Method Of Mechanical Quadratures Research Articles

Flexural vibrations of a thin circular elastic inclusion in an unbounded body under the action of a plane harmonic wave

The interaction of a plane harmonic longitudinal wave with a thin circular elastic inclusion is considered. The wave front is assumed to be parallel to the inclusion plane. Since the inclusion is thin, the matrix-inclusion interface conditions (perfect bonding) are formulated on the mid-plane of the inclusion. The bending displacements of the inclusion are determined from the bending equation for a thin plate. The problem is solved using discontinuous Lame solutions for harmonic vibrations. Therefore, the problem can be reduced to the Fredholm equation of the second kind for a function associated with the discontinuity of normal stresses on the inclusion. The equation obtained is solved by the method of mechanical quadratures using Gaussian quadrature formulas. Approximate formulas for the stress intensity factors are derived. Results from a numerical analysis of the dependence of the SIFs on the dimensionless wave number and the stiffness of the inclusion are presented

SPLITTING EXTRAPOLATIONS FOR SOLVING BOUNDARY INTEGRAL EQUATIONS OF MIXED BOUNDARY CONDITIONS ON POLYGONS BY MECHANICAL QUADRATURE METHODS

To solve the boundary integral equations (BIE) of mixed boundary conditions, we propose the mechanical quadrature methods (MQM) using specific quadrature rule to deal with weakly singular integrals. Denote $h_{m}$ as the mesh width of a curved edge $\Gamma_{m}$ ($m=1,...,d)$ of polygons. Then the multivariate asymptotic expansions of solution errors are found to be $O(h^{3}),$ where $h=\max_{1\leq m\leq d}h_{m}.$ Hence, by using the splitting extrapolation methods (SEM), the high convergence rates as $O(h^{5})$ can be achieved. Moreover, numerical examples are provided to support our theoretical analysis.

Mechanical quadrature methods and their extrapolations for solving boundary integral equations of the conductivity problem with discontinuous media

Mixed boundary value problems of the conductivity equation ∇ . ( γ ∇ u ) = 0 with discontinuous media are converted into boundary integral equations by using the single layer potential. For solving the induced boundary integral equations, a mechanical quadrature method is proposed for periodic Fredholm integral equations with logarithmic singularity, which possesses a high order accuracy O ( h 3 ) , less computational complexity and asymptotic expansion of the errors. By means of Richardson extrapolation, an approximation with a higher accuracy order O ( h 5 ) is obtained. Moreover, a posteriori error estimate for the algorithm is derived, which can be used to constructed adaptive algorithm. Several numerical examples show that the accuracy order of approximation is very high, the extrapolation and a posteriori error estimates are also very effective.

Permeance computation using indirect boundary integral equation method

An approach using indirect boundary integral equation method is proposed to determine the permeance between ferromagnetic poles in axisymmetric and three-dimensional magnetic systems. A generalised mathematical model is given for both types of magnetic systems. It consists of Fredholm integral equations of the first kind with respect to fictitious magnetic charge density sought in the form of simple layer potential. The system of boundary integral equations is solved using the method of mechanical quadratures. The approach is implemented in its own computer code. Results are presented for axisymmetric poles of electromagnets (cylinders, cones and frustum cones) and for a three-dimensional clapper-type system. Comparisons with known formulas are made and their accuracies are estimated. The approach presented is useful at the stage of preliminary design of magnetic systems. It is also applicable to computation of capacitances and electrical conductances.

Quadrature formulas of the highest trigonometric accuracy grade and their applications

In many branches of mathematics and in applications the quadrature formulas of the highest algebraic accuracy grade (HAAG) are widely used. The first formula of this type was first proposed and studied by Gauss for the unit weight function and the standard segment. Later many papers were devoted to the quadrature formulas of the HAAG (see, e. g., [1–5] and references therein). During the last decades the quadrature formulas of the highest trigonometric accuracy grade (HTAG) were also put into use. The first results in this realm, evidently, were obtained in the works of V. I. Krylov and I. P. Mysovskikh (see, e. g., [1, 3]). However, the quadrature formulas of the HAAG and those of the HTAG differ in principle. As distinct from the quadrature formulas of the HAAG, the formulas of the HTAG are studied insufficiently; most papers devoted to them are only illustrative. The objective of this paper is to fill the mentioned gap (to certain extent). We establish approximative and structural properties of the quadrature formulas of the HTAG. In addition, we derive the following rather unexpected conclusion: The well-known classical quadrature formulas obtained by Hermite and Gauss with the Chebyshev weight of the II nd kind, as well as the Clenshaw–Curtis formula represent special cases of a quadrature formula of the HTAG under the proper choice of its parameters. The second part of our paper is dedicated to the application of these formulas in the numerical analysis of the Fredholm integral equations with periodic coefficients. We pay the most attention to the functional-theoretic substantiation of the method of mechanical quadratures based on quadrature formulas of the HTAG.

Quadrature solution methods for nonlinear singular integral equations

Here y(t) and h(t, τ, u) are given functions defined for −1 ≤ t, τ ≤ 1, −∞ < u < ∞; λ is a numerical parameter; x(t) is the desired function; the singular integral is understood in the sense of the Cauchy principal value ([1], Chap. 2, § 1). Many problems important for applications can be reduced to these NSIE on an open contour (see, e. g., [1–3] and references therein). In what follows, we consider several computational schemes of the method of mechanical quadratures. These schemes are based on the approximation of the singular operators by finite-dimensional ones generated by the quadrature formulas. The quadrature methods are most simple for the numerical realization, but their theoretical substantiation is rather difficult. To this end, we apply certain results from nonlinear functional analysis (for instance, [4]) and from the theory of singular integral equations [5, 6, 2].

Approximate solution of singular integro- differential equations in generalized Hölder spaces

We have elaborated the numerical schemes of collocation methods and mechanical quadrature methods for approximate solution of singular integro- differential equations with kernels of Cauchy type. The equations are defined on the arbitrary smooth closed contours of complex plane. The researched methods are based on Fejer points. Theoretical background of collocation methods and mechanical quadrature methods has been obtained in Generalized Holder spaces.

Integral-projection method and singularities of its application in mating shells with inconsistent parameters of approximation

We describe a numerical integral-projection method used by the authors for the approximate solution of systems of interrelated two-dimensional linear boundary-value problems in mechanics of composite shell systems. The method is based on discretization in each shell substructure of a two-dimensional problem along one of coordinates using a projection-grid variant of the Galerkin-Petrov method and its subsequent transformation to a system of ordinary differential equations; by integration and introduction of sought functions as unknown derivatives, the system is reduced to a system of integral equations being solved by the method of mechanical quadratures. The method is characterized by the fact that its application requires no additional conditions of conformity with discretization parameters of substructures being mated.

Boundary-value problem of electroelasticity for a piezoceramic elliptic cylinder and a layer with tunnel hollow

We consider a skew-symmetric stressed state of a piezoceramic layer weakened by a through elliptic hollow filled with air. The surfaces of the layer (or a finite elliptic piezoceramic cylinder) operating in contact with air are covered with a diaphragm rigid in its plane and flexible in the perpendicular direction. The stress vector {X1, X2, X3} acts on the lateral surface. The boundary-value problem of electroelasticity is reduced to a system of singular integral equations. This system is numerically solved by the method of mechanical quadratures. The integral representations of the solutions are constructed on the basis of the corresponding matrix of Φ-solutions. The distribution of normal circular stresses over the elliptic contour subjected to the action of normal or tangential stresses is investigated.

Plane problem of diffraction of elastic harmonic waves on periodic curvilinear inserts

Using the method of integral equations, we solved the plane problem of diffraction of elastic waves on a periodic system of rigid tunnel inserts. The obtained system of singular integral equations was solved numerically by the method of mechanical quadratures. We present the diagrams of stress distribution near the insert tips.

Stress concentration near a thin elastic inclusion under interaction with harmonic waves in the case of smooth contact

We solve the problem on the interaction of plane harmonic waves with a thin elastic plate-shaped inclusion. The ambient medium is assumed to be in plane strain. The smooth contact conditions are satisfied on both sides of the inclusion. The bending displacements of the inclusion are determined from the corresponding differential equation. In the statement of boundary conditions for this equation, one should take into account the transverse forces and bending moments applied to the lateral edges of the inclusion, while the boundary conditions are posed on the midplane of the inclusion. Using the discontinuous solution method, we reduce the problem to a system of two singular integral equations, which are solved numerically by the mechanical quadrature method. We obtain approximate formulas for the stress intensity coefficients near the ends of the inclusion and for the transverse forces and moments applied to the inclusion.

Influence of the coupling of mechanical and thermal fields on the amplitude-frequency characteristics of a cylinder

A dynamic boundary-value problem of coupled thermoelasticity for a finite cylinder with mixed boundary conditions is solved. The problem is reduced to a system of four singular integral equations solved by the mechanical-quadrature method. A numerical experiment is conducted to obtain amplitude-frequency characteristics for finite cylinders with different cross sections. The effect of thermoelastic coupling on stress distribution is assessed

Mechanical quadrature methods and their extrapolation for solving first kind Abel integral equations

This paper presents high accuracy mechanical quadrature methods for solving first kind Abel integral equations. To avoid the ill-posedness of problem, the first kind Abel integral equation is transformed to the second kind Volterra integral equation with a continuous kernel and a smooth right-hand side term expressed by weakly singular integrals. By using periodization method and modified trapezoidal integration rule, not only high accuracy approximation of the kernel and the right-hand side term can be easily computed, but also two quadrature algorithms for solving first kind Abel integral equations are proposed, which have the high accuracy O ( h 2 ) and asymptotic expansion of the errors. Then by means of Richardson extrapolation, an approximation with higher accuracy order O ( h 3 ) is obtained. Moreover, an a posteriori error estimate for the algorithms is derived. Some numerical results show the efficiency of our methods.

Mechanical quadrature methods and their splitting extrapolations for solving boundary integral equations of axisymmetric Laplace mixed boundary value problems

The mechanical quadrature methods (MQM) and splitting extrapolation methods (SEM) are applied to the boundary integral equations (BIE) of axisymmetric mixed boundary value problems governed by Laplace's equation. By ring potential theory, the double integral equations of axisymmetric Laplace problems can be converted into the single integral equations. For solving the BIE, the MQM based on the quadrature rules for computing the singular periodic functions are presented, which possesses a high order accuracy O ( h 0 3 ) and low computing complexities. Moreover, using the SEM based on the multi-parameter asymptotic error expansion, we cannot only improve the accuracy order of approximation, but also give a posteriori error estimate. Several numerical examples show that the accuracy order of approximation is very high, and the SEM and a posteriori error estimate are also very effective.

Modeling of the scattering of electromagnetic waves by a thin dielectric coating on a cylinder

Double-sided boundary conditions containing only tangential components of a diffracted field are used to model the interaction of electromagnetic waves with a cylinder of arbitrary cross section covered with a thin dielectric layer. The obtained boundary-value problem is reduced to a system of two singular integral equations of the second kind with kernels whose structure is similar to the kernels of integral equations of the first kind for a perfectly conducting scatterer. The numerical solution of the integral equations of the problem is obtained by the method of mechanical quadratures. The scattering properties of an elliptic cylinder with different dielectric coatings are studied in the superhigh-frequency band. It is shown that the coating strongly affects the diffraction properties of the cylinder.

On the numerical solution of two-dimensional singular integral equation

The mechanical quadrature method described for two-dimensional nonlinear singular integral equation with Hilbert kernel in Banach space. The results are compared with the exact solution of the integral equation.

Heating of a half space containing an inclusion and a crack

Two-dimensional stationary problems of heat conduction and thermoelasticity for an elastic half space containing an elastic cylindrical macroinclusion and a thermally insulated crack are investigated. The problems are reduced to systems of two singular integral equations on closed (the boundary of the inclusion) and nonclosed (crack) contours. The numerical solutions of these systems are obtained by the method of mechanical quadratures for an inclusion in the form of an elliptic cylinder and a rectilinear crack in a half space heated by a friction heat flow uniformly distributed over a part of the surface of the half pace. For the problem posed for two cases of matrix-inclusion compositions (steel-aluminium and aluminium-steel), the numerical solutions are presented in the form of the plots of the stress intensity factors as functions of the dimensions of the inclusion and the distance between the inclusion and the crack.

Longitudinal Shear of an Infinite Body with a System of Polygonal Cracks

On the basis of singular integral equations, we propose a new method for the solution of antiplane problems of elasticity theory for bodies with a system of polygonal cracks with regard for the singularities of stresses at the corner points. The modified singular integral equations have continuous regular kernels and right-hand sides, i.e., belong to the same type as in the case of smooth boundary contours. A numerical solution of these equations is found by the method of mechanical quadratures. The values of the stress intensity factors at the corner points and tips of both a three-link polygonal crack and a system of two two-link polygonal cracks in an infinite body are presented. The limiting case of two cracks which form a rhombic hole is also analyzed.

Application of the Method of Singular Integral Equations to the Determination of the Contours of Equistrong Holes in Plates

We propose a procedure for the determination of the contours of equistrong holes in elastic plates based on the method of singular integral equations. The problem is reduced to the solution of inverse two-dimensional problems of the theory of elasticity with unknown boundaries. The singular integral equations of the direct problems are solved numerically by the method of mechanical quadratures. For plates with one, two, or a system of periodic holes loaded by biaxial tension at infinity and internal pressure on the contours of holes, we determine the parameters of the contours of equistrong holes. Some numerical results are compared with the available data obtained by using different methods. Good accuracy of numerical results is attained even for small numbers of the parameters of shape of the holes and nodal points on their contours, which confirms the efficiency of the proposed procedure.

Elastic Equilibrium of an Anisotropic Half Plane with Periodic System of Holes and Cracks

We present integral equations of a problem of the theory of elasticity for a half plane weakened by holes and curvilinear cracks (including the case of their periodic arrangement). We consider the cases where the boundary of the half plane is stress-free or rigidly fixed. The integral equations are solved numerically by the method of mechanical quadratures.