A highly-efficient method for stationary response of multi-degree-of-freedom nonlinear stochastic systems

Analytical and numerical studies of multi-degree-of-freedom(MDOF) nonlinear stochastic or deterministic dynamic systems have long been a technical challenge.This paper presents a highly-efficient method for determining the stationary probability density functions(PDFs) of MDOF nonlinear systems subjected to both additive and multiplicative Gaussian white noises. The proposed method takes advantages of the sufficient conditions of the reduced Fokker-Planck-Kolmogorov(FPK) equation when constructi...     

Selection of finite-element mesh parameters in modeling the growth of hydraulic fracturing cracks

The effect of the mesh geometry on the accuracy of solutions obtained by the finite-element method for problems of linear fracture mechanics is investigated. The guidelines have been formulated for constructing an optimum mesh for several routine problems involving elements with linear and quadratic approximation of displacements. The accuracy of finite-element solutions is estimated based on the degree of the difference between the calculated stress-intensity factor (SIF) and its value obtained analytically. In problems of hydrofracturing of oil-bearing formation, the pump-in pressure of injected water produces a distributed load on crack flanks as opposed to standard fracture mechanics problems that have analytical solutions, where a load is applied to the external boundaries of the computational region and the cracks themselves are kept free from stresses. Some model pressure profiles, as well as pressure profiles taken from real hydrodynamic computations, have been considered. Computer models of cracks with allowance for the pre-stressed state, fracture toughness, and elastic properties of materials are developed in the MSC.Marc 2012 finite-element analysis software. The Irwin force criterion is used as a criterion of brittle fracture and the SIFs are computed using the Cherepanov–Rice invariant J-integral. The process of crack propagation in a linearly elastic isotropic body is described in terms of the elastic energy release rate G and modeled using the VCCT (Virtual Crack Closure Technique) approach. It has been found that the solution accuracy is sensitive to the mesh configuration. Several parameters that are decisive in constructing effective finite-element meshes, namely, the minimum element size, the distance between mesh nodes in the vicinity of a crack tip, and the ratio of the height of an element to its length, have been established. It has been shown that a mesh that consists of only small elements does not improve the accuracy of the solution.     

The Osher scheme for non-equilibrium reacting flows

An extension of the Osher upwind scheme to non-equilibrium reacting flows is presented, Owing to the presence of source terms, the Riemann problem is no longer self-similar and therefore its approximate solution becomes tedious. With simplicity in mind, a linearized approach which avoids an iterative solution is used to define the intermediate states and sonic points. The source terms are treated explicitly. Numerical computations are presented to demonstrate the feasibility, efficiency and accuracy of the proposed method. The test problems include a ZND (Zeldovich-Neumann-Doring) detonation problem for which spurious numerical solutions which propagate at mesh speed have been observed on coarse grids. With the present method, a change of limiter causes the solution to change from the physically correct CJ detonation solution to the spurious weak detonation solution.     

Development of an adaptive discontinuity‐capturing hyperbolic finite element model

In this paper we present a five‐parameter Taylor–Galerkin finite element model to simulate Euler equations in a domain of two dimensions. The introduced free parameters are theoretically determined by employing M‐matrix theory to obtain a physically correct and non‐oscillatory solution in regions containing a sharp solution profile. To improve the computational efficiency and solution accuracy, grids are adaptively added to obtain solutions with fewer mesh points. The discontinuity‐capturing finite element model has been validated against test cases, reproducing analytical solutions to the gas dynamic problems under the current investigation. Copyright © 2004 John Wiley & Sons, Ltd.     

Comparing analytical and numerical solution of nonlinear two and three‐dimensional hydrostatic flows

New test cases for frictionless, three‐dimensional hydrostatic flows have been derived from some known analytical solutions of the two‐dimensional shallow water equations. The flow domain is a paraboloid of revolution and the flow is determined by the initial conditions, the nonlinear advective terms, the Coriolis acceleration and by the hydrostatic pressure. Wetting and drying is also included. Some specific properties of the exact solutions are discussed under different hypothesis and relative importance of the forcing terms. These solutions are proposed for testing the stability, the accuracy and the efficiency of numerical models to be used for simulating environmental hydrostatic flows. The computed solutions obtained with a semi‐implicit finite difference—finite volume algorithm on unstructured grid are compared with the corresponding analytical solutions in both two and three space dimension. Excellent agreement are obtained for the velocity and for the resulting water surface elevation. Comparison of the computed inundation area also shows a good agreement with the analytical solution with degrading accuracy observed when the inundation area becomes relatively large and for long simulation time. Copyright © 2006 John Wiley & Sons, Ltd.     

桁架结构截面优化设计的改进模拟退火算法

将模拟退火算法应用于桁架结构截面尺寸优化设计,提出若干方法改进了算法的鲁棒性、计算效率和求解精度。通过一批经典问题,同时与传统结构优化算法和遗传算法进行了比较。数值结果表明,本文的改进模拟退火算法具有很高的优化求解精度,计算效率有显著提高且优于遗传算法,有望在结构优化设计问题中发挥其特点。     

Determination of optimum aspect ratio for laminar flow heat transfer of dilute viscoelastic solutions in flattened tube heat exchangers

Heat transfer of viscoelastic liquids in five flattened tubes with aspect ratios ranging from 1.4 to 5.7 were presented. Water was used as the heating medium; and solutions of polyacrylamide were used as the viscoelastic solutions. Heat transfer increase from flattening was 101% higher while secondary flow contributed a maximum increase of 65% for the 250?ppm solution and about 85% for the 500?ppm solutions at an aspect ratio of 1.6.     

A revised Cattaneo-Christov micropolar viscoelastic nanofluid model with combined porosity and magnetic effects

The dynamics of non-Newtonian fluids along with nanoparticles is quite interesting with numerous industrial applications. The current predominately predictive modeling deals with the flow of the viscoelastic micropolar fluid in the presence of nanoparticles. A progressive amendment in the heat and concentration equations is made by exploiting the Cattaneo-Christov(C-C) heat and mass flux expressions. Besides, the thermal radiation effects are contributed in the energy equation and aspect of the ...     

Method of constructing estimates of the solution of equations of filtration of an aerated liquid

The exact solutions of the nonlinear equations of filtration of an aerated liquid have been obtained in [1–3]. In [4] the system of equations of an aerated liquid have been reduced to the heat-conduction equation under certain assumptions. An approximate method of computing the nonsteady flow of an aerated liquid is given in [5], where the real flow pattern is replaced by a computational scheme of successive change of stationary states. In [6] the same problem is solved by the method of averaging. In the present article estimates of the solution of the equations for nonstationary filtration of an aerated liquid in one-dimensional layer are constructed under certain conditions imposed on the desired functions. These estimates can be used as approximate solutions with known error or for the verification of the accuracy of different approximate methods. We note that the use of comparison theorem for the estimate of solutions of equations of nonlinear filtration is discussed in [7–9]. The methods of constructing estimates of solutions of various problems of heat conduction are given in [10, 11]     

The method of matched asymptotic expansions for the periodic solution of the van der pol equation

The purpose of this paper is to apply the method of intermediate matching for connecting the four local asymptotic solutions of the Van der Pol equation, given by Dorodnicyn [1]. It turns out that for the approximation of the periodic solution a fifth local solution is needed. The present approach results in a reduction of the computational work. The amplitude of the periodic solution is determined up to a higher order accuracy in v than has been done so far.     

Recursive geometric integrators for wave propagation in a functionally graded multilayered elastic medium

The differential equations governing transfer and stiffness matrices and acoustic impedance for a functionally graded generally anisotropic magneto-electro-elastic medium have been obtained. It is shown that the transfer matrix satisfies a linear 1st order matrix differential equation, while the stiffness matrix satisfies a nonlinear Riccati equation. For a thin nonhomogeneous layer, approximate solutions with different levels of accuracy have been formulated in the form of a transfer matrix using a geometrical integration in the form of a Magnus expansion. This integration method preserves qualitative features of the exact solution of the differential equation, in particular energy conservation. The wave propagation solution for a thick layer or a multilayered structure of inhomogeneous layers is obtained recursively from the thin layer solutions. Since the transfer matrix solution becomes computationally unstable with increase of frequency or layer thickness, we reformulate the solution in the form of a stable stiffness-matrix solution which is obtained from the relation of the stiffness matrices to the transfer matrices. Using an efficient recursive algorithm, the stiffness matrices of the thin nonhomogeneous layer are combined to obtain the total stiffness matrix for an arbitrary functionally graded multilayered system. It is shown that the round-off error for the stiffness-matrix recursive algorithm is higher than that for the transfer matrices. To optimize the recursive procedure, a computationally stable hybrid method is proposed which first starts the recursive computation with the transfer matrices and then, as the thickness increases, transits to the stiffness matrix recursive algorithm. Numerical results show this solution to be stable and efficient. As an application example, we calculate the surface wave velocity dispersion for a functionally graded coating on a semispace.     

On the plastic zone size of solids containing doubly periodic rectangular-shaped arrays of cracks under longitudinal shear

Multiple cracks interaction plays an important role in fracture behavior of materials. A number of studies have been devoted to analytical and numerical analyses of the doubly periodic arrays of cracks. A very natural and highly accurate solution procedure is proposed to describe the interaction effect among the doubly periodic rectangular-shaped arrays of cracks. The proposed solution is implemented in the framework of continuously distributed dislocation model and singular integral equation approach. The accuracy of this solution is proved through a comparison of results from the present simulation and known closed form solutions. Further, the interaction effects among the periodic cracks on the plastic zone size and crack tip opening displacement are studied. It is found that the interaction distance among the vertical and horizontal periodic cracks is quite different.     

Nonlinear wave propagation and reflection — Comparing the numerics with the analytics

The accuracy of numerical methods needs always a special attention. In this paper, analytical and numerical methods have been compared to describe the initial stage of nonlinear propagation and reflection of longitudinal ultrasonic waves. The perturbation method has been used to derive the analytical solution and the finite difference scheme to find the numerical solution for multiple free-boundary reflections of a harmonic burst at ultrasonic frequencies. The comparison of results at relatively small nonlinearities reveals a good qualitative and quantitative agreement between the analytical and numerical solutions. The method for determining analytically the exact region of interaction for counter-propagating waves is outlined in detail. At higher frequencies and larger nonlinear effects some quantitative differences between analytical and numerical results appear. The results are applicable in modelling nonlinear wave motion, including NDT and nonlinear one-dimensional vibrations.     

The lowest order solution of the crack–inhomogeneity interaction for mode I crack

The lowest order solution to crack–inhomogeneity interaction is derived for mode I crack. The basic solution evaluates the variation of near-tip stress intensity factors induced by an inhomogeneity of arbitrary shape. A set of simplified forms of the basic solution is also obtained for several special inhomogeneity shapes. As validated by numerical examples, the approximate solutions have good accuracy.     

Shortcomings of discontinuous‐pressure finite element methods on a class of transient problems

Past studies that have compared LBB stable discontinuous‐ and continuous‐pressure finite element formulations on a variety of problems have concluded that both methods yield solutions of comparable accuracy, and that the choice of interpolation is dictated by which of the two is more efficient. In this work, we show that using discontinuous‐pressure interpolations can yield inaccurate solutions at large times on a class of transient problems, while the continuous‐pressure formulation yields solutions that are in good agreement with the analytical solution. Copyright © 2009 John Wiley & Sons, Ltd.     

The residual formulation of the method of manufactured solutions for computationally efficient solution verification

The method of manufactured solutions (MMS) is a solution verification methodology for determining whether the implementation of a discretization method is achieving its theoretical order of accuracy. This methodology combines the advantages of grid refinement studies and comparison with exact solution, by modifying the governing equations solved within a code by adding a source term to drive the solution towards a predetermined analytic function. By solving the modified equations on a sequence of grids and comparing the differences between the converged solution and manufactured solution, the order of accuracy of the implementation can be investigated. However, in its current form, converged solutions on a sequence of grids are required which can be quite costly and difficult to obtain. In this paper, by comparing the MMS to the method for determining the theoretical order of accuracy of a discretization method, the residual formulation of the MMS is developed. This new formulation only requires that the residual of the discretized governing equations to be calculated and not the solution to the discretized equations, thus avoiding the computational cost and difficulties inherent in obtaining converged solutions. Furthermore, since only the residuals are interrogated, individual components of the flow solver can be tested, in isolation, allowing the MMS to be used more effectively in locating errors within the code. This new approach is demonstrated to yield the same order of accuracy as the original MMS using three different cases—one-dimensional porous media equation, one-dimensional St Venant equations and two-dimensional unstructured Navier–Stokes simulations.     

随机激励下滞迟系统的稳态响应闭合解

滞迟系统属于一类典型的强非线性系统,滞迟力不仅取决于系统的瞬时变形,还与变形历程有关.虽然滞迟系统的随机振动问题已被广泛研究,但至今尚未得到滞迟系统随机响应概率密度函数的精确闭合解.本文运用迭代加权残值法获得了高斯白噪声激励下Bouc-Wen滞迟系统稳态响应概率密度函数的近似闭合解.首先,运用等效线性化法求出系统的稳态高斯概率密度函数;然后以此构造权函数,应用加权残值法求得了系统指数多项式形式的非高斯概率密度函数;最后引入迭代的过程,逐步优化权函数,提高计算所得结果的精度.以随机地震激励下钢纤维陶粒混凝土结构的稳态响应作为算例,其中Bouc-Wen模型的参数是基于拟静力学试验数据,并应用最小二乘法辨识获得.与Monte Carlo模拟结果相比,等效线性化法得到的结果精度较差;由加权残值法得到的结果能够表现出非线性特征,但其精度依然无法令人满意;采用迭代加权残值法得到的近似闭合解与Monte Carlo模拟的结果吻合非常好;对于较强随机激励情形,采用渐进迭代加权残值法具有较高的求解效率,所获得的理论解析解具有较高的精度.结果表明,所获得的近似闭合解不仅对于土木工程领域具有重要的实际应用价值,而且还可作为检验其他非线性系统随机响应预测方法的精度的标准.     

Embedding formulae for wave diffraction by a circular arc

For certain wave diffraction problems, embedding formulae can be derived, which represent the solution (or far-field behaviour of the solution) for all plane wave incident angles in terms of solutions of a (typically small) set of other auxiliary problems. Thus a complete characterisation of the scattering properties of an obstacle can be determined by only determining the solutions of the auxiliary problems, and then implementing the embedding formula. The class of scatterers for which embedding formulae can be derived has previously been limited to obstacles with piecewise linear boundaries; here this class is extended to include a simple curved obstacle, consisting of a thin circular arc. Approximate numerical calculations demonstrate the accuracy of the new embedding formulae.     

弹性力学问题的一阶有限元解法

求解弹性力学问题的应力时,如果采用常规的位移有限元法,需要先求得单元的节点位移,再经过求导运算得到。为了解决这种求解方式引起的应力精度下降的问题,提出了弹性力学问题的一阶多变量形式,使得应力与位移精度同阶,并推导了弱形式。采用有限元方法,对弹性力学问题给出了一阶解法的二维、三维数值算例,并且将一阶解法的结果与常规位移有限元法的解进行了比较。数值计算结果表明,一阶解法有效提高了应力的精度,并且应力的误差和节点位移的误差具有相同的收敛阶,验证了本文方法的有效性,为提高有限元法的应力精度提供了新的思路。     

Calculation of the reflected wave from a pipe with a nozzle end by the Lax-Wendroff method

The solution of gas flow problems in pipes with nozzle ends is discussed. The Lax-Wendroff method, with a hybrid boundary condition approximation, is used to compute the numerical solutions to some test problems. The accuracy of the solutions obtained by this method is assessed by a comparison with theoretical solutions.