Piecewise Smooth Boundary Research Articles

The hot spots problem in planar domains with one hole

There exists a planar domain with piecewise smooth boundary and one hole such that the second eigenfunction for the Laplacian with Neumann boundary conditions attains its maximum and minimum inside the domain.

Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation

In this paper, the boundary knot method is extended to the solution of inhomogeneous equations, and it is applied to the Cauchy problem associated with the inhomogeneous Helmholtz equation. Here, we assume that the boundary condition is specified only on a part of the boundary, and the boundary conditions on the remaining part of the boundary are to be determined with the assistance of additional data. Since the resulting matrix equation is highly ill-conditioned, a regularized solution is obtained by employing the truncated singular value decomposition to solve the matrix equation arising from the boundary knot method, with the regularization parameter determined by the L-curve method. Numerical results are presented for several examples with smooth and piecewise smooth boundaries. The numerical verification shows that the proposed numerical scheme is accurate, stable with respect to data noise, and convergent with respect to decreasing the amount of noise in the data.

A new stereological principle for test lines in three‐dimensional space

A new principle is presented to generate isotropic uniform random (IUR) test lines hitting a geometric structure in three-dimensional space (3D). The principle therefore concerns the estimation of surface area, volume, membrane thickness, etc., of arbitrary structures with piecewise smooth boundary. The principle states that a point-sampled test line on an isotropic plane through a fixed point in 3D is effectively an invariant test line in 3D. Particular attention is devoted to the stereology of particles, where an alternative to the surfactor method is obtained to estimate surface area. An interesting case arises when the particle is convex. The methods are illustrated with synthetic examples.

The 3D Inverse Problem of the Wave Equation for a General Multi-connected Vibrating Membrane with a Finite Number of Piecewise Smooth Boundary Conditions

The trace of the wave kernel \( \hat{\mu }{\left( t \right)} = {\sum\nolimits_{\omega = 1}^\infty {\exp {\left( { -{\text{\bf i}}tE^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}_{\omega } } \right)}} },\;{\text{where}}\;{\left\{ {E_{\omega } } \right\}}^{\infty }_{{\omega = 1}} \) are the eigenvalues of the negative Laplacian \( - \nabla ^{2} = - {\sum\nolimits_{k = 1}^3 {{\left( {\frac{\partial } {{\partial x^{k} }}} \right)}} }^{2} \) in the (x1, x2, x3)–space, is studied for a variety of bounded domains, where −∞ < t < ∞ and \({\text{\bf i}} = {\sqrt { - 1} } \). The dependence of \( \hat{\mu } \) (t) on the connectivity of bounded domains and the Dirichlet, Neumann and Robin boundary conditions are analyzed. Particular attention is given for a multi–connected vibrating membrane Ω in R3 surrounded by simply connected bounded domains Ω j with smooth bounding surfaces Sj (j = 1, . . . , n), where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components \( S^{ * }_{i} \) (i = 1+kj−1, . . . , kj) of the bounding surfaces S j are considered, such that \( S_{j} = \cup ^{{k_{j} }}_{{i = 1 + k_{{j - 1}} }} S^{ * }_{i} \) , where k0 = 0. The basic problem is to extract information on the geometry Ω by using the wave equation approach from a complete knowledge of its eigenvalues. Some geometrical quantities of Ω (e.g. the volume, the surface area, the mean curvuture and the Gaussian curvature) are determined from the asymptotic expansion of \( {\hat{\mu }} \) (t) for small |t|.

Generalized Solutions of Nonlocal Elliptic Problems

An elliptic equation of order 2m with general nonlocal boundary-value conditions, in a plane bounded domain G with piecewise smooth boundary, is considered. Generalized solutions belonging to the Sobolev space W 2 m (G) are studied. The Fredholm property of the unbounded operator (corresponding to the elliptic equation) acting on L 2(G), and defined for functions from the space W 2 m (G) that satisfy homogeneous nonlocal conditions, is established.

Splitting as an approach to constructing local exact artificial boundary conditions

Abstract We employ the technique of splitting for constructing artificial boundary conditions (ABCs) for the linear advection–diffusion–reaction equation when the computational domain is an nD open set with a piecewise smooth artificial boundary. The splitting is performed both by the physical processes and by coordinates. The former permits to construct ABCs for each of the processes separately, which provides local exact boundary conditions; the latter leads to ABCs much less exigent to the shape of artificial boundary in comparison with many others. We also prove that the corresponding boundary value problems are well-posed, and present results of the numerical experiments that confirm the theoretical study.

Quantum Ergodicity of Boundary Values of Eigenfunctions: A Control Theory Approach

AbstractConsider M, a bounded domain in ℝd, which is a Riemanian manifold with piecewise smooth boundary and suppose that the billiard associated to the geodesic flow reflecting on the boundary according to the laws of geometric optics is ergodic. We prove that the boundary value of the eigen-functions of the Laplace operator with reasonable boundary conditions are asymptotically equidistributed in the boundary, extending previous results by Gérard and Leichtnam as well as Hassell and Zelditch, obtained under the additional assumption of the convexity of M.

Strong Summability of Faber Series and Estimates for the Rate of Convergence of a Group of Deviations in a Closed Domain with Piecewise-Smooth Boundary

We establish estimates for groups of deviations of Faber series in closed domains with piecewise-smooth boundary.

&lt;i&gt;C&lt;/i&gt;&lt;sup&gt;&amp;infin;&lt;/sup&gt;-REGULARITY OF THE TANGENTIAL CAUCHY-RIEMANN EQUATIONS ON LEVI-FLAT SUBMANIFOLDS OF C&lt;sup&gt;&lt;i&gt;n&lt;/sup&gt;&lt;/i&gt;

Regularity of solutions of the tangential Cauchy-Riemann (CR) equation is shown on domains in Levi-flat CR manifolds for the class of C∞ forms up to the boundary. These domains are transversal intersections of the CR submanifold with pseudoconvex domains with piecewise smooth boundary.

Problem of Conjugation of Solutions of the Lame Wave Equation in Domains with Piecewise-Smooth Boundaries

We study the problem of conjugation of solutions of the Lame wave equation in domains containing singular lines (sets of angular points) and conic points. We show that solutions of the Lame wave equation have power-type singularities near nonsmoothnesses of boundary surfaces and determine their asymptotics. Taking these asymptotics into account and using the introduced simple-layer, double-layer, and volume elastic retarded potentials, we reduce the problem to a system of functional equations and formulate conditions for its solvability.

Smooth surface extension with curvature bound

For a given surface with a piecewise smooth boundary, a new method to extend the surface across its boundary is suggested. The extended surface is C 2 -continuous along the old boundary, and some extra conditions can be imposed on the new boundary. Furthermore, it is shown that normal curvature of the extended part of the surface can be controlled by predetermined conditions. In order to test the effectiveness of our algorithm, it is shown that surfaces with rectangular and circular boundaries can be extended with all necessary conditions satisfied.

Short-time Asymptotics of the Heat Kernel on Bounded Domain with Piecewise Smooth Boundary Conditions and Its Applications to an Ideal Gas

The asymptotic expansion of the heat kernel $$ \Theta {\left( t \right)} = {\sum\limits_{j = 1}^\infty {\exp {\left( { - t\lambda _{j} } \right)}} } $$ where $$ {\left\{ {\lambda _{j} } \right\}}^{\infty }_{{j = 1}} $$ are the eigenvalues of the negative Laplacian $$ - \Delta _{n} = - {\sum\limits_{k = 1}^n {{\left( {\frac{\partial } {{\partial x^{k} }}} \right)}} }^{2} $$ in R n (n = 2 or 3) is studied for short-time t for a general bounded domain Ω with a smooth boundary $$ \partial \Omega $$ . In this paper, we consider the case of a finite number of the Dirichlet conditions φ = 0 on Γ i (i = 1, ..., J) and the Neumann conditions $$ \frac{{\partial \phi }} {{\partial \upsilon _{i} }} = 0 $$ on Γ i (i = J +1, · · · , k) and the Robin conditions $$ {\left( {\frac{\partial } {{\partial \upsilon _{i} }} + \gamma _{i} } \right)}\phi = 0 $$ on Γ i (i = k+1, · · ·,m) where γ i are piecewise smooth positive impedance functions, such that $$ \partial \Omega $$ consists of a finite number of piecewise smooth components Γ i (i = 1, · · ·,m) where $$ \partial \Omega = {\bigcup\limits_{i = 1}^m {\Gamma _{i} } } $$ . We construct the required asymptotics in the form of a power series over t. The senior coefficients in this series are specified as functionals of the geometric shape of the domain Ω. This result is applied to calculate the one-particle partition function of a “special ideal gas”, i. e., the set of non-interacting particles set up in a box with Dirichlet, Neumann and Robin boundary conditions for the appropriate wave function. Calculation of the thermodynamic quantities for the ideal gas such as the internal energy, pressure and specific heat reveals that these quantities alone are incapable of distinguishing between two different shapes of the domain. This conclusion seems to be intuitively clear because it is based on a limited information given by a one-particle partition function; nevertheless, its formal theoretical motivation is of some interest.

Dynamical approach to some problems in integral geometry

As is well known, the main problem in integral geometry is to reconstruct a function in a given domain D D , where its integrals over a family of subdomains in D D are known. Such a problem is interesting not only as an object of pure analysis, but also in connection with various applications in practical disciplines. The most remarkable example of such a connection is the Radon problem and tomography. In this paper we solve one of these problems when D D is a bounded domain in R 2 {\mathbb {R}}^2 with a piecewise smooth boundary. Some intermediate results related to dynamical systems with two generators and to some functional-integral equations are new and interesting per se. As an application of the results obtained we briefly study a boundary problem for a general third order hyperbolic partial differential equation in a bounded domain D ⊂ R 2 D\subset {\mathbb {R}}^2 with data on the whole boundary ∂ D \partial D .

Restriction matrices for SGBEM applications

In the context of linear elasticity, we consider a symmetric boundary integral formulation associated with a mixed boundary value problem defined on a domain 2 IR m , m = 2, 3, with piecewise smooth boundary . We assume that is mapped onto itself by a finite groupG of congruences having at least two distinct elements. It is easy to take advantage of this when the source fields and the boundary conditions share the symmetry, as it is the case for an even or odd parity with respect to a reflection plane, but the intuitive reasoning fails for complex geometry. However, it is still possible to take advantage of the symmetry owing to the group representation theory and a decomposition theorem. From the latter, using the symmetry one can reduce the original problem to a family of smaller ones defined on a reduced geometry. The aim of this paper is to present a systematic technique for exploiting symmetry in the numerical treatment of boundary integral equations with the symmetric Galerkin BEM (SGBEM). This technique will be based upon suitable restriction matrices strictly related to the mesh defined on the boundary. Hence, we can decompose the related symmetric Galerkin BEM problem into independent subproblems of reduced dimension with respect to the complete one. Shape functions for each subproblem can be obtained from classical BEM basis, ordered as a vector, applying suitable restriction matrices constructed starting from group representation theory.

Inverse Problems of Boundary Value Problems for Annular Vibrating Membranes with Piecewise Smooth Positive Functions in the Boundary Conditions

The asymptotic expansions of the trace of the heat kernel <artwork name="GAPA31035ei1"> for small positive t, where λν are the eigenvalues of the negative Laplacian <artwork name="GAPA31035ei2"> in Rn (n=2 or 3), are studied for a general annular bounded domain Ω with a smooth inner boundary ∂Ω1 and a smooth outer boundary ∂Ω2 where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the components Γ j (j=1,…,m) of ∂Ω1 and on the components <artwork name="GAPA31035ei3"> of ∂Ω2 are considered such that <artwork name="GAPA31035ei4"> and <artwork name="GAPA31035ei5"> and where the coefficients <artwork name="GAPA31035ei6"> in the Robin boundary conditions are piecewise smooth positive functions. Some applications of Θ (t) for an ideal gas enclosed in the general annular bounded domain Ω are given.

Système de Stokes avec flux de vitesse et pression imposés

In this Note, we study the Stokes equations with imposed velocity fluxes and pressure, in a bounded domain, with a piecewise smooth boundary. To cite this article: P. Ciarlet, Jr., C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Steady supersonic flow past an almost straight wedge with large vertex angle

This paper studies the problem on the steady supersonic flow at the constant speed past an almost straight wedge with a piecewise smooth boundary. It is well known that if each vertex angle of the straight wedge is less than an extreme angle determined by the shock polar, the shock wave is attached to the tip of the wedge and constant states on both side of the shock are supersonic. This paper is devoted to generalizing this result. Under the hypotheses that each vertex angle is less than the extreme angle and the total variation of tangent angle along each edge is sufficiently small, a sequence of approximate solutions constructed by a modified Glimm scheme is proved to be convergent to a global weak solution of the steady problem. A sequence of the corresponding approximate leading shock fronts issuing from the tip is shown to be convergent to the leading shock front of the obtained solution. The regularity of the leading shock front is established and the asymptotic behaviour of the obtained solution at infinity is also studied.

On the Upper Bound of Second Eigenvalues for Uniformly Elliptic Operators of any Orders

Let $$\Omega \subset R^m (m\ge 1)$$ be a bounded domain with piecewise smooth boundary ∂Ω. Let t and r be positive integers with t > r + 1. We consider the eigenvalue problems (1.1) and (1.2), and obtain Theorem 1 and Theorem 2, which generalize the results in [1,2,5].

Efficient method for solving the problems of wave diffraction by scatterers with broken boundaries

The pattern equations method is extended to solving the problems of wave scattering by bodies with piecewise smooth boundaries. The method is based on the reduction of the initial boundary-value problem to an integro-operator equation of the second kind in the scattering pattern of a body. With the use of the series expansion of the scattering pattern in angular spherical harmonics, the problem is ultimately reduced to solving an infinite algebraic system of equations in the expansion coefficients of the scattering pattern. The conditions at which this system can be solved by the method of reduction are formulated. Examples of solving the problems of wave scattering by bodies with impedance boundaries are considered. Essential advantages of the proposed method over other known methods are demonstrated.

ICONE11-36501 ADVANCES IN THE SOLUTION OF THREE-DIMENSIONAL NODAL NEUTRON TRANSPORT EQUATION

In this paper we study the three-dimensional nodal discrete-ordinates approximations of neutron transport equation in a convex domain with piecewise smooth boundaries. We use the combined collocation method of the angular variables and nodal approach for the spatial variables^1. By nodal approach we mean the iterated transverse integration of the S_N equations. This procedure leads to the set of one-dimensional averages angular fluxes in each spatial variable. The resulting system of equations is solved with the LTS_N method, first applying the Laplace transform to the set of the nodal S_N equations and then obtaining the solution by symbolic computation. We include the LTS_N method by diagonalization to solve the nodal neutron transport equation and then we outline the convergence of these nodal-LTS_N approximations with the help of a norm associated to the quadrature formula used to approximate the integral term of the neutron transport equation^<(2,3)> We give numerical results obtained with an algebraic computer system (for N up to 8) and with a code for higher values of N. We compare our results for the geometry of a box with a source in a vertex and a leakage zone in the opposite vertex with others techniques used in this problem. code for higher values of N. We compare our results for the geometry of a box with a source in a vertex and a leakage zone in the opposite vertex with others techniques used in this problem.