Complex hypersingular integrals and integral equations in plane elasticity

Linkov, A. M.; Mogilevskaya, Sofia G.

doi:10.1007/bf01183951

Cited by 78 publications

(51 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Numerical results for cracks [9], and recently produced calculations for two and four interacting blocks, confirm efficiency and accuracy of the method employing complex hypersingular equations.…”

Section: Complex Hypersingular Equation For Plane Elasticity Problemsmentioning

confidence: 75%

“…A new hypersingular equation follows from these results [9]. It serves for cracks and blocky inhomogeneous systems; for internal and external problems.…”

Section: Complex Hypersingular Equation For Plane Elasticity Problemsmentioning

confidence: 99%

“…The way to do this is quite clear from the methodology described above. It was used in [9], where presented all the subsequent details.…”

Section: Complex Hypersingular Equation For Plane Elasticity Problemsmentioning

confidence: 99%

“…All the coefficients are expressed by analytical formulae. For tip elements we also have analytical quadrature formulae for any rational exponent in asymptotic [9].…”

Section: Complex Hypersingular Equation For Plane Elasticity Problemsmentioning

confidence: 99%

See 3 more Smart Citations

Real and Complex Hypersingular Integrals and Integral Equations in Computational Mechanics*

Linkov¹

1995

Demonstratio Mathematica

Self Cite

View full text Add to dashboard Cite

Hypersingular integrals and integral equations became very popular last decade in computational mechanics. The reason is quite clear: they provide a natural and effective means to solve problems involving discontinuities. These are problems of cracks and interacting blocks in elasticity; thin wings in fluid dynamics; shields in electroplating; low permeability walls and geotextile layers in groundwater studies, etc.But these integrals and their direct values have a rather long history. Direct values of hypersingular integrals became generally known as a result of publication in 1923 of the famous Hadamard lectures on Cauchy problem for hyperbolic equations (supplemented French edition was published in 1932 [1]). J. Hadamard termed the direct values as "finite part integrals". They are now also widely known as Hadamard's integrals. Many years later, in his book "Psychology of invention in mathematical field" edited in 1954 [2], Hadamard wrote (my back translation from Russian): "I could no more avoid this method than the prisoner in Edgar Allen Poe's poem 'The pit and the pendulum' could avoid the pit in the center of his dungeon."These integrals were a great invention and became an important stimulus to the development of distribution theory. R. Courant in his course "Partial differential equations" [3] wrote (again my back translation): "Actually,

show abstract

Section: Complex Hypersingular Equation For Plane Elasticity Problemsmentioning

confidence: 75%

“…A new hypersingular equation follows from these results [9]. It serves for cracks and blocky inhomogeneous systems; for internal and external problems.…”

Section: Complex Hypersingular Equation For Plane Elasticity Problemsmentioning

confidence: 99%

“…The way to do this is quite clear from the methodology described above. It was used in [9], where presented all the subsequent details.…”

Section: Complex Hypersingular Equation For Plane Elasticity Problemsmentioning

confidence: 99%

“…All the coefficients are expressed by analytical formulae. For tip elements we also have analytical quadrature formulae for any rational exponent in asymptotic [9].…”

Section: Complex Hypersingular Equation For Plane Elasticity Problemsmentioning

confidence: 99%

See 2 more Smart Citations

Real and Complex Hypersingular Integrals and Integral Equations in Computational Mechanics*

Linkov¹

1995

Demonstratio Mathematica

Self Cite

View full text Add to dashboard Cite

show abstract

“…If one extends one's search to Fredholm equations of the first kind it is easier to find candidates. In fact, many popular integral equation used for fracture mechanics problems [8,37,44] (hypersingular extensions of the Somigliana identity [34], also known as standard BEM) are equations of this type. Schemes based on first kind equations retain some of the advantages of second kind integral equation schemes, such as ease of meshgeneration.…”

Section: Why Are Integral Equation Methods Needed?mentioning

confidence: 99%

Stress computations on perforated polygonal domains

Englund

Helsing

2003

Engineering Analysis with Boundary Elements

View full text Add to dashboard Cite

A high order accurate and fast algorithm is constructed for 2D stress problems on multiply connected finite domains. The algorithm is based on a Fredholm integral equation of the second kind with non-singular operators. The unknown quantity is the limit of an analytic function. On polygonal domains there is a trade-off between stability and rate of convergence. A moderate amount of precomputation in higher precision arithmetic increases the stability in difficult situations. Results for a loaded single edge notched specimen perforated with 1170 holes are presented. The general usefulness of integral equation methods is discussed. q

show abstract

The complex one-sided integrals of Cauchy and Hadamard and application to boundary element method

Mogilevskaya

1998

Int. J. Numer. Anal. Meth. Geomech.

Self Cite

View full text Add to dashboard Cite

SUMMARYThe definitions of complex integrals of Cauchy and Hadamard with the singular point coinciding with the end point of the integration curve are proposed. It is shown that the new integrals satisfy most of the properties of the regular ones, including the change of variables. It is also shown that the Cauchy principal value (CPV) and Hadamard finite-part (HFP) integrals can be considered as a sum of the new type integrals. The application to numerical solution by the boundary element method (BEM) and the complex hypersingular integral equation (CHSIE) for the multiregions of interacting elastic bodies and bodies with cracks and holes is discussed. The different ways to place the collocation points are considered. The numerical results for the problems of circular hole and circular elastic inclusion in infinite plate indicated that the appropriate choice of the approximating functions leads to a high accuracy of the calculation. Applications of the new technique to geomechanics problems are discussed.1998 John Wiley & Sons, Ltd.

show abstract

Complex hypersingular integrals and integral equations in plane elasticity

Cited by 78 publications

References 11 publications

Real and Complex Hypersingular Integrals and Integral Equations in Computational Mechanics*

Real and Complex Hypersingular Integrals and Integral Equations in Computational Mechanics*

Stress computations on perforated polygonal domains

The complex one-sided integrals of Cauchy and Hadamard and application to boundary element method

Contact Info

Product

Resources

About