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The Behaviour of Elastic Fields and Boundary Integral Mellin Techniques Near Conical Points
Abstract
For the computation of the local singular behaviour of an homogeneous anisotropic clastic field near the three-dimensional vertex subjected to displacement boundary conditions, one can use a boundary integral equation of the first kind whose unkown is the boundary stress. Mellin transformation yields a one - dimensional integral equation on the intersection curve 7 of the cone with the unit sphere. The Mellin transformed operator defines the singular exponents and Jordan chains, which provide via inverse Mellin transformation a local expansion of the solution near the vertex. Based on Kondratiev's technique which yields a holomorphic operator pencil of elliptic boundary value problems on the cross - sectional interior and exterior intersection of the unit sphere with the conical interior and exterior original cones, respectively, and using results by Maz'ya and Kozlov, it can be shown how the Jordan chains of the one-dimensional boundary integral equation are related to the corresponding Jordan chains of the operator pencil and their jumps across γ. This allows a new and detailed analysis of the asymptotic behaviour of the boundary integral equation solutions near the vertex of the cone. In particular, the integral equation method developed by Schmitz, Volk and Wendland for the special case of the elastic Dirichlet problem in isotropic homogeneous materials could be completed and generalized to the anisotropic case.
... There are several other types of applications, and the single layer potential is a rather well studied object. We refer to Kozlov, Wendland, and Goldberg [7], Costabel [1], and references found therein for applications and properties of the single layer potential. Moreover, we note that in the case that the surface is the hyperplane x N +1 = 0, the single layer potential is reduced to a Riesz potential of order one. ...
This article considers two weight estimates for the single layer potential
--- corresponding to the Laplace operator in --- on
Lipschitz surfaces with small Lipschitz constant. We present conditions on the
weights to obtain solvability and uniqueness results in weighted Lebesgue
spaces and weighted homogeneous Sobolev spaces, where the weights are assumed
to be radial and doubling. In the case when the weights are additionally
assumed to be differentiable almost everywhere, simplified conditions in terms
of the logarithmic derivative are presented, and as an application, we prove
that the operator corresponding to the single layer potential in question is an
isomorphism between certain weighted spaces of the type mentioned above.
Furthermore, we consider several explicit weight functions. In particular, we
present results for power exponential weights which generalize known results
for the case when the single layer potential is reduced to a Riesz potential,
which is the case when the Lipschitz surface is given by a hyperplane.
... Thus, the need to analyze and characterize singular forms and regularity results for BVPs is crucial and much work has been done in this area (cf. [16] [21], [4,5,8,11,12,15,22], [23] [26]). In this paper we consider the homogeneous Dirichlet problem for the Lamé system in axisymmetric domainsˆΩdomainsˆ domainsˆΩ ⊂ R 3 with conical points on the rotation axis, i.e. ...
Partial Fourier series expansion is applied to the Dirichlet problem for the Lamé equations in axisymmetric domains ⊂ℝ3 with conical points on the rotation axis. This leads to dimension reduction of the three-dimensional boundary value problem resulting to an infinite sequence of two-dimensional boundary value problems on the plane meridian domain Ωa⊂ℝ of with solutions un(n=0,1,2,…) being the Fourier coefficients of the solution û of the 3D BVP. The asymptotic behaviour of the Fourier coefficients un (n=0,1,2,…) near the angular points of the meridian domain Ωa is fully described by singular vector-functions which are related to the zeros αn of some transcendental equations involving Legendre functions of the first kind. Equations which determine the values of αn are given and a numerical algorithm for the computation of αn is proposed with some plots of values obtained presented. The singular vector functions for the solution of the 3D BVP is obtained by Fourier synthesis. Copyright © 2004 John Wiley & Sons, Ltd
This paper is devoted to some transmission problems for the Laplace and linear elasticity operators in two- and three-dimensional nonsmooth domains. We investigate the behaviour of harmonic and linear elastic fields near geometrical singularities, especially near corner points or edges where the interface intersects with the boundaries. We give a short overview about the known results for 2-D problems and add new results for 3-D problems. Numerical results for the calculation of the singular exponents in the asymptotic expansion are presented for both two- and three-dimensional problems. Some spectral properties of the corresponding parameter depending operator bundles are also given. Furthermore, we derive boundary integral equations for the solution of the transmission problems, which lead finally to "local" pseudo-differential operator equations with corresponding Steklov–Poincaré operators on the interface. We discuss their solvability and uniqueness. The above regularity results are used in order to characterize the regularity of the solutions of these integral equations.
This article considers two weight estimates for the single layer potential — corresponding to the Laplace operator in R
N+1 — on Lipschitz surfaces with small Lipschitz constant. We present conditions on the weights to obtain solvability and uniqueness results in weighted Lebesgue spaces and weighted homogeneous Sobolev spaces, where the weights are assumed to be radial and doubling. In the case when the weights are additionally assumed to be differentiable almost everywhere, simplified conditions in terms of the logarithmic derivative are presented, and as an application, we prove that the operator corresponding to the single layer potential in question is an isomorphism between certain weighted spaces of the type mentioned above. Furthermore, we consider several explicit weight functions. In particular, we present results for power exponential weights which generalize known results for the case when the single layer potential is reduced to a Riesz potential, which is the case when the Lipschitz surface is given by a hyperplane.
This paper considers to the equation [\int_{S} \frac{U(Q)}{|P-Q|^{N-1}} dS(Q)
= F(P), P \in S,] where the surface S is the graph of a Lipschitz function \phi
on R^N, which has a small Lipschitz constant. The integral in the left-hand
side is the single layer potential corresponding to the Laplacian in R^{N+1}.
Let \Lambda(r) be a Lipschitz constant of \phi on the ball centered at the
origin with radius 2r. Our analysis is carried out in local L^p-spaces and
local Sobolev spaces, where 1 < p < \infty, and results are presented in terms
of \Lambda(r). Estimates of solutions to the equation are provided, which can
be used to obtain knowledge about the behaviour of the solutions near a point
on the surface. The estimates are given in terms of seminorms. Solutions are
also shown to be unique if they are subject to certain growth conditions. Local
estimates are provided and some applications are supplied.
The simple and double layer potentials for second order linear strongly elliptic differential operators on Lipschitz domains are studied and it is shown that in a certain range of Sobolev spaces, results on continuity and regularity can be obtained without using either Calderon’s theorem on the -continuity of the Cauchy integral on Lipschitz curves [J. L. Journe, “Calderon-Zygmuno operators, pseudo-differential operators and the Cauchy integral of Calderon,” in Lecture Notes in Math. 994, Springer-Verlag, Berlin, 1983] or Dahlberg’s estimates of harmonic measures [“On the Poisson integral for Lipschitz and domains,” Studio Math., 66 (1979), pp. 7–24]. The operator of the simple layer potential and of the normal derivative of the double layer potential are shown to be strongly elliptic in the sense that they satisfy Carding inequalities in the respective energy norms. As an application, error estimates for Galerkin approximation schemes for integral equations of the first kind are derived.
The question is: Under what conditions does the congruence of the projections of two compacta on every two-dimensional plane imply the congruence of the compacta themselves? The results are extended to the cases of (n – 2)-visible and (n – 2)-convex compacta, and even to the case when it is known the projections of the compacta are merely similar.
The advent of high-speed computers has made it possible for the first time to calculate values from models accurately and rapidly. Researchers and engineers thus have a crucial means of using numerical results to modify and adapt arguments and experiments along the way. Every facet of technical and industrial activity has been affected by these developments. The objective of the present work is to compile the mathematical knowledge required by researchers in mechanics, physics, engineering, chemistry and other branches of application of mathematics for the theoretical and numerical resolution of physical models on computers. Since the publication in 1924 of the "Methoden der mathematischen Physik" by Courant and Hilbert, there has been no other comprehensive and up-to-date publication presenting the mathematical tools needed in applications of mathematics in directly implementable form.
The first boundary value problem for the Stokes, Navier-Stokes, Lamé sytems and for the Laplace equation in a bounded domain is studied. The boundary of contains singularities, such as conic points, edges or polyhedral angles. Theorems on solvability in spaces, supplied with weighted and norms () are proved. Coercive estimates of solutions in these spaces as well as pointwise estimates of the Green functions are obtained. The change of properties of generalized solutions under the change of right-hand sides is observed.
Boundary integral equations of the method of potentials for the Laplace differential equation in a domain containing a conic point are considered. For the density of a double-layer (single-layer) potential in the case of Dirichlet’s (Neumann’s) problem, the asymptotic expansion in a neighbourhood of the conic point is derived and the corresponding coefficients are calculated. In the case of a circular cone, the summands of the asymptotic expansion and the dependence of the power exponent of its principal term on the apex angle of the cone are given.
CONTENTS ??1. Introduction. Basic notation ??2. Korn's inequalities in bounded domains ??3. Korn-type inequalities in unbounded domains ??4. Uniqueness theorems of solutions of the main boundary-value problems with finite energy or with a finite Dirichlet integral ??5. Uniqueness theorems under restrictions on the absolute value of the solution in a neighbourhood of infinity ??6. The Dirichlet problem. Uniqueness theorems. Examples ??7. Existence theorems References
CONTENTS Introduction Chapter I. General elliptic boundary-value problems § 1. The solubility of general elliptic boundary-value problems in domains with conic points § 2. The asymptotic behaviour of the solutions of a general boundary-value problem in the neighbourhood of a conic boundary point § 3. General boundary-value problems in non-smooth domains Chapter II. Boundary-value problems for the equations of mathematical physics in non-smooth domains § 1. Boundary-value problems for the system of elasticity theory § 2. Problems of hydrodynamics in domains with a non-smooth boundary § 3. The biharmonic equation Chapter III. Second-order elliptic equations in domains with a non-smooth boundary § 1. Boundary-value problems for second-order elliptic equations in an arbitrary domain § 2. Boundary-value problems in domains with isolated non-regular points on the boundary § 3. Second-order elliptic equations in domains with edges § 4. Boundary-value problems in domains that are diffeomorphic to a polyhedron Chapter IV. Parabolic and hyperbolic equations and systems in non-smooth domains § 1. Parabolic equations and systems in non-smooth domains § 2. Hyperbolic equations and systems in domains with singular points on the boundary References
The asymptotics near a conical point of the solution of an elliptic boundary-value problem contains linear combinations of the special solutions of the “model” homogeneous problem in the cone. One gives formulas for the coefficients of these linear combinations under the assumption that the domain is a cone.
We obtain the operator generalization of the theorem on the logarithmic residue for meromorphic operator-functions. The proof of the generalization is based on a theorem concerning a special factorization of a meromorphic operator-function at a point. This theorem also allows us to generalize, to the case of meromorphic operator-functions, the formula of M. V. Keldyš for the principal part of the resolvent as well as several other theorems.
A definition is given for the multiplicity of a pole for a meromorphic operator-function. The basic properties of the multiplicity of a pole are proved, and also a generalization of the Rouché theorem.
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The operator pencil whose eigenvalues determine singularities of solutions to the Dirichlet problem at the vertex of a cone is studied. First the Dirichlet-Sobolev problem with data on the ray is considered, and the eigenvalues and eigenfunctions of the corresponding operator pencil are described. Then this information is used to show that the result of [2] is best possible in a sense.
For the computation of the singular behavior of an elastic field near a three-dimensional vertex subject to displacement boundary conditions we use a boundary integral equation of the first kind whose unknown is the boundary stress. Localization at the vertex and Mellin transformation yield a one-dimensional integral equation on a piecewise circular curve γ in IR3 depending holomorphically on the complex Mellin parameter. The corresponding spectral points and packets of generalized eigenvectors characterize the desired stress field and are computed by a spline-Galerkin method with graded meshes at the corner points of the curve γ. © 1993 John Wiley & Sons, Inc.
If T is a holomorphic Fredholm operator function on a domain in IE, the inverse of which exists at least at one point, then this inverse is a meromorphic function whose poles are exactly the eigenvalues of T. The operators in the singular part of T- a are finitedimensional and representable by canonical systems of eigenvectors and associated vectors of T and the adjoint operator function T*. The eigenvectors and the associated vectors belonging to these canonical systems fulfil certain biorthogonal relationships. This theorem is fundamental for expansions in series of eigenvectors and associated vectors; it has important applications to boundary eigenvalue problems that are nonlinear in the parameter. It yields a representation of the Green functions in terms of eigenvectors and associated vectors. The theorem is due to Keldyg for holomorphic operator functions of the type T(2) = I + K (2), where I is the identity operator and K (2) is compact for 2, cf. [7], [8]. In [2] Gohberg and Sigal established a factorization theorem for meromorphic operator functions. As an application they proved Keldy~'s representation theorem (apart from the biorthogonality relationships for the eigenvectors and associated vectors) in the general setting. Instead of eigenvectors and associated vectors they used the notion of root functions, which considerably simplified the formulas and made the theorem as well as its proof become more transparent. In the present paper we give a direct proof of Keldy~'s representation theorem which we state in the (simpler) notation of root functions as well as in the original form with respect to eigenvectors and associated vectors. Our proof does not make use of Gohberg and Sigal's factorization theorem; it is extremely elementary and omits all rearrangements of multiple sums.
This paper discusses a finite element approximation for a class of singular integral equations of the first kind. These integral equations are deduced from Dirichlet problems for strongly elliptic differential equations in two independent variables. By a variation of technique due to Aubin, it is shown that the Galerkin method with finite elements as trial functions leads to an optimal rate of convergence.
. This paper is concerned with the effective numerical treatment of elliptic boundary value problems when the solutions contain singularities. The paper deals first with the theory of problems of this type in the context of weighted Sobolev spaces and covers problems in domains with conical vertices and non-intersecting edges, as well as polyhedral domains with Lipschitz boundaries. Finite element schemes on graded meshes for second order problems in polygonal/polyhedral domains are then proposed for problems with the above singularities. These schemes exhibit optimal convergence rates with decreasing mesh size. Finally, we describe numerical experiments which demonstrate the efficiency of our technique in terms of "actual" errors for specific (finite) mesh sizes in addition to the asymptotic rates of convergence. Key Words. Elliptic boundary value problem, singularities, finite element method, mesh grading. AMS(MOS) subject classification. 65N30 1 Introduction This paper is concern...
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