Piecewise Smooth Boundary Research Articles

Characterization of the angular coefficient method on 2D and 3D piecewise smooth boundaries

The angular coefficient method represents a valid and efficient strategy to estimate the distribution of molecules in ultra-high vacuum systems. The problem is described through a Fredholm integral equation of the second kind that is usually solved with standard numerical methods, e.g., the finite element method or the Nyström quadrature method. In this work, we aim to rigorously study the underlying integral equation in order to verify some fundamental mathematical properties and justify the application and behaviour of such numerical methods. In particular, we address to the general scenario where domains are not globally smooth. In such context, boundary corners entail poorly regular integral kernels, which in turn lead to non-Lipschitz solutions and require the adoption of non-standard analysis techniques. By introducing the concept of vacuum-connection, we can establish a methodology to prove the well-posedness of the underlying problem. Furthermore, the undermined regularity of the analytical solution and the consequent lower numerical convergence rate are proved analytically and verified through simple numerical tests.

TetraFEM: Numerical Solution of Partial Differential Equations Using Tensor Train Finite Element Method

In this paper, we present a methodology for the numerical solving of partial differential equations in 2D geometries with piecewise smooth boundaries via finite element method (FEM) using a Quantized Tensor Train (QTT) format. During the calculations, all the operators and data are assembled and represented in a compressed tensor format. We introduce an efficient assembly procedure of FEM matrices in the QTT format for curvilinear domains. The features of our approach include efficiency in terms of memory consumption and potential expansion to quantum computers. We demonstrate the correctness and advantages of the method by solving a number of problems, including nonlinear incompressible Navier–Stokes flow, in differently shaped domains.

On the Existence of Global Compactly Supported Weak Solutions of the Vlasov–Poisson System with an External Magnetic Field

We consider the first mixed problem for the Vlasov–Poisson system with an external magnetic field in a domain with piecewise smooth boundary. This problem describes the kinetics of a two-component high-temperature plasma under the influence of a self-consistent electric field and an external magnetic field. The existence of global weak solutions is proved. In the case of a cylindrical domain, sufficient conditions are obtained for the existence of global weak solutions with supports in a strictly internal cylinder; this corresponds to the confinement of high-temperature plasma in a mirror trap.

Oscillations of the total kinematic momentum vector for viscous fluid dynamics in pipes of arbitrary section

We derive the oscillatory exact solutions to the Navier–Stokes equations that describe z-invariant viscous fluid flows in a general cylindrical pipe C=D×ℝ1 with section D having an arbitrary geometry and a piece-wise smooth boundary ∂D. Exact viscous fluid flows are presented that satisfy the no-slip boundary condition at the pipe's boundary ∂D×ℝ1 and have an arbitrary number of oscillations of the total kinematic momentum vector on any given interval of time [T1,T2]. The stability of the oscillations with respect to small perturbations of infinitely many parameters is proven. The new method for the generation of a hierarchy of exact solutions with oscillating kinematic momentum is developed. Exact solutions to the Navier–Stokes equations without external forces (in addition to the friction forces at the pipes's boundary ∂D×ℝ1) are derived, which have any number of oscillations of the average shift of viscous fluid.

Limits of an increasing sequence of complex manifolds

Let M be a complex manifold which admits an exhaustion by open subsets $$M_j$$ each of which is biholomorphic to a fixed domain $$\Omega \subset \mathbb C^n$$ . The main question addressed here is to describe M in terms of $$\Omega $$ . Building on work of Fornaess–Sibony, we study two cases, namely M is Kobayashi hyperbolic and the other being the corank one case in which the Kobayashi metric degenerates along one direction. When M is Kobayashi hyperbolic, its complete description is obtained when $$\Omega $$ is one of the following domains—(i) a smoothly bounded Levi corank one domain, (ii) a smoothly bounded convex domain, (iii) a strongly pseudoconvex polyhedral domain in $$\mathbb C^2$$ , or (iv) a simply connected domain in $$\mathbb C^2$$ with generic piecewise smooth Levi-flat boundary. With additional hypotheses, the case when $$\Omega $$ is the minimal ball or the symmetrized polydisc in $$\mathbb C^n$$ can also be handled. When the Kobayashi metric on M has corank one and $$\Omega $$ is either of (i), (ii) or (iii) listed above, it is shown that M is biholomorphic to a locally trivial fibre bundle with fibre $$\mathbb C$$ over a holomorphic retract of $$\Omega $$ or that of a limiting domain associated with it. Finally, when $$\Omega = \Delta \times \mathbb B^{n-1}$$ , the product of the unit disc $$\Delta \subset \mathbb C$$ and the unit ball $$\mathbb B^{n-1} \subset \mathbb C^{n-1}$$ , a complete description of holomorphic retracts is obtained. As a consequence, if M is Kobayashi hyperbolic and $$\Omega = \Delta \times \mathbb B^{n-1}$$ , it is shown that M is biholomorphic to $$\Omega $$ . Further, if the Kobayashi metric on M has corank one, then M is globally a product; in fact, it is biholomorphic to $$Z \times \mathbb C$$ , where $$Z \subset \Omega = \Delta \times \mathbb B^{n-1}$$ is a holomorphic retract.

Spontaneous multi-scale void formation and closure in densifying epithelial and fibroblast monolayers from the sub-confluent state.

Using time-lapse phase contrast microscopy, the formation and closure of spontaneously generated voids in the densifying monolayers of isotropic epithelial cells (ECs) and elongated fibroblast cells (FCs) through proliferation from the sub-confluent state are investigated. It is found that, in both types of monolayers after forming a connected network composed of nematic patches with different orientations, numerous multi-scale voids can be spontaneously formed and gradually close with increasing time. The isotropic fluctuations of deformation and crawling of ECs and the anisotropic axial motion/alignment polarizations of FCs are the two keys leading to the following different generic dynamical behaviors. In EC monolayers, voids exhibit irregular boundary fluctuations and easier cell re-orientation of front layer cells (FLCs) surrounding void boundaries. Void closures are mainly through pinching the gap between the opposite fluctuating void boundaries, and the inward crawling of FLCs to reduce void area associated with topological rearrangement to reduce FLC number. In FC monolayers, large voids have piecewise smooth convex boundaries, and cusp-shaped concave boundaries with cells orienting toward the void at cusp tips. The extension of a thin cell bridge from the cusp tip can bisect a large void into smaller voids. For smaller FC voids dominated by convex boundaries, along which cell alignment prohibits inward crawling, the reduction of FLC number through successive outward squeezing of single FLCs by neighboring FLCs sliding along the void boundary plays an important role for topological rearrangement and void closure. Unlike those surrounding artificial wounds in dense EC monolayers, the absence of ring-like purse-strings surrounding EC and FC voids allows topological rearrangements for reducing void perimeter and void area.

Some Results on Theory of Numbers, Partial Differential Equations and Numerical Analysis

In this article, given a number <img src=image/13427314_01.gif> that ends in one and assuming that there are integer solutions <img src=image/13427314_02.gif> for the equations <img src=image/13427314_03.gif> or <img src=image/13427314_04.gif> or <img src=image/13427314_05.gif>, the straight line was used passing through the center of gravity of the triangle bounded by the vertices <img src=image/13427314_06.gif>. Considering A ≥ 25, we manage to divide the domain of the curve <img src=image/13427314_03.gif> into two disjoint subsets, and using Theorem (2.2) of this article, we find the subset where the integer solution of the equation <img src=image/13427314_03.gif> is found. Similar process is done when <img src=image/13427314_05.gif>, in case P is of the form <img src=image/13427314_04.gif> or <img src=image/13427314_07.gif>. These curves are different and to obtain a process similar to the one carried out previously, we proceeded according to Observation 2.2. Our results allow minimizing the number of operations to perform when our problem requires to be implemented computationally. Furthermore, we obtain some conditions to find the solution of the equations: <img src=image/13427314_08.gif> <img src=image/13427314_09.gif>, where <img src=image/13427314_10.gif> is of class <img src=image/13427314_11.gif>, <img src=image/13427314_12.gif> and <img src=image/13427314_13.gif> is a bounded open domain of <img src=image/13427314_14.gif> with piecewise smooth boundary <img src=image/13427314_15.gif>. All the operations carried out to find the solution have been carried out assuming that these exist, and we have found the conditions that <img src=image/13427314_16.gif> must satisfy for the coefficients <img src=image/13427314_17.gif>. We finish by finding an optimal domain for the real solution of a given polynomial of degree five. This process carried out on said given polynomial can also be carried out to reduce the degree of a given polynomial and thus obtain information about its roots.

Mikhlin’s Integral Equation and the Integral Equation with the Generalized Neumann Kernel on Simply Connected Domains

Mikhlin’s integral equation is a classical integral equation for solving boundary value problems for Laplace’s equation. The kernel of the integral equation is known as the Neumann kernel. Recently, an integral equation for solving the Riemann–Hilbert problem was derived. The kernel of the new integral equation is a generalization of the Neumann kernel, and hence, it is called the generalized Neumann kernel. The objective of this paper is to present a detailed comparison between these two integral equations with emphasis on their similarities and differences. This comparison is done through applying both equations to solve Laplace’s equation with Dirichlet boundary conditions in simply connected domains with smooth and piecewise smooth boundaries.

Harmonic Measure of Arcs of Fixed Length

We consider Jordan domains Ω with piece-wise smooth boundaries such that all arcs α ⊂ ∂Ω having fixed length l, 0 < l < length(∂Ω), have equal harmonic measures ω(z0, α, Ω) evaluated at some point z0 ∈ Ω. It is proved that Ω is a disk centered at z0 if the ratio l/length(∂Ω) is irrational and that Ω possesses rotational symmetry by some angle 2π/n, n ≥ 2, around the point z0, if this ratio is rational.

Differences Between Robin and Neumann Eigenvalues

Let Omega {subset } {mathbb {R}}^2 be a bounded planar domain, with piecewise smooth boundary partial Omega . For sigma >0, we consider the Robin boundary value problem -Δf=λf,∂f∂n+σf=0on∂Ω\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} -\\Delta f =\\lambda f, \\qquad \\frac{\\partial f}{\\partial n} + \\sigma f = 0 \\text{ on } \\partial \\Omega \\end{aligned}$$\\end{document}where frac{partial f}{partial n} is the derivative in the direction of the outward pointing normal to partial Omega . Let 0<lambda ^sigma _0le lambda ^sigma _1le ldots be the corresponding eigenvalues. The purpose of this paper is to study the Robin–Neumann gaps dn(σ):=λnσ-λn0.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} d_n(\\sigma ):=\\lambda _n^\\sigma -\\lambda _n^0 . \\end{aligned}$$\\end{document}For a wide class of planar domains we show that there is a limiting mean value, equal to 2{text {length}}(partial Omega )/{text {area}}(Omega )cdot sigma and in the smooth case, give an upper bound of d_n(sigma )le C(Omega ) n^{1/3}sigma and a uniform lower bound. For ergodic billiards we show that along a density-one subsequence, the gaps converge to the mean value. We obtain further properties for rectangles, where we have a uniform upper bound, and for disks, where we improve the general upper bound.

Continuous operator method application for direct and inverse scattering problems

Abstract. We describe the continuous operator method for solution nonlinear operator equations and discuss its application for investigating direct and inverse scattering problems. The continuous operator method is based on the Lyapunov theory stability of solutions of ordinary differential equations systems. It is applicable to operator equations in Banach spaces, including in cases when the Frechet (Gateaux) derivative of a nonlinear operator is irreversible in a neighborhood of the initial value. In this paper, it is applied to the solution of the Dirichlet and Neumann problems for the Helmholtz equation and to determine the wave number in the inverse problem. The internal and external problems of Dirichlet and Neumann are considered. The Helmholtz equation is considered in domains with smooth and piecewise smooth boundaries. In the case when the Helmholtz equation is considered in domains with smooth boundaries, the existence and uniqueness of the solution follows from the classical potential theory. When solving the Helmholtz equation in domains with piecewise smooth boundaries, the Wiener regularization is carried out. The Dirichlet and Neumann problems for the Helmholtz equation are transformed by methods of potential theory into singular integral equations of the second kind and hypersingular integral equations of the first kind. For an approximate solution of singular and hypersingular integral equations, computational schemes of collocation and mechanical quadrature methods are constructed and substantiated. The features of the continuous method are illustrated with solving boundary problems for the Helmholtz equation. Approximate methods for reconstructing the wave number in the Helmholtz equation are considered.

Local equilibrium of particle density in planar Lorentz processes

Particles are injected into a large planar domain through the boundary and perform a random or sufficiently chaotic deterministic motion inside the domain. Our main example is the Sinai billiard, which periodically extended to our large planar domain, is referred to as the Lorentz process. Assuming that the particles move independently from one another and the boundary is also absorbing, we prove the emergence of local equilibrium of the particle density in the diffusive scaling limit in two scenarios. One scenario is an arbitrary domain with piece-wise smooth boundary and a carefully chosen injection rule; the other scenario is a rectangular domain and a much more general injection mechanism. We study the latter scenario in an abstract framework that includes Lorentz processes and random walks and hopefully allows for more applications in the future.

On functions with the boundary Morera property in domains with piecewise-smooth boundary

The problem of holomorphic extension of functions defined on the boundary of a domain into this domain is actual in multidimensional complex analysis. It has a long history, starting with the proceedings of Poincaré and Hartogs. This paper considers continuous functions defined on the boundary of a bounded domain $ D $ in $ \mathbb C ^ n $, $ n&gt; 1 $, with piecewise-smooth boundary, and having the generalized boundary Morera property along the family of complex lines that intersect the boundary of a domain. Morera property is that the integral of a given function is equal to zero over the intersection of the boundary of the domain with the complex line. It is shown that such functions extend holomorphically to the domain $ D $. For functions of one complex variable, the Morera property obviously does not imply a holomorphic extension. Therefore, this problem should be considered only in the multidimensional case $ (n&gt; 1) $. The main method for studying such functions is the method of multidimensional integral representations, in particular, the Bochner-Martinelli integral representation.

On Convergence of Polynomial Approximate Solutions of Minimal Surface Equations in Domains Satisfying the Cone Condition

In this paper we consider the polynomial approximate solutions of the minimal surface equation. It is shown that under certain conditions on the geometric structure of the domain the absolute values of the gradients of the solutions are bounded as the degree of these polynomials increases. The obtained properties imply the uniform convergence of approximate solutions to the exact solution of the minimal surface equation. In numerical solving of boundary value problems for equations and systems of partial differential equations, a very important issue is the convergence of approximate solutions.The study of this issue is especially important for nonlinear equations since in this case there is a series of difficulties related with the impossibility of employing traditional methods and approaches used for linear equations. At present, a quite topical problem is to determine the conditions ensuring the uniform convergence of approximate solutions obtained by various methods for nonlinear equations and system of equations of variational kind (see, for instance, [1]). For nonlinear equations it is first necessary to establish some a priori estimates of the derivatives of approximate solutions.In this paper, we gave a substantiation of the variational method of solving the minimal surface equation in the case of multidimensional space.We use the same approach that we used in [3] for a two-dimensional equation. Note that such a convergence was established in [3] under the condition that a certain geometric characteristic Δ(Ω) in the domain Ω, in which the solutions are considered, is positive. In particular, domain with a smooth boundary satisfied this requirement. However, this characteristic is equal to zero for a fairly wide class of domains with piecewise-smooth boundaries and sufficiently "narrow" sections at the boundary. For example, such a section of the boundary is the vertex of a cone with an angle less than π/2. In this paper, we present another approach to determining the value of Δ(Ω) in terms of which it is possible to extend the results of the work [3] in domain satisfying cone condition.

On the properties of topological discretization of solutions to boundary value problems

This paper seems to show for the first time that Packed block elements used in solving a boundary value problem by the block element method are elements of a discrete topological space. Since the solutions to boundary value problems belong to a discrete topological space instead of equations, it is possible to obtain a solution in a new coordinate system without having to study the boundary value problem. The block element method used in problems of continuum mechanics, which has a connection with topology, can become more effective in applications if the General properties of topological spaces studied in detail are used more deeply. One of the important properties of topology is the existence of discrete topological spaces. Their characteristic property is that an element that represents the Union of any set of elements in a topological space belongs to a discrete topological space. Belonging of discrete block elements to a topological space means that it is possible to completely cover any area with a piecewise smooth boundary and, thus, an exact solution of the boundary problem in it. In topological space, continuous geometric transformations and transitions to new coordinate systems are possible. In this paper, it is shown that Packed block elements generated by the boundary value problem for the Helmholtz equation, are elements of a discrete topological space. Given that scalar solutions of the Helmholtz equation can describe solutions to a fairly wide set of vector boundary value problems, this property also applies to solutions of more complex boundary value problems. The constructions made for the homogeneous Helmholtz equation remain valid for the inhomogeneous one.

Hierarchical Lossy Bilevel Image Compression Based on Cutset Sampling.

We consider lossy compression of a broad class of bilevel images that satisfy the smoothness criterion, namely, images in which the black and white regions are separated by smooth or piecewise smooth boundaries, and especially lossy compression of complex bilevel images in this class. We propose a new hierarchical compression approach that extends the previously proposed fixed-grid lossy cutset coding (LCC) technique by adapting the grid size to local image detail. LCC was claimed to have the best rate-distortion performance of any lossy compression technique in the given image class, but cannot take advantage of detail variations across an image. The key advantages of the hierarchical LCC (HLCC) is that, by adapting to local detail, it provides constant quality controlled by a single parameter (distortion threshold), independent of image content, and better overall visual quality and rate-distortion performance, over a wider range of bitrates. We also introduce several other enhancements of LCC that improve reconstruction accuracy and perceptual quality. These include the use of multiple connection bits that provide structural information by specifying which black (or white) runs on the boundary of a block must be connected, a boundary presmoothing step, stricter connectivity constraints, and more elaborate probability estimation for arithmetic coding. We also propose a progressive variation that refines the image reconstruction as more bits are transmitted, with very small additional overhead. Experimental results with a wide variety of, and especially complex, bilevel images in the given class confirm that the proposed techniques provide substantially better visual quality and rate-distortion performance than existing lossy bilevel compression techniques, at bitrates lower than lossless compression with the JBIG or JBIG2 standards.

Rigidity Properties of the Blum Medial Axis

We consider the Blum medial axis of a region in $$\mathbb R^n$$ with piecewise smooth boundary and examine its “rigidity properties,”by which we mean properties preserved under diffeomorphisms of the regions preserving the medial axis. There are several possible versions of rigidity depending on what features of the Blum medial axis we wish to retain. We use a form of the cross ratio from projective geometry to show that in the case of four smooth sheets of the medial axis meeting along a branching submanifold, the cross ratio defines a function on the branching sheet which must be preserved under any diffeomorphism of the medial axis with another. Second, we show in the generic case, along a Y-branching submanifold, that there are three cross ratios involving the three limiting tangent planes of the three smooth sheets and each of the hyperplanes defined by one of the radial lines and the tangent space to the Y-branching submanifold at the point, which again must be preserved. Moreover, the triple of cross ratios then locally uniquely determines the angles between the smooth sheets. Third, we observe that for a diffeomorphism of the region preserving the Blum medial axis and the infinitesimal directions of the radial lines, the second derivative of the diffeomorphism at points of the medial axis must satisfy a condition relating the radial shape operators and hence the differential geometry of the boundaries at corresponding boundary points.

О некоторых классах корректных задач мембранной теории выпуклых оболочек

On the basis of the theory of the modified Riemann-Hilbert problem for generalized analytic functions, a geometric description is given of a fairly wide family of correct by I. N. Vekua of boundary value problems of the membrane theory of convex hulls with a piecewise smooth boundary. Solutions to the corresponding Riemann-Hilbert problem for an elliptic system of equilibrium equations are found in the classes of N.I. Muskhelishvili and realize a state of tense equilibrium under the condition of stress concentration in corner points. An effective formula is given for calculating the index of the boundary condition, which allows us to formulate the results in a visible form. Families of shells are found for which the solvability picture of the main boundary-value problem coincides with the solvability picture of the Vekua problem for shells with a smooth border.

Топологическая дискретизация решений граничных задач механики сплошных сред

This paper seems to show for the first time that Packed block elements used in solving a boundary value problem by the block element method are elements of a discrete topological space. Since the solutions to boundary value problems belong to a discrete topological space instead of equations, it is possible to obtain a solution in a new coordinate system without having to study the boundary value problem. The block element method used in problems of continuum mechanics, which has a connection with topology, can become more effective in applications if the General properties of topological spaces studied in detail are used more deeply. One of the important properties of topology is the existence of discrete topological spaces. Their characteristic property is that an element that represents the Union of any set of elements in a topological space belongs to a discrete topological space. Belonging of discrete block elements to a topological space means that it is possible to completely cover any area with a piecewise smooth boundary and, thus, an exact solution of the boundary problem in it. In topological space, continuous geometric transformations and transitions to new coordinate systems are possible. In this paper, it is shown that Packed block elements generated by the boundary value problem for the Helmholtz equation, are elements of a discrete topological space. Given that scalar solutions of the Helmholtz equation can describe solutions to a fairly wide set of vector boundary value problems, this property also applies to solutions of more complex boundary value problems.

On Solutions of Elliptic Systems with a Jump at the Boundary

The Dirichlet problem for a strongly elliptic system of the second order with constant coefficients in the domain with a piecewise smooth boundary and piecewise continuous boundary data is considered. The behavior near the jump point of the boundary function is shown.