Numerical manifold method for thermo-mechanical coupling simulation of fractured rock mass

As a calculation method based on the Galerkin variation,the numerical manifold method(NMM)adopts a double covering system,which can easily deal with discontinuous deformation problems and has a high calculation accuracy.Aiming at the thermo-mechanical(TM)coupling problem of fractured rock masses,this study uses the NMM to simulate the processes of crack initiation and propagation in a rock mass under the influence of temperature field,deduces related system equations,and proposes a penalty function method to deal with boundary conditions.Numerical examples are employed to confirm the effectiveness and high accuracy of this method.By the thermal stress analysis of a thick-walled cylinder(TWC),the simulation of cracking in the TWC under heating and cooling conditions,and the simulation of thermal cracking of the SwedishÄspöPillar Stability Experiment(APSE)rock column,the thermal stress,and TM coupling are obtained.The numerical simulation results are in good agreement with the test data and other numerical results,thus verifying the effectiveness of the NMM in dealing with thermal stress and crack propagation problems of fractured rock masses.

Delineating homogeneous domains of fractured rocks using topological manifolds and deep learning

Determining homogeneous domains statistically is helpful for engineering geological modeling and rock mass stability evaluation.In this text,a technique that can integrate lithology,geotechnical and structural information is proposed to delineate homogeneous domains.This technique is then applied to a high and steep slope along a road.First,geological and geotechnical domains were described based on lithology,faults,and shear zones.Next,topological manifolds were used to eliminate the incompatibility between orientations and other parameters(i.e.trace length and roughness)so that the data concerning various properties of each discontinuity can be matched and characterized in the same Euclidean space.Thus,the influence of implicit combined effect in between parameter sequences on the homogeneous domains could be considered.Deep learning technique was employed to quantify abstract features of the characterization images of discontinuity properties,and to assess the similarity of rock mass structures.The results show that the technique can effectively distinguish structural variations and outperform conventional methods.It can handle multisource engineering geological information and multiple discontinuity parameters.This technique can also minimize the interference of human factors and delineate homogeneous domains based on orientations or multi-parameter with arbitrary distributions to satisfy different engineering requirements.

A STABILITY RESULT FOR TRANSLATINGSPACELIKE GRAPHS IN LORENTZ MANIFOLDS

In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piecewise smooth boundary,and ℝ denotes the Euclidean 1-space.We prove an interesting stability result for translating spacelike graphs in M^(n)×ℝ under a conformal transformation.

KAEHLER SUBMANIFOLDS IN A LOCALLY SYMMETRIC BOCHNER-KAEHLER MANIFOLD

In our preceding paper, we gave some Pinching conditions of sectional curvature and holomorphic sectional curvature for a compact Kaehler submanifold Mn in a locally symmetric Bochner-Kaehler manifold （?）n+p to be totally geodesic. As a continuation, this letter gives the Pinching conditions of Ricci curvature and scalar curvature for Mn in （?）n+p to be totally geodesic. The main results are as follows.

KAEHLER SUBMANIFOLDS IN A LOCALLY SYMMETRIC BOCHNER-KAEHLER MANIFOLD

By a locally symmetric Bochner-Kaehler manifold is meant a Kaehler manifold with parallel Riemannian curvature tensor and vanishing Bochner curvature tensor. Complex space forms are special locally symmetric Bochner-Kaehler manifolds.This letter gives some sufficient conditions for a compact Kaehler submanifold in a locally symmetric Bochner-Kaehler manifold to be totally geodesic. Main results are as follows.

KAEHLER SUBMANIFOLDS IN A LOCALLY SYMMETRIC BOCHNER-KAEHLER MANIFOLD

This paper gives some sufficient conditions for a compact Kaehler submanifold Mn in a locally symmetric Bochner-Kaehler manifold n+p to be totally geodesic. The conditions are given by inequalities which are established between. the sectional curvature（resp, holomorphic sectional curvature） of Mn and the Ricci curvature of n+p. In particular, similar results in the case where n+p is a complex projective spathe are contained.

Galerkin-based quasi-smooth manifold element(QSME)method for anisotropic heat conduction problems in composites with complex geometry

The accurate and efficient analysis of anisotropic heat conduction problems in complex composites is crucial for structural design and performance evaluation. Traditional numerical methods, such as the finite element method(FEM), often face a trade-off between calculation accuracy and efficiency. In this paper, we propose a quasi-smooth manifold element(QSME) method to address this challenge, and provide the accurate and efficient analysis of two-dimensional(2D) anisotropic heat conduction problems in composites with complex geometry. The QSME approach achieves high calculation precision by a high-order local approximation that ensures the first-order derivative continuity.The results demonstrate that the QSME method is robust and stable, offering both high accuracy and efficiency in the heat conduction analysis. With the same degrees of freedom(DOFs), the QSME method can achieve at least an order of magnitude higher calculation accuracy than the traditional FEM. Additionally, under the same level of calculation error, the QSME method requires 10 times fewer DOFs than the traditional FEM. The versatility of the proposed QSME method extends beyond anisotropic heat conduction problems in complex composites. The proposed QSME method can also be applied to other problems, including fluid flows, mechanical analyses, and other multi-field coupled problems, providing accurate and efficient numerical simulations.

Isometric Immersions of Lightlike Warped Product Manifolds

In this paper, we deal with isommetric immersions of globally null warped product manifolds into Lorentzian manifolds with constant curvature c in codimension k≥3. Under the assumptions that the globally null warped product manifold has no points with the same constant sectional curvature c as the Lorentzian ambient, we show that such isometric immersion splits into warped product of isometric immersions.

局部对称Bochner-Kaehler流形及其Kaehler子流形

本文给出局部对称的Bochner-Kaehler流形的Riemann结构以及它的Kaehler子流形为全测地子流形的几个Pinching条件,推广了关于复射影空间的Kaehler子流形的相应定理。

局部对称Bochner-Kaehler流形中的全实极小子流形

讨论了局部对称Bochner Kaehler流形中的紧致全实极小子流形 ,并证明了如果第二基本形式长度的平方‖σ‖2 <16 (n + 1) C ,则Mn

局部对称的Bochner-Kaehler流形的全实极小子流形

本文拓广Urbano F.关于复射影空间的全实极小子流形的定理到局部对称的Bochner-Kaehler流形的全实极小子流形。

Bochner-Kaehler子流形的余维数

设 M^n(n≥4)是复 n 维浸入在复 n+p 维 Bochner-Kaehler 流形■^(n+p)的完备 Bochner-Kaehler子流形时,H 是 M^n 在任意点 C∈M^n 的全纯截面曲率,■表示■^(n+p)在同点的全纯截面曲率的下确界.若 H<■,则 M^n 的余维数 p 不小于 n(n+1)/2.证明了在上述特殊情形下,Ogiue 猜想是正确的.

关于局部对称Bochner-Kaehler流形的K-子流形的注记

本文给出了局部对称Bochner-Kaehler流形的Kaehler子流形为全测地子流形的一个充分条件,推广了Koichi Ogiue的一个结果。

Secret Sharing Scheme Based on the Differential Manifold

In this paper, the concepts of topological space and differential manifold are introduced, and it is proved that the surface determined by function F (x2, x2, …, xt) of class Cr in Euelidean Rt is a differential manifold. Using the intersection of the tangent plane and the hypernormal of the differential manifold to construct the shared master key of participants, an intuitive, secure and complete (t,n)-threshold secret sharing scheme is designed. The paper is proved to be safe, and the probability of successful attack of attackers is only 1/pt-1. When the prime number p is sufficiently large, the probability is almost 0. The results show that this scheme has the characteristics of single-parameter representation of the master key in the geometric method, and is more practical and easy to implement than the Blakley threshold secret sharing scheme.

SOME CONVERGENCE PROBLEMS REGARDING THE FRACTIONAL SCHRODINGER PROPAGATOR ON NONCOMPACT MANIFOLDS

Let L be the Laplace-Beltrami operator.On an n-dimensional(n≥2),complete,noncompact Riemannian manifold M,we prove that if 0<α<1,s>α/2 and f∈Hs(M),then the fractional Schr?dinger propagator e(it|L|α/2)(f)(x)→f(x)a.e.as t→0.In addition,for when M is a Lie group,the rate of the convergence is also studied.These results are a non-trivial extension of results on Euclidean spaces and compact manifolds.

Invariant manifold growth formula in cylindrical coordinates and its application for magnetically confined fusion

For three-dimensional vector fields,the governing formula of invariant manifolds grown from a hyperbolic cycle is given in cylindrical coordinates.The initial growth directions depend on the Jacobians of Poincarémap on that cycle,for which an evolution formula is deduced to reveal the relationship among Jacobians of different Poincarésections.The evolution formula also applies to cycles in arbitrary finite n-dimensional autonomous continuous-time dynamical systems.Non-Möbiusian/Möbiusian saddle cycles and a dummy X-cycle are constructed analytically as demonstration.A real-world numeric example of analyzing a magnetic field timeslice on EAST is presented.

Fluid-Structure Coupled Analysis of the Transient Thermal Stress in an Exhaust Manifold

The development of thermal stress in the exhaust manifold of a gasoline engine is considered.The problem is addresses in the frame of a combined approach wherefluid and structure are coupled using the GT-POWER and STAR-CCM+software.First,the external characteristic curve of the engine is compared with a one-dimen-sional simulation model,then the parameters of the model are modified until the curve matches the available experimental values.GT-POWER is then used to transfer the inlet boundary data under transient conditions to STAR-CCM+in real-time.The temperature profiles of the inner and outer walls of the exhaust manifold are obtained in this way,together with the thermal stress and thermal deformation of the exhaust manifold itself.Using this information,the original model is improved through the addition of connections.Moreover,the local branch pipes are optimized,leading to significant improvements in terms of thermal stress and thermal deforma-tion of the exhaust manifold(a 7%reduction in the maximum thermal stress).

A Discussion on the Establishment That a Fibre Metric on the Positive Definite Real Inner-Product of a Properly Embedded Smooth Submanifold Be Always Extended to a Riemannian Metric on the Positive Definite Real Inner-Product

Let M be a smooth manifold and S ⊆ M a properly embedded smooth submanifold. Suppose that we have a fibre metric on TM|s i.e. a positive definite real inner-product on TpM for all p ∈ S, which depends smoothly on p ∈ S. The purpose of this article is to figure out that the fibre metric on TM|s can always be extended to a Riemannian metric on TM from a special perspective.

Experimental Investigation of Regular or Wavy Two-Phase Flow in a Manifold

An experimental study was conducted to investigate the properties of stratified regular or wavy two-phase flow in two parallel separators located after a manifold.A total of 103 experiments with various gas and liquid velocity combinations in three inlet pipes were conducted,including 77 groups of outlet pipe resistance symmetry and 26 groups of outlet pipe resistance asymmetry trials.The experimental results have revealed that when the gas-liquid flow rate is low,the degree of uneven splitting is high,and“extreme”conditions are attained.When the superficial gas velocity is greater than that established in the extreme case,the direction of the liquid-phase displacement is reversed,while that of the gas remains unchanged.Thus,the degree of gas phase bias tends to be mitigated with an increase in the gas velocity,while the uneven splitting degree of liquid approaches 10%.Finally,varying the gas-phase outlet pipe resistance is shown to effectively change the gas-liquid two-phase flow distribution.

Representation of Physical Fields as Einstein Manifold

In this work we investigate the possibility to represent physical fields as Einstein manifold. Based on the Einstein field equations in general relativity, we establish a general formulation for determining the metric tensor of the Einstein manifold that represents a physical field in terms of the energy-momentum tensor that characterises the physical field. As illustrations, we first apply the general formulation to represent the perfect fluid as Einstein manifold. However, from the established relation between the metric tensor and the energy-momentum tensor, we show that if the trace of the energy-momentum tensor associated with a physical field is equal to zero then the corresponding physical field cannot be represented as an Einstein manifold. This situation applies to the electromagnetic field since the trace of the energy-momentum of the electromagnetic field vanishes. Nevertheless, we show that a system that consists of the electromagnetic field and non-interacting charged particles can be represented as an Einstein manifold since the trace of the corresponding energy-momentum of the system no longer vanishes. As a further investigation, we show that it is also possible to represent physical fields as maximally symmetric spaces of constant scalar curvature.