Three-dimensional parabolic equation model for seismo-acoustic propagation: Theoretical development and preliminary numerical implementation

A three-dimensional（3D） parabolic equation（PE） model for sound propagation in a seismo-acoustic waveguide is developed in Cartesian coordinates, with x, y, and z representing the marching direction, the longitudinal direction, and the depth direction, respectively. Two sets of 3D PEs for horizontally homogenous media are derived by rewriting the 3D elastic motion equations and simultaneously choosing proper dependent variables. The numerical scheme is for now restricted to the y-independent bathymetry. Accuracy of the numerical scheme is validated, and its azimuthal limitation is analyzed. In addition, effects of horizontal refraction in a wedge-shaped waveguide and another waveguide with a polyline bottom are illustrated. Great efforts should be made in future to provide this model with the ability to handle arbitrarily irregular fluid-elastic interfaces.

Alternating Direction Finite Volume Element Methods for Three-Dimensional Parabolic Equations

This paper presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes, one is analogous to Douglas finite difference scheme with second-order splitting error, the other two schemes have third-order splitting error, and the last one is an extended LOD scheme. The L2 norm and H1 semi-norm error estimates are obtained for the first scheme and second one, respectively. Finally, two numerical examples are provided to illustrate the efficiency and accuracy of the methods.

CHEBYSHEV PSEUDOSPECTRAL-HYBRID FINITE ELEMENT METHOD FOR THREE-DIMENSIONAL VORTICITY EQUATION

In this paper,Chebyshev pseudospectral-finite element schemes are proposed for solving three dimensional vorticity equation.Some approximation results in nonisotropic Sobolev spaces are given.The generalized stability and the convergence are proved strictly.The numerical results show the advantages of this method.The technique in this paper is also applicable to other three-dimensional nonlinear problems in fluid dynamics.

FOURIER-CHEBYSHEV PSEUDOSPECTRAL METHOD FOR THREE-DIMENSIONAL VORTICITY EQUATION WITH UNILATERALLY PERIODIC BOUNDARY CONDITION

A Fourier-Chebyshev pseudospectral scheme is proposed for three-dimensionalvorticily equation with unilaterally periodic boundary condition. The generalized stability and convergence are analysed. The numerical results are presented.

INTERFACIAL CRACK ANALYSIS IN THREE-DIMENSIONAL TRANSVERSELY ISOTROPIC BI-MATERIALS BY BOUNDARY INTEGRAL EQUATION METHOD

The integral-differential equations for three-dimensional planar interfacial cracks of arbitrary shape in transversely isotropic bimaterials were derived by virtue of the Somigliana identity and the fundamental solutions, in which the displacement discontinuities across the crack faces are the unknowns to be determined. The interface is parallel to both the planes of isotropy. The singular behaviors of displacement and stress near the crack border were analyzed and the stress singularity indexes were obtained by integral equation method. The stress intensity factors were expressed in terms of the displacement discontinuities. In the non-oscillatory case, the hyper-singular boundary integral-differential equations were reduced to hyper-singular boundary integral equations similar to those of homogeneously isotropic materials.

OpenMP-Based PCG Solver for Three-Dimensional Heat Equation

As one of the most important mathematics-physics equations, heat equation has been widely used in engineering area and computing science research. Large-scale heat problems are difficult to solve due to computational intractability. The parallelization of heat equation is available to improve the simulation model efficiency. In order to solve the three-dimensional heat problems more rapidly, the OpenMP was adopted to parallelize the preconditioned conjugate gradient （PCG） algorithm in this paper. A numerical experiment on the three-dimensional heat equation model was carried out on a computer with four cores. Based on the test results, it is found that the execution time of the original serial PCG program is about 1.71 to 2.81 times of the parallel PCG program executed with different number of threads. The experiment results also demonstrate the available performance of the parallel PCG algorithm based on OpenMP in terms of solution quality and computational performance.

Numerical Solutions of Three-Dimensional Coupled Burgers’ Equations by Using Some Numerical Methods

In this paper, we found the numerical solution of three-dimensional coupled Burgers’ Equations by using more efficient methods: Laplace Adomian decomposition method, Laplace transform homotopy perturbation method, variational iteration method, variational iteration decomposition method and variational iteration homotopy perturbation method. Example is examined to validate the efficiency and accuracy of these methods and they reduce the size of computation without the restrictive assumption to handle nonlinear terms and it gives the solutions rapidly.

A New Fourth Order Difference Approximation for the Solution of Three-dimensional Non-linear Biharmonic Equations Using Coupled Approach

This paper deals with a new higher order compact difference scheme, which is, O(h4) using coupled approach on the 19-point 3D stencil for the solution of three dimensional nonlinear biharmonic equations. At each internal grid point, the solution u(x,y,z) and its Laplacian Δ4u are obtained. The resulting stencil algo-rithm is presented and hence this new algorithm can be easily incorporated to solve many problems. The present discretization allows us to use the Dirichlet boundary conditions only and there is no need to discretize the derivative boundary conditions near the boundary. We also show that special treatment is required to handle the boundary conditions. Convergence analysis for a model problem is briefly discussed. The method is tested on three problems and compares very favourably with the corresponding second order approximation which we also discuss using coupled approach.

A NEW HIGH-ORDER ACCURACY EXPLICIT DIFFERENCE SCHEME FOR SOLVING THREE-DIMENSIONAL PARABOLIC EQUATIONS

In this paper, a new three-level explicit difference scheme with high-order accuracy is proposed for solving three-dimensional parabolic equations. The stability condition is r = Delta t/Delta x(2) = Delta t/Delta gamma(2) = Delta t/Delta z(2) less than or equal to 1/4, and the truncation error is O(Delta t(2) + Delta x(4)).

Conical Sonic-Supersonic Solutions for the 3-D Steady Full Euler Equations

This paper concerns the sonic-supersonic structures of the transonic crossflow generated by the steady supersonic flow past an infinite cone of arbitrary cross section.Under the conical assumption,the three-dimensional(3-D)steady Euler equations can be projected onto the unit sphere and the state of fluid can be characterized by the polar and azimuthal angles.Given a segment smooth curve as a conical-sonic line in the polar-azimuthal angle plane,we construct a classical conical-supersonic solution near the curve under some reasonable assumptions.To overcome the difficulty caused by the parabolic degeneracy,we apply the characteristic decomposition technique to transform the Euler equations into a new degenerate hyperbolic system in a partial hodograph plane.The singular terms are isolated from the highly nonlinear complicated system and then can be handled successfully.We establish a smooth local solution to the new system in a suitable weighted metric space and then express the solution in terms of the original variables.

Three-dimensional nonlinear analysis of creep in concrete filled steel tube columns

This paper proposes a based on 3D-VLE （three-dimensional nonlinear viscoelastic theory） three-parameters viscoelastic model for studying the time-dependent behaviour of concrete filled steel tube （CFT） columns. The method of 3D-VLE was developed to analyze the effects of concrete creep behavior on CFT structures. After the evaluation of the parameters in the proposed creep model, experimental measurements of two prestressed reinforced concrete beams were used to investigate the creep phenomenon of three CFT columns under long-term axial and eccentric load was investigated. The experimentally obtained time-dependent creep behaviour accorded well with the cu~＇es obtained from the proposed method. Many factors （such as ratio of long-term load to strength, slenderness ratio, steel ratio, and eccentricity ratio） were considered to obtain the regularity of influence of concrete creep on CFT structures. The analytical results can be consulted in the engineering practice and design.

A Creep Equation Based on Thermal Activation

The creep equation proposed in so called θ projection concept is developed in the concept of thermal activation of creep. The measured creep curves of A286 alloy are fitted by the equation. The activation energy in theequation is evaluated, and creep rupture lives are predicted. The results are confirmed by creep tests of up to 13years.

Steady State Creep Rate Equation of Inconel 718 Superalloy

The steady state creep rate equdtion of a nickel base superalloy Inconel 718, strengthened by coherent ordered disc-shaped bct γ^(11) phase and coherent spherical fcc γ~1 phase precipitates, has been established in the stress and temperature ranges of 620-840 MN m^(-2) and 853-943K, respecti- vely. Constant stress tensile creep tests were used to medsure the values of steady state creep rate, ε_s, and the consecutive stress reduction method was used to measure the back stress during creep deformation. The values of effective stress exponent, n_e, were detemined from the slopes of the lgε_s vs. lg(σ_a-σ_0)/G plots. The effect of grain size, d, on steady state creep rdte has been also studied in this investigation, and the grain size sensitive exponents m were detemined from the slopes of lgε_s vs. lg(b/d) plots. The creep rate equations of Inconel 718, in the above stress and temperature ranges, have been proposed to be ε_s=1.6×10^(-5)(D_1Gb/KT) (b/d )^(0.19)[(σ_a-σ_0)/G]^(1.35) in diffusional creep region, and ε_s =75(D_1Gb/KT) (b/d)^(-0.42)[(σ_a-σ_0)/G]^(5.5) in dislocation power law creep region.

APPLICABILITY OF CREEP CONSTITUTIVE EQUATIONS TOCREEP CURVES UNDER CONSTANT LOAD

Applicability of the following creep constitutive equations was investigated for normal type creep curves of Ni-18.5Cr alloy and tertiary creep dominant curves of Ni-18.5Cr16W alloy under constant load: the θ projection method describing a creep curse by a sum of two exponential terms, modified θ method describing a primary creep stage by an exponential term and a tertiary creep stage by a logarithmic term, modified Ω method describing a creep curve by a sum of two logarithmic term, 2θ method with only a tertiary creep component and Ω method. The θ, modified θ and modified Ω methods can describe normal type and tertiary creep dominant curves. Tertiary creep dominant curves of Ni-18.5Cr-16W alloy at 900℃ are also described using 2θ and Ω methods. Applicability of the modified θ and modified Ω methods is superior for constant load creep curves because they can predict creep curves up to rupture and rupture life accurately and conservatively.

Creep properties and a creep equation of delay outburst coal and its adjacent mudstone

The study of the creep properties of coal and its adjacent mudstone is very important for understanding the mechanism of delay outburst coal. The samples of delay outburst coal and its adjacent mudstone collected from Yongshanqiao mine were used to carry out triaxial creep tests. The influence of confining pressure and axial compression on the creep test was analyzed. An accelerated creep model was constructed in parallel with a nonlinear viscous component and plastic component. It is connected with the traditional Burges creep model in series. A creep model which can describe the nonlinear viscoelastic-plastic creep model of rock was established and the corresponding creep equation was derived.According to the results of the creep test, the related parameters of the equation were fitted. The results show that, under the same confining pressure, instantaneous creep strain, creep strain of deceleration phase and constant rate creep of the coal and its adjacent mudstone are increased with an increase in the deviatoric stress. But at the same axial pressure, all of the above decrease with an increase of confining pressure. The duration time of the deceleration creep phase increases with the increase in the deviatoric stress. The theoretical values of the creep equation are in good agreement with the experimental results. It indicates that the creep properties of the delayed outburst coal and its adjacent mudstone can be well described by the creep model established in this paper.

Numerical simulation of standing wave with 3D predictor-corrector finite difference method for potential flow equations

A three-dimensional （3D） predictor-corrector finite difference method for standing wave is developed. It is applied to solve the 3D nonlinear potential flow equa- tions with a free surface. The 3D irregular tank is mapped onto a fixed cubic tank through the proper coordinate transform schemes. The cubic tank is distributed by the staggered meshgrid, and the staggered meshgrid is used to denote the variables of the flow field. The predictor-corrector finite difference method is given to develop the difference equa- tions of the dynamic boundary equation and kinematic boundary equation. Experimental results show that, using the finite difference method of the predictor-corrector scheme, the numerical solutions agree well with the published results. The wave profiles of the standing wave with different amplitudes and wave lengths are studied. The numerical solutions are also analyzed and presented graphically.

A CLASS OF TWO-LEVEL EXPLICIT DIFFERENCE SCHEMES FOR SOLVING THREE DIMENSIONAL HEAT CONDUCTION EQUATION

A class of two-level explicit difference schemes are presented for solving three-dimensional heat conduction equation. When the order of truncation error is 0(Deltat + (Deltax)(2)), the stability condition is mesh ratio r = Deltat/(Deltax)(2) = Deltat/(Deltay)(2) = Deltat/(Deltaz)(2) less than or equal to 1/2, which is better than that of all the other explicit difference schemes. And when the order of truncation error is 0((Deltat)(2) + (Deltax)(4)), the stability condition is r less than or equal to 1/6, which contains the known results.

A STUDY ON STRESS INTENSITY FACTORS AND SINGULAR STRESS FIELDS OF THREE-DIMENSIONAL INTERFACE CRACK

By using the finite-part integral concepts and limit technique,the hypersingular inte- grodifferential equations ofthree-dimensional(3D)planar interface crack were obtained; then thedominant-part analysis of 2D hypersingular integral was further usedto investigate the stress fields near the crack front theoretically,and the accurate formulae were obtained for the singular stressfields and the complex stress intensity factors.

STUDY OF THE NONLINEAR THREE-DIMENSIONAL DEBYE SCREENING IN PLASMAS

The nonlinear three-dimensional Debye screening in plasmas is investigated. A new kind of analytical equation, which is in agreement with the three-dimensional Poisson equation for the nonlinear Debye potential, is obtained. It is shown that the symmetry distribution of the Debye screening in plasmas can be described by the equation.

Three-Dimensional Numerical Model for Tsunami Propagation Passing Through An Obstacle

A three-dimensional numerical tsunami model is developed to analyze the nonlinear behavior of flow around obstacles with the Marker and Cell （MAC） method based on the Navier-Stokes equations. Tnrough a comparison with experimental data for the cases of dam break and solitary wave propagation, verification of the three-dimensional numerical model is given. Numerical experiment is performed for the analysis of the nonlinear behavior of flow around obstacles and compared with experimental data. The velocity and pressure around obstacles are presented with sufficient accuracy for tstmami propagation passing through an obstacle.