An inversion integral for crack-scattering data

The inverse problem of determining the size, shape and orientation of a flat crack from high-frequency far-field elastic waves scattered by the crack is investigated. The results show that desired information on a crack can be obtained from the first arriving scattered longitudinal waves only. It is shown that an approximate high-frequency solution to the direct problem, based on physical elastodynamics, yields an expression for the scattered far-field of longitudinal motion which suggests a solution to the inverse problem by application of Fourier-type inversion integrals to scattering data. Two kinds of inversion integrals are examined. The inversion problem becomes relatively simple if some a-priori information is available, either on the orientation of the plane of the crack or on a plane of symmetry. The method of inversion is verified for a flat crack of elliptical shape. Some computational technicalities are discussed, and the method is also applied to experimental scattering data.     

Mapping similarity between parallel and serial architecture kinematics

First the principles of mapping spatial points to surfaces is introduced in the context of the inverse kinematics of a general six revolute serial wrist partitioned robot. Then the advantage of choosing ideal frames is illustrated by showing that in the case of some architectures an image space formulation, though possible, may be an impediment to clear geometric insight and a satisfactory and much simpler solution. After showing how the general point mapping transformation is reduced to classical Blaschke-Grünwald planar mapping a novel three legged planar robot??s direct kinematics is solved in image space and then using conventional ??distance?? constraints. The purpose is to show why the latter approach yields spurious solutions and how the displacement pole rotation performed with kinematic mapping reliably avoids this problem. In conclusion certain other new and/or interesting reduced mobility parallel robots are discussed briefly to point out some advantages and insights gained with an image space approach. Particular effort is made to expose in detail how mapping simplifies and extends the solution of direct kinematics pertaining to Calvel??s ??Delta?? 3D translational robot.     

Three-dimensional surface waves propagating over long internal waves

A non-uniform current, such as may be generated by long internal waves, interacts with short surface waves and causes patterns on the sea surface that are of interest. In particular, regions of steep breaking waves may be relevant to specular radar scattering.A simple approach to modelling this problem is to take a set of short, surface waves of uniform wavenumber on the sea surface, as may be caused by a gust of wind. The direction of propagation of the surface waves is firstly taken to be the same as that of the current, and surface tension and viscous effects are neglected. We have a number of methods of solution at our disposal: linear (one-dimensional) ray theory is simple to apply to the problem, a nonlinear Schrödinger equation for the modulated wave amplitude, modified to include to effect of the current, can be used and solutions can be found using a fully nonlinear irrotational flow solver. Comparisons between the ‘exact’ nonlinear calculations for two dimensions (which are too complicated/ computationally intensive to be extended to three dimensions) compare well with the two approximate methods of solution, both of which can be extended, within their limitations, to model the full three-dimensional problem; here we present three-dimensional results from the linear ray theory.By choosing such a simple (although we consider physically realistic) initial state of uniform wavenumber short waves and assuming a sinusoidal surface current, we can reduce the two-dimensional problem to dependence on three non-dimensional parameters.In three-dimensions, we consider an initial condition with a uniform wavetrain at an angle α say, to the propagating current, thus introducing a fourth parameter into the problem. Extension of the linear ray theory from one space to two space dimensions is numerically quite simple since we maintain uniformity in the direction perpendicular to the current, and the only difficulty lies with the presentation of results, due to the large number of variables now present in the problem such as initial wavenumber, angle of propagation, position in (x, y, t) space etc. In this paper we present just one solution in detail where waves are strongly refracted and form two distinct foci in space-time. There is a collimation of the short waves with the direction of the propagating current.     

The time domain elastodynamic Kirchhoff approximation for cracks: The inverse problem

Obtaining the size, shape, and orientation of a crack from ultrasonic elastic wave scattering information is one example of the solution of an inverse problem. Here, we obtain the formal solution to this inverse problem for ideal, flat cracks using the Kirchhoff approximation for the scattered elastodynamic wavefield. Time and frequency domain verisions of the solution will be given, both in the general case and, in reduced form, for circular cracks. The time domain inverse formulation, in particular, will be shown to be equivalent to the method of projections, leading to a classical two-dimensional Radon transform. A method is also demonstrated for performing a constrained inversion to obtain the parameters of a flat crack that is assumed a priori to be of elliptical shape.     

Parametric design of the band structure for lattice materials

Lattice materials are often investigated to determine how small parameter variations in the periodic microstructrure can influence the elastic wave propagation. A general hierarchical scheme, based on asymptotic perturbation techniques, is outlined to analytically assess the parametric sensitivity of the material band structure to a generic multi-parametric perturbation (direct problem). Modeling refinements, parameters updates, microstructural damages and manufacturing irregularities can be treated indifferently and simultaneously. According to a converse strategy, based on the inversion of the sensitivity problem, a hierarchical scheme is sketched to identify the parameter combinations which realize a design band structure (inverse problem). The direct and inverse problem are applied to the sensitivity analysis and band structure design of the anti-tetrachiral lattice material. Despite the high spectral density and the high-dimensional parameter space, the multi-parameter perturbation technique demonstrates its suitability in, first, analytically—although asymptotically—describe the material spectrum and, second, designing the material microstructure to obtain the desired spectral components. The inverse problem solution is discussed in terms of existence, uniqueness, asymptotic consistency and physical admissibility.     

Two-dimensional piezoelectricity. Part II: general solution,Green’s function and interface cracks

The complete solution space of a piezoelectric material is the direct sum of several orthogonal eigenspaces, one for each distinct eigenvalue. Each one of the 14 different classes of piezoelectric materials has a distinct form of the general solution, expressed in terms of the eigenvectors of the zeroth and higher orders and a kernel matrix containing analytic functions. When these functions are chosen to be logarithmic, one obtains, in a unified way, Green’s function of the infinite space as a single 8 × 8 matrix function G_∞ for the various load cases of concentrated line forces, dislocations, and a line charge. This expression of Green’s function is valid for all classes of nondegenerate and degenerate materials. With an appropriate choice of the parameters, it reduces to the solution of a half space with concentrated (line) forces at a boundary point, and with dislocations in the displacements. As another application, eigenvalues and eigensolutions are obtained for the bimaterial interface crack problem.     

PRECISE INTEGRAL ALGORITHM BASED SOLUTION FOR TRANSIENT INVERSE HEAT CONDUCTION PROBLEMS WITH MULTI-VARIABLES

IntroductionIHCPs (InverseHeatConductionProblems)arecloselyassociatedwithmanyengineeringaspects,andwelldocumentedintheliteratures,coveringtheidentificationsofthermalparameters[1,2 ],boundaryshapes[3],boundaryconditions[4 ]andsource_relatedterms[5 ,6 ]etc .Howeveritseemsthatonlylittleworkisdirectlyconcernedwithmulti_variablesidentificationsbyauthors’knowledge.Tsengetal.presentedanapproachtodeterminingtwokindsofvariables[7],butonlygavefewnumericalexamplestodeterminethemsimultaneously .Thesol…     

The L-pseudo-solution using stochastic algorithm of Landweber

In this paper, we consider a linear equation Ax=u. A is an operator with an unbounded inverse in a Hilbert space. The right side u does not belong to the range of A. Obviously, a solution in classical sense does not exist and A ^?1 u does not have a sense. To solve this problem arising from many experimental fields of science, where the second member u stems from measurements, we propose a recurrent procedure which converges almost completely and in quadratic mean to L-pseudo-solution and for which we build up a confidence interval. To check the validity of our results, a numerical example which is standard in rheology is proposed.     

New approximate solution for N-point correlation functions for heterogeneous materials

Statistical N-point correlation functions are used for calculating properties of heterogeneous systems. The strength and the main advantage of the statistical continuum approach is the direct link to statistical information of microstructure. Two-point correlation functions are the lowest order of correlation functions that can describe the morphology and the microstructure-properties relationship. Experimentally, statistical pair correlation functions are obtained using SEM or small x-ray scattering techniques. Higher order correlation functions must be calculated or measured to increase the precision of the statistical continuum approach. To achieve this aim a new approximation methodology is utilized to obtain N-point correlation functions for non-FGM (functional graded materials) heterogeneous microstructures. Conditional probability functions are used to formulate the proposed theoretical approximation. In this approximation, weight functions are used to connect subsets of (N?1)-point correlation functions to estimate the full set of N-point correlation function. For the approximation of three and four point correlation functions, simple weight functions have been introduced. The results from this new approximation, for three-point probability functions, are compared to the real probability functions calculated from a computer generated three-phase reconstructed microstructure in three-dimensional space. This three-dimensional reconstruction was based on an experimental two-dimensional microstructure (SEM image) of a three-phase material. This comparison proves that our new comprehensive approximation is capable of describing higher order statistical correlation functions with the needed accuracy.     

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.     

Stress state of an elastic ball in a spherically symmetric gravitational field

We consider a spherically symmetric static problem of general relativity whose solution was obtained in 1916 by Schwarzschild for a metric form of a special type. This solution determines the metric coefficients of the exterior and interior Riemannian spaces generated by a gravitating solid ball of constant density and includes the so-called gravitational radius r _g. For a ball of outer radius R=r _g, the metric coefficients are singular, and hence the radius r _g is traditionally assumed to be the radius of the event horizon of an object called a black hole. The solution of the interior problem obtained for an incompressible ideal fluid shows that the pressure at the ball center increases without bound for R=9/8r _g, which is traditionally used for the physical justification of the existence of black holes. The discussion of Schwarzschild’s traditional solution carried out in this paper shows that it should be generalized with respect to both the geometry of the Riemannian space and the elastic medium model. In this connection, we consider the general metric form of a spherically symmetric Riemannian space and prove that the solution of the corresponding static problem exists for a broad class of metric forms. A special metric form based on the assumption that the gravitation generating the Riemannian space inside a fluid ball or an elastic ball does not change the ball mass is singled out from this class. The solution obtained for the special metric form is singular with respect to neither the metric coefficients nor the pressure in the fluid ball and the stresses in the elastic ball. The obtained solution is compared with Schwarzschild’s traditional solution.     

Optimal error bound in a Sobolev space of regularized approximation solutions for a sideways parabolic equation

The inverse heat conduction problem （IHCP） is a severely ill-posed problem in the sense that the solution （ if it exists） does not depend continuously on the data. But now the results on inverse heat conduction problem are mainly devoted to the standard inverse heat conduction problem. Some optimal error bounds in a Sobolev space of regularized approximation solutions for a sideways parabolic equation, i. e. , a non-standard inverse heat conduction problem with convection term which appears in some applied subject are given.     

Global solution of the inverse problem for a class of nonlinear evolution equations of dispersive type

The inverse problem for a class of nonlinear evolution equations of dispersive type was reduced to Cauchy problem of nonlinear evolution equation in an abstract space. By means of the semigroup method and equipping equivalent norm technique, the existence and uniqueness theorem of global solution was obtained for this class of abstract evolution equations, and was applied to the inverse problem discussed here. The existence and uniqueness theorem of global solution was given for this class of nonlinear evolution equations of dispersive type. The results extend and generalize essentially the related results of the existence and uniqueness of local solution presented by YUAN Zhong-xin. Contributed by Chen Yu-shu Foundation item: the National Natural Science Foundation of China (Significance 199990510); the National Key Basic Research Special Foundation of China (G1998020316); Liuhui Center for Applied Mathematics, Nankai University & Tianjin University Biography: Chen Fang-qi (1963-)     

Structure and properties of the solution space of general anisotropic laminates

The algebraic structure of the solution space of all types of anisotropic laminates is determined. The full space is shown to be the direct sum of a number of orthogonal eigenspaces, one for each simple or multiple eigenvalue, whose dimension equals the multiplicity. There are eight different types of eigenvalues, which combine to yield eleven distinct types of laminates with peculiar representations of the general solution. All such representations are explicitly obtained, along with the pseudo-metrics based on the binary product of the eigenvectors. This leads to the projection operators in the solution space, spectral sums and intrinsic tensors analogous to the Stroh–Barnett–Lothe tensors in 2-D elasticity. The present theoretical results are obtained by adopting a mixed formulation involving the deflection function and Airy’s stress function, and by using new laminate elasticity matrices different from the conventional stiffness matrices A, B and D. The new formulation also discloses an isomorphism relating each anisotropic laminate to an image laminate, such that every equilibrium solution of the former directly yields an image solution of the latter by interchanging the kinematical and kinetic variables and the in-plane and out-of-plane variables. This implies, in particular, that the classical bending theory of homogeneous plates and symmetric laminates is not a distinct subject, despite its historical development and pedagogical recognition, but is mathematically identical to the plane stress problem of anisotropic elasticity.     

Propagation of electroacoustic waves in the transversely isotropic piezoelectric medium reinforced by randomly distributed cylindrical inhomogeneities

The propagation of electroacoustic waves in a piezoelectric medium containing a statistical ensemble of cylindrical fibers is considered. Both the matrix and the fibers consist of piezoelectric transversely isotropic material with symmetry axis parallel to the fiber axes. Special emphasis is given on the propagation of an electroacoustic axial shear wave polarized parallel to the axis of symmetry propagating in the direction normal to the fiber axis.The scattering problem of one isolated continuous fiber (“one-particle scattering problem”) is considered. By means of a Green’s function approach a system of coupled integral equations for the electroelastic field in the medium containing a single inhomogeneity (fiber) is solved in closed form in the long-wave approximation. The total scattering cross-section of this problem is obtained in closed form and is in accordance with the electroacoustic analogue of the optical theorem.The solution of the one-particle scattering problem is used to solve the homogenization problem for a random set of fibers by means of the self-consistent scheme of effective field method. Closed form expressions for the dynamic characteristics such as total cross-section, effective wave velocity and attenuation factor are obtained in the long-wave approximation.     

Scattering of rayleigh lamb waves from a 2d-cavity in an elastic plate

The T-matrix method is applied to the problem of scattering of Rayleigh-Lamb modes from a twodimensional cavity in an elastic plate. A formal solution is obtained which is valid also for non-planar surfaces. Explicit expressions and numerical results are given for a plate with plane surfaces.     

Explicit dynamics equations of the constrained robotic systems

Recursive matrix relations concerning the kinematics and the dynamics of a constrained robotic system, schematized by several kinematical chains, are established in this paper. Introducing frames and bases, we first analyze the geometrical properties of the mechanism and derive a general set of relations. Kinematics of the vector system of velocities and accelerations for each element of robot are then obtained. Expressed for every independent loop of the robot, useful conditions of connectivity regarding the relative velocities and accelerations are determined for direct or inverse kinematics problem. Based on the general principle of virtual powers, final matrix relations written in a recursive compact form express just the explicit dynamics equations of a constrained robotic system. Establishing active forces or actuator torques in an inverse dynamic problem, these equations are useful in fact for real-time control of a robot.     

The comparison model method: A new arithmetic approach to the discrete inverse problem of groundwater hydrology

Let the steady-state pressure z(·) of a fluid in a one-dimensional domain be governed by the equation d _x (a d _x z) = f subject to Dirichlet boundary conditions. We consider the identification of the transmissivity a (·), given f(·), and measured pressure z(·) by the comparison model method, a direct method which has been known and applied for some time but lacked theoretical background. With reference to a domain in one spatial dimension, we examine both the infinite-(‘continuous’) and finite-(discrete) dimensional cases. In the former, the method is based on the solution p(·) of an auxiliary flow equation, where f(·) and the two-point boundary conditions are unchanged with respect to the original or z(·) equation, whereas a tentative constant value b is assigned to the auxiliary transmissivity. The ratio of the first derivatives of p(·) and z(·) multiplied by b yields a solution ã(·) to the inverse problem. We examine in detail the nonuniqueness of ã(·) as a function of b, some cases where existence implies uniqueness, the role of positivity constraints, and a special feature: self-identifiability. We then translate all available results into the discrete case, where the good unknowns for the inverse problem are the internode coefficients. Several algebraic and numerical examples are presented.     

Analysis of fractional anomalous diffusion caused by an instantaneous point source in disordered fractal media

A theoretical analysis of fractional anomalous diffusion caused by an instantaneous point source in disordered fractal media is studied. Using the method of symmetry group of scaling transformations and the H-function, the analytical solutions of concentration distribution are given. At the same time we derive the expressions of scattering function spectrum.The result shows that the scattering function spectra still have the properties of scaling function. The scattering functions of point source, line source and area source in regular Euclidean space can be regarded as particular cases of this paper and are included in this paper. At the end of the paper we discuss the asymptotic behaviors of the solution in detail. The results of this paper can be taken to be the fundamental solutions for every kind of boundary value problems of fractional anomalous diffusion in disordered fractal media.     

Theory of reconstructing the spatial distribution of the filtration coefficient in vascularized soft tissues: Exact and approximate inverse solutions

We formulate and solve an inverse poroelastic problem to reconstruct the spatial distribution of the filtration coefficient for soft vascularized tissue from a collection of displacement fields obtained during its relaxation. We present two solutions for the inverse problem, both developed using direct non-iterative approach. The first is a simple closed form approximate solution. It depends upon the approximation that the interstitial pressure is spatially homogeneous. The second solution relaxes this assumption. It requires the solution of a Poisson equation to reconstruct the pressure distribution. The inversion thus obtained is exact in the limit of negligible percolation. We present inversion results from computational experiments to validate and compare the two approaches. The closed form solution provides accurate results in favorable circumstances. The exact-pressure approach accommodates inhomogeneous loading easily. Both approaches are somewhat sensitive to noise. Our results suggest that it may be possible to image the filtration coefficient using this approach. Future work would include further test with noisy data and experimental validation.