Quadrature Formula Method Research Articles

Modeling of dynamic and spectral viscoelastic characteristics of composite materials

When designing products made of composite materials intended for use in difficult conditions of inhomogeneous deformations and temperatures, it is important to take into account viscoelastic, including spectral and dynamic, properties of the binder and fillers. The article considers dynamic characteristics (complex modulus, complex malleability, their real and imaginary parts, loss angle tangent) and spectral characteristics of relaxation and creep and their dependence on each other. The abovementioned characteristics were found for all known types of creep kernel and relaxation kernel. To find the spectral characteristics, one of the numerical methods of reversing the Laplace transformation was used - the method of quadrature formulas. Algorithms and computer programs have been compiled to implement this method.

Numerical methods of Laplace transform inversion in the problem of determination of viscoelastic characteristics of composite materials

sWhen designing products made of composite materials intended for operation in difficult conditions of inhomogeneous deformations and temperatures, it is important to consider the viscoelastic properties of the binder and fillers. The article analyzes the relationship between relaxation and creep characteristics. All known creep and relaxation nuclei in the literature are considered. The problem of transformation of creep characteristics into relaxation characteristics and Vice versa is discussed in detail. There is a simple relationship between the creep and relaxation functions in the Laplace image space. However, when returning to the space of the originals, in many cases there are great difficulties in reversing the Laplace transform. Two numerical methods for inverting the Laplace transform are considered: the use of the Fourier series in sine and the method of quadrature formulas. Algorithms and computer programs for realization of these methods are made. It is shown that the operating time of a computer program implementing the Fourier method by sine is almost 2 times less than the operating time of a computer program implementing the quadrature formula method. However, the first method is inferior to the latter method in accuracy of calculations: the relaxation functions and relaxation rates, it is advisable to find the first method, since the computational error is almost indiscernible, and the functions of creep and creep speed, the second way, because for most functions, the result obtained by the second method is much more accurate than the result obtained by the first method.

Axially Symmetric Temperature Field of a Truncated Conic Shell with Variable Heat-Transfer Coefficients

A method for reducing the heat-conduction problem for a truncated conic shell to the solution of a system of integral equations with Volterra and Fredholm integral operators of the second kind is proposed for the case of coordinate-dependent heat-transfer coefficients and ambient temperature. The system is solved numerically by the method of quadrature formulas. The numerical analyses of the distributions of mean temperature and temperature moment are performed.

Thermostressed state of an infinite cylindrical shell with coordinate-dependent heat exchange coefficients and ambient temperature

We investigate the stress-strain state of an infinite cylindrical shell caused by a difference between the temperatures of the surrounding medium on the faces and the dependence of the coefficients of heat exchange from them on the coordinate. The contour of the shell at infinity is free from forces and moments, and the influence of the curvature of the shell on the temperature distribution and stress distribution is also taken into account. The problem of heat conduction is reduced to a system of Fredholm integral equations of the second kind, which is solved numerically by the method of quadrature formulas. We present results of a numerical analysis of distributions of temperature characteristics, the deflection, stresses, force, and bending moments.

Thermal bending of a strip and a rectangular plate with coordinate-dependent heat exchange coefficients

The thermal bending of a simply supported strip and a rectangular plate caused by a difference between the temperatures of the surroundings on the faces and the coordinate-dependent heat exchange coefficients on them are investigated. A method of reducing the boundary-value problem of heat conduction for these bodies to an interrelated system of Fredholm integral equations of the second kind is proposed. We construct a numerical scheme for the solution of integral equations by the method of quadrature formulas. Results of a numerical analysis of distributions of the temperature moment and bending caused by it for a rectangular plate are presented.

Numerical analysis of a non‐singular boundary integral method: Part II: The general case

AbstractIn order to numerically solve the interior and the exterior Dirichlet problems for the Laplacian operator, we have presented in a previous paper a method which consists in inverting, on a finite element space, a non‐singular integral operator for circular domains. This operator was described as a geometrical perturbation of the Steklov operator, and we have precisely defined the relation between the geometrical perturbation and the dimension of the finite element space, in order to obtain a stable and convergent scheme in which there are non‐singular integrals. We have also presented another point of view under which the method can be considered as a special quadrature formula method for the standard piecewise linear Galerkin approximation of the weakly singular single‐layer potential.In the present paper, we extend the results given in the previous paper to more general cases for which the Laplace problem is set on any 𝒞∞ domains. We prove that the properties of stability and convergence remain valid. Copyright © 2002 John Wiley & Sons, Ltd.

Numerical analysis of a non‐singular boundary integral method: Part I. The circular case

AbstractIn order to numerically solve the interior and the exterior Dirichlet problems for the Laplacian operator, we present here a method which consists in inverting, on a finite element space, a non‐singular integral operator. This operator is a geometrical perturbation of the Steklov operator, and we precisely define the relation between the geometrical perturbation and the dimension of the finite element space, in order to obtain a stable and convergent scheme. Furthermore, this numerical scheme does not give rise to any singular integral.The scheme can also be considered as a special quadrature formula method for the standard piecewise linear Galerkin approximation of the weakly singular single layer potential, the special quadrature formula being defined by the introduction of a neighbouring curve.In the present paper, we prove stability and we give error estimates of our numerical scheme when the Laplace problem is set on a disk. We will extend our results to any domains by using compact perturbation arguments, in a second paper. Copyright © 2001 John Wiley & Sons, Ltd.

On the numerical solution of the direct scattering problem for an open sound-hard arc

We derive a hypersingular integral equation, which is equivalent to the scattering problem. Using the cosine substitution, we obtain an integral equation which is closely related to the integral equation for the case of a closed boundary. We solve this equation approximately by a quadrature formula method and obtain pointwise error estimates in Hölder norms by using a perturbation argument.

Approximative methods for nonlinear equations (two approaches to the convergence problem)

THIS survey paper gives a functional-analytical treatment of discretization methods such as quadrature formula method for nonlinear integral equations, difference method for nonlinear boundary value problems, etc. Two approaches to the convergence problem have been developed. The first of them (Section 3) is applicable to an equation with differentiable operator and rests on a remark that such an operator is locally almost linear. The second, less traditional approach (Section 4) is based on a topological concept, namely the invariance of the fixed point index under suitable approximations of an operator. As regards the approximation concepts, the paper is built on a relatively novel principle of regular convergence of operators (Section 2). In our fixed opinion, this concept is rather appropriate to applications, and we hope that the reader agrees with us familiarizing himself with the proof ideology of Sections 5-7. Another methodological prop of the paper is the concept of discrete convergence (Section 1). In Sections 5-7 the abstract results of Sections l-4 have been applied to the quadrature formula method for nonlinear integral equations and to the collocation, subregion, Galerkin and difference methods for nonlinear boundary value problems. Only ordinary differential equations are considered. For partial differential equations our approaches are still weakly developed : first works (e.g. [l-3]) concern linear equations. Sections l-3 contain more material than is urgently needed for applications, our significant goal. By stars are labelled the sections, propositions etc. that can be omitted if one wishes to get to applications more quickly. The main text contains only few references. For the reference notes, see the end of the paper.

Implementation of non-linear estimators using quadrature formulae†

Abstract A new method of non-linear estimator implementation called quadrature formula method is presented. It is different from other methods in that emphasis is placed directly upon approximating the weighted integral of the conditional density function rather than the function itself. Valuable perspective is developed by relating this method to certain former methods based on density function approximation. In particular, it is shown that the point mass representation method is included as a special ease of the quadrature formula method with an accuracy consistent with that of a rectangular rule. The fundamental spline approximation method when formulated in the context of quadrature formula method turns out to be comparable to a trapezoidal rule. Of significant practical interest is the fact that improved accuracy can he achieved with essentially the same amount of labour required by the point mass representation method, an amount which practically is the minimum of all the existing methods. In additi...