev spaces for Dirac operators

In this paper we consider the Cauchy problem as a typical example of ill-posed boundary-value problems. We obtain the necessary and (separately) sufficient conditions for the solvability of the Cauchy problem for a Dirac operator A in Sobolev spaces in a bounded domain D ⊂ ℝn with a piecewise smooth boundary. Namely, we reduce the Cauchy problem for the Dirac operator to the problem of harmonic extension from a smaller domain to a larger one.

ASIFT: A New Framework for Fully Affine Invariant Image Comparison

If a physical object has a smooth or piecewise smooth boundary, its images obtained by cameras in varying positions undergo smooth apparent deformations. These deformations are locally well approximated by affine transforms of the image plane. In consequence the solid object recognition problem has often been led back to the computation of affine invariant image local features. Such invariant features could be obtained by normalization methods, but no fully affine normalization method exists for the time being. Even scale invariance is dealt with rigorously only by the scale-invariant feature transform (SIFT) method. By simulating zooms out and normalizing translation and rotation, SIFT is invariant to four out of the six parameters of an affine transform. The method proposed in this paper, affine-SIFT (ASIFT), simulates all image views obtainable by varying the two camera axis orientation parameters, namely, the latitude and the longitude angles, left over by the SIFT method. Then it covers the other four parameters by using the SIFT method itself. The resulting method will be mathematically proved to be fully affine invariant. Against any prognosis, simulating all views depending on the two camera orientation parameters is feasible with no dramatic computational load. A two-resolution scheme further reduces the ASIFT complexity to about twice that of SIFT. A new notion, the transition tilt, measuring the amount of distortion from one view to another, is introduced. While an absolute tilt from a frontal to a slanted view exceeding 6 is rare, much higher transition tilts are common when two slanted views of an object are compared (see Figure hightransitiontiltsillustration). The attainable transition tilt is measured for each affine image comparison method. The new method permits one to reliably identify features that have undergone transition tilts of large magnitude, up to 36 and higher. This fact is substantiated by many experiments which show that ASIFT significantly outperforms the state-of-the-art methods SIFT, maximally stable extremal region (MSER), Harris-affine, and Hessian-affine.

Strong Superconvergence of Finite Element Methods for Linear Parabolic Problems

We study the strong superconvergence of a semidiscrete finite element scheme for linear parabolic problems on , where is a bounded domain in with piecewise smooth boundary. We establish the global two order superconvergence results for the error between the approximate solution and the Ritz projection of the exact solution of our model problem in and with and the almost two order superconvergence in and . Results of the case are also included in two space dimensions ( or 2). By applying the interpolated postprocessing technique, similar results are also obtained on the error between the interpolation of the approximate solution and the exact solution.

Numerical Conformal Mapping via a Boundary Integral Equation with the Generalized Neumann Kernel

We present a unified boundary integral method for approximating the conformal mappings from any bounded or unbounded multiply connected region $G$ onto the five classical canonical slit domains. The method is based on a uniquely solvable boundary integral equation with the generalized Neumann kernel. Using the proposed method, the approximate mapping functions onto the five canonical slit domains can be computed in a unified way by solving linear systems with a common coefficient matrix. The method can be also used for calculating the conformal mappings of simply and doubly connected regions. The performance of the method is illustrated by several examples for regions with smooth boundaries and with piecewise smooth boundaries.

Bases of singular solutions in problems of mechanics of cracks

The mechanics of cracks serves as a remarkable example of a consistent application of the general results of the theory of elliptic boundary value problems to domains with piecewise smooth boundaries (see key papers [1–3] and also monographs [4, 5]). In the first place, this applies to the analysis of the behavior of stress-strain state in the vicinity of the crack tip and to operations involving the most important characteristics of this state, because explicit formulas that enable a reasonable prediction of the fracture process are available only for isotropic elastic materials. Meanwhile, the modern engineering practice involves the study of damages in anisotropic and composite bodies.

A boundary integral method for the Riemann–Hilbert problem in domains with corners

This article presents a boundary integral method for the Riemann–Hilbert problems in simply connected regions with piecewise smooth boundaries. The method is based on two boundary integral equations with the generalized Neumann kernel. To illustrate the results, several numerical examples are presented where the integral equations are solved using a Nyström method based on the trapezoidal rule with a graded mesh.

Billiards and Two-Dimensional Problems of Optimal Resistance

A body moves in a medium composed of noninteracting point particles; the interaction of the particles with the body is completely elastic. The problem is: find the body’s shape that minimizes or maximizes resistance of the medium to its motion. This is the general setting of the optimal resistance problem going back to Newton. Here, we restrict ourselves to the two-dimensional problems for rotating (generally non-convex) bodies. The main results of the paper are the following. First, to any compact connected set with piecewise smooth boundary $${B \subset \mathbb{R}^2}$$ we assign a measure ν B on ∂(conv B)×[ − π/2, π/2] generated by the billiard in $${\mathbb{R}^2 \setminus B}$$ and characterize the set of measures {ν B }. Second, using this characterization, we solve various problems of minimal and maximal resistance of rotating bodies by reducing them to special Monge–Kantorovich problems.