A New Quasi-Boundary Scheme for Three-Dimensional Backward Heat Conduction Problems

In this study, we employ a semi-analytical scheme to resolve the three-dimensional backward heat conduction problem (BHCP) by utilizing a quasi-bound -ary concept. First, the Fourier series expansion method is used to estimate the temperature field u(x, y, z, t) at any time t < T. Second, we ponder a direct regularization by adding an extra term a(x, y, z, 0) to transform a second-kind Fredholm integral equation for u(x, y, z, 0). The termwise separable property of the kernel function allows us to acquire a closed-form regularized solution. In addition, a tactic to determine the regularization parameter is recommended. We find that the proposed method is robust and applicable to the three-dimensional BHCP when several numerical experiments are examined.     

A Fictitious Time Integration Method for Multi-Dimensional Backward Wave Problems

We address a new numerical approach to deal with these multi-dimensional backward wave problems (BWPs) in this study. A fictitious time τ is utilized to transform the dependent variable u(x, y, z, t) into a new one by (1+τ)u(x, y, z, t)=: v(x, y, z, t, τ), such that the original wave equation is written as a new hyperbolic type partial differential equation in the space of (x, y, z, t, τ). Besides, a fictitious viscous damping coefficient can be employed to strengthen the stability of numerical integration of the discretized equations by using a group preserving scheme. Several numerical instances demonstrate that the present scheme can be utilized to retrieve the initial wave very well. Even though the noisy final data are very large, the fictitious time integration method is also robust against disturbance.     

The Fictitious Time Integration Method to Solve the Space- and Time-Fractional Burgers Equations

We propose a simple numerical scheme for solving the space- and time-fractional derivative Burgers equations: D_t^αu + εuu_x = vu_xx + ηD_x^βu, 0 < α, β ≤ 1, and u_t + D_*^β(D_*^1-βu)²/2 = vu_xx, 0 < β ≤ 1. The time-fractional derivative D_t^αu and space-fractional derivative D_x^βu are defined in the Caputo sense, while D_*^βu is the Riemann-Liouville space-fractional derivative. A fictitious time τ is used to transform the dependent variable u(x,t) into a new one by (1+τ)^γu(x,t) =: v(x,t,τ), where 0 < γ ≤ 1 is a parameter, such that the original equation is written as a new functional-differential type partial differential equation in the space of (x,t,τ). When the group-preserving scheme is used to integrate these equations under a semi-discretization of u(x,t,τ) at the spatial-temporal grid points, we can achieve rather accurate solutions.     

A Quasi-Boundary Semi-Analytical Method for Backward in Time Advection-Dispersion Equation

In this paper, we take the advantage of an analytical method to solve the advection-dispersion equation (ADE) for identifying the contamination problems. First, the Fourier series expansion technique is employed to calculate the concentration field C(x, t) at any time t< T. Then, we consider a direct regularization by adding an extra term αC(x,0) on the final condition to carry off a second kind Fredholm integral equation. The termwise separable property of the kernel function permits us to transform itinto a two-point boundary value problem. The uniform convergence and error estimate of the regularized solution C^α(x,t) are provided and a strategy to select the regularized parameter is suggested. The solver used in this work can recover the spatial distribution of the groundwater contaminant concentration. Several numerical examples are examined to show that the new approach can retrieve all past data very well and is good enough to cope with heterogeneous parameters’ problems, even though the final data are noised seriously.     

A Quasi-Boundary Semi-Analytical Approach for Two-Dimensional Backward Advection-Dispersion Equation

In this study, we employ a semi-analytical approach to solve a two-dimensional advection-dispersion equation (ADE) for identifying the contamination problems. First, the Fourier series expansion technique is used to calculate the concentration field C(x, y, t) at any time t < T. Then, we ponder a direct regularization by adding an extra termaC(x, y, 0) on the final time data C(x, y, T), to reach a second-kind Fredholm integral equation. The termwise separable property of kernel function allows us obtaining a closed-form solution of the Fourier coefficients. A strategy to choose the regularization parameter is offered. The solver utilized in this work can retrieve the spatial distribution of the groundwater contaminant concentration. Several numerical examples are scrutinized to display that the new method can recover all the past data very well, and is good enough to deal with heterogeneous parameters, even though the final time data are noised seriously.     

A Lie-Group Adaptive Method to Identify the Radiative Coefficients in Parabolic Partial Differential Equations

We consider two inverse problems for estimating radiative coefficients α(x) and α(x, y), respectively, in T_t(x, t) = T_xx(x, t)-α(x)T(x, t), and T_t(x, y, t) = T_xx(x, y, t) + T_yy(x, y, t)-α(x, y)T(x, y, t), where a are assumed to be continuous functions of space variables. A Lie-group adaptive method is developed, which can be used to find a at the spatially discretized points, where we only utilize the initial condition and boundary conditions, such as those for a typical direct problem. This point is quite different from other methods, which need the overspecified final time data. Three-fold advantages can be gained by the present Lie-group adaptive method (LGAM): (i) no a priori information of radiative coefficients is required, (ii) no extra data are measured, and (iii) no complicated procedure is involved. The accuracy and efficiency of present method are confirmed by comparing the estimated results with some exact solutions for 1-D and 2-D cases.     

Mechanical wave interaction at a diffuse interface

An investigation is made of the reflection coefficientR of an ultrasonic wave impinging on a boundary at which there is a gradual transition of material properties from one material to another. The study is confined to the one-dimensional case of compression waves incident at 90° on the boundary. Both analytical and numerical solutions forR are presented for various interfacial profiles, and are seen in general to be strongly dependent on wave frequency. Measurements are made on diffusion bonded samples of nickel to copper, and the results compared with theory. Significant scatter is found in the experimental results due primarily to the migration of oxygen into the diffusion zone. A second set of measurements on diffusion bonded samples of nickel to carbon steel show general agreement with theory.Nomenclature A _i ,B _i ,C _i ,i=1, 2 coefficients in the expression forX (x) - c _i concentration of speciesi - c _L compression wave velocity - D _i diffusion coefficient for speciesi - diffusion coefficient for a binary mixture - E(x) Young's modulus - stiffness for the case of one-dimensional strain, =(1–v)E(x)/(1+v)(1–2v) - one-dimensional stiffness within bluk materiali - J _i ,i=1, 2 flux of speciesi - k(x) wave number at spatial co-ordinatex, equal to 2/ - k _i ,i=1, 2 wave number within bulk materiali - K(x) pseudo-wavenumber, defined by Eq. (8). - k _B Boltzmann's constant - L thickness of interfacial zone - f wave frequency - m, N material profile parameters, used to specifyK(x) - Q activation energy for diffusion - R reflection coefficient - t time - T transmission coefficient - u(x, t) displacement of molecules due to wave action,u(x, t)=X(x) (t) - X(x) spatial portion of functionu(x, t) - x spatial variable - Y(x, t) (x, t)/ - Z _i ,i=1, 2 mechanical impedance of bulk materiali - strain - wavelength of sound - Poisson's ratio - material density - (x, t) stress - (t) temporal portion of functionu(x, t) - _i (x),i=1, 2 functional form ofX(x) within interfacial region - angular frequency of wave     

Blended infinite elements for paraboblic boundary value problems

A common method for numerically approximating two-point parabolic boundary value problems of the form u_t = L[u]+f(u) defined of the semi-infinite strip S = [0, 1]×[0, ∞] is to first discretize the spatial operator in the differential equation and then solve for the time evolution. Such an approach typically involves solving a system of algebriaic equations at a sequence of time steps. In this paper we take a different approach and subdivide S into a collection of semi-infinite substrips S_i = [x_i, x_i+1]×[0, ∞], and use blending function techniques to derive finite parameter functions e_i(x, t) defined on S_i. Spectral matching methods are used in deriving e_i to ensure that (u ? e_i) can be made small on S_i. Galerkin's method, with associated integration sover the entire space-time domain S, is then used to generate approximations to u(x, t) based upon the so defined infinite element (e_i, S_i). Approximations are hence found for all (x, t) in S by solving one well structed system of algebraic equations. We apply the method to several linear and non-linear problms.     

Quasi-static mode III fracture in a nonhomogeneous viscoelastic body

Summary An analysis is presented of the quasi-static propagation of a semi-infinite mode III crack in an inhomogeneous isotropic viscoelastic body. A shear modulus is assumed of the formG(t,y)=(t)(y) where |y| denotes the distance measured from the plane of the crack and (t) is a positive, nonincreasing, convex function of time. A closed form expression is derived for the energy release rate (ERR) when a Barenblatt type failure zone is incorporated into the crack model. Numerical and asymptotic results illustrate the combined effects of viscoelastic properties, material inhomogeneity, and the failure zone at the crack tip upon the ERR.     

Physical model of flow with anisotropic layer impregnation

Isothermal filling of a plane cavity by a Newtonian fluid with simultaneous anisotropic impregnation of a reinforcing layer is considered.Notation p, p _c fluid pressure in porous medium and in cavity, respectively - u, longitudinal and transverse components of filtration velocity - u _c,_c longitudinal and transverse components of flow velocity in cavity - mean velocity in cavity - h half-height of cavity - H porous layer height - l ₀,l length of sprue channel and cast mould, respectively - fluid viscosity - y ₀(t) coordinate of the impregnation front - x ₀(t) axial coordinate at whichy ₀=H - impregnated material porosity - k _x ,k _y longitudinal and transverse coefficients of permeability I. M. Gubkin State Academy of Oil and Gas, Scientific-Research Institute of Plastics of the Scientific Production Association Plastmassy, Moscow. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 5, pp. 845–851, September–October, 1995.     

An inverse problem of simultaneously estimating contact conductance and heat transfer coefficient of exhaust gases between engine's exhaust valve and seat

The conjugate gradient method using two search step sizes is used to solve the inverse problem of simultaneously estimating the periodic thermal contact conductance, h_c(t), and the heat transfer coefficient of the exhaust gases, h_g(t), between the exhaust valve and seat in an internal combustion engine. The importance of the determination of h_c(t) and h_g(t) lie in that they are the critical factors for designing the cooling system and the insulation of the exhaust valve. The inverse analysis is based on the temperature measurements taken from the sensors placed in both the valve and seat regions during the transient process of operation. In this study two unknown timewise-varying functions h_c(t) and h_g(t) are to be estimated at the same time, thus two search step sizes with each one corresponding to each unknown function are derived. The results show that the CPU time for the inverse solutions using two search step sizes are greatly reduced than using just one search step size¹ for the determination of two unknowns, besides, it also shows that the inverse solutions are reliable even when the measurement errors are considered. The advantage of the conjugate gradient method is that no a priori information is needed on the variation of the unknown quantities, since the solution automatically determines the functional form over the domain specified. The successful development of the present technique can be applied to any kind of two-dimensional periodic contact problems, such as the determination of a two-dimensional contact conductance problem² and the temperature or heat flux behaviour on the inside wall of internal combustion engines³.     

The propagation and basal solidification of two-dimensional and axisymmetric viscous gravity currents

The effect of basal solidification on viscous gravity currents is analysed using continuum models. A Stefan condition for basal solidification is incorporated into the Navier-Stokes equations. A simplified version of this model is determined in the lubrication and large-Bond-number limit. Asymptotic solutions are obtained in three parameter régimes. (i) A similarity solution is possible in the following cases: the two-dimensional problem when volume per unit length (V) is proportional to time (t) raised to the power 7/4(V = qt ^7/4) and the Julian number (v ³ g ² /q ⁴ ) is large, where v is kinematic viscosity, q is a constant of proportionality and g is the acceleration due to gravity; the axisymmetric problem when volume is proportional to time raised to the power 3 (V = Qt ³) and the dimensionless group vg/Q is large, where Q is a constant of proportionality. In both cases, the front is found to depend on time raised to the power 5/4, as it does in the absence of solidification, but the constant of proportionality satisfies a modified system of equations. (ii) In the case of large Stefan number and small modified Peclet number (Pe ² 1, where Pe is the Peclet number and is the aspect ratio), asymptotic and numerical methods are combined to produce the most revealing results. The temperature of the fluid approaches the melting point over a short time-scale. Over the long time-scale, the solid/liquid interface is determined from the conduction of latent heat into the solid. Strong coupling is observed with the basal solidification modifying the flow at leading order. The solidification may retard and eventually arrest the front motion long before complete phase change has taken place. (iii) In the case of constant volume and large modified Peclet number (Pe ² 1), similarity solutions are found for the solidification at the base of the gravity current on the short time-scale. The coupling is weak on this time-scale with the solidification being dependent on the front position but not influencing the fluid motion at leading order. Over the long time-scale, the drop completely solidifies. Analytical solutions are not obtained on this time-scale, but scalings are deduced.     

Structure,growth and morphology polyphenylene sulphide

The crystaIIinity, particle size and morphology of polyphenylene sulphide synthesized under various conditions have been investigated by X-ray diffraction and scanning electron microscopy. It was found that crystaIIinity decreased from 71 to 66% with increase of reaction time. The growth of particle size as well as total polymer mass followed a time dependence of the formX =X ₀ (1 -e ^–t ) The particle size distribution curve was noted to be sharp centring at 3 µn for short reaction time, high speed of stirring and also for low concentration of reactants. The particle morphology showed very strong dependence on various reaction parameters. Intricate sheaf-like morphology was noted for the particles at long reaction times or low stirring speeds while oblong platelet type two-dimensional morphology was noted when a low concentration of reactants was used.NCL communication No. 3674     

Probing the Vorticity in a Superconductor Rapidly Quenched from the Normal State

Topological defects such as vortices subsist for some time in a rapidly frozen superconducting film (W. H. Zurek, Phys. Rep. 276, 177, 1996). We propose to use as a probe of the vortex density the finite delay T _d, which, in narrow strips, exists between a current step and the voltage response. Technically, this amounts to driving a bridge into the localized hot-spot regime by means of a pump pulse (laser or electrical). Cooling of such films as epitaxial YBCO-on-MgO, or niobium-on-sapphire, requires only a few nanoseconds at low T. A time t later, a probe pulse is applied to measure T _d. The dependence t T _d is interpreted as a witness of the fossil vorticity, long after quenching into the zero-resistance regime.     

Fatigue of metallic glasses

The fatigue behaviour of Ni₄₉Fe₂₉P₁₄B₆Si₂, Ni₄₈Fe₂₉P₁₄B₆Al₃ and Pd_77.5Cu₆Si_16.5 metallic glasses is examined. In the finite lifetime regime the relationship between stress amplitude ( _a), fracture stress ( _f), mean stress ( _m) and cycles to failure (N _f) is _a=A( _f–_m) (2N _f) ^b , whereA andb are 16.9 and –0.40 respectively for reduced gauge section Ni₄₉ strips (for _m 140 kg mm^–2) and 27.0 and –0.44 for Pd base wires. These results are unusual in thatA 1. Consequently, a sharp discontinuity exists near _a( _f– _m) –1. In a simple tensile test failure occurs at _f(=_y) and 2N_f=1; for peak stresses only a percent or so less than _f the sample will withstand hundreds of cycles of stress. For uniform cross-section glassy metal filaments, a fatigue limit is observed at stress ratios ( _a/ _f) in the vicinity of 0.07 to 0.15. The fatigue limit for reduced section specimens is a factor of 2 higher. Fatigue failure of the Ni-Fe strips may occur under partially or fully plane stress or plane strain conditions, depending on sample thickness and stress. Final failure of the Pd_77.5Cu₆Si_16.5 wires always occurs by general yielding of the remaining section.     

Strontium Substitution For Barium In Mercury Superconductors

HgBa _2-x Sr _x CuO ₄₊ (x = 0.06 - 0.45) and Hg _1-y V _y Ba _2-x Sr _x CuO ₄₊ (x = 0.16-1.1, y = 0.2 - 0.35) series were prepared by the encapsulation method. In the former series the optimum doping level is achieved by annealing under oxygen for x = 0.06 and by annealing under nitrogen for x = 0.45. This behaviour is explained by different equilibrium oxygen pressures over phases with different Sr content. The decrease of T _c is –8 K/x. The latter series is doped by replacing (HgO ₂ ) ^2- groups by (VO ₄ ) ^3- tetrahedra and optimum doping is achieved by argon annealing.     

Sparse shifts for univariate polynomials

Letf(x) be a polynomial of degreed with rational coefficients and lett be a positive integer deg(f). We consider the problem of finding at-sparse shift forf(x). The problem is to find an a, if one exists (in some algebraic extension of the rationals), such that in the representation off(x) in the basis 1,x – , (x – )²,..., i.e., at most t of the coefficients f_i are non-zero. We derive explicit conditions for the uniqueness and rationality of at-sparse shift forf(x) and provide an efficient algorithm for computing a sparse shift when one exists. We also point out an application of our result to the problem of constructing sparse decompositions of univariate polynomials.Work by Y. N. Lakshman was supported by NSF grant CCR-9203062Work by B. D. Saunders was supported by NSF grant CCR-9123666     

Electron-phonon coupling as a mechanism for high-temperature superconductivity

We have solved the Eliashberg gap equations which are valid for arbitrary phonon frequency, (_ph), electron-phonon coupling constant, (), and screened Coulomb interaction, ( ^*). We have used values of (_ph),, and ( ^*) appropriate to the cuprate superconductors, and calculated the density of states, the pair potential., and the value of the gap at T=0 K. Using the linearized Eliashberg equations in the matrix representation, we have calculatedT _c and 2/k T _c . We have found that we can account for the highT _c 's in the cuprates with reasonable values for, ^*, and _ph.     

Superconductivity in amorphous Bi and Ga

The electron-phonon ²()F() has been measured by Chenet al. in films of amorphous Bi and Ga. This was accomplished by inversion of low-temperature quasiparticle tunneling data. From ²()F(), the phonon renormalization of the electronic effective mass was calculated and was found to be considerably greater than it is in the case of the well-known strong-coupling superconductors Pb and Hg. These amorphous materials should, therefore, be ideal for a study of strong-coupling effects, i.e., deviations from BCS laws. In this paper, we present results of a theoretical study of select thermodynamic properties of these materials based on numerical solutions, at finite temperatures, of the Eliashberg equations.Work performed under the auspices of the U.S. Atomic Energy Commission, and supported in part by the National Research Council of Canada.     

The failure of fibre composites and adhesively bonded fibre composites under high rates of test

The dynamic effects which are commonly encountered during high-rate DCB tests with fibre composite and adhesively bonded fibre composite arms have been studied in detail. This paper, Part II of the series, follows Part I, which described the experimental aspects of the high-rate testing. Part III will report the results from mode II and mixed-mode I/II tests on the fibre-composite materials.Nomenclature a crack length - a ₀ initial crack length - a crack speed - ä crack acceleration - c longitudinal wave speed - h thickness of single arm of test specimen - p crack length perturbation (i.e. the measured value of the crack length minus the value predic ted by steady-state theory) - p crack velocity perturbation - crack acceleration perturbation - t time - t ₀ time taken for crack to initiate during the mode I test - u ₀ load-line vertical displacement of single arm of test specimen (/2 in Part I) - u(x) vertical displacement of specimen at distance x from the load-line - u(x) vertical displacement rate of specimen at distance x from the load-line - x distance along the test specimen from the load-line - A constant relating the steady state crack length to root time - B width of specimen - C compliance of the specimen (u ₀/P) - E ₁₁ axial modulus of the fibre-composite beam - G mode I energy release rate - G _Ic mode I critical energy release rate or fracture toughness - G ₁ half the value of G _Ic during steady-state propagation (i.e. calculated for half the beam as shown in Fig. 1) - G ₂ half the value of G _Ic at crack initiation - P end load applied to specimen - U _ext external work done - U _s strain energy - U _k kinetic energy - V velocity of a single arm of test specimen (i.e. half the measured test velocity) - dynamic term, governed by the ratio of the energy to initiate versus that to propagate a crack - _I mode I crack shear deflection and root rotation correction term - crack length correction term, evaluated by the negative intercept on the a versus t ^1/2 plot - dynamic term controlling the form of the computed perturbations - Poisson's ratio for the fibre-composite beams density of the fibre-composite beams - time, normalized by the initiation time, t ₀ and thus equivalent to (t/t ₀) - values of at which crack arrest occurs. n = 1,2,3... - ratio of distance along beam to crack length (x/a)