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The pile responses induced by adjacent tunnel excavation have been a hot research topic in geotechnical engineering. Tunnel excavation may exert disturbance to the surrounding soil mass and then influence the adjacent pile foundations. In this paper, the random finite difference analysis considering the spatial variations of soil properties is conducted to explore the effect of tunnel excavation on the adjacent pile response by varying the distance of pile away from the tunnel centerline (D), the pile length (L), the pile diameter (dp), the tunnel depth (h), and the anisotropic ratio between the horizontal and the vertical scales of fluctuation (δx/δy). A set of pile response curves are developed to assess the influence of tunnel excavation on the adjacent passive pile foundations in spatially variable clays and provide guidelines for the tunnel excavation in the complex constructed environment.

Both cryosurgery and radiofrequency ablation (RFA) for solid liver tumor treatment are available as minimally invasive procedures that induce changes to the tumor’s thermal environment in order to destroy cancer cells. However, one of the critical factors that impedes cryosurgery and RFA’s successful outcomes is the relatively high recurrence rate caused by the inability to ablate a large damaged tissue zone that envelopes targeted tumors, resulting in therapy failure. To overcome these challenges, a hybrid cryo-RFA system under thermal stress control is proposed in this study. A three-dimensional finite difference analysis is employed to simulate the combined cryosurgery and RF heating protocol. Based on the data acquired from measured experiments, the simulated results derived have demonstrated close agreement with experimental data, with a maximum deviation of 4.8%. We investigated the impacts of varying cryoprobe’s holding temperature and cooling rates on tissue damage. Results have revealed that the tissue damage region is enhanced by properly increasing the cryoprobe’s cooling rate, but the increase of cryoprobe’s cooling rate cannot enlarges the tissue damage area without limit in the freezing process. In addition, the employment of a hybrid cryo-RFA system markedly promotes the destruction of cancer tissue in contrast to conventional stand-alone RF heating or cryosurgery.

An enhanced cohesion weakening and friction strengthening model is proposed to simulate the progressive failure of hard rock pillars using two- and three-dimensional finite difference analysis. The documented behavior of 85 pillars in two mines was used for model validation. The numerical models successfully separated unstable/failed pillars from stable pillars and also captured the observed mechanism of progressive failure in hard rock pillars. Pillar stress-strain curves indicated that while pillars with W/H

Two different approaches are employed to predict the magnitude of lateral spreading induced by liquefaction using a two-dimensional finite difference analysis with an advanced plasticity constitutive model for the liquefiable sand. One approach uses the advanced plasticity model to directly predict lateral spreading. The other approach is a hybrid approach that first uses the advanced plasticity model to calculate the time to triggering of liquefaction and then uses a Newmark-type displacement analysis to predict the magnitude of lateral spreading. To evaluate the accuracy of these methods for predicting lateral spreading displacements, both approaches were applied to five well-documented case histories of lateral spreading. Comparison of the calculated and reported displacements shows that the hybrid approach that calculates lateral spreading with a Newmark analysis that employs the free field ground surface motions starting from the time at which liquefaction is triggered was, on the average, the most accurate approach. The two-dimensional finite difference analysis gave generally conservative results, yielding larger displacement than the reported value. While the uncertainty associated with both methods is very high, based upon the five case histories analyzed herein it appears that the two-dimensional finite difference analysis can be used with good confidence to predict a conservative value for the magnitude of liquefaction-induced lateral spreading if the parameters that govern lateral spreading are well known.

Radiofrequency ablation (RFA) for liver tumors is a minimally invasive procedure that uses electrical energy and heat to destroy cancer cells. One of the critical factors that impedes its successful outcome is the thermal heat sink effects from complex vascular systems that give rise to incomplete destruction of the target tumor tissue, resulting in therapy failure. To better understand the thermal influence of the complex vascular system during RFA, this work proposes the employment of two 3D fractal tree-like branched networks to investigate which key factors of the tree-like vascular system impact heating process. A three-dimensional finite difference analysis is employed to simulate the RFA treatment. Based on the data acquired from the measured experiments, the simulated results derived from combining the Pennes bioheat model and the boundary condition-enforced immersed boundary method (IBM) have demonstrated close agreement with experimental data with a maximum discrepancy of ±8.3%. We employed the orthogonal design approach to analyze 3 factors, namely, the blood vessel's volume, the average distance between probe center and the blood vessel system and the number of the selected part's branches at three different levels. Results have revealed that the distance between RFA probe and blood vessel plays a major role during the heating process compared with the other two factors. In addition, both the ablating rates and the volume of damaged tissue are slightly reduced with increasing number of blood vessel branches.

Radiofrequency ablation (RFA) for liver tumors is a minimally invasive procedure that uses electrical energy and heat to destroy cancer cells. One of the critical factors that impedes its successful outcome is the use of inappropriate radiofrequency levels that will not completely destroy the target tumor tissues, resulting in therapy failure. Additionally, the surrounding healthy tissues may suffer from serious damage due to excessive ablation. To address these challenges, this work proposes the employment of injected nanoparticles to thermally promote the ablation efficacy of conventional RFA. A three-dimensional finite difference analysis is employed to simulate the RFA treatment. Based on the data acquired from measured experiments, the simulation results have demonstrated close agreement with experimental data with a maximum discrepancy of within ±8.7%. Several types of nanoparticles were selected to evaluate their influences on liver tissue's thermal and electrical properties. We analysed the effects of nanoparticles on liver RFA via a tumor rending process incorporating several clinically-extracted tumor profiles and vascular systems. Simulations were conducted to explore the temperature difference responses between conventional RFA treatment and one with the inclusion of assisted nanoparticles on several irregularly-shaped tumors. Results have indicated that applying selected nanoparticles with high thermal conductivity and electrical conductivity on the targeted tissue zone promotes heating rate while sustaining a similar ablation zone that experiences lower maximum temperature when compared with the conventional RFA treatment. In sum, incorporating thermally-enhancing nanoparticles promotes heat transfer during the RFA treatment, resulting in improved ablation efficiency.

The performances of 2 K Joule–Thomson (J-T) heat exchanger are related to the operation efficiency and economy of the superfluid helium cryogenic system, and the helically finned-tube heat exchanger (HFHE) is designed and tested. Heat transfer characteristics of the HFHE are investigated by a finite difference analysis method, and the design optimization of the HFHE is performed by the orthogonal test and variance analysis method. Meanwhile, the influences of the geometrical parameters on the performances are discussed. The experimental investigation of the HFHE is carried out in the superfluid helium cryogenic system, and the performances of the HFHE are verified experimentally. The results are shown that the calculated performances of the HFHE cause by heat transfer are in reasonable agreement with the experimental results, which indicates that the thermal calculation model of the HFHE is more accurate. Moreover, the deviations of pressure drop of cold fluid between the calculation and measurement are analyzed and the friction factor is modified in this paper.

This paper investigates the use of stability factors to estimate tunnel heading pressures. The primary method adopted is a conventional equation based on the soil property and stability factors, analogous to the bearing capacity factors (Nc, Ns and Nγ) of strip footings. The finite element limit analysis (FELA) is employed to determine rigorous upper bound (UB) and lower bound (LB) solutions of stability factors (Fc, Fs and Fγ), which are functions of the depth ratio (C/D) and soil internal friction angle (O). The obtained results are compared and validated by using the finite difference analysis as well as other available published results in the literature. A number of examples are illustrated on how to use the factors to estimate tunnel heading pressures.

Although face bolting has been used as a stabilisation technique in open-face tunnelling for decades, there is still a lack of systematic ways for determining the optimum parameters of face bolts. To optimise design for face bolting in soft ground, it is necessary to understand the influences of each parameter associated with face bolting on ground response. In this note, five series of numerical parametric studies are carried out, to investigate the effects of length, density, reinforcement area, axial rigidity of face bolts and strength of soil on tunnel face stability and deformation in soft rock. Based on the ground condition, geometries of tunnel and configurations of face bolts simulated, the optimum length, density and axial rigidity of face bolts are found to be 0.6 H ( H = height of tunnel), 1 bolt/m 2 and 195 MN, respectively. The optimum axial rigidity of face bolt appears to be independent of the bolt density. The computed results also reveal that it is more effective to reduce face deformation by installing face bolts around the tunnel periphery, than installing them near the central area of the tunnel face.

A finite difference analysis was performed to investigate the behavior of one-way reinforced concrete slabs exposed to fire. The objective of the study was to investigate the fire resistance and the fire risk after extinguishing the fire. Firstly, the fire resistance was obtained using the ISO834 standard fire without cooling phase. Secondly, the ISO834 parametric fire with cooling phase was applied to study the effect of cooling time. Accordingly, the critical time for cooling was identified and the corresponding failure time was calculated. Moreover, the maximum risk time which is the time between the fire extinguishing and the collapse of slab was obtained. Sixteen one-way reinforced concrete slabs were considered to study the effect of important parameters namely: the concrete cover thickness; the plaster; and the live load ratio. Equations for heat transfer through the slab thickness were used in the fire resistance calculations. Studying the cooling time revealed that the slabs are still prone to collapse although they were cooled before their fire resistance. Moreover, increasing the concrete cover thickness and the presence of plaster led to an increase in the maximum risk time. However, the variation in the live load ratio has almost no effect on such time.

A numerical analysis of pile driving for tapered piles is presented in this paper. A three-dimensional finite difference analysis for tapered angle and geometry effects has been used on pile driving response of tapered piles. The simulation considers an idealization for pile–soil system in drivability. The vertical pile is assumed to have linear and elastic behavior. It is also assumed that the soil is elasto-plastic material and its failure stage is controlled using the Mohr–Coulomb failure criterion. At the soil–pile contact surfaces along the pile shaft and pile toe, slip is allowed to occur during the driving procedure using interface elements. Quiet boundaries are used to prevent waves traveling in the lateral and vertical directions for the soil. Cylindrical, fully tapered, and semi-tapered piles were analyzed. The results obtained from numerical analyses were compared with those obtained from available laboratory tests and also other available numerical data, resulting in a satisfactory agreement. The results have shown that among piles of the same length and material volume, with increasing the taper angle from zero (representing a cylindrical pile), the driving stresses decrease and the permanent pile toe settlement (set) increases. These are interesting in pile driving and are on the safe side for driven piles and increasing the driving efficiency. It has also been found that the geometry of the pile can generally influence the pile drivability. Generally speaking, tapered and partially tapered piles offer better drivability performance than cylindrical piles of the same volume and length.

An important component of any CO2 sequestration project is seismic monitoring for tracking changes in subsurface physical properties such as velocity and density. Reservoir conditions and CO2 inj ...

A finite difference procedure that can account for strain-rate effects, both shear and flexural deformations, permits variations in cross-section geometry and strength and loading over the length of a component is proposed to accurately and efficiently analyze the dynamic response of a simply supported structural member under blast loads. A section-based layered analysis model that accommodates varying strain rates across a member’s cross-section is used to derive sectional moment–curvature relationships. A formula is derived to estimate the distribution of strain rate over the depth of a cross-section along the length of the member, and the corresponding strain rate effects are incorporated into the section-based layered analysis model. The Timoshenko beam equations that include both shear deformations and rotational inertia are solved numerically using an explicit finite difference scheme. The accuracy of the proposed finite difference analysis model is part validated using results of blast testing of reinforced concrete slabs with combinations of explosive weights and standoff distances. The results are also compared with those obtained by conventional single-degree-of-freedom (SDOF) analysis and finite element (FE) analysis using solid elements. The finite difference analysis procedure is both fast-running and accurate and most suitable for design office application, combining the speed of SDOF analysis and the detail and accuracy of FE analysis.

We have studied in detail the theoretical and numerical properties of a finite-difference algorithm for image-wave time-remigration. For a number of synthetic models, numerical experiments have been performed. For these examples, we obtained perfect agreement between the theoretical predictions and numerical results. The examples also prove the computational efficiency of the algorithm. An application to ground-penetrating-radar (GPR) data demonstrates that image-wave remigration can be used to estimate models with laterally varying velocities. The quality of the latter is confirmed by a final zero-offset time migration.

The performance of ground-penetrating (GPR) radar is determined fundamentally by the soil electromagnetic (EM) properties and the target characteristics. In this paper, we predict the effects of such properties on the antipersonnel (AP) landmine detection performance of GPR in a Colombian scenario. Firstly, we use available soil geophysical information in existing pedotransfer models to calculate soil EM properties. The latter are included in a two-dimensional (21)), finite-difference time-domain (FDTD) modeling program in conjunction with the characteristics of AP landmines to calculate the buried target reflection. The approach is applied to two soils selected among Colombian mine-affected areas, and several local improvised explosive devices (IEDs) and AP landmines are modeled as targets. The signatures from such targets buried in the selected soils are predicted, considering different conditions. Finally, we show how the GPR can contribute in detecting Iow- and non-metallic targets in these Colombian soils. Such a system could be quite adequate for complementing humanitarian landmine detection by metal detectors. (c) 2007 Elsevier B.V. All rights reserved.

The objective of this research was to develop methods for simulating the heat transfer process during pasteurization of frankfurter in single layer packages by hot water immersion to inactivate Listeria monocytogenes . A computer simulation program based on finite difference analysis was developed to simulate the temperature distributions within the frankfurter packages. This program was first used to estimate the apparent thermal diffusivity and surface heat transfer coefficients during the heating and cooling of frankfurters. Then this computer program was used to simulate the heat transfer processes during hot water pasteurization of frankfurters. The results of computer simulation indicated that the simulated temperature histories were in close agreement with the experimentally observed curves, both at the center and on the surface of the packages. The computer simulation model was biologically validated with frankfurters surfaced-inoculated with freshly grown L. monocytogenes. The observed bacterial reductions were generally 1–2 logs higher than those calculated by the General Method when the heating time was short. The theoretical calculation was more accurate when the heating time was long and the total inactivation was >5 logs. In general, the result of computer simulation for calculating the log reduction was more conservative than the experimental observations, therefore suitable for designing processes to ensure the safety of products and to reduce the incidence of foodborne listeriosis associated with RTE meats.

Seismic stability, liquefaction, and deformation of earth structures are critical issues in geotechnical earthquake engineering practice. At present, the equivalent linear approach is considered the ‘state of practice’ in common use. More recently, dynamic analyses incorporating nonlinear, effective-stress-based soil models have been used more frequently in engineering applications. This paper describes a bounding surface hypoplasticity model for sand [Wang ZL. Bounding surface hypoplasticity model for granular soils and its applications. PhD Dissertation for the University of California at Davis, U.M.I. Dissertation Information Service, Order No. 9110679; 1990; Wang ZL, Dafalias YF, Shen CK. Bounding surface hypoplasticity model for sand. ASCE, J Eng Mech 1990;116(5):983–1001; Wang ZL, Makdisi FI. Implementing a bounding surface hypoplasticity model for sand into the FLAC program. In: Proceedings of the international symposium on numerical modeling in geomechanics. Minnesota, USA; 1999. p. 483–90] incorporated into a two-dimensional finite difference analysis program [Itasca Consulting Group, Inc. FLAC (Fast Lagrangian Analysis of Continua), Version 4. Minneapolis, MN; 2000] to perform nonlinear, effective-stress analyses of soil structures. The soil properties needed to support such analyses are generally similar to those currently used for equivalent linear and approximate effective-stress analyses. The advantages of using a nonlinear approach are illustrated by comparison with results from the equivalent linear approach for a rockfill dam. The earthquake performance of a waterfront slope and an earth dam were evaluated to demonstrate the model's ability to simulate pore-pressure generation and liquefaction in cohesionless soils.

The study covered by this paper was focused on the historical case of the Calitri landslide, which was repeatedly reactivated by earthquakes, as reported since 1694. The town of Calitri (Southern Italy) is located on a ridge whose southern slope, from its top to the Ofanto river valley floor, has been historically affected by major landsliding. The last record of recurrence of the Calitri landslide leads back to the 1980 Irpinia earthquake, which caused significant damage to the town and had pervasive and visible ground effects. Based on a detailed historical reconstruction of landsliding and seismicity at Calitri, the study analysed the current static and dynamic stability of the landsliding slope by means of a finite-difference numerical analysis, taking into account the various factors of landslide initiation and kinematics. The results of the numerical analysis: (i) were consistent with the roto-translational mechanism observed upon the latest reactivation of the landslide; (ii) demonstrated that excess pore pressure redistribution caused a lag between the seismic trigger and the initiation of landsliding; and (iii) showed the impact of seismic input frequency on propagation and depth of slope instability.

The vulnerability of hydraulic fill dams under strong earthquake shaking has long been recognized. When located in areas of high seismic hazard, seismic upgrading of these types of dams is often required to meet current dam safety standards. Selection of an appropriate design concept for seismic upgrading of such dams requires careful consideration of seismically induced deformations when the hydraulic fill is to remain as part of the dam. This paper presents a case history of the seismic upgrade of Butt Valley Dam, a hydraulic fill dam located in Northern California. The dam was strengthened to withstand the Maximum Credible Earthquake (MCE) by buttressing of its upstream and downstream slopes. The paper discusses the evaluation of alternatives to upgrade the dam, the design criteria, and the design and analysis of the seismic upgrade. It is shown that a conservative and robust design was developed based on well-established engineering principles and multiple lines of defense, and sound use of analysis procedures including finite-difference non-linear dynamic deformation analyses.

A number of high-resolution schemes have been recently developed to solve the homogeneous form of the shallow water equations. However, most approximate Riemann solvers experience difficulties with natural river applications if the irregular bed topography is not handled correctly. Based on the finite-difference flux-limited total variation diminishing (TVD) scheme, this paper develops a simple approach to handle the source terms for the one-dimensional open channel flow simulation with rapidly varying bed topography. Conclusions on the validity of the operator-splitting approach, the eigenvector-projection approach, and the proposed approach are presented. Analytical solution, experimental data, and available numerical result comparisons are shown to demonstrate the accuracy, robustness, stability, simplicity, and applicability of the proposed model.

Themildslopeequationisusedtodescribewavepropagatinginthenearshoreregion.Inthispaper,thefinitedifferencemethod isusedtodiscretizethegoverningellipticequationandthediscretizedlinearequationissolvedusingGPBiCG(m,n)method.Two cases of wave propagating are used to test the model, and reasonable agreements have been achieved. It is shown that the present algorithm: GPBiCG(m,n)withaflexibleanddiverseform,hasafastandrelativelyrobust convergencerate,andcanbeefficiently and economically run in either the linear or nonlinear wave model and easily used to a complicated region. D 2004 Elsevier B.V. All rights reserved.

This paper treats the finite-difference (FD) analysis of passive MMIC components including finite conductivity. Quasi-static field simulations are used to estimate the field and current distribution in miniaturized structures. The results are used to determine so-called correction factors, which are incorporated into the FD equations of the hybrid method's dynamic part. This allows to reduce computational efforts while maintaining accuracy of the conventional FD method. Verification is done by applying the hybrid approach to coplanar transmission-line elements. Results from conventional FD analysis and mode matching simulations serve as a reference.

Recent observations from Soufriere Hills volcano in Montserrat reveal a wide variety of low-frequency seismic signals. We discuss similarities and differences between hybrid earthquakes and long-period events, and their role in explosions and rockfall events. These events occur usually in swarms, and occasionally merge into tremor, an observation that can shed further light on the generation and composition of harmonic tremor. We use a 2D finite difference method to model major features of low-frequency seismic signatures and compare them with the observations. A depth-dependent velocity model for a fluid-filled conduit is introduced which accounts for the varying gas-content in the magma, and the impact on the seismic signals is discussed. We carefully analyse episodes of tremor that show shifting spectral lines and model those in terms of changes in the gas content of the magma as well as in terms of a time-dependent triggering mechanism of low-frequency resonances. In this way we explain the simultaneous occurrence of low-frequency events and tremor with a spectral content comprising integer harmonics.

The two-dimensional equation of a track wall was rewritten in a form suitable for the finite difference method and numerically solved using the MATHEMATICA software. The results obtained were compared with the ones obtained using other computer software for calculation of the track parameters and profile, as well as, with experimental data, and good agreement was found.

Modelling of 2D resistivity imaging was done in order to understand the principle resolution of the technique in different geological situations, and for assessing the behaviour of the interpretation methods under controlled circumstances. The Wenner array was used throughout. The results show that the 1D approximation only provides reasonable results in environments with very gradual lateral resistivity changes, otherwise the result may be strongly misleading. Inversion using the 2D quasi-Newton technique results in adequate resolution of the structures in moderately complex environments, but the Gauss–Newton method holds a significant advantage in some complicated cases. The data density can also be of crucial importance for the resolution capability, notably of narrow structures.

This special issue is concerned with the present state of the art in methods of numerical modelling of geo-electromagnetic fields in inhomogeneous media. A theoretical overview is followed by specific applications of the various modelling methods and computer programs (developed throughout the world) to the geo-electric test models of the international project on the Comparison Of Modelling Methods for ElectroMagnetic Induction problems (COMMEMI). Numerous tables and diagrams provide a comparison of the results obtained by these different approaches. This material is intended for geophysicists dealing with the modelling and interpretation of geo-electromagnetic fields, for scientists involved in the testing of related software, for specialists in the field of computational geophysics, and for graduate and senior undergraduate students studying this branch of geophysics.

Numerical simulation of fluid explusion from geopressured sediments using a one-dimensional finite difference model shows that large volumes of fluid can be transported through a fractured seal during short-lived (

Waveguides with gyrotropic materials are of large interest for the realization of optical isolators and circulators in integrated optical devices. This paper presents a finite difference analysis of such waveguides. The method is applicable to both planar and rib structures. From the calculated results some guidelines for optimizing the non-reciprocal effect are derived. We propose a new geometry for an effective gyrotropic waveguide.

A numerical study has been carried out to investigate the problem of convective heat transfer in a gap formed by vertical concentric cylinders. Numerical results of a finite difference scheme of velocities and temperature distribution are presented here. The study covers the case of stationary cylinders and the influences of the flow direction and the inlet temperature when both walls are at constant temperatures. The results were obtained for axial Reynolds number ( Re a ⩽ 250 and 2.5 × 10 4 ⩽ Gr ⩽ × 10 4 .

The nonlinear Dupuit-Forchheimer theory and two linearized versions are considered as to their applicability for giving accurate results on recharge mound shapes beneath strip and square basins. Results on mound rises from the approximate theories are compared with those from the nonlinear theory to determine criteria for the use of the approximate theories.

A hybrid boundary element—finite-difference program, FFFLOW, has been developed for the modelling of fluid flow in deformable, jointed rock. The model has been used to simulate reservoirs with variable joint orientations, injection rates, joint dilation angles and variable far-field stresses. The formulation and example results are presented.

1. 1. The b- and c-type cytochromes of Tubifex mitochondria were resolved by fourth-order finite difference analysis of difference spectra measured at liquid nitrogen temperature. 2. 2. Tubifex has a higher ratio of cytochromes b and c to cytochrome aa 3 than found in vertebrates. 3. 3. Tubifex mitochondria have three b-type cytochromes ( b 554 , b 557.5 and b 560 ) and two c-type cytochromes ( c and c 1 ) which are similar to cytochromes found in mitochondria from other species. 4. 4. Tubifex mitochondria lack cytochrome b 562.5 which is found in mitochondria from other species.

A set of numerical modelling is carried out for studying the behaviour of the Parkinson arrows near the anomalous conductor-host medium interface. In these models the conductivity contrast between the conducting plate and the host medium is taken to be 5700. Our results indicate that the imaginary Parkinson arrow and the real Parkinson arrow do not change their signs directly over the plate-host interface.

An improved finite difference analysis has been used to calculate self-consistent axisymmetric cellular shapes. The results obtained are compared with the analytical models which have been proposed to describe array growth. It is concluded that the analytical models should be used with extreme caution in the cellular region. A preliminary examination of the stability of the steady state shapes has been made numerically. The results are compared with experiment where it is found that neither the minimum undercooling condition nor marginal stability analysis as applied, correctly predicts the growth conditions. The model includes solute diffusion in the solid so that the extent of microsegregation is fully described for the first time.

A secondary flow model for back-arc spreading is developed in this paper that shows some of the characteristics of observed back-arc spreading. Back-arc spreading has formed marginal seas around the west and southwest rim of the Pacific. The episodic spreading and different directions of opening are not completely understood; however, there does appear to be a limited lifetime (

Finite difference analysis was used to model the three dimensional transient conduction heat transfer inside foods packaged in flexible retort pouches. Tests were conducted to validate the model by monitoring the response of a model conduction heating product which was subjected to a thermal process in a weir type pressurized water retort. The thermal properties of a bentonite paste were measured as a function of temperature to provide data for the simulation test. Heat transfer coefficients were measured within the retort and the values obtained were also used as input to the computer program. The thermal resistance of the layers which lie between the food product and the fluid heating medium was found to be a significant factor affecting model accuracy.

Bound states of an electron in the field of a proton and a magnetic monopole, separated by a distance R , are studied in the non-relativistic Pauli approximation. The wave functions and the energy levels are obtained by a finite difference analysis of the corresponding Schrodinger equation. The agreement of the results with previous calculations is discussed.

Centrifugal filtration through porous matermls, under both saturated as well as unsaturated flow condmons, plays an important role in diversified fields of science and technology, such as soft physics, groundwater hydrology, sludge dewaterlng, pharmacology, and petroleum engmeenng It is of specml interest in certain industrial processes that involve fluid extraction and drying, e g , in the sugar and paper pulp industries To meet the needs to evaluate centrifugal filtraUon in all these cases, studies have been conducted leading to formulas that express relations among the many variables that describe the process Most of these studies have been expenmental ones, leading to empirical relations that describe the filtration capacity In some cases, results have been obtained by using very simple models of the filtration phenomenon Because of the nature of these investigations, each result could be apphed only to a specific case, or to a hmlted range of cases The absence of a umform methodology for the design of centrifuges has forced engineers to use pdot-scale evaluation procedures In what follows, we shall show how the theory of flmd flow through a deformable porous medmm, in the presence ofa centnfugal force, may serve as a basis for the construction of a rather general model of centrifugal filtration

The equilibrium of a grillage is expressed by finite difference equations, assuming that the bars are rigidly interconnected at the joints. Solutions of these equations are given and a rapid method of obtaining linearly independent solutions from known solutions is evolved. A method due to Bessel is adapted to the analysis of problems where the solutions express the loading by sinusoidal functions. Numerical examples of circular and rectangular grillages are given. Further justification of Lightfoot's1 method, for analysing plates by the use of substitute grillages, is found on examining the above solutions as the grillage meshes become infinitely fine.

The stress distributions in two thick circular plates with diameter to thickness ratios of 5 and 7 are obtained from a finite difference analysis and a three-dimensional frozen stress study. The results are compared with those obtained from a rational use of thin-plate theory and also with those given by the British Standard B.S. 1500 Design Code. It is shown that the finite difference study employing a relatively coarse net yields comparative field stress values to those provided by the frozen stress study and that a rational use of thin-plate theory gives a closer approximation to the peak stresses than those predicted by the design code. The assumption of linear bending stress employed in the thin-plate theory, however, is shown to be only approximately true for the plates of practical proportions studied.

A finite difference analysis is presented for five coupled non-linear equations governing the behavior of a partially buried cable. A three-dimensional loading is assumed for the cable and foundation modulus is variable in both the vertical and transverse coordinate directions. Numerical results are presented for a specific problem, indicating the effect of variations in the foundation modulus.

A control volume finite difference analysis has been used to calculate self-consistent cellular growth shapes. The results obtained are compared with the analytical models which have been proposed to describe array growth. It is concluded that the analytical models should be used with extreme caution in the cellular region. A preliminary examination of the stability of the steady state shapes has been made numerically. The results are compared with experiment and it is found that neither the minimum undercooling condition nor marginal stability analysis as applied, correctly predicts the growth conditions.