A combined numerical and theoretical study on the penetration of a jacketed rod into semi-infinite targets

A combined numerical and theoretical study is conducted herein on the penetration of semi-infinite targets by jacketed rods with different r_j0/r_c0 ratios where r_j0 and r_c0 are the radii of the jacket and the core, respectively. The numerical results show that for smaller r_j0/r_c0 ratios the u–v relationship changes only a little compared to that of unitary long rod penetrator of the same core material, hence, the u–v relationship of unitary (homogeneous) long rod penetration is also applicable for jacketed rod penetration. Model for cratering in semi-infinite targets by jacketed rods is then suggested by using the laws of conversation of mass, momentum and energy, together with the u–v relationship of unitary (homogeneous) long rod penetration and an analytical model for predicting the depth of penetration has also been given for jacketed long rods penetrating semi-infinite targets in co-erosion mode. A new criterion for transition from bi-erosion to co-erosion is proposed. It transpires that the present model is in good agreement with the experimental observations for EN24 steel jacketed tungsten alloy long rods penetrating semi-infinite armor steel targets in terms of crater diameter and penetration depth.     

ON THE USE OF FFT ALGORITHM FOR THE CIRCUMFERENTIAL CO-ORDINATE IN BOUNDARY ELEMENT FORMULATION OF AXISYMMETRIC PROBLEMS

A new numerical method is proposed for the boundary element analysis of axisymmetric bodies. The method is based on complex Fourier series expansion of boundary quantities in circumferential direction, which reduces the boundary element equation to an integral equation in (r–z) plane involving the Fourier coefficients of boundary quantities, where r and z are the co-ordinates of the (r, θ, z) cylindrical co-ordinate system. The kernels appearing in these integral equations can be computed effectively by discrete Fourier transform formulas together with the fast Fourier transform (FFT) algorithm, and the integral equations in (r–z) plane can be solved by Gaussian quadrature, which establishes the Fourier coefficients associated with boundary quantities. The Fourier transform solution can then be inverted into (r, θ, z) space by using again discrete Fourier transform formulas together with FFT algorithm. In the study, first we present the formulation of the proposed method which is outlined above. Then, the method is assessed by using three sample problems. A good agreement is observed in the comparisons of the predictions of the method with those available in the literature. It is further found that the proposed method provides considerable saving in computer time compared to existing methods of literature. © 1997 by John Wiley & Sons, Ltd.     

Maximum density effects on natural convection in a porous cavity under thermal non-equilibrium conditions

Summary The problem of natural convection flow in a cavity filled with a water near its maximum density saturated porous medium and subjected to thermal non-equilibrium condition is investigated numerically in the present article. The natural convection flow in the horizontally heated rectangular cavity is assumed to be two-dimensional. A parabolic relationship of the density-temperature is used in Darcy's model. The dimensionless governing equations were solved using the finite volume method, and the results are presented to show the effect of the governing parameters. The numerical results are presented in the form of variations of the average Nusselt number with the Rayleigh number with different values of the heat transfer coefficient parameter H, and the thermal conductivity parameter K _r . It is found that by increasing H and K _r the shape of the isotherms of the solid phase appear to be similar to those of the water due to the enhancement of the thermal communications between the two phases. The results for the average Nusselt number of the thermal equilibrium model, which is the maximum possible value, can be achieved for high values of H×K _r . The numerical results reveal the dependence of the total (solid + fluid) average Nusselt number on the aspect ratio, and the maximum values of the average Nusselt number are found for the cavities of aspect ratio A≈0.5.     

A mixed semi analytical solution for functionally graded (FG) finite length cylinders of orthotropic materials subjected to thermal load

A simplified and accurate analytical cum numerical model is presented here to investigate the behavior of functionally graded (FG) cylinders of finite length subjected to thermal load. A diaphragm supported FG cylinder under symmetric thermal load which is considered as a two dimensional (2D) plane strain problem of thermoelasticity in (r, z) direction. The boundary conditions are satisfied exactly in axial direction (z) by taking an analytical expression in terms of Fourier series expansion. Fundamental (basic) dependent variables are chosen in the radial coordinate of the cylinder. First order simultaneous ordinary differential equations are obtained as mathematical model which are integrated through an effective numerical integration technique by first transforming the boundary value problem into a set of initial value problems. For FG cylinders, the material properties have power law dependence in the radial coordinate. Effect of non homogeneity parameters and orthotropy of the materials on the stresses and displacements of FG cylinder are studied. The numerical results obtained are also first validated with existing literature for their accuracy. Stresses and displacements in axial and radial directions in cylinders having various l/r _i and r _o/r _i ratios parameter are presented for future reference.     

Adaptive isogeometric analysis based on a combined r-h strategy

In the present work, an r-h adaptive isogeometric analysis is proposed for plane elasticity problems. For performing the r-adaption, the control net is considered to be a network of springs with the individual spring stiffness values being proportional to the error estimated at the control points. While preserving the boundary control points, relocation of only the interior control points is made by adopting a successive relaxation approach to achieve the equilibrium of spring system. To suit the noninterpolatory nature of the isogeometric approximation, a new point-wise error estimate for the h-refinement is proposed. To evaluate the point-wise error, hierarchical B-spline functions in Sobolev spaces are considered. The proposed adaptive h-refinement strategy is based on using De-Casteljau’s algorithm for obtaining the new control points. The subsequent control meshes are thus obtained by using a recursive subdivision of reference control mesh. Such a strategy ensures that the control points lie in the physical domain in subsequent refinements, thus making the physical mesh to exactly interpolate the control mesh and thereby allowing the exact imposition of essential boundary conditions in the classical isogeometric analysis (IGA). The combined r-h adaptive refinement strategy results in better convergence characteristics with reduced errors than r- or h-refinement. Several numerical examples are presented to illustrate the efficiency of the proposed approach.     

Mapped infinite elements for exterior wave problems

Recently, a new type of infinite elements which uses r^?1 decay was proposed. They were applied to exterior wave problems and good results were obtained. In two-dimensional problems, however, it was necessary to move the origin of the r^?1 decay in order to model the outgoing wave more accurately, because it decays roughly as r^?1/2. In this paper, the mapped infinite elements with r^?1/2 decay and the necessary numerical integration procedure are presented. These elements do not require any artificial movement of the origin. Several example problems are solved. The results show that the infinite elements with r^?1/2 decay here give much more accurate values than the infinite elements with exponential decay and any damper elements.     

On the computation of nearly singular integrals in 3D BEM collocation

In this paper, we propose an efficient strategy to compute nearly singular integrals over planar triangles in R ³ arising in boundary element method collocation. The strategy is based on a proper use of various non‐linear transformations, which smooth or move away or quite eliminate all the singularities close to the domain of integration. We will deal with near singularities of the form 1/r, 1/r² and 1/r³, r=∥ x ? y ∥ being the distance between a fixed near observation point x and a generic point y of a triangular element. Extensive numerical tests and comparisons with some already existing methods show that the approach proposed here is highly efficient and competitive. Copyright © 2007 John Wiley & Sons, Ltd.     

Two-dimensional stress intensity factor computations using the boundary element method

A multidomain boundary element formulation for the analysis of general two-dimensional plane strain/stress crack problems is presented. The numerical results were accurate and efficient. The analyses were performed using traction singular quater-point boundary elements on each side of the crack tip(s) with and without transition elements. Traction singular quarter-point boundary elements contain the correct √r displacement and 1/√r traction variations at the crack tip. Transition elements are appended to the traction singular elements to model the √r displacement variation. The 1/√r traction singularity is not represented with these elements. Current research studies for the crack propagation analysis of quasi-static and fatigue fracture problems are discussed.     

Radial transport in a porous medium with Dirichlet, Neumann and Robin-type inhomogeneous boundary values and general initial data: analytical solution and evaluation

The analytical solution is presented to the convection–diffusion equation describing the concentration of solutes in a radial velocity field due to extracting groundwater from or injecting water into an aquifer with arbitrary initial concentration data F(r), with r the radial distance, and an inhomogeneous mixed boundary condition G(t), with t the time, at the well radius r = r ₀. The analytical solution is obtained with a generalized Hankel transformation or with a Laplace transformation. The Hankel transformation turns out to be easier for G = 0, F ≠ 0, while the Laplace transformation is easier for F = 0, G ≠ 0. Both techniques can, however, deal with the full problem. The representation found by the generalized Hankel transform can also be found by the Laplace transform, through modification of the contour through the complex plane in the Bromwich integral for the inverse Laplace transform to the real axis. In practice, the numerical evaluation of the integral representation is difficult, due to the oscillating behavior of the integrands. A more appropriate numerical inversion procedure is also suggested, which circumvents the integration of the oscillating integrands, by an alternative modification of the contour in the Bromwich integral such that the new contour follows the steepest descent path starting from a saddle point at the real axis.     

Effect of curvature at the crack tip on the stress intensity factor for curved cracks

In this paper, the numerical solution of the hypersingular integral equation using the body force method in curved crack problems is presented. In the body force method, the stress fields induced by two kinds of standard set of force doublets are used as fundamental solutions. Then, the problem is formulated as a system of integral equations with the singularity of the form r ^–2. In the numerical calculation, two kinds of unknown functions are approximated by the products of the fundamental density functions and power series. The calculation shows that the present method gives rapidly converging numerical results for curved cracks under various geometrical conditions. In addition, a method of evaluation of the stress intensity factors for arbitrary shaped curved cracks is proposed using the approximate replacement to a simple straight crack.     

The onset of steady Marangoni convection in a spherical geometry

In this paper we use a combination of analytical and numerical techniques to analyse the onset of steady Marangoni convection in a spherical shell of fluid with an outer free surface surrounding a rigid sphere. In so doing we correct the formulation of the problem and the results presented by Cloot & Lebon (Microgravity sci. technol. 3 (1) 1990: 44–46). We find that if the free surface of the layer is non-deformable then the layer is always stable when heated from the outside and is unstable when heated from the inside if the magnitude of the (positive) non-dimensional Marangoni number is sufficiently large. If the free surface of the layer is deformable then the layer is always unstable when heated from the inside. It is stable when heated from the outside if C _r < r ₂/4, but if C _r > r ₂/4 then it is unstable if the magnitude of the (negative) Marangoni number is sufficiently large, where C _r is the non-dimensional Crispation number and r ₂ the non-dimensionalradius of the undisturbed outer free surface of the fluid.     

Numerical implementation of the symmetric Galerkin boundary element method in 2D elastodynamics

The symmetric Galerkin boundary element method (SGBEM) employs both the displacement integral equation and the traction integral equation which lead to a symmetric system of equations. A two‐dimensional SGBEM is implemented in this paper, with emphasis on the special treatments of singular integrals. The integrals in the time domain are carried out by an analytical method. In order to evaluate the strong singular double integrals and the hypersingular double integrals in the space domain which are associated with the fundamental solutions G ^pu and G ^pp, artificial body forces are introduced which can be used to indirectly derive the singular terms. Thus, those singular integrals which behave like 1/r and 1/r² are all avoided in the proposed SGEBM implementation. An artificial body force scheme is proposed to evaluate the body force term effectively. Two numerical examples are presented to assess the accuracy of the numerical implementation, and show similar accuracy when compared with the FEM and the analytical solutions. Copyright © 2003 John Wiley & Sons, Ltd.     

A CCC‐r chart for high‐yield processes

The cumulative count of a conforming (CCC) chart is used to monitor high‐quality processes and is based on the number of items inspected until observing r non‐conforming ones. This charting technique is known as a CCC‐r chart. The function of the CCC‐r chart is the sensitive detection of an upward shift in the fraction defectives of the process, p. As r gets larger, the CCC‐r chart becomes more sensitive to small changes of upward shift in p. However, since many observations are required to obtain a plotting point on the chart, the cost is fairly high. For this trade‐off problem it is necessary to determine the optimal number of non‐conforming items observed before a point is plotted, the sampling (inspection) interval, and the lower control limit for the chart. In this paper a simplified optimal design method is proposed. For illustrative purposes, some numerical results for the optimal design parameter values are provided. The expected profits per cycle obtained using the proposed optimal design method are compared with those obtained using other misspecified parameter values. The effects of changing these parameters on the profit function are shown graphically. Copyright © 2001 John Wiley & Sons, Ltd.     

Self-consistent forms of the chemical rate theory of Ostwald ripening

The chemical rate theory of Ostwald ripening introduced by A. D. Brailsford and P. Wynblatt (Act. Metall. 27 (1979) 498) determines the mean growth rate of particles of a particular size class by solving the diffusion equations for a representative particle (radius r) surrounded by a shell of matrix (the averaging sphere, radius r _A) outside which there is a homogeneous effective medium averaging the emission and absorption of solute atoms by the remainder of the particles. Brailsford and Wynblatt set r = r _A, in effect removing the matrix shell. It is argued herein that the feature of the theory so omitted is a very important one and we therefore use it to develop and extend the theory to make it self-consistent in the sense that the mean ratio of the particle and averaging sphere volumes is equal to the volume fraction of particles. Three self-consistent versions are developed, two of which have r _A relatively constant for small particles and slowly increasing for particles greater than approximately average size. These were motivated by the observation from numerical simulations that small particles are little influenced by their neighbours whereas larger particles are much more strongly affected by the environment. Analytical expressions in terms of experimentally observable variables are given for the probability distributions for particle sizes, and tables of the parameters required to evaluate the distribution functions as a function of volume fraction are provided. It is concluded that the properties of the Brailsford and Wynblatt effective medium are closely reproduced by the alternative analytical theories, but that the idea of a matrix shell round the representative particle is unique to the chemical rate theory. It is argued that this feature makes the theory flexible and adaptable. This adaptability could be used to reproduce the results of sophisticated numerical simulations in a form which would be computationally efficient to include in wider simulations involving, say, the effect of particle growth on long term mechanical properties.     

Energy determination of cosmic ray showers in surface arrays using signal inference at a single distance from the core

In most high energy cosmic ray surface arrays, the primary energy is currently determined from the value of the lateral distribution function at a fixed distance from the shower core, r₀. The value of r₀ is mainly related to the geometry of the array and is, therefore, considered as fixed independently of the shower energy or direction. We argue, however, that the dependence of r₀ on energy and zenith angle is not negligible. Therefore, in the present work we propose a new characteristic distance, which we call r_opt, specifically determined for each individual shower, with the objective of optimizing the energy reconstruction. This parameter may not only improve the energy determination, but also allow a more reliable reconstruction of the shape and position of rapidly varying spectral features. We show that the use of a specific r_opt determined on a shower-to-shower basis, instead of using a fixed characteristic value, is of particular benefit in dealing with the energy reconstruction of events with saturated detectors, which are in general a large fraction of all the events detected by an array as energy increases. Furthermore, the r_opt approach has the additional advantage of applying the same unified treatment for all detected events, regardless of whether they have saturated detectors or not.     

A reappraisal of transition elements in linear elastic fracture mechanics

This article offers a detailed comparison of the transition elements described by P.P. Lynn and A.R. Ingraffea [International Journal for Numerical Methods in Engineering 12,1031–1036] and C. Manu[Engineering Fracture Mechanics 24,509–512]. The source of a numerical phenomenon in using Manu's transitionelement (TE) is explained. The effect of eight-noded TEs with differentquarter-point elements (QPE) on the calculated stress intensity factors (SIFs) isinvestigated. Strain at the crack tip is shown to be singular for any ray emanating from the crack tip within an eight-noded TE, but strain has bothr ^–1/2andr ^–1singularities, withr ^–1/2dominating for large TEs. Semi-transition elements (STEs) are defined and shown to have a marginal effect on the calculated SIFs. Nine-nodedtransition elements are formulated whose strain singularity is shown to be the same as that of eight-noded TEs. Then the effect of eight-noded and nine-noded TEs with collapsed triangular QPEs, and rectangular and nonrectangular quadrilateral eight-noded and nine-noded QPEs, is studied, and nine-noded TEs are shown to behave exactly like eight-noded TEs with rectangular eight-noded and nine-noded QPEs and to behave almost the same with other QPEs. The layered transition elements proposed by V. Murti and S.Valliapan [Engineering Fracture Mechanics 25, 237–258] areformulated correctly. The effect of layered transition elements is shown by two numerical examples.     

A mapping method for numerical evaluation of two-dimensional integrals with 1/r singularity

Singular integrals occur commonly in applications of the boundary element method (BEM). A simple mapping method is presented here for the numerical evaluation of two-dimensional integrals in which the integrands, at worst, are O(1/r) (r being the distance from a source to a field point). This mapping transforms such integrals over general curved triangles into regular 2-D integrals. Over flat and curved quadratic triangles, regular line integrals are obtained, and these can be easily evaluated by standard Gaussian quadrature. Numerical tests on some typical singular integrals, encountered in BEM applications, demonstrate the accuracy and efficacy of the method.     

A finite element investigation of quasi-static and dynamic asymptotic crack-tip fields in hardening elastic-plastic solids under plane stress

The asymptotic structures of crack-tip stress and deformation fields are investigated numerically for quasi-static and dynamic crack growth in isotropic linear hardening elastic-plastic solids under mode I, plane stress, and small-scale yielding conditions. An Eulerian type finite element scheme is employed. The materials are assumed to obey the von Mises yield criterion and the associated flow rule. The ratio of the crack-tip plastic zone size to that of the element nearest to the crack tip is of the order of 1.6 × 10⁴. The results of this study strongly suggest the existence of crack-tip stress and strain singularities of the type r ^s (s < 0) at r=0, where r is the distance to the crack tip, which confirms the asymptotic solutions of variable-separable type by Amazigo and Hutchinson [1] and Ponte Castañeda [2] for quasi-static crack growth, and by Achenbach, Kanninen and Popelar [3] for dynamic crack propagation. Both the values of the parameter s and the angular stress and velocity field variations from the present full-field finite element analysis agree very well with those from the analytical solutions. It is found that the dominance zone of the r ^s-singularity is quite large compared to the size of the crack-tip active plastic zone. The effects of hardening and inertia on the crack-tip fields as well as on the shape and size of the crack-tip active plastic zone are also studied in detail. It is discovered that as the level of hardening decreases and the crack propagation speed increases, a secondary yield zone emerges along the crack flank, and kinks in stress and velocity angular variations begin to develop. This dynamic phenomenon observable only for rapid crack growth and for low hardening materials may explain the numerical difficulties, in obtaining solutions for such cases, encountered by Achenbach et al. who, in their asymptotic analysis, neglected the existence of reverse yielding zones along the crack surfaces.     

Some General Remarks on Consulting in Statistics

This paper deals with obtaining a prediction interval on a future observation X, in an ordered sample of size n from a two-parameter exponential distribution for the situation where some or all the first r observations X ₁ < X ₂ < … < X _r , 1 ≤ r < s ≤ n, have been observed. The intervals are based on the statistic Z = (X _s , – X _r )/S _v , where S _v , is a function of the observations X ₀ ≡ A < X ₁ < X ₂ < … < X _r , such that X _s – X _r , and S _v , are independent variables and 2vS_v /σ has the distribution χ²(2v). The expressions for the quantiles z_p are given and some problems of numerical determination of z_p 's are discussed. The results can be also applied to related distributions.     

Stress-intensity factors of r-cracks in fiber-reinforced composites under thermal and mechanical loading

This paper is concerned with the problem of the calculation of stress-intensity factors at the tips of radial matrix cracks (r-cracks) in fiber-reinforced composites under thermal and/or transverse uniaxial or biaxial mechanical loading. The crack is either located in the immediate vicinity of a single fiber or it terminates at the interface between the fiber and the matrix. The problem is stated and solved numerically within the framework of linear elasticity using Erdogan's integral equation technique. It is shown that the solutions for purely thermal and purely mechanical loading can simply be superimposed in order to obtain the results of the combined loading case. Stress-intensity factors (SIFs) are calculated for various lengths and distances of the crack from the interface for each of these loading conditions. The behavior of the SIFs for cracks growing towards or away from the interface is examined. The role of the elastic mismatch between the fibers and the matrix is emphasized and studied extensively using the so-called Dundurs' parameters. It is shown that an r-crack, which is remotely located from the fiber, can either be stabilized or destabilized depending on both the elastic as well as the thermal mismatch of the fibrous composite. Furthermore, Dundurs' parameters are used to predict the exponent of the singularity of the crack tip elastic field and the behavior of the corresponding SIFs for cracks which terminate at the interface. An analytical solution for the SIFs is derived for all three loading conditions under the assumption that the elastic constants of the matrix and the fiber are equal. It is shown that the analytical solution is in good agreement with the corresponding numerical results. Moreover, another analytical solution from the literature [15], which is based upon Paris' equation for the calculation of stress-intensity factors, is compared with the numerical results and it is shown to be valid only for extremely short r-cracks touching the interface. The numerical results presented are valid for practical fiber composites with r-cracks close to or terminating at the interface provided the matrix material is brittle and the crack does not interact with other neighboring fibers. They may be applied to predict the transverse mechanical behavior of high strength fiber composites.