A biem optimization method for fracture dynamics inverse problem

In the present paper, based on the theory of dynamic boundary integral equation, an optimization method for crack identification is set up in the Laplace frequency space, where the direct problem is solved by the author's new type boundary integral equations and a method for choosing the high sensitive frequency region is proposed. The results show that the method proposed is successful in using the information of boundary elastic wave and overcoming the ill-posed difficulties on solution, and helpful to improve the identification precision. The project supported by Foundation of the National Post-Doctoral Committee.     

有限散射信号下二维缺陷形状识别的罚函数方法

研究在有限照射角度和频带宽度下二维缺陷的形状识别问题。首先,通过引进介质参数扰动函数,建立介质参数扰动函数和弹性波散射场之间的非线性关系,并将所关心的缺陷的形状识别问题转化为关于扰动函数的反演;然后,利用变分技术和优化方法求解,为了弥补散射数据的不足,在总的目标函数中,采用附加度量函数作为罚函数;最后,对后场散射远场测量时有限照射角度和频带宽度下几种典型缺陷进行了模拟识别,表明了;表明了罚函数法的有效性。     

Fundamental solutions of the streamfunction–vorticity formulation of the Navier–Stokes equations

A new boundary element procedure is developed for the solution of the streamfunction–vorticity formulation of the Navier–Stokes equations in two dimensions. The differential equations are stated in their transient version and then discretized via finite differences with respect to time. In this discretization, the non-linear inertial terms are evaluated in a previous time step, thus making the scheme explicit with respect to them. In the resulting discretized equations, fundamental solutions that take into account the coupling between the equations are developed by treating the non-linear terms as in homogeneities. The resulting boundary integral equations are solved by the regular boundary element method, in which the singular points are placed outside the solution domain.     

Solution of a mixed dynamic problem on the displacements given on the boundary of a half-plane lying on the surface of an isotropic homogeneous elastic half-space

We consider the problem on the motion of an isotropic elastic body occupying the half-space z ≥ 0 on whose boundary, along the half-plane x ≥ 0, the horizontal components of displacement are given, while the remaining part of the boundary is stress-free. We seek the solution by the method of integral Laplace transforms with respect to time t and Fourier transforms with respect to the coordinates x, y; the problem is reduced to a system of Wiener-Hopf equations, which can be solved by the methods of singular-integral equations and circulants. We invert the integral transforms and reduce the solution to the Smirnov-Sobolev form. We calculate the tangential stress intensity coefficients near the boundary z = 0, x = 0, |y| < ∞ of the half-plane. The circulant method for solving the Wiener-Hopf system was proposed in [1]. A static problem similar to that considered in the present paper was solved earlier. The Hilbert problem was reduced to a system of Fredholm integral equations in [2]. In the present paper, we solve the above problem by reducing the solution to quadratures and a quasiregular system of Fredholm integral equations. We give a numerical solution of the Fredholm equations and calculate the integrals for the tangential stress intensity coefficients.     

Wiener-Hopf analysis of an acoustic plane wave in a trifurcated waveguide

In this paper, we have made Wiener-Hopf analysis of an acoustic plane wave by a semi-infinite hard duct that is placed symmetrically inside an infinite soft/hard duct. The method of solution is integral transform and Wiener-Hopf technique. The imposition of boundary conditions result in a 2 × 2 matrix Wiener-Hopf equation associated with a new canonical scattering problem which is solved by using the pole removal technique. In the solution, two infinite sets of unknown coefficients are involved that satisfy two infinite systems of linear algebraic equations. These systems of linear algebraic equations are solved numerically. The graphs are plotted for sundry parameters of interest. Kernel functions are also factorized.     

Numerical studies on non‐linear free surface flow using generalized Schwarz–Christoffel transformation

This paper deals with a technique to transform a free surface flow problem in the physical domain with an unknown boundary to a standard domain that has a fixed boundary. All the difficulties in the physical domain are reduced to finding an unknown mapping function that can be solved iteratively in a standard domain. A derivation is first presented to express an analytic function in terms of the boundary value of its imaginary part. Using a relationship between boundaries of the standard and the physical domains, a formula for the generalized Schwarz–Christoffel transformation is then developed. Based on the generalized Schwarz–Christoffel integral and the Hilbert transform, a pair of non‐linear boundary integro‐differential equations in an infinite strip is formulated for solving fully non‐linear free surface flow problems. The boundary integral equations are then discretized with quadratic elements in an untruncated standard domain and solved by the Levenberg–Marquardt algorithm. Several examples of supercritical flow past obstructions are provided to demonstrate the flexibility and the accuracy of the proposed numerical scheme. Copyright © 2000 John Wiley & Sons, Ltd.     

A 2.5-D dynamic model for a saturated porous medium. Part II: Boundary element method

The 3-D boundary integral equation is derived in terms of the reciprocal work theorem and used along with the 2.5-D Green’s function developed in Part I [Lu, J.F., Jeng, D.S., Williams, S., submitted for publication. A 2.5-D dynamic model for a saturated porous medium: Part I. Green’s function. Int. J. Solids Struct.] to develop the 2.5-D boundary integral equation for a saturated porous medium. The 2.5-D boundary integral equations for the wave scattering problem and the moving load problem are established. The Cauchy type singularity of the 2.5-D boundary integral equation is eliminated through introduction of an auxiliary problem and the treatment of the weakly singular kernel is also addressed. Discretisation of the 2.5-D boundary integral equation is achieved using boundary iso-parametric elements. The discrete wavenumber domain solution is obtained via the 2.5-D boundary element method, and the space domain solution is recovered using the inverse Fourier transform. To validate the new methodology, numerical results of this paper are compared with those obtained using an analytical approach; also, some numerical results and corresponding analysis are presented.     

Investigation of the scattering of antiplane shear waves by two griffith cracks using the non-local theory

In this paper, the scattering of harmonic antiplane shear waves by two finite cracks is studied using the non-local theory. The Fourier transform is applied and a mixed boundary value problem is formulated. Then a set of triple integral equations is solved using a new method, namely Schmidt's method. This method is more exact and more reasonable than Eringen's for solving this kind of problem. The result of the stress near the crack tip was obtained. Contrary to the classical elasticity solution, it is found that no stress singularity is present at the crack tip, which can explain the problem of macroscopic and microscopic mechanics.     

The Gelfand-Levitan, the Marchenko, and the Gopinath-Sondhi integral equations of inverse scattering theory, regarded in the context of inverse impulse-response problems

In this paper, we are concerned with interpreting some classical inverse-scattering theories so that they are relevant to the one-dimensional inverse impulse-response problem which arises in reflection seismology

First the Gelfand-Levitan integral equations (which arise in the inverse scattering theory for the Schrödinger equation) are derived strictly in the time domain. Originally these celebrated equations were derived as a means of solving an inverse spectral problem, which is naturally posed in the frequency domain.

We show not only that these equations have a time-dependent formulation, but that their derivation is actually simpler here than in its original context. Next we give a similar derivation for the Marchenko integral equation.

We then obtain, by an independent method, the integral equation of Gopinath and Sondhi, which is not unlike the Gelfand-Levitan linear equation. It was used by them as a means of solving a time-dependent inverse problem arising in speech synthesis. A new integral equation, similar to the Marchenko equation is also derived.

Finally the integral equations are related to each other by direct transformation independent of the corresponding differential equations.

The present paper opens with a section in which the equation of one-dimensional elastic waves and the corresponding seismic inverse impulse-response problem are transformed into a form to which the Gelfand-Levitan theory applies, and then into the equations which arose in Gopinath and Sondhi's work.

We regard the integral equation of Gopinath and Sondhi as being directly applicable to the interpretation of seismic reflection data. The Gelfand-Levitan theory is also applicable but only after considerable transformation.

Our calculations are entirely in the time domain. The resulting equations have the merit that in order to recover the unknown coefficient on a finite interval (0, L) we need to use the boundary data also only on a finite segment of the reflection time series. The length of the record is just the two-way travel time associated with length L. By contrast, frequency-domain approaches require that long records be Fourier transformed. We distinguish results which are true only when the unknown coefficient is continuous from those true when it is bounded but only piecewise continuous.

In this work we have not addressed the important problem of deconvolution.     

A wideband fast multipole boundary element method for half-space/plane-symmetric acoustic wave problems

This paper presents a novel wideband fast multipole boundary element approach to 3D half-space/planesymmetric acoustic wave problems.The half-space fundamental solution is employed in the boundary integral equations so that the tree structure required in the fast multipole algorithm is constructed for the boundary elements in the real domain only.Moreover,a set of symmetric relations between the multipole expansion coefficients of the real and image domains are derived,and the half-space fundamental solution is modified for the purpose of applying such relations to avoid calculating,translating and saving the multipole/local expansion coefficients of the image domain.The wideband adaptive multilevel fast multipole algorithm associated with the iterative solver GMRES is employed so that the present method is accurate and efficient for both lowand high-frequency acoustic wave problems.As for exterior acoustic problems,the Burton-Miller method is adopted to tackle the fictitious eigenfrequency problem involved in the conventional boundary integral equation method.Details on the implementation of the present method are described,and numerical examples are given to demonstrate its accuracy and efficiency.     

Radiative heat transfer in absorbing, emitting, and anisotropically scattering boundary-layer flows with reflecting boundary

The heat transfer in absorbing, emitting, and anisotropically scattering boundary-layer flows with reflecting boundary over a flat plate, over a 90-deg wedge, and in stagnation flow is solved by application of the Galerkin method with the particular solution boundary condition I _p (τ₀,ξ,?μ) of the equation of radiative transfer for an inhomogeneous term and the Box method. The exact integral expressions for the radiation part of this problem are developed. The coupling between convective and radiative heat transfer in boundary-layer flows is described by a set of nonlinear simultaneous equations including differential equations and integrodifferential equations. The Galerkin method and the particular solution boundary condition I _p (τ₀,ξ,?μ) are used to analyze the radiation part of the problem. The nonsimilar boundary-layer equations are solved by the Box method. The present numerical procedure solutions are compared in tables with the other exact treating results, the P-3, and P-1 approximation methods for the case of isotropically scattering boundary-layer flows. The effects of linearly anistropically scattering and reflecting surface are taken into account. It is found that the present method is a reliable and efficient numerical procedure and scattering leads to a reduction in the total heat flux. The influence of the forward-backward scattering parameter on the total heat flux decreases with the increase of the surface reflectivity.     

On the dynamic behaviour of a piezoelectric laminate with multiple interfacial collinear cracks

This paper investigates the dynamic behaviour of a piezoelectric laminate containing multiple interfacial collinear cracks subjected to steady-state electro-mechanical loads. Both the permeable and impermeable boundary conditions are examined and discussed. Based on the use of integral transform techniques, the problem is reduced to a set of singular integral equations, which can be solved using Chebyshev polynomial expansions. Numerical results are provided to show the effect of the geometry of interacting collinear cracks, the applied electric fields, the electric boundary conditions along the crack faces and the loading frequency on the resulting dynamic stress intensity and electric displacement intensity factors.     

A boundary collocation method for the solution of a flow problem in a complex three-dimensional porous medium

A flow problem in a complex three-dimensional domain with a free surface and mixed-type boundary conditions is solved by the boundary collocation method. The solution is expressed as a combination of source functions distributed all around the domain close to the boundary, plus a special basis function to take care of a corner singularity. The resulting procedure is compared with the boundary integral elements method and is found to be simpler and more flexible to implement and faster to compute.     

非均质问题中的无网格边界单元法

介绍了一种不需要内部网格计算非均匀介质问题的边界元算法.该算法是建立在一种能将任何区域积分转换成边界积分的径向积分转换法基础上,首先用对应各向同性问题的基本解来建立以正规化位移表示的非均质问题的积分方程,然后用径向积分转换法将出现在积分方程中的区域积分转换成边界积分,从而形成不需要使用内部网格来计算区域积分的纯边界元算法.与其它无网格法相比,此方法需要很少的内部点,有些问题甚至不需要内部点都能得到满意的结果,因此,可以计算大型的三维非均匀介质工程问题.由于此方法继承了边界元和无网格算法的优点,因而具有广阔的发展前景.     

Investigation of the Scattering of Harmonic Elastic Waves by Two Collinear Symmetric Cracks Using the Non-Local Theory

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｌａｓｔｆｏｕｒｄｅｃａｄｅｓｈａｖｅｗｉｔｎｅｓｓｅｄｔｈｅｉｎａｕｇｕｒａｔｉｏｎｏｆａｎｏｖｅｌｔｈｅｏｒｙｏｆｍａｔｅｒｉａｌｂｏｄｉｅｓ,ｎａｍｅｄｔｈｅｎｏｎ＿ｌｏｃａｌｍｅｃｈａｎｉｃｓ.ＴｈｉｓｗａｓｄｏｎｅｐｒｉｍａｒｉｌｙｄｕｅｔｏｔｈｅｅｆｆｏｒｔｓｏｆＥｄｅｌｅｎ[1],Ｅｒｉｎｇｅｎ[2 ],ＧｒｅｅｎａｎｄＲｉｖｌｉｎ[3].Ａｃｃｏｒｄｉｎｇｔｏｔｈｅｎｏｎ＿ｌｏｃａｌｔｈｅｏｒｙ ,ｔｈｅｓｔｒｅｓｓａｔａｐｏｉｎｔＸｉｎａｂｏｄｙｄｅｐｅｎｄｓｎｏ…     

功能梯度材料动态断裂力学的径向积分边界元法

采用径向积分边界元法分析功能梯度材料动态断裂力学问题. 该方法使用与弹性模量无关的弹性静力学开尔文基本解作为问题的基本解,在导出的边界-域积分方程中含有由材料的非均质性和惯性项引起的域积分,通过径向积分法将域积分转化为等效的边界积分,得到只含边界积分的纯边界积分方程;从而建立只需边界离散的无内部网格边界元算法. 采用候博特方法求解关于时间二阶导数的系统离散的常微分方程组. 最后通过数值算例验证本文方法的精度和有效性.      

Computational model for short-fiber composites with eigenstrain formulation of boundary integral equations

A computational model is proposed for short-fiber reinforced materials with the eigenstrain formulation of the boundary integral equations(BIE)and solved with the newly developed boundary point method(BPM).The model is closely derived from the concept of the equivalent inclusion of Eshelby tensors.Eigenstrains are iteratively determined for each short.fiber embedded in the matrix with various properties via the Eshelby tensors,which can be readily obtained beforehand either through analytical or numerical means.As unknown variables appear only on the boundary of the solution domain,the solution scale of the inhomogeneity problem with the model is greatly reduced.This feature is considered significant because such a traditionally time-consuming problem with inhomogeneity can be solved most cost-effectively compared with existing numerical models of the FEM or the BEM.The numerical examples are presented to compute the overall elastic properties for various short-fiber reinforced composites over a representative volume element(RVE),showing the validity and the effectiveness of the proposed computational modal and the solution procedure.     

Computational model for short-fiber composites with eigenstrain formulation of boundary integral equations

A computational model is proposed for short-fiber reinforced materials with the eigenstrain formulation of the boundary integral equations （BIE） and solved with the newly developed boundary point method （BPM）. The model is closely derived from the concept of the equivalent inclusion Of Eshelby tensors. Eigenstrains are iteratively determined for each short-fiber embedded in the matrix with various properties via the Eshelby tensors, which can be readily obtained beforehand either through analytical or numerical means. As unknown variables appear only on the boundary of the solution domain, the solution scale of the inhomogeneity problem with the model is greatly reduced. This feature is considered significant because such a traditionally time-consuming problem with inhomogeneity can be solved most cost-effectively compared with existing numerical models of the FEM or the BEM. The numerical examples are presented to compute the overall elastic properties for various short-fiber reinforced composites over a representative volume element （RVE）, showing the validity and the effectiveness of the proposed computational modal and the solution procedure.     

Low Reynolds number hydrodynamic interaction of a solid particle with a planar wall

The slow viscous flow problem of an arbitrary solid particle in motion near a planar wall is recast into a boundary integral formulation. The present formulation employs the Green function appropriate to the planar wall problem and is developed in sufficient generality to allow calculations for arbitrary particles in any base flow which satisfies Stokes equations and no-slip on the wall. The resulting integral equations are easily discretized and solved for the particle surface tractions. Calculations are performed for axisymmetric motions of a variety of ellips?ids near the planar wall. Agreement with existing theory is excellent.     

Thermomagnetic effect with microtemperature in a semiconducting photothermal excitation medium

The main goal of this paper is to focus on the investigation of interaction between a magnetic field and elastic materials with microstructure, whose microelements possess microtemperatures with photothermal excitation. The elastic-photothermal problem in one-dimension is solved by introducing photothermal excitation at the free surface of a semi-infinite semiconducting medium (semiconductor rod). The integral transform technique is used to solve the governing equations of the problem under the effect of the microtemperature field. The analytical expressions for some physical quantities in the physical domain are obtained with the heating boundary surface and free traction. The numerical inversion technique is used to obtain the resulting quantities in the physical domain. The obtained numerical results with some comparisons are discussed and shown graphically.