应用数学和力学

Edition-in-Chief Comment

ZHONG Wan-xie

2016, 37(1): .

Abstract(912) PDF(849)

Abstract:

The Constrained Hamilton Variational Principle for Shallow Water Problems and the Zu-Type Symplectic Algorithm

WU Feng, ZHONG Wan-xie

2016, 37(1): 1-13. doi: 10.3879/j.issn.1000-0887.2016.01.001

Abstract(1166) PDF(678)

Abstract:
The shallow water problems were addressed. With the incompressible condition as the constraint, a constrained Hamilton variational principle was proposed for the shallow water problems. Based on the constrained Hamilton variational principle, the corresponding shallow water equations based on the displacement and pressure (SWE-DP) were developed. A hybrid numerical method combining the finite element method for the spatial discretization and the Zu-type symplectic method for the time integration was proposed to solve the SWE-DP. The correctness of the proposed SWE-DP is verified through the numerical comparisons of the present results with those from 2 sets of existing shallow water equations. The feasibility of the hybrid numerical method proposed for the SWE-DP is also proved through the numerical experiments. Moreover, the numerical experiments demonstrate the excellent performance of the Zu-type method for the simulation of the long time evolution of the shallow water motion.

Study of Contact Angle Hysteresis at Moving Contact Lines Based on CFD Simulation and Mechanical Analysis

JIAO Yun-long, LIU Xiao-jun, PANG Ming-hua, LIU Kun

2016, 37(1): 14-26. doi: 10.3879/j.issn.1000-0887.2016.01.002

Abstract(1391) PDF(1106)

Abstract:
Contact angle hysteresis (CAH) is one of the significant phenomena in the process of droplet spreading on solid surfaces. It plays an important role in mechanical lubrication and other industrial applications. CAH on both smooth and rough surfaces was studied through parametric analysis based on numerical simulation and ISO 25178. The changes of kinetic energy of molecules, liquid pressure and flow velocity near the moving contact line (MCL) related to the mechanism of CAH were investigated with the Flow-3D software. In addition, quantitative tests of the relationship between surface roughness and CAH were conducted with several key parameters according to ISO 25178. The results show that the contact angle is not usually equal to the Young’s contact angle because of the influence of surface roughness; the dynamic balance between sliding force (caused by Laplace pressure and droplet gravity) and viscous resistance (caused by surface roughness) at the MCL is the main reason for the occurrence of CAH. Moreover, CAH is closely related to surface topography, the CAH results on various substrates with different micro-texture types show great differences. The present study enhances the understanding of the CAH mechanism from a micro-perspective and provides some guidance for the practical manufacture of surfaces with better lubrication property and wettability.

Simulation of 6-DOF Rigid Bodies Moving in Supersonic Flow

LI Tao, SUI Jing-xia, WU Chui-jie

2016, 37(1): 27-47. doi: 10.3879/j.issn.1000-0887.2016.01.003

Abstract(989) PDF(526)

Abstract:
Simulation of bodies moving in fluid has very broad application areas. A method for solving unsteady compressible supersonic flow with freely moving rigid bodies of 6 degrees of freedom was presented. The fluid solver dealt with the large-eddy simulation turbulence model, which was a stretched vortex subgrid model in the current work. The WENO scheme was used in the discontinuous flow regions (the shock waves and the contact surfaces) and the tuned center difference scheme was applied in the smooth flow regions. An optimal 3rd-order strong-stability preserving Runge-Kutta scheme was used for the time integration. The model for the rigid bodies was of 6 degrees of freedom and its orientation was tracked with a quaternion. Several numerical examples were presented to verify the correctness and accuracy of the solvers and the results were satisfactory.

Analysis on Deformation Properties of Marine Risers Under Fluid Lift Forces

FENG Yu-qin, AI Zhi-jiu, AI Yu

2016, 37(1): 48-59. doi: 10.3879/j.issn.1000-0887.2016.01.004

Abstract(939) PDF(545)

Abstract:
In order to analyze the riser deformation in complex sea conditions, the energy method and variational principle were adopted to establish the mechanical behavior model, and numerical analysis was made with the central difference quotient. The 3D deformation diagrams of the marine risers were obtained, and the main factors influencing the marine riser deformation were investigated by numerical tests. The results show that the fluid lift force partly affects the riser deformation. The floating drilling platform offset has a significant effect on the riser deformation, i.e. the riser deformation increases dramatically with the offset. The installation of buoyancy blocks improves the stress condition of the marine riser, although that partly increases the riser deformation. Furthermore, the riser deformation decreases along with the top tension and drilling fluid density to some extent.

Uncertain Inversion of Crack Parameters for Plates Based on the SmXFEM

ZHANG Li-xuan, QING Hong-jun, HU De-an

2016, 37(1): 60-72. doi: 10.3879/j.issn.1000-0887.2016.01.005

Abstract(1068) PDF(562)

Abstract:
The crack parameters of positions and sizes are very important information for engineering monitoring. The smoothed extended finite element method (SmXFEM) was an effective method developed for the simulation of crack problems in recent years. The SmXFEM works well without high demand on the element quality, and gives accurate simulation results even with extremely irregular elements. The great advantages of the SmXFEM make it very suitable for automatic mesh generation of crack models in the real time calculation of crack inversion. An approach of uncertain inversion based on the SmXFEM was proposed to indentify the positions and sizes of straight cracks in elastic plane plates. In this approach, the SmXFEM, used to solve the forward problem of the crack model under tension, was called repeatedly by the genetic algorithm. Then an optimization model was established through measurement of the displacements of selected key nodes at the edge of the plate. Finally, with the elastic modulus and Poisson’s ratio as uncertain interval variables, the 1storder Taylor formula was used for the identification of crack parameters in the plates. The results show the correctness and applicability of the present method.

Simulation of Multi-Hydrofracture Horizontal Wells in Shale Based on the Extended Finite Element Method

CHEN Jun-bin, WEI Bo, XIE Qing, WANG Han-qing, LI Tao-tao, WANG Hao

2016, 37(1): 73-83. doi: 10.3879/j.issn.1000-0887.2016.01.006

Abstract(1429) PDF(705)

Abstract:
The cluster spacing optimization of segmented multi-cluster hydrofracture of horizontal wells makes a key point for the hydrofracturing technology in shale reservoir. The fluid-solid coupling mathematical model for the horizontal well hydrofracture was established. Based on the extended finite element method, the propagation process of multiple cracks was simulated. The turning law of simultaneously propagating multiple cracks, as well as the relationships between the stress interference, horizontal principal stress difference, fracture spacing and the crack turning angle, were studied. The results show that the stress interference has restrictive effects on the crack width, and the opening width of 1 single crack is 1.3 times that of 2 concomitant cracks. The crack turning angle increases with the decrease of the stress difference and the lengthening of the fracturing time. The smaller the cluster spacing is, the stronger the stress interference is and the greater the crack turning angle is. For the sake of uniform propagation of the primary fracture, easy packing of the proppant and effective formation of the complex crack network, the optimal cluster spacing is determined as 30 m to 40 m. In the case of the multiple simultaneously propagating cracks, the middle cracks are restricted by those on both sides. The smaller the cluster spacing is, the stronger the restriction is, which results in a longer time of crack development and a lower propagation rate.

Analytical Stiffness Matrixes for Biot Consolidation of Multilayered Viscoelastic Foundations in the Cartesian Coordinate System

KOU Lei, BAI Yun

2016, 37(1): 84-96. doi: 10.3879/j.issn.1000-0887.2016.01.007

Abstract(942) PDF(542)

Abstract:
Based on the basic viscoelastic equations of Biot consolidation in the Cartesian coordinate system and in view of the viscoelasticity of soft soil skeleton, the analytical solutions to 3D problems and plane strain problems of Biot consolidation in the integral transform domain were obtained through the FourierLaplace transform and the decoupling transform according to the differential equation theory and the matrix theory, and in turn the corresponding element stiffness matrixes were derived. The global stiffness matrixes for Biot consolidation 3D problems and plane strain problems of multilayered viscoelastic foundations were assembled with the matrix matching method, and the solutions to the corresponding problems of multilayered viscoelastic foundations in the transform domain were obtained in the solution of the algebraic equations for the global stiffness matrixes. The solutions in the physical domain were acquired through the inverse FourierLaplace transform. The validity of the proposed method was examined in the comparison of the present results of 2 examples, where viscoelastic Biot consolidation was reduced to elastic Biot consolidation, with the previous reference solutions. The analytical stiffness matrixes provide a theoretical base for Biot consolidation of multilayered viscoelastic foundations.

Effects of the Number of Representative Points on the Analysis Accuracy of the Probability Density Evolution Method

MEI Zhen, GUO Zi-xiong, HUANG Qun-xian, LIU Yang

2016, 37(1): 97-106. doi: 10.3879/j.issn.1000-0887.2016.01.008

Abstract(1123) PDF(500)

Abstract:
3 sets of random ground motions with different-size samples were generated based on a physical stochastic ground motion model and a strategy of selecting points via sphere of contact. Then the stochastic response analysis of a model structure subjected to the generated ground motions was carried out with the probability density evolution method (PDEM). Through comparison of the results from the stochastic seismic response analysis, the effects of the number of representative points on the analysis accuracy of the PDEM were investigated. Numerical results show that a relatively small number of representative points selected under the strategy of sphere of contact yield fairly accurate results of the 1st- and 2nd-order statistical characteristics of structural responses. At the same time, it is pointed out that a small sample size may lead to a certain analytical errors of the response probability distribution at each time point. Therefore, the number of representative points in the PDEM should be reasonably determined according to the number of random variables of the related stochastic dynamic system, the method of selecting representative points and the expected calculation accuracy.

A Reduced-Order Extrapolating Finite Difference Algorithm Based on the POD Method for Sobolev Equations

LUO Zhen-dong, ZHANG Bo

2016, 37(1): 107-116. doi: 10.3879/j.issn.1000-0887.2016.01.009

Abstract(1222) PDF(574)

Abstract:
The singular value decomposition technique and the proper orthogonal decomposition (POD) method were applied to establish a reduced-order extrapolating finite difference algorithm for Sobolev equations. Firstly, the absolutely stable fully 2nd-order accurate Crank-Nicolson (C-N) scheme for Sobolev equations was built, and the C-N reduced-order extrapolating finite difference algorithm was constructed based on the POD method, where the number of unknowns in numerical computation was greatly reduced. Secondly, the error estimates of the reduced-order finite difference solutions were provided. Finally, a numerical example was used to verify the feasibility and efficiency of the proposed reduced-order extrapolating finite difference algorithm.

2016 Vol. 37, No. 1