Free boundary problem of non-steady state seepage flow

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A VARIATIONAL INEQUALITY APPROACH FOR NON-STEADY SEEPAGE FLOW THROUGH THREE-DIMENSIONAL FRACTURE NETWORK

为了求解裂隙岩体有自由面非稳定渗流问题,将Darcy定律延拓至整个研究区域,使得潜在溢出边界条件满足Signorini型边界条件,建立了三维裂隙网络非稳定渗流问题的抛物型变分不等式（parabolic variational inequality,PVI）提法,并证明其与偏微分方程（partial differential equation,PDE）提法的等价性,从而将自由面上的流量条件以及潜在溢出边界上的互补条件转化成自然边界条件,降低该问题求解难度。同时给出了基于PVI提法的有限元数值求解方法,通过与交叉裂隙模型理论解的对比分析,证明了该方法的正确性。最后将该方法对含复杂三维裂隙网络的边坡进行非稳定渗流分析,计算结果表明该方法对于复杂裂隙网络求解具有较强的可靠性和适应性。     

Variational formulation of the static Levinson beam theory

In this communication, we provide a consistent variational formulation for the static Levinson beam theory. First, the beam equations according to the vectorial formulation by Levinson are reviewed briefly. By applying the Clapeyron's theorem, it is found that the stresses on the lateral end surfaces of the beam are an integral part of the theory. The variational formulation is carried out by employing the principle of virtual displacements. As a novel contribution, the formulation includes the external virtual work done by the stresses on the end surfaces of the beam. This external virtual work contributes to the boundary conditions in such a way that artificial end effects do not appear in the theory. The obtained beam equations are the same as the vectorially derived Levinson equations. Finally, the exact Levinson beam finite element is developed.     

A variational formulation for a boundary value problem considering an electro-sensitive elastomer interacting with two bodies

We present a variational formulation for an electro-elastic body in contact with two semi-infinite rigid bodies, which are electric conductors and have a distribution of free charge. These three bodies are surrounded by free space, where far away we have a given electric displacement and an electric potential on disjoint surfaces.     

岩土中弹塑性渗透固结问题的参变量变分原理

本文给出渗透固结过程的参变量变分原理,可以用来处理岩土弹塑性渗透固结问题,并不受 Drucker 假设的限制,对于关连或非关连塑性流动情形皆可。     

Boundary Element Analysis of Vibro-Acoustic Interaction between Vibrating Membrane and Thin Fluid Layer

This paper presents a variational formulation for the study of the acoustic propagation and radiation of a vibrating membrane coupled to a cavity filled with a visco-thermal fluid. This formulation is obtained by combining a variational formulation by integral equations of both internal and external fluids, which takes account of acoustic and entropic waves coupling, with a classical variational formulation of the membrane. Numerical results obtained by this new formulation are compared to those obtained when the viscous and thermal effects are not considered. This revised version was published online in July 2006 with corrections to the Cover Date.     

Seepage from a channel: Solution structure and flow-rate estimation

Using functional analysis methods, problems that arise in calculating underground water seepage from an earth channel are investigated. The problem of estimating the seepage loss is considered. Novel methods which make it possible to overcome the difficulties in solving variational problems of channel seepage are proposed. An exact (unimprovable) estimate is found for the seepage loss from earth channels. The channel shape optimal among channels with a given cross-section contour length is obtained.     

Novel variational formulations for nonlocal plasticity

A nonlocal structural model of softening plasticity is considered in the framework of the internal variable theories of inelastic behaviours of associative type. The finite-step nonlocal structural problem in a geometrically linear range is formulated according to a backward difference scheme for time integration of the flow rule. The related finite-step variational formulation in the complete set of local and nonlocal state variables is recovered. A family of mixed nonlocal variational formulations, with different combinations of state variables, is provided starting from the general variational formulation. The specialization of a mixed variational formulation to existing nonlocal models of softening plasticity, assuming both linear and nonlinear constitutive behaviour, is provided to show the effectiveness of the theory.     

Effects of Entropic Wave in Vibroacoustic Problem Using Boundary Element Analysis

A variational formulation for a vibroacoustic problem of a membrane and a viscothermal fluid is investigated in this paper. The formulation combines a variational formulation by integral equations of the fluid, that takes into account the acoustic and entropic waves coupling, with a variational formulation of the membrane. The formulation has been implemented numerically for the problems with axisymmetric geometry. The numerical results are compared to the analytical solution for a circular membrane coupled to a cylindrical cavity filled with air. These results show the validity of numerical implementation and illustrate the thermal effects of air on the membrane-cavity system modes in the micro cavities cases.     

Acoustics in anisotropic poroelasticity: space–time variational formulation

A variational space–time formulation to treat acoustics problems in porous medium is developed. At first, no separation of the space and time has been applied in the approximation of the field variables. This produces a matrix variational formulation for more general numerical treatments. Finally, separation of variables has been applied to produce a simpler space–time formulation, in which the numerical treatment of the space domain and the time integration algorithms can be identified. The numerical treatment has been oriented to find symmetry in the space and time domains. Numerical simulation in the frequency domain and in the time domain have been performed to validate the variational formulation with separation of variables.     

Variational principles of elastic-viscous dynamics in laplace transformation form,F. E. M. formulation and numerical method

The author gives variational principles of elastic-viscous dynamics in spectral resolving form^[1], it will be extended to Laplace transformation form in this paper, mixed variational principle of shell dynamics and variational principle of dynamics of elastic-viscous-porous media are concerned, for the latter, F. E. M. formulation has been worked out.Variational principles in Laplace transformation form have concise forms, for the sake of utilizing F. E. M. conveniently it is necessary to find values of preliminary time function at some instants, when values of Laplace transformation at some points are known, but there are no efficient methods till now. In this paper, a numerical method for finding discrete values of preliminary function is presented, from numerical example we see such a method is efficient.By combining both methods stated above, variational principles in Laplace transformation form and numerical method, a quite wide district of solid dynamic problems can be solved by ths aid of digital computers.     

极坐标系下的Legendre谱元方法求解Poisson-型方程

r=0处的坐标奇异性是求解极坐标下Poisson-型方程的关键。本文提出一种极坐标系下基于Galerkin变分的Legendre谱元方法用于求解圆形区域内的Poisson-型方程,物理区域的径向和周向划分若干单元,计算单元均采用Legendre多项式展开;圆心所在单元的径向使用LGR(Legendre Gauss Radau)积分点,其他单元径向使用LGL(Legendre Gauss Lobatto)积分点,从而避免了极点处1/r坐标奇异性,周向单元均采用LGL积分点。利用区域分解技术,可以避免节点在极点附近聚集;最后求解了多个Dirichlet或Neumann边界条件下的Poisson-型方程算例。数值结果表明,谱元方法具有很高的精度。     

On the derivation of well posed boundary value problems in structural mechanics

The paper starts with a discussion of a number of published boundary value problems in structural mechanics, whose intuitive formulation led to incorrect boundary or matching conditions. It is then shown on three examples, using variational methods, how to obtain well posed formulations for problems of this type. For uniformity of presentation, all examples deal with continuously supported beams. The first two examples exhibit boundary and matching points whose position is fixed along the beam axis. In the third example it is shown how to use the method of variational calculus for a variable end point to formulate structural problems with variable matching points. This is demonstrated on the problem of a beam which rests on, but is not attached to, a Pasternak base; thus where parts of the beam may lift off the base. Each example concludes with a comparison and discussion of related formulations published in the literature.     

On normal modes in classical Hamiltonian systems

Normal modes of Hamittonian systems that are even and of classical type are characterized as the critical points of a normalized kinetic energy functional on level sets of the potential energy functional. With the aid of this constrained variational formulation the existence of at least one family of normal modes is proved and, for a restricted class of potentials, bifurcation of modes is investigated. Furthermore, a conjecture about a lower bound for the number of normal modes in case the potential is homogeneous, is proved.     

FREE BOUNDARY PROBLEM OF THE 2D SEEPAGE FLOW

ＦＲＥＥＢＯＵＮＤＡＲＹＰＲＯＢＬＥＭＯＦＴＨＥ２ＤＳＥＥＰＡＧＥＦＬＯＷ￥ＳｈｅＹｉｎｇｈｅ（佘颖禾）ＳｕｎＹｉｎｇ（孙鹰）ＧｕｏＸｉａｏｍｉｎｇ（郭小明）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓａｎｄＭｅｃｈａｎｉｃｓ，Ｓｏｕｔｈｅａｓｔ...     

A Universal Variational Formulation for Two Dimensional Fluid Mechanics

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ 50’ｓ,ｇｒｅａｔｐｒｏｇｒｅｓｓｈａｄｂｅｅｎｍａｄｅｏｎｔｈｅｒｅｓｅａｒｃｈｏｆｖａｒｉａｔｉｏｎａｌｐｒｉｎｃｉｐｌｅｉｎｓｏｌｉｄｍｅｃｈａｎｉｃｓｗｉｔｈＨｕ＿Ｗａｓｈｉｚｕｇｅｎｅｒａｌｉｚｅｄｖａｒｉａｔｉｏｎａｌｐｒｉｎｃｉｐｌｅ(ＧＶＰ) [1]ａｓｍｉｌｅｓｔｏｎｅ ,ａｎｄｉｎｆｌｕｉｄｍｅｃｈａｎｉｃｓｗｉｔｈＣ .Ｃ .Ｌｉｎ’ｓｃｏｎｓｔｒａｉｎｓ[2 ]ａｓｍｉｌｅｓｔｏｎｅ .Ｉｎ 80’ｓ,ＣｈｉｅｎＷｅｉ＿ｚａｎｇｄｅｄｕｃｅｄａｍｏｒｅｇｅｎｅｒａｌ…     

Variational base and solution strategies for non-linear force-based beam finite elements

This paper presents the variational bases for the non-linear force-based beam elements. The element state determination of these elements is obtained exactly from a two-field functional with independent stress and strain fields. The variational base of the non-linear force-based beam elements implemented in a general purpose displacement-based finite element program requires the inclusion of independent displacement field in the formulation. For this purpose, a three-field functional is considered with independent displacement, stress, and strain fields. Various local and global solution strategies come out from the mixed formulation of the beam element, and these are shown to yield the algorithms presented for non-linear force formulation beam elements in literature; thus removing any doubts on their variational bases. The presented numerical examples demonstrate the accuracy and robustness of the solution algorithms adapted for mixed formulation elements over popularly used displacement-based beam finite elements even for large structural systems.     

Variational formulation of a rod under torsional vibration for crack identification

A variational formulation is developed for the torsional vibration of a cylindrical shaft with a circumferential crack. The work is compared with existing methods. The Hu–Washizu–Barr variational formulation was used to develop the differential equation and the boundary conditions of the cracked rod. The general variational principle and the independent assumptions about the displacement, the momentum, the strain and the stress fields of the cracked rod, and the equations of motion for a uniform rod in torsional vibration, are derived. The crack was modelled as a continuous flexibility using the displacement field in the vicinity of the crack, found with fracture mechanics methods. Rayleigh quotient was used to approximate the natural frequencies of the cracked rod. Independent evaluations of crack identification methods in rotating shafts are reported and compared with methods using the continuous crack flexibility theory.     

稳定节点积分伽辽金无网格法的应力计算方法研究

应力计算是基于稳定节点积分的伽辽金无网格法的重要组成部分.该文着重研究稳定节点积分伽辽金无网格法的应力计算方法,对稳定节点积分方法的变分一致条件进行了讨论.证明当节点代表域内的应变采用非局郎光滑应变时,相应的应力在节点代表域内为常数,稳定节点积分伽辽金无网格离散方程是变分一致的.文中提出了三种节点应力计算方法,研究表明,基于位移梯度的节点应力计算方法不满足变分一致性要求,而采用光滑应变的节点应力计算方法和一致形心应力计算方法满足变分一致性要求.典型数值算例的误差分析表明,满足变分一致性不一定确保得到更为精确的结果.而基于光滑应变的一致形心应力计算方法总是较其它两种方法更为精确.     

A nonlocal model with strain-based damage

A thermodynamically consistent formulation of nonlocal damage in the framework of the internal variable theories of inelastic behaviours of associative type is presented. The damage behaviour is defined in the strain space and the effective stress turns out to be additively splitted in the actual stress and in the nonlocal counterpart of the relaxation stress related to damage phenomena. An important advantage of models with strain-based loading functions and explicit damage evolution laws is that the stress corresponding to a given strain can be evaluated directly without any need for solving a nonlinear system of equations. A mixed nonlocal variational formulation in the complete set of state variables is presented and is specialized to a mixed two-field variational formulation. Hence a finite element procedure for the analysis of the elastic model with nonlocal damage is established on the basis of the proposed two-field variational formulation. Two examples concerning a one-dimensional bar in simple tension and a two-dimensional notched plate are addressed. No mesh dependence or boundary effects are apparent.