PARTITION OF UNITY FINITE ELEMENT METHOD FOR SHORT WAVE PROPAGATION IN SOLIDS

IntroductionIt is known that standard finite element procedure is unable to simulate the wavepropagation with high oscillations or gradients in space in the media with reasonableefficiency and accuracy due to the nature of polynomial interpolation approxi…     

Path-independentJ-integral and its dual form in elastic-plastic solids with finite displacements

In this paper, based on energy variational principles of elastic-plastic solids, the path-independentJ-integral and its dual form in elastic-plastic solids with finite displacements are presented. Whose testification is given there after.     

Shot peening and peen forming finite element modelling – Towards a quantitative method

Peen forming is commonly used on aluminium alloys in the aerospace industry for wing skin shaping. Numerous analytical, numerical, and experimental studies have been made to better understand the effects of various peening parameters on the final material state and to predict deformed shapes, but conclusions were often limited to trends. The purpose of this study is therefore to develop and verify experimentally quantitative numerical tools for peen forming applications by studying the simple case of peening an Almen-sized AA-2024 aluminium strip in an Almen holder. The first step consisted in improving an existing random dynamic model by determining optimal dimensions. The AA-2024 target mechanical behaviour was characterized experimentally and a combined isotropic-kinematic hardening law was selected to model the material behaviour. The dynamic impact model and material constitutive law provided good prediction of peening-induced stresses in thick AA-2024 for two shot velocities. The sequence-sensitive aspect of the forming process was also investigated and a new shell-based finite element model was proposed. Numerical and experimental results for three shot velocities were compared to evaluate the validity of this numerical simulation method and promising agreement was observed.     

A two‐phase flow interface capturing finite element method

A new interface capturing algorithm is proposed for the finite element simulation of two‐phase flows. It relies on the solution of an advection equation for the interface between the two phases by a streamline upwind Petrov–Galerkin (SUPG) scheme combined with an adaptive mesh refinement procedure and a filtering technique. This method is illustrated in the case of a Rayleigh–Taylor two‐phase flow problem governed by the Stokes equations. Copyright © 2006 John Wiley & Sons, Ltd.     

Vibration modelling of structural networks using a hybrid finite element/wave and finite element approach

The vibration modelling of waveguide structures is considered. These structures comprise waveguides connected via joints. Traditionally, analytical models of the wave behaviour of such structures can be developed if they are simple (beams or rods connected at point joints, etc.). However, if the waveguides are of complicated constructions (truss-like, layered media, etc.) or the joints are complicated (e.g. of significant physical dimensions), obtaining the wave characteristics might be a formidable task. In this paper, such structures are modelled using a hybrid finite element/wave and finite element (FE/WFE) approach. The waveguides are modelled using the WFE method and thus their wave characteristics are obtained regardless of the complexity of their cross-section. The joints are modelled using standard FE, and the WFE and FE models are coupled to yield the scattering properties of the joints. The propagation and scattering models are assembled to describe the behaviour of the structure using relatively small models, while also providing information for other applications such as structure-borne sound, statistical energy analysis, etc. Numerical examples are presented to illustrate the approach.     

Exact finite element method

In this paper,a new method,exact element method for constructing finite element,ispresented.It can be applied to solve nonpositive definite or positive definite partialdifferential equation with arbitrary variable coefficient under arbitrary boundarycondition.Its convergence is proved and its united formula for solving partial differentialequation is given.By the present method,a noncompatible element can be obtained and thecompatibility conditions between elements can be treated very easily.Comparing the exactelement method with the general finite element method with the same degrees of freedom,the high convergence rate of the high order derivatives of solution can be obtained.Threenumerical examples are given at the end of this paper,which indicate all results canconverge to exact solution and have higher numerical precision.     

Some examples of electron microscopy studies of microstructures and phase transitions in solids

In this contribution the results of some electron microscopy studies on microstructures related with phase transitions in a variety of materials will be presented. The materials include binary and ternary alloys, high T_C superconductors as well as C₆₀ and C₇₀ fullerenes, while the transitions can be diffusional, displacive or both.

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Sommario Si presentano i risultati di alcuni studi fatti attraverso la microscopia elettronica sulle microstrutture relative a transizioni di fase in una varietà di materiali. I casi comprendono leghe binarie e ternarie, superconduttori T_C e materiali C₆₀ e C₇₀; le transizioni esaminate sono diffusionali, displacive o di entrambi i tipi.

     

固体中短波传播单位分解有限元法的解析积分

针对固体中短波传播数值模拟的单位分解有限元法中单元矩阵积分的被积函数的强烈振荡特性,应用直角坐标系下标准有限元形函数和单元内的波动方向知识提出了一种单元矩阵的解析积分方案。它对于平面三,六,四,八和九节点的直边单位分解有限单元是完全解析的,对于与这些单元相应的曲边单元则是半解析的。数值结果显示所提出的积分方案在计算效率上比高斯-勒让德积分有大幅度提高。     

Model of the motion and shock waves in two-phase solids with phase transitions

The one-velocity and one-temperature model of the motion of a two-phase solid, in which each phase occupies a certain part of the volume, is considered. The investigation is carried out in Lagrangian variables, which offers certain advantages in solving one-dimensional nonstationary problems. The stress tensor for the mixture is decomposed into two parts -hydrostatic pressure, common to the two phases, associated with the three-term equation of state, and the deviator, which varies elastically up to a certain value and then remains constant. A certain relation, determined by the characteristic reaction time, is proposed for the phase transition kinetics. Then a solution is obtained for the problem of the nonstationary one-dimensional motion of a metal (iron) resulting from the impact of a plate against a target. The phase transitions (FeFe) behind the wave and their characteristic time have an important effect on the damping of the disturbance and on the zone in which these transitions go to completion. A method is proposed for determining the coefficient in the relation for the phase transition rate from the residual effect (hardening) after impact.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 1, pp. 88–95, January–February, 1970.     

The construction of large element in finite element method

In the usual finite element method, the order of the interpolation in an element is kept unchanged, and the accuracy is raised by subdividing the grid denser and denser. Alternatively, in the large element method, the grid is kept unchanged, and the terms of approximate series in the element are increased to raise the accuracy.In this paper, a method for constructing large elements is presented. When using this method, two sets of variables, one set defined inside the element, and the other defined on the boundary of the element, are adopted. Then, these two sets of variables are combined by the hybrid-penalty function method. This method can be applied to any elliptic equations in a domain with arbitrary shape and arbitrary complex boundary condition. It is proved with strict mathematical method in this paper, that in general cases, the accuracy of this method is much higher than that of the usual element and the large element method presented in [7]. Therefore, the degrees of freedom needed in this method are much fewer than those in the two methods if the same accuracy is preserved.     

Density-contrast instabilities in finite element modelling of slow viscous flow

Finite element methods are often used to model Earth processes involving slow viscous or viscoelastic flow. Inertial terms of the Navier-Stokes equations are neglected in very slow flows, so timestep size is not limited by the Courant instability. However, where there is advection of density contrasts in a gravitational field, over-advection can lead to numerically induced flow oscillations. We derive analytic results for the maximum stable timestep size in two cases: a free surface over a fluid of uniform density, and a free surface kept level by sedimentation/erosion, but with a density gradient in the underlying medium. Using parameters appropriate to the Earth's crust we show that the density-contrast instability occurs for timesteps larger than 3000 years for the constant-density case. For a fluid with a density gradient of 10 kg/m³ per km the solution is stable for timesteps up to about 200,000 years if full erosion/sedimentation is implemented.     

Canonical dual finite element method for solving post-buckling problems of a large deformation elastic beam

This paper presents a canonical dual mixed finite element method for the post-buckling analysis of planar beams with large elastic deformations. The mathematical beam model employed in the present work was introduced by Gao in 1996, and is governed by a fourth-order non-linear differential equation. The total potential energy associated with this model is a non-convex functional and can be used to study both the pre- and the post-buckling responses of the beams. Using the so-called canonical duality theory, this non-convex primal variational problem is transformed into a dual problem. In a proper feasible space, the dual variational problem corresponds to a globally concave maximization problem. A mixed finite element method involving both the transverse displacement field and the stress field as approximate element functions is derived from the dual variational problem and used to compute global optimal solutions. Numerical applications are illustrated by several problems with different boundary conditions.     

Nonlinear quasi-conforming finite element method

The nonlinear quasi-conforming FEM is presented based on the basic concept of the quasiconforming finite element. First, the incremental principle of stationary potential energy is discussed. Then, the formulation process of the nonlinear quasi-conforming FEM is given. Lastly, two computational examples of shells are given.     

随机结构有限元分析的递推求解方法

提出了一种新的谱随机有限元分析方法——递推求解方法。该方法将随机结构的随机响应表示成非正交多项式展式,建立了和摄动法类似的一系列确定的递推方程,并通过确定性有限元方法对这些递推方程进行静力问题求解。算例表明,当随机量出现较大涨落时,计算结果相对于传统摄动法有不小的改进。     

Finite element modelling of natural-convection-controlled change of phase

The problem of phase change in the presence of natural convection has been investigated. A model has been proposed based on the treatment of the release/absorption of latent heat as a heat source/sink in combination with the standard Galerkin finite element method with a primitive variable formulation on a fixed grid. To demonstrate the capabilities of the model, three cases of phase change of an aluminium alloy in the presence of natural convection arc considered, i.e. solidification, melting and combined solidification and melting. The solidification of water in a square cavity is modelled as another example, taking into account the density extremum, and the results are compared with a previously published work.     

应力函数及其对偶理论在有限元中的应用

借助于Cosserat连续介质模型，探讨了应力函数和位移对避免有限元C$^{1}$ 连续性困难的互补性作用. 通过对应力函数对偶理论的深入分析，为将应力函数列式得到的 余能单元转化为具有一般位移自由度的势能单元提供了严格的理论基础，在此基础上， 给出应用应力函数构造有限元的一般方法.