3D J‐integral evaluation using the computation of line and surface integrals

In the analysis of fracture mechanics of structures using three‐dimensional (3D) J‐integral, an integral evaluation of line and surface is required. However, because surface integral evaluation requires the calculation of the second derivative of displacement field and commercial finite element codes cannot calculate it, then this portion of the integral is neglected in some research. In this paper, a method for computing 3D J‐integral is presented using finite element analysis. In the analysis, the second derivative evaluation of displacement field is employed. The method is implemented in calculating the J‐integral of some 3D cracks and results are compared to well‐known reference values. The results show that the method is reliable and is suitable for applications in engineering. The portion of 3D J‐integral, namely the surface integral value is investigated and it is shown that neglecting this portion can introduce considerable error in the final results.     

带积分边界条件的Poisson方程边值问题的近似解

对带积分形式边界条件的Ｐｏｉｓｓｏｎ方程边值问题，当它的边界充分小时，其边值问题的解可用不带积分形式边界条件的边值问题的解来近似。本文将用泛函分析方法证明这个结论     

Some applications for Kramer's generalized sampling theorem

Summary Kramer's generalization of Shannon's sampling theorem takes us from a signal represented by a finite Fourier transform to a signal represented by another and more general finite integral transform. In this paper we will attempt to show that the already obtained results for Kramer's theorem are of use in the field of finite integral transforms. Also by introducing such transforms one can treat some communications problems. An example is the case of representing a signal which is the output of time variant filter.     

An interface integral equation method for solving general multi‐medium mechanics problems

In this paper, based on the general stress–strain relationship, displacement and stress boundary‐domain integral equations are established for single medium with varying material properties. From the established integral equations, single interface integral equations are derived for solving general multi‐medium mechanics problems by making use of the variation feature of the material properties. The displacement and stress interface integral equations derived in this paper can be applied to solve non‐homogeneous, anisotropic, and non‐linear multi‐medium problems in a unified way. By imposing some assumptions on the derived integral equations, detailed expressions for some specific mechanics problems are deduced, and a few numerical examples are given to demonstrate the correctness and robustness of the derived displacement and stress interface integral equations. Copyright © 2015 John Wiley & Sons, Ltd.     

ON THE CHOICE OF A DERIVATIVE BOUNDARY ELEMENT FORMULATION USING HERMITE INTERPOLATION

This paper reports on some problems that can arise with the use of regularized derivative boundary integral equations. It concentrates on developing a formulation for the simple Laplace equation using a cubic Hermite interpolation and shows how certain combinations of derivative and conventional boundary integral equations can result in a solution scheme severely lacking in stability. With some simple two- and three-dimensional geometries, the derivative equations on their own do not provide enough information to solve a Dirichlet problem. Even combinations of the conventional and derivative equations fail for some simple geometries. We conclude that the only consistently successful combination is that of the conventional equation with the tangential derivative equation, which showed cubic convergence of results with mesh refinement. Numerical results are presented for this scheme in both two and three dimensions.     

Curved computational integral imaging reconstruction technique for resolution-enhanced display of three-dimensional object images

A novel curved computational integral imaging reconstruction (C-CIIR) technique for the virtually curved integral imaging (VCII) system is proposed, and its performances are analyzed. In the C-CIIR model, an additional virtual large-aperture lens is included to provide a multidirectional curving effect in the reconstruction process, and its effect is analyzed in detail by using the ABCD matrix. With this method, resolution-enhanced 3D object images can be computationally reconstructed from the picked-up elemental images of the VCII system. To confirm the feasibility of the proposed model, some experiments are carried out. Experiments revealed that the sampling rate in the VCII system could be kept at a maximum value within some range of the distance z, whereas in the conventional integral imaging system it linearly decreased as the distance z increased. It is also shown that resolutions of the object images reconstructed by the C-CIIR method have been significantly improved compared with those of the conventional CIIR method.     

A new iterative integral formulation for semilinear equations based on the generalized quasilinearization theory

This paper presents a new iterative integral approach for solving semilinear equations. The integral formulation is derived based on the generalized quasilinearization theory in which nonlinear equations are replaced by a set of iterative linear equations. An advantage of the new formulation is that its convergence is guaranteed under a given condition and the convergence rate can be quadratic. The effectiveness of the new approach has been demonstrated on several examples of the nonlinear Poisson type. Comparisons with some existing methods and a study of the convergence rate have also been conducted in this work.     

Boundary integral Equations Method in two- and three-dimensional problems of elastodynamics

In this paper the boundary integral equations method (BIEM) are considered for elastodynamic initial boundary value problems. It's known two approaches are discerned for account time. First of one is a combination of BIEM with Laplace (Fourier) transformation. This approach was suggested and realized by Cruse T.E. and Rizzo F. J. By them BIE in Laplace transformation space were obtained, investigated and some concrete problems were solved. This method was developed also by Manolis G. D., Beskos D. and other scholars for some dynamic problems solving.The second approach using retarding potentials was considered by Brebbia C. A., Fujiki K., Fukui T., Kato S., Kishima T., Kobayashi S., Nishimura N., Niwa Y., Manolis G. D. Mansur W.J. (for 2D elastodynamics), Chutoryansky N.M. (for 3D elastodynamics). Detailed review of abroad scholars elaborating BIEM was made by Beskos D. [7].This paper discusses BIEM for 2 and 3D elastodynamics on the base of the second approach. The fundamental solutions, integral representations and boundary integral equations are constructed by means distributions theory for the general case of anisotropic elastic media. It's suggested some new results concerning special regularization of singularities on the wave fronts of the integral equations kernels. The illustrative numerical examples concern the scattering of elastic waves on cavities embedded in an infinite isotropic medium. So, it's shown the numerical results of waves diffraction on the one and two cavities of arched and rectangular forms in 2 and 3D cases. These results show quite stability of the elaborating algorithm.     

Some improvements of accuracy and efficiency in three dimensional direct boundary element method

Numerical analysis with the Boundary Element Method (BEM) has been used more and more in various engineering fields in recent years. In numerical techniques, however, there are some problems which have not been fully solved even now. The most essential one is the drop in the accuracy of results for internal points near the boundary of the structure, where the singularity of integrands in the boundary integral equation is too strong to be evaluated with the normal numerical method. For the boundary integral equation of stress, this problem became more serious, and the accuracy can be improved only partly, even though very refined boundary elements are used. In this paper, the boundary integral equation is newly formulated using a relative quantity of displacement. In this way, the singularity of boundary integrals is reduced by the order of 1/r, and the accuracy of solution is improved significantly. Furthermore, in order to integrate it more accurately, two kinds of numerical integral methods are newly developed. By using these methods, both displacement and stress can be obtained with excellent accuracy at almost any point in the structure without any numerical difficulty, although the discretization may be comparatively coarse. The generality and practicability of the present formulation and integral methods are confirmed through some examples of three dimensional elastic problems.     

A comparison of integral formulations for the analysis of low Reynolds number flows

Integral formulations for the analysis of low Reynolds number flows have been developed over the past 25 years. These formulations can typically be categorized as being either direct or indirect and either velocity integral equations or traction integral equations. Depending on the boundary conditions imposed for a given problem, the resulting integral formulation will result in either a Fredholm integral equation of the first kind, second kind or mixed. In general, Fredholm integral equations of the first kind lead to unstable numerical schemes based upon discretization and can result in low-order accuracy. For most practical problems, the direct velocity integral equation results in a Fredholm equation of the first kind. Nevertheless, many researchers have used this integral formulation with great success and any potential matrix ill-conditioning seems to have little influence on the accuracy of the numerical solutions. In this research, three integral formulations are compared, namely, the direct velocity integral equation, the indirect velocity integral equation, and the direct traction integral equation. Three benchmark problems are chosen for which there are analytic solutions. Although the discretized matrix condition numbers are in some cases orders of magnitude larger for the direct velocity integral formulation, there is little to distinguish between the three methods in terms of accuracy. In fact, the results for the direct velocity integral equation were slightly more accurate in most cases for the three benchmark problems considered in this research.     

The application of the differential transform technique to point source problems of elasticity

The differential transform, being a formal integral transform in which integrations have been eliminated, is used to represent some point sources in operational form. The sources can also be represented by integrals or integral transforms. However, in order to solve some boundary value problems, it is simpler to use the operational representation than the integral or integral transform representation. Three examples illustrate the application of the differential transform and the operational representations.     

The general type of finite-part singular integrals and integral equations with logarithmic singularities used in fracture mechanics

Summary A new method is proposed, by using some special quadrature rules, for the numerical evaluation of the general type of finite-part singular integrals and integral equations with logarithmic singularities. In this way the system of such equations can be numerically solved by reduction to a system of linear equations. For this reduction, the singular integral equation is applied to a number of appropriately selected collocation points on the integration interval, and then a numerical integration rule is used for the approximation of the integrals in this equation. An application is given, to the determination of the intensity of the logarithmic singularity in a simple crack inside an infinite, isotropic solid.With 1 Figure     

Numerical expansion-iterative method for analysis of integral equation models arising in one- and two-dimensional electromagnetic scattering

Most integral equations of the first kind are ill-posed, and obtaining their numerical solution needs often to solve a linear system of algebraic equations of large condition number. So, solving this system may be difficult or impossible. Since many problems in one- and two-dimensional scattering from perfectly conducting bodies can be modeled by Fredholm integral equations of the first kind, this paper presents an effective numerical expansion-iterative method for solving them. This method is based on vector forms of block-pulse functions. By using this approach, solving the first kind integral equation reduces to solve a recurrence relation. The approximate solution is most easily produced iteratively via the recurrence relation. Therefore, computing the numerical solution does not need to directly solve any linear system of algebraic equations and to use any matrix inversion. Also, the method practically transforms solving of the first kind Fredholm integral equation which is inherently ill-posed into solving second kind Fredholm integral equation. Another advantage is low cost of setting up the equations without applying any projection method such as collocation, Galerkin, etc. To show convergence and stability of the method, some computable error bounds are obtained. Test problems are provided to illustrate its accuracy and computational efficiency, and some practical one- and two-dimensional scatterers are analyzed by it.     

Integral analysis of a magnetic field for an arbitrary geometry coil with rectangular cross section

The integral method can effectively analyze magnetic fields, but the traditional integral method can analyze only coils with regular geometries. Therefore, a new integral method was developed to calculate the three-dimensional (3-D) magnetic field created by an arbitrary geometry coil with a rectangular cross section using the local coordinate method and a 3-D coordinate transformation. However, when the field points are on the surface of the coil or the basic segment is the right angle trapezoidal prism, singularities occur that make the numerical analysis of the magnetic field more difficult. Thus, we present here some mathematical methods to eliminate the singularities to allow accurate numerical analysis of the magnetic field. We validate the integral method by comparing it with the analytical solutions for regular geometry coils.     

Wide-viewing-angle integral three-dimensional imaging system by curving a screen and a lens array

A wide-viewing-angle integral three-dimensional imaging system made by curving a screen and a lens array is described. A flexible screen and a curved lens array are incorporated into an integral imaging system in place of a conventional flat display panel and a flat lens array. One can effectively eliminate flipped images by adopting barriers. As a result, the implemented system permits the limitation of viewing angle to be overcome and the viewing angle to be expanded remarkably. Using the proposed method, we were able to achieve a viewing angle of 33 degrees (one side) for real integral imaging and 40 degrees (one side) for virtual integral imaging, which is four times wider than that of the currently used conventional techniques. The principle of the implemented system is explained, and some experimental results are presented.     

Evaluating J‐integral from Displacement Fields Measured by Digital Image Correlation

Path integral, domain integral and least squares methods for evaluating J‐integral from measured displacement fields for a power‐law hardening material are described in this paper. The values of the J‐integral are evaluated by applying the path and domain integral methods to the displacement fields obtained by elastoplastic finite element analysis and the displacement fields obtained through the measurement using digital image correlation. Results show that the values obtained by the domain integral method are slightly better than those by the path integral method, because the domain integral method efficiently uses the full‐field measurement data. The values of the J‐integral are also evaluated by the least squares method with the Hutchinson, Rice, and, Rosengren displacement fields. Results show that the J‐integral can be obtained by the least squares method simply and easily without any calculation of the integration. The J‐integral values obtained by the least squares method agree well with the values obtained using other methods. Because J‐integral can be evaluated easily by any method described in this paper, it is expected that these methods are applicable to various fracture problems during experimental evaluation of structural components.     

Reconfigurable supply chain: the X-network

Research on supply chain (SC) digitalization, resilience, sustainability and leagility has remarkably progressed, most of it focused on the individual contributions of these four major frameworks. However, a lack of integration spanning these individual frameworks can be observed. In this conceptual paper, we hypothesize that reconfigurability can be considered such an integral perspective that consolidates the research in SC adaptation to ever changing environments. We theorize a new notion – a Reconfigurable SC or the X-network – that exhibits some crucial design and control characteristics for complex value-adding systems in highly vulnerable environments. We support our argumentation and conceptual viewpoints by a literature analysis along with tertiary studies to review and structure contextual factors of designing the X-networks. We propose respective frameworks and discuss the implementation principles and technologies at the macro and micro levels. Two novel concepts – dynamic SC meta-structures and dynamic autonomous services – are introduced. Distinctively, we go beyond the existing knowledge to predict proactively the future directions in the reconfigurable SCs. Our results can be of value for decision-makers to decipher chances and barriers in contemporary SC transformations.     

压力容器分析设计中基于“处理面” 的等效线性化方法

从球形压力容器在内压作用下的解析解出发进行分析,指出在现有著作和规范中,直接对沿壳体厚度变化的应力表达式沿壳体厚度的一条“处理线”进行一维积分求解厚壁球壳壁上的薄膜应力和弯曲应力的方法在概念上不够准确,会导致一定的额外误差。由此提出了在壳体的环形截面区域的一个“处理面”上进行二维积分来求解壳壁上的薄膜应力和弯曲应力的方法,在概念上是准确的,能够避免“处理线”法导致的额外误差。通过一个受内压厚壁球壳的弹性解析计算,说明新方法比原方法有明显的改进。     

Calculation of multivalued potentials in exterior regions

In magnetostatic and magnetodynamic problems, Ampere's law leads to a multivalued scalar magnetic potential. A method is proposed to calculate this potential through a finite elements program or, in a more efficient way, through a program using a boundary integral method in nonconducting exterior regions. In this case, one has only to define "cutting lines" on the boundary instead of cutting surfaces. Reported here are the results obtained with the three-dimensional eddy-current code Trifou, using the finite element method inside the conductors coupled with an integral method outside, in which the method has been incorporated, for a test model where some global values can be obtained by hand and compared with those obtained by the code. A study of the influence of mesh refinement and of the position of cutting lines is given. Good agreement and numerical stability indicate that the method is operational.     

An efficient method to evaluate hypersingular and supersingular integrals in boundary integral equations analysis

An efficient algorithm is employed to evaluated hyper and super singular integral equations encountered in boundary integral equations analysis of engineering problems. The algorithm is based on multiple subtractions and additions to separate singular and regular integral terms in the polar transformation domain, primarily established in Refs. (Guiggiani M, Krishnasamy G, Rudolphi TJ, Rizzo FJ. A general algorithm for the numerical solution of hypersingular boundary integral equations. Trans ASME 1992;59:604–614; Guiggiani M, Casalini P. Direct computation of Cauchy principal value integral in advanced boundary element. Int J Numer Meth Engng 1987;24:1711–1720. Guiggiani M, Gigante A. A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method. J Appl Mech Trans ASME 1990;57:906–915). It can be proved that the regular terms have finite analytical solutions in the range of integration, and the singular terms will be replaced by special periodic kernels in the integral equations. The subtractions involve to multiple derivatives of analytical kernels and the additions require some manipulation to separate the remaining regular terms from singular ones. The regular terms are computed numerically. Three examples on numerical evaluation of singular boundary integrals are presented to show the efficiency and accuracy of the algorithm. In this respect, strongly singular and hypersingular integrals of potential flow problems are considered, followed by a supersingular integral which is extracted from the partial differentiation of a hypersingular integral with respect to the source point.