Integral Equation Methods and Numerical Solutions of Crack and Inclusion Problems in Planar Elastostatics

Helsing, Johan; Peters, Gunnar

doi:10.1137/s0036139998332938

Cited by 16 publications

(5 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Several second kind singular integral equations can be derived for the elastostatic crack problem using potential representations due to Muskhelishvili [17,24]. We pick two equations and only state results.…”

Section: Two Integral Equationsmentioning

confidence: 99%

“…Let qðzÞ be a weight given by qðzÞ ¼ ððz À c s Þðz À c e ÞÞ À 1 2 ; ð19Þ whose value for z 2 C is defined as the limit from the right of the branch given by a branch cut along C and zqðzÞ ¼ 1 at infinity [17]. The two Muskhelishvili integral equations then are…”

Section: Two Integral Equationsmentioning

confidence: 99%

“…The problem of simultaneously resolving the density and the kernel across a corner is difficult. One can use intense panel refinement or product integration with the densities represented in terms of tailor-made basis functions [7,15,17]. While tailor-made basis functions can yield accurate results with few discretization points, their construction relies on analytical methods which are hard to apply general situations.…”

Section: Regularization and Special-purpose Interpolatory Quadraturementioning

confidence: 99%

“…For complex f ðzÞ belonging to a certain weighted L 2 space and by subtracting and adding f ðzÞqðsÞ or f ðzÞ=qðsÞ in the numerator of the kernel one can show the regularizations [17] …”

Section: Regularization and Special-purpose Interpolatory Quadraturementioning

confidence: 99%

See 3 more Smart Citations

Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning

Helsing

Ojala

2008

Journal of Computational Physics

Self Cite

114

170

View full text Add to dashboard Cite

a b s t r a c tWe take a fairly comprehensive approach to the problem of solving elliptic partial differential equations numerically using integral equation methods on domains where the boundary has a large number of corners and branching points. Use of non-standard integral equations, graded meshes, interpolatory quadrature, and compressed inverse preconditioning are techniques that are explored, developed, mixed, and tested on some familiar problems in materials science. The recursive compressed inverse preconditioning, the major novelty of the paper, turns out to be particularly powerful and, when it applies, eliminates the need for mesh grading completely. In an electrostatic example for a multiphase granular material with about two thousand corners and triple junctions and a conductivity ratio between phases up to a million we compute a common functional of the solution with an estimated relative error of 10 À12 . In another example, five times as large but with a conductivity ratio of only a hundred, we achieve an estimated relative error of 10 À14 .

show abstract

Section: Two Integral Equationsmentioning

confidence: 99%

Section: Two Integral Equationsmentioning

confidence: 99%

Section: Regularization and Special-purpose Interpolatory Quadraturementioning

confidence: 99%

Section: Regularization and Special-purpose Interpolatory Quadraturementioning

confidence: 99%

See 2 more Smart Citations

Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning

Helsing

Ojala

2008

Journal of Computational Physics

Self Cite

114

170

View full text Add to dashboard Cite

show abstract

“…Many problems in elasticity (e.g., elastic materials with inclusions) [4][5][6] and crack problems in non-homogeneous materials with stress singularity [7][8][9] of solid mechanics, the electrostatics and magnetostatics (e.g., calculation characteristic parameters of microstrip lines in TEM approximation) [10,11], aerodynamics [12,13], electrodynamical (e.g., dispersion characteristics of 2D (two-dimensional) and 3D (three-dimensional) structures) [14,15], give rise to singular integrals with the Cauchy kernel. Since most Singular Integral Equations (SIE) arising in applications do not have analytical solutions, there is considerable interest in the numerical solution to these problems.…”

Section: Introductionmentioning

confidence: 99%

Method of Singular Integral Equations for Analysis of Strip Structures and Experimental Confirmation

et al. 2021

View full text Add to dashboard Cite

This paper presents a rigorous solution of the Helmholtz equation for regular waveguide structures with the finite sizes of all cross-section elements that may have an arbitrary shape. The solution is based on the theory of Singular Integral Equations (SIE). The SIE method proposed here is used to find a solution to differential equations with a point source. This fundamental solution of the equations is then applied in an integral representation of the general solution for our boundary problem. The integral representation always satisfies the differential equations derived from the Maxwell’s ones and has unknown functions μe and μh that are determined by the implementation of appropriate boundary conditions. The waveguide structures under consideration may contain homogeneous isotropic materials such as dielectrics, semiconductors, metals, and so forth. The proposed algorithm based on the SIE method also allows us to compute waveguide structures containing materials with high losses. The proposed solution allows us to satisfy all boundary conditions on the contour separating materials with different constitutive parameters and the condition at infinity for open structures as well as the wave equation. In our solution, the longitudinal components of the electric and magnetic fields are expressed in the integral form with the kernel consisting of an unknown function μe or μh and the Hankel function of the second kind. It is important to note that the above-mentioned integral representation is transformed into the Cauchy type integrals with the density function μe or μh at certain singular points of the contour of integration. The properties and values of these integrals are known under certain conditions. Contours that limit different materials of waveguide elements are divided into small segments. The number of segments can determine the accuracy of the solution of a problem. We assume for simplicity that the unknown functions μe and μh, which we are looking for, are located in the middle of each segment. After writing down the boundary conditions for the central point of every segment of all contours, we receive a well-conditioned algebraic system of linear equations, by solving which we will define functions μe and μh that correspond to these central points. Knowing the densities μe, μh, it is easy to calculate the dispersion characteristics of the structure as well as the electromagnetic (EM) field distributions inside and outside the structure. The comparison of our calculations by the SIE method with experimental data is also presented in this paper.

show abstract

Scattering of SH waves by an elastic thin-walled rigidly supported inclusion

Emets

Kunets

Матус

2004

Archive of Applied Mechanics (Ingenieur Archiv)

View full text Add to dashboard Cite

Explicit forms of the first-order approximate boundary conditions are derived for a 2D problem of SH waves scattering by a thin, curvilinear, elastic, rigidly supported inclusion in a uniform background. The effects of varying elastic modulus and geometrical forms of the inclusion on the stress and strain states of the body near and far from the ends of the inhomogeneity are examined. The method of investigation is based on the matching of asymptotic expansions with the thickness-to-length ratio as the perturbation parameter. IntroductionThe study of wave propagation in inhomogeneous media consisting of thin-walled layers, confronts with the problem of appropriate approximations for boundary conditions and equilibrium equations of thin-walled inhomogeneities. The general purpose of the approximated boundary conditions, usually referred to as effective boundary conditions, [1, 2], or imperfect interface conditions, [3], is to simplify the analytical or numerical solutions of the wave scattering problem involving complex structures by, e.g., converting a three-media problem into a two-media problem. Such boundary conditions have been widely used in problems of wave propagation and diffraction in inhomogeneous layered media, [4, 5, 6], antenna and radar cross-section studies, [2,7], nondestructive testing of materials, [8-10], modeling of interphases in fiber-reinforced composites, [3,11,12], etc. In the latter case, the interphases constitute thin elastic layers between the fibers and the matrix, where the fiber is usually much stiffer that the matrix material. For this class of structures, only cylindrical or plane unit cell models have been used, see, e.g., [11,12].The goal of the present investigation is to obtain effective boundary conditions for elastodynamic interactions between a thin-walled, curvilinear, elastic, rigidly supported inclusion and the surrounding matrix that would hold for arbitrary values of mechanical parameters of the matrix and the inclusion in the 2D case of SH-waves scattering. The displacement asymptotic near the ends of the scatterer and the approximation of displacements over the scatterer thickness are also considered.

show abstract

Integral Equation Methods and Numerical Solutions of Crack and Inclusion Problems in Planar Elastostatics

Cited by 16 publications

References 15 publications

Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning

Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning

Method of Singular Integral Equations for Analysis of Strip Structures and Experimental Confirmation

Scattering of SH waves by an elastic thin-walled rigidly supported inclusion

Contact Info

Product

Resources

About