Extended finite element-based cohesive zone method for modeling simultaneous hydraulic fracture height growth in layered reservoirs

In this study,a fully coupled hydromechanical model within the extended finite element method(XFEM)-based cohesive zone method(CZM)is employed to investigate the simultaneous height growth behavior of multi-cluster hydraulic fractures in layered porous reservoirs with modulus contrast.The coupled hydromechanical model is first verified against an analytical solution and a laboratory experiment.Then,the fracture geometry(e.g.height,aperture,and area)and fluid pressure evolutions of multiple hydraulic fractures placed in a porous reservoir interbedded with alternating stiff and soft layers are investigated using the model.The stress and pore pressure distributions within the layered reservoir during fluid injection are also presented.The simulation results reveal that stress umbrellas are easily to form among multiple hydraulic fractures’tips when propagating in soft layers,which impedes the simultaneous height growth.It is also observed that the impediment effect of soft layer is much more significant in the fractures suppressed by the preferential growth of adjoining fractures.After that,the combined effect of in situ stress ratio and fracturing spacing on the multi-fracture height growth is presented,and the results elucidate the influence of in situ stress ratio on the height growth behavior depending on the fracture spacing.Finally,it is found that the inclusion of soft layers changes the aperture distribution of outmost and interior hydraulic fractures.The results obtained from this study may provide some insights on the understanding of hydraulic fracture height containment observed in filed.

AN ANZAHL THEOREM IN SINGULAR PSEUDO-SYMPLECTIC GEOMETRY OVER FINITE FIELDS

This paper obtains formulas for the number of subspaces of a given type contained in a given subspace in singular pseudo-symplectic geometry.

Viscoacoustic prestack reverse time migration based onthe optimal time-space domain high-order finite-difference method

Prestack reverse time migration （RTM） is an accurate imaging method ofsubsurface media. The viscoacoustic prestack RTM is of practical significance because itconsiders the viscosity of the subsurface media. One of the steps of RTM is solving thewave equation and extrapolating the wave field forward and backward; therefore, solvingaccurately and efficiently the wave equation affects the imaging results and the efficiencyof RTM. In this study, we use the optimal time-space domain dispersion high-order finite-difference （FD） method to solve the viscoacoustic wave equation. Dispersion analysis andnumerical simulations show that the optimal time-space domain FD method is more accurateand suppresses the numerical dispersion. We use hybrid absorbing boundary conditions tohandle the boundary reflection. We also use source-normalized cross-correlation imagingconditions for migration and apply Laplace filtering to remove the low-frequency noise.Numerical modeling suggests that the viscoacoustic wave equation RTM has higher imagingresolution than the acoustic wave equation RTM when the viscosity of the subsurface isconsidered. In addition, for the wave field extrapolation, we use the adaptive variable-lengthFD operator to calculate the spatial derivatives and improve the computational efficiencywithout compromising the accuracy of the numerical solution.

Channel Coding at Finite Blocklength and Its Application in 6G URLLC

To support mission-critical applications, such as factory automation and autonomous driving, the ultra-reliable low latency communications (URLLC) is adopted in the fifth generation (5G) mobile communications network, which requires high level of reliability and low latency. Naturally, URLLC in the future 6G is expected to have a better capability than its 5G version which poses an unprecedented challenge to us. Fortunately, the potential solution can still be found in the well-known classical Shannon information theory. Since the latency constraint can be represented equivalently by blocklength, channel coding at finite blocklength plays an important role in the theoretic analysis of URLLC. Applying these achievements in rapidly development of massive MIMO techniques gives rise to a new theory on space time exchanging. It tells us that channel coding can also be performed in space domain, since it is capable of providing the same coding rate as that in time domain. This space time exchanging theory points out an exciting and feasible direction for us to further reduce latency in 6G URLLC. .

New Pooling Design Constructed with Pseudo-symplectic Space

Pooling design is a mathematical tool in many application areas. In this paper, we give a new construction of pooling design with subspaces of the pseudo-symplectic space and discuss its properties. We define the design parameters of a d^2-disjunct matrix. Then we discuss the change law of the design parameters in our construction along with their variables.

Finite Element Orthogonal Collocation Approach for Time Fractional Telegraph Equation with Mamadu-Njoseh Polynomials

Finite element method (FEM) is an efficient numerical tool for the solution of partial differential equations (PDEs). It is one of the most general methods when compared to other numerical techniques. PDEs posed in a variational form over a given space, say a Hilbert space, are better numerically handled with the FEM. The FEM algorithm is used in various applications which includes fluid flow, heat transfer, acoustics, structural mechanics and dynamics, electric and magnetic field, etc. Thus, in this paper, the Finite Element Orthogonal Collocation Approach (FEOCA) is established for the approximate solution of Time Fractional Telegraph Equation (TFTE) with Mamadu-Njoseh polynomials as grid points corresponding to new basis functions constructed in the finite element space. The FEOCA is an elegant mixture of the Finite Element Method (FEM) and the Orthogonal Collocation Method (OCM). Two numerical examples are experimented on to verify the accuracy and rate of convergence of the method as compared with the theoretical results, and other methods in literature.

THE SPACE-TIME FINITE ELEMENT METHOD FOR PARABOLIC PROBLEMS

Adaptive space-time finite element method, continuous in space but discontinuous in time for semi-linear parabolic problems is discussed. The approach is based on a combination of finite element and finite difference techniques. The existence and uniqueness of the weak solution are proved without any assumptions on choice of the spacetime meshes. Basic error estimates in L-infinity (L-2) norm, that is maximum-norm in time, L-2-norm in space are obtained. The numerical results are given in the last part and the analysis between theoretic and experimental results are obtained.

OUTER OPERATORS FOR THE NONCOMMUTATIVE SYMMETRIC HARDY SPACES ASSOCIATED WITH FINITE SUBDIAGONAL ALGEBRA

In this article, we extended main results on outer operators of [6] to the symmetric Hardy spaces, when associated subdiagonal algebra is finite.

Weak KKM theorems in generalized finitely continuous space with applications

Some new weak Knaster-Kuratouski-Mazurkiewicz （KKM） theorems are proved under the noncompact situation in the generalized finitely continuous space （GFC-space） without any convexity. As applications, the minimax inequalities of the Ky Fan type are also given under some suitable conditions. The results unify and generalize some known results in recent literatures.

Mixed time discontinuous space-time finite element method for convection diffusion equations

A mixed time discontinuous space-time finite element scheme for secondorder convection diffusion problems is constructed and analyzed. Order of the equation is lowered by the mixed finite element method. The low order equation is discretized with a space-time finite element method, continuous in space but discontinuous in time. Stability, existence, uniqueness and convergence of the approximate solutions are proved. Numerical results are presented to illustrate efficiency of the proposed method.

Almost Injective Mappings of Totally Bounded Metric Spaces into Finite Dimensional Euclidean Spaces

Let χ= be a metric space and let ε be a positive real number. Then a function f: X→Y is defined to be an ε-map if and only if for all y∈Y, the diameter of f-1(y)?is at most ε. In Theorem 10 we will give a new proof for the following well known fact: if χ is totally bounded, then for all ε there exists a finite number n and a continuous ε-map fε: X→Rn (here Rn is the usual n-dimensional Euclidean space endowed with the Euclidean metric). If ε is “small”, then fε is “almost injective”;and still exists even if χ has infinite covering dimension (in this case, n depends on ε, of course). Contrary to the known proofs, our proof technique is effective in the sense, that it allows establishing estimations for n in terms of ε and structural properties of χ.

COARSE ISOMETRIES BETWEEN FINITE DIMENSIONAL BANACH SPACES

Assume that X and Y are real Banach spaces with the same finite dimension.In this paper we show that if a standard coarse isometry f:X→Y satisfies an integral convergence condition or weak stability on a basis,then there exists a surjective linear isometry U:X→Y such that∥f(x)−Ux∥=o(∥x∥)as∥x∥→∞.This is a generalization about the result of Lindenstrauss and Szankowski on the same finite dimensional Banach spaces without the assumption of surjectivity.As a consequence,we also obtain a stability result forε-isometries which was established by Dilworth.

A Finite Volume Unstructured Mesh Method for Fractional-in-space Allen-Cahn Equation

Fractional-in-space Allen-Cahn equation containing a very strong nonlinear source term and small perturbation shows metastability and a quartic double well potential.Using a finite volume unstructured triangular mesh method, the present paper solves the twodimensional fractional-in-space Allen-Cahn equation with homogeneous Neumann boundary condition on different irregular domains. The efficiency of the method is presented through numerical computation of the two-dimensional fractional-in-space Allen-Cahn equation on different domains.

Finite Element Method for a Kind of Two-Dimensional Space-Fractional Diffusion Equation with Its Implementation

In this article, we consider a two-dimensional symmetric space-fractional diffusion equation in which the space fractional derivatives are defined in Riesz potential sense. The well-posed feature is guaranteed by energy inequality. To solve the diffusion equation, a fully discrete form is established by employing Crank-Nicolson technique in time and Galerkin finite element method in space. The stability and convergence are proved and the stiffness matrix is given analytically. Three numerical examples are given to confirm our theoretical analysis in which we find that even with the same initial condition, the classical and fractional diffusion equations perform differently but tend to be uniform diffusion at last.

Construction of PBIB Desings by Using Subspace of Vector Space over Finite Fields

A new transitivity theorem of the general linear group GLn(Fq)is proved. A kind of new PBIB desings is constructed.

SPACE-TIME FINITE ELEMENT METHOD FOR SCHRDINGER EQUATION AND ITS CONSERVATION

Energy conservation of nonlinear Schrodinger ordinary differential equation was proved through using continuous finite element methods of ordinary differential equation; Energy integration conservation was proved through using space-time continuous fully discrete finite element methods and the electron nearly conservation with higher order error was obtained through using time discontinuous only space continuous finite element methods of nonlinear Schrodinger partial equation. The numerical results are in accordance with the theory.

Amplitude-squared squeezing of the generalized odd-even coherent states of the anharmonic oscillator in a finite-dimensional Hilbert space

This paper discusses the properties of amplitude-squared squeezing of the generalized odd-even coherent states of anharmonic oscillator in finite-dimensional Hilbert space. It demonstrates that the generalized odd coherent states do exhibit strong amplitude-squared squeezing effects in comparison with the generalized even coherent states.

Modeling of Surface Waves in a Fluid Saturated Poro-Elastic Medium under Initial Stress Using Time-Space Domain Higher Order Finite Difference Method

In this present context, mathematical modeling of the propagation of surface waves in a fluid saturated poro-elastic medium under the influence of initial stress has been considered using time dependent higher order finite difference method (FDM). We have proved that the accuracy of this finite-difference scheme is 2M when we use 2nd order time domain finite-difference and 2M-th order space domain finite-difference. It also has been shown that the dispersion curves of Love waves are less dispersed for higher order FDM than of lower order FDM. The effect of initial stress, porosity and anisotropy of the layer in the propagation of Love waves has been studied here. The numerical results have been shown graphically. As a particular case, the phase velocity in a non porous elastic solid layer derived in this paper is in perfect agreement with that of Liu et al. (2009).

On m-ovoids of finite classical polar spaces with an irreducible transitive automorphism group

In this paper, we classify the m-ovoids of finite classical polar spaces that admit a transitive automorphism group acting irreducibly on the ambient vector space. In particular, we obtain several new infinite families of transitive m-ovoids.

A locking-free anisotropic nonconforming rectangular finite element approximation for the planar elasticity problem

This paper deals with a new nonconforming anisotropic rectangular finite element approximation for the planar elasticity problem with pure displacement boundary condition. By use of the special properties of this element, and by introducing the complementary space and a series of novel techniques, the optimal error estimates of the energy norm and the L^2-norm are obtained. The restrictions of regularity assumption and quasi-uniform assumption or the inverse assumption on the meshes required in the conventional finite element methods analysis are to be got rid of and the applicable scope of the nonconforming finite elements is extended.