应用数学和力学

2014 Vol. 35, No. 9

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Application of Fractional Calculus to Simulate the Isolation Effects of Discontinuous Pile Barriers on Viscoelastic SH Waves

LI Yuan, CHEN Wen, PANG Guo-fei

2014, 35(9): 949-958. doi: 10.3879/j.issn.1000-0887.2014.09.001

Abstract(1299) PDF(1212)

Abstract:
Based on the 3-dimensional (3D) fractional order constitutive equation of viscoelastic body waves, the frequency dispersion characteristics of viscoelastic P- and S-waves were analyzed. Also, the isolation effects of the discontinuous rigid pile barrier and the elastic pile barrier in soft clay on viscoelastic SH waves were comparatively studied. With the finite difference method (FDM), an array of vibration amplitude reduction factors for different pile spacing-to-diameter ratios, different fractional orders and different frequencies of incident waves were obtained, and the isolation effect of the elastic pile isolation system in comparison with the rigid was analyzed. The results exhibit that the smaller pile spacing-to-diameter ratio is, or the larger the fractional order is, the better isolation effect of the rigid barrier will be. In contrast, the elastic barrier has better isolation effect in some special target area when the fractional order becomes smaller.

Research on the Configuration of the Deepwater Steel Lazy-Wave Riser Under Effects of Ocean Currents

WANG Jin-long, DUAN Meng-lan, TIAN Kai

2014, 35(9): 959-968. doi: 10.3879/j.issn.1000-0887.2014.09.002

Abstract(1642) PDF(902)

Abstract:
As a kind of improved SCR, the steel lazy-wave riser (SLWR) gained more and more application in the development of deepwater oil and gas field. The configuration of the SLWR is of great significance in its design process. Under ocean current, the configuration and mechanical parameters of the SLWR vary a lot. A model of multipoint boundary-value differential equations based on the nonlinear large deformation beam theory was proposed to simulate the responses of the SLWR under ocean current. Numerical program was built to get the approximate solution to the theoretical model and parametric analysis was conducted to study the influences of different ocean currents on the configuration of the SLWR. Results show that the axial tension force at the touchdown point of the SLWR increases distinctly with the rise of the ocean current velocity, which deserves full consideration in the configuration design of the SLWR. The proposed model makes an important reference for the performance analysis of the deepwater SLWR.

Convergence of an Adaptive Finite Element Method for 2D Elasticity Problems

LIU Chun-mei, ZHONG Liu-qiang, SHU Shi, XIAO Ying-xiong

2014, 35(9): 969-978. doi: 10.3879/j.issn.1000-0887.2014.09.003

Abstract(1857) PDF(868)

Abstract:
For 2D linear elasticity problems, firstly, a standard adaptive finite element method (AFEM) was developed based on the newest vertex bisection grid refinement, which was marked only according to the error estimators without special treatment of the oscillation terms and intended conformance to the interior node properties. Secondly, through analysis of the numerical solution functions and error indicators at all the grid levels, the AFEM was strictly proved to be convergent by means of the orthogonality between the numerical solution functions at adjacent grid levels, the upper bound estimation of the energy errors between the true solution functions and the numerical solution functions, and the approximate compressibility of the error indicators between adjacent grid levels. Finally, several numerical experiments confirm that the presented AFEM is convergent and robust.

On the Fracture Modeling Method for Crack Tips Penetrating Elastic Interfaces

CHEN Chang-rong

2014, 35(9): 979-985. doi: 10.3879/j.issn.1000-0887.2014.09.004

Abstract(1702) PDF(897)

Abstract:
The theoretical defects of the linear elastic fracture mechanics in modeling crack tips passing through elastic interfaces were analyzed; for an idealized layered material, the cohesive zone model was applied to study the effects of the material cohesive strength ahead of the interface on the behavior of a crack perpendicularly approaching and penetrating an elastic interface; based on the finite element calculation results, the difference between the cohesive zone model and the linear elastic fracture mechanics in simulating a perpendicular crack near an elastic interface was discussed. The results show that the material cohesive strength ahead of the interface is the key factor causing the simulation difference between the cohesive zone model and the linear elastic fracture mechanics. The study gives the conclusion that, to model the crack growth in complex materials, the strength criterion is needed in addition to the traditional energy-based fracture criterion, and the cohesive zone model theoretically satisfies this requirement.

A Reduced-Order Stabilized CNFVE Extrapolating Model for Non-Stationary Stokes Equations

TENG Fei, LUO Zhen-dong

2014, 35(9): 986-1001. doi: 10.3879/j.issn.1000-0887.2014.09.005

Abstract(1846) PDF(631)

Abstract:
A reduced-order stabilized Crank-Nicolson finite volume element (SCNFVE) extrapolating model with sufficiently high accuracy and few degrees of freedom for non-stationary Stokes equations was established by means of the SCNFVE method and the proper orthogonal decompostion (POD) technique. The error estimates of the reduced-order approximate solutions and the algorithm implementation for the reduced-order SCNFVE extrapolating model were provided. Finally, a numerical example of conduit flow indicates that the results of the proposed model are consistent with those of the theoretical solution. Moreover, the advantages of lower computation complexity and higher calculation accuracy of the reduced-order SCNFVE extrapolating model are shown in comparison with the classical methods.

Bifurcations of Solitary Wave Solutions to the Shallow Water Equation of Moderate Amplitude

QIN Yu-yue, DENG Zi-chen, HU Wei-peng

2014, 35(9): 1002-1010. doi: 10.3879/j.issn.1000-0887.2014.09.006

Abstract(1378) PDF(664)

Abstract:
The qualitative behavior and solitary wave solutions to the model equation for shallow water waves of moderate amplitude were studied with the bifurcation method for dynamic systems. The phase portraits of the system were given under different parametric conditions. The implicit expressions of the smooth solitary waves, cuspons and periodic wave solutions were obtained. Numerical simulations were conducted for the smooth solitary waves, cuspons and periodic wave solutions to the model equation. The results show that the presented findings improve the related previous conclusions.

A Preconditioned Parallel Method for Solving Saddle Point Problems

JIANG Xiao-lin, Lü Quan-yi, XIE Gong-nan

2014, 35(9): 1011-1019. doi: 10.3879/j.issn.1000-0887.2014.09.007

Abstract(1849) PDF(788)

Abstract:
A parallel algorithm with preconditioned modified conjugate gradient method for solving saddle point problems was studied. It is a model that by using iterative method for preconditioning and applying modified conjugate gradient method for solving the problems. Firstly the approximate inverse of the coefficient matrix’s polynomial expressions is constructed and become the inverse matrix of the preconditioned matrix, secondly the modified conjugate gradient method is used for parallel solving the preconditioned linear equations. In order to reduce the amount of calculation, we have to parallel compute the polynomials and vector multiplication by using iterative method. By adjusting the number of iterations and polynomials to exam the effect of preconditioning. The results show that our algorithm is superior to the modified conjugate gradient method and it has the best effect when the number of iterations is four.

D-η-E-Semipreinvex Vector Mappings and Vector Optimization

PENG Zai-yun, LI Ke-ke, ZHANG Shi-sheng

2014, 35(9): 1020-1032. doi: 10.3879/j.issn.1000-0887.2014.09.008

Abstract(1962) PDF(880)

Abstract:
A class of new vector valued generalized convex mappings—D- η -E-semipreinvex mappings, as a true generalization of the E-preinvex mappings and the D-η-semipreinvex mappings, were given. First, several examples were presented to show the existence of the E-semi-invex sets and the D- η -E-semipreinvex mappings. Second, a decision criterion for the D- η -E-semipreinvex mappings was introduced, and the relationships among the D- η -E-semipreinvexity, the D- η -E-strict semipreinvexity and the D- η -E-semistrict semipreinvexity were discussed.Finally, an important application of the D- η -E-semistrictly semipreinvexity to vector optimization with implicit constraint was discussed, then some examples were given to prove the main conclusions.

Approximate Solutions to the Nonlinear Compartmental Model for Extravascular Administration

HU Xiao-hu, TANG San-yi

2014, 35(9): 1033-1045. doi: 10.3879/j.issn.1000-0887.2014.09.009

Abstract(1801) PDF(1097)

Abstract:
The analytical solution to the pharmacokinetics model plays a key role in the design of new drugs, especially in determining the pharmacokinetic parameters. In recent years, the analytical formulae for most of the pharmacokinetics models decided by the nonlinear Michaelis-Menten elimination process, were investigated and solved. However, the pharmacokinetics model with nonlinear Michaelis-Menten elimination rate for extravascular administration was a non-autonomous system, which resulted in difficulties in seeking its analytical solutions. Therefore, the problem of approximation to the solutions to the non-autonomous nonlinear pharmacokinetics models in the cases of single or periodic extravascular administrations was addressed. Different upper and lower bounds were given based on the comparison theorems for differential equations and impulsive differential equations, with the definition and related properties of the Lambert W function employed. Numerical simulations show the effectiveness of the proposed approximation method.

Solutions to a Class of Nonlinear Strong-Damp Disturbed Evolution Equations

SHI Juan-rong, SHI Lan-fang, MO Jia-qi

2014, 35(9): 1046-1054. doi: 10.3879/j.issn.1000-0887.2014.09.010

Abstract(1324) PDF(865)

Abstract:
Widely emerging in the fields of mathematics and mechanics, a class of 3rd-order nonlinear strong-damp disturbed partial differential evolution equations were studied. Firstly, a functional homotopic mapping was constructed to express the solution to the evolution equation in a form of power series with artificial parameters, which was substituted into the homotopic mapping to build a method of successive iteration for the solution to the nonlinear disturbed equation. Then the corresponding non-disturbed strong-damp evolution equation was analyzed with exact solution based on the theory of Fourier transform. Secondly, the found exact solution was used as the initial function of the homotopic mapping iteration, and the iteration expansion of the nonlinear disturbed equation was applied to solve the related equations with the Fourier transform method. Finally, both the exact and arbitrary-order approximate analytic solutions to the nonlinear strong-damp disturbed evolution equation were obtained. The proposed homotopic mapping method is proved to have the advantages of convenience and accuracy.

Dynamics of a Complex-Valued Heat Equation

XIONG Hui, YANG Guang

2014, 35(9): 1055-1062. doi: 10.3879/j.issn.1000-0887.2014.09.011

Abstract(1138) PDF(819)

Abstract:
The Cauchy problem for a parabolic system which was derived from a complexvalued heat equation with inverse nonlinearity was studied. Some criteria for the global existence and quenching of the solutions were provided. Through transformation of the invariant subset of the solution plane, it was proved that, for the initial values which are asymptotically constants, whether the solution quenches at spatial infinity or exists globally at any time, depends on the asymptotic limits of the initial values.

An Interior-Point Method With a New Iterative Scheme

YANG Xi-mei, LIU Hong-wei, ZHANG Yin-kui

2014, 35(9): 1063-1070. doi: 10.3879/j.issn.1000-0887.2014.09.012

Abstract(1217) PDF(770)

Abstract:
A 2nd-order Mehrotra-type predictor-corrector interior-point method was proposed for linear programming, in which the predictor direction and corrector direction were computed with the Newton method and the search direction was obtained through a new form of combination of the predictor direction and corrector direction. At each step of the iteration, the step size parameter was calculated with the iteration restricted to a wide neighborhood of the central path. Analysis indicates the proposed algorithm converges to the optimal solution after finite times of iteration and has the polynomial iteration complexity O(√nL), which is the best complexity result for the current interior-point methods. Numerical experiment proves the high efficiency of the proposed algorithm.