Search results for "Line integral"
showing 10 items of 15 documents
Arbitrarily shaped plates analysis via Line Element-Less Method (LEM)
2018
Abstract An innovative procedure is introduced for the analysis of arbitrarily shaped thin plates with various boundary conditions and under generic transverse loading conditions. Framed into Line Element-less Method, a truly meshfree method, this novel approach yields the solution in terms of the deflection function in a straightforward manner, without resorting to any discretization, neither in the domain nor on the boundary. Specifically, expressing the deflection function through a series expansion in terms of harmonic polynomials, it is shown that the proposed method requires only the evaluation of line integrals along the boundary parametric equation. Further, minimization of appropri…
Indefinite integrals involving complete elliptic integrals of the third kind
2017
ABSTRACTA method developed recently for obtaining indefinite integrals of functions obeying inhomogeneous second-order linear differential equations has been applied to obtain integrals with respect to the modulus of the complete elliptic integral of the third kind. A formula is derived which gives an integral involving the complete integral of the third kind for every known integral for the complete elliptic integral of the second kind. The formula requires only differentiation and can therefore be applied for any such integral, and it is applied here to almost all such integrals given in the literature. Some additional integrals are derived using the recurrence relations for the complete …
Exact and approximate analytical solutions for nonlocal nanoplates of arbitrary shapes in bending using the line element-less method
2021
AbstractIn this study, an innovative procedure is presented for the analysis of the static behavior of plates at the micro and nano scale, with arbitrary shape and various boundary conditions. In this regard, the well-known Eringen’s nonlocal elasticity theory is used to appropriately model small length scale effects. The proposed mesh-free procedure, namely the Line Element-Less Method (LEM), only requires the evaluation of simple line integrals along the plate boundary parametric equation. Further, variations of appropriately introduced functionals eventually lead to a linear system of algebraic equations in terms of the expansion coefficients of the deflection function. Notably, the prop…
Henstock type integral in harmonic analysis on zero-dimensional groups
2006
AbstractA Henstock type integral is defined on compact subsets of a locally compact zero-dimensional abelian group. This integral is applied to obtain an inversion formula for the multiplicative integral transform.
Mutual inductance of thick coils for arbitrary relative orientation and position
2017
An exact solution method has been developed recently which gives the mutual inductance of two thin cylindrical coils in terms of line integrals of a new kind of vector potential, induced by the primary coil, around the two circular edges of the secondary coil. This paper describes the extension of this method to thick coils, by wrapping two radial integrations around these line integrals. Results are presented for two pairs of conventional coils and a combination of a superconducting coil and a Bitter coil. Excellent agreement with existing results for non coaxial coils was obtained. The trade-off between accuracy and computing time is also examined.
CVBEM solution for De Saint-Venant orthotropic beams under coupled bending and torsion
2014
The aim of this paper is to provide a solution for the coupled flexure–torsion De Saint Venant problem for orthotropic beams taking full advantage of the complex variable boundary element method (CVBEM) properly extended using a complex potential function whose real and imaginary parts are related to the shear stress components, the orthotropic ratio and the Poisson coefficients. The proposed method returns the complete stress field and the unitary twist rotation of the cross section at once by performing only line integrals. Numerical applications have been reported to show the validity and the efficiency of the proposed modified CVBEM to handle shear stress problems in the presence of ort…
Complex Potential Function in Elasticity Theory: shear and torsion solution through line integrals
2012
Aim of this paper is to introduce a basis formulation framed into complex analysis valid to solve shear and torsion problems. Solution, in terms of a complex function related to the complete tangential stress field, may be evaluated performing line integrals only. This basis formulation framed into elasticity problems may be a useful support for a boundary method to verify the accuracy of an approximation of function solution. The numerical applications stress the latter point and show the validity of these formulas since exact solutions may be reached for sections where the exact solution is known.
CVBEM application to a novel potential function providing stress field and twist rotation at once
2013
AbstractIn this paper, complex variable boundary element method (CVBEM) is used for the solution of de Saint-Venant’s torsion problem in homogenous isotropic elastic beams with a generic cross section, considering a complex potential function related to the stress field. Generally, CVBEM, when used for torsion problems, leads to evaluation of the stress field divided by the twist rotation. The latter has been evaluated by performing a domain integral. In this paper, taking advantage of the aforementioned potential function, it is possible, by applying CVBEM, to evaluate the complete stress distribution and the twist rotation of the cross section and the torsional stiffness factor, performin…
Analytical solution for the solid angle subtended at any point by an ellipse via a point source radiation vector potential
2010
An axially symmetric radiation vector potential is derived for a spherically symmetric point source. This vector potential is used to derive a line integral for the solid angle subtended at a point source by a detector of arbitrary shape and location. An equivalent line integral given previously by Asvestas for optical applications is derived using this formulation. The line integral can be evaluated in closed form for important cases, and the analytical solution for the solid angle subtended by an ellipse at a general point is presented. The solution for the ellipse was obtained by considering sections of a right elliptic cone. The general solution for the ellipse requires the solution of …
Calculations for a disk source and a general detector using a radiation vector potential
2008
A closed form expression for a radiation vector potential is derived for a generalized disk radiation source. By applying Stokes's theorem the surface integral for the radiation flux into a general detector is converted into a much simpler line integral of the vector potential around the edge of the detector. This line integral can be easily evaluated for general detector geometry and general location and angular orientation relative to the disk source. For a number of cases the line integral reduces to integrals of Bessel functions which give various generalizations of Ruby's formula. Explicit formulas and numerical results for the geometric efficiency are given for circular and elliptical…