An EBE finite element method for simulating nonlinear flows in rotating spheroidal cavities

Many planetary and astrophysical bodies are rotating rapidly, fluidic and, as a consequence of rapid rotation, in the shape of an ablate spheroid. We present an efficient element‐by‐element (EBE) finite element method for the numerical simulation of nonlinear flows in rotating incompressible fluids that are confined in an ablate spheroidal cavity with arbitrary eccentricity. Our focus is placed on temporal and spatial tetrahedral discretization of the EBE finite element method in spheroidal geometry, the EBE parallelization scheme and the validation of the nonlinear spheroidal code via both the constructed exact nonlinear solution and the special resonant forcing in the inviscid limit. Copyright © 2009 John Wiley & Sons, Ltd.     

Analysis of the electromechanical behavior of ferroelectric ceramics based on a nonlinear finite element model

A nonlinear finite element (FE) model based on domain switching was proposed to study the electromechanical behavior of ferroelectric ceramics. The incremental FE formulation was improved to avoid any calculation instability. The problems of mesh sensitivity and convergence, and the efficiency of the proposed nonlinear FE technique have been assessed to illustrate the versatility and potential accuracy of the said technique. The nonlinear electromechanical behavior, such as the hysteresis loops and butterfly curves, of ferroelectric ceramics subjected to both a uniform electric field and a point electric potential has been studied numerically. The results obtained are in good agreement with those of the corresponding theoretical and experimental analyses. Furthermore, the electromechanical coupling fields near (a) the boundary of a circular hole, (b) the boundary of an elliptic hole and (c) the tip of a crack, have been analyzed using the proposed nonlinear finite element method (FEM). The proposed nonlinear electromechanically coupled FEM is useful for the analysis of domain switching, deformation and fracture of ferroelectric ceramics.The project supported by the National Natural Science Foundation of China (10025209, 10132010 and 90208002), the Research Grants of the Council of the Hong Kong Special Administrative Region, China (HKU7086/02E) and the Key Grant Project of the Chinese Ministry of Education (0306)     

An optical method to assess electromechanical coupling in ferroelectric ceramics

A new technique is described, which allows the assessment of elastic and inelastic regions around a macroscopic defect in ferroelectric-ferroelastic ceramics. The accuracy and robustness of the method are demonstrated on a PZT plate with a centered hole subjected to uni-axial compressive stresses. From the electrical potential distribution on the sample surface, the mechanical response of the material is obtained at different load levels.     

An Euler-Bernoulli-like finite element method for Timoshenko beams

In this paper a new finite element approach for the solution of the Timoshenko beam is shown. Similarly to the Euler-Bernoulli beam theory, it has been considered a single fourth order differential equation governs the equilibrium of the Timoshenko beam. The results obtained by this approach are very good, both in terms of accuracy and computational effort.     

An adaptive finite element method for forced convection

This paper presents an adaptive finite element method to solve forced convective heat transfer. Solutions are obtained in primitive variables using a high-order finite element approximation on unstructured grids. Two general-purpose error estimators are developed to analyse finite element solutions and to determine the characteristics of an improved mesh which is adaptively regenerated by the advancing front method. The adaptive methodology is validated on a problem with a known analytical solution. The methodology is then applied to heat transfer predictions for two cases of practical interest. Predictions of the Nusselt number compare well with measurements and constitute an improvement over previous results. © 1997 John Wiley & Sons, Ltd.     

A dynamic finite element method for inhomogeneous deformation and electromechanical instability of dielectric elastomer transducers

We present a three-dimensional nonlinear finite element formulation for dielectric elastomers. The mechanical and electrical governing equations are solved monolithically using an implicit time integrator, where the governing finite element equations are given for both static and dynamic cases. By accounting for inertial terms in conjunction with the Arruda–Boyce rubber hyperelastic constitutive model, we demonstrate the ability to capture the various modes of inhomogeneous deformation, including pull-in instability and wrinkling, that may result in dielectric elastomers that are subject to various forms of electrostatic loading. The formulation presented here forms the basis for needed computational tools that can elucidate the electromechanical behavior and properties of dielectric elastomers that are used for engineering applications.     

一种适用于板料成形过程分析的自适应有限元方法

本文主要讨论在板料成形过程的动态显式有限元分析中,有限元网格的丙分割技术。单元为四节点壳单元,板料的材料特性假定满足Ｈｉｌｌ各向异性的弹塑性准则,采用自适应ｈ－方案,通过细化和聚合单元调整网格的疏密,模具被假定为由刚性小平面组成,与同一单元的节点发生接触处的模具表面的法线之间的夹角作为单元再分的判据。通过对方表盒冲压成形过程的计算,说明该方法是有效的。     

A finite element model for rate-dependent behavior of ferroelectric ceramics

In this article, materials within a crystallite are modeled by continuum particles consisting of various types of ferroelectric variants which are characterized by their mass fractions. The constitutive behavior of each type of variant is characterized by a proposed Helmholtz free energy potential. Polarization switching is modeled by continuous changes of mass fractions which are governed by a onset criterion and a kinetic relation. A finite element algorithm is developed using the virtual work principle. The simulated results on the rate dependence in the polarization and strain responses to applied alternating electric field of different frequencies are in qualitative consistence with experimental observations. The rate-dependent behavior is explained in terms of changes of mass fractions of the variants that polarization switching involves, in response to the loading programs of different loading rates.     

An iterative parallel algorithm of finite element method

In this paper, a parallel algorithm with iterative form for solving finite element equation is presented. Based on the iterative solution of linear algebra equations, the parallel computational steps are introduced in this method. Also by using the weighted residual method and choosing the appropriate weighting functions, the finite element basic form of parallel algorithm is deduced. The program of this algorithm has been realized on the ELXSI-6400 parallel computer of Xi'an Jiaotong University. The computational results show the operational speed will be raised and the CPU time will be cut down effectively. So this method is one kind of effective parallel algorithm for solving the finite element equations of large-scale structures.     

A 3D multi-field element for simulating the electromechanical coupling behavior of dielectric elastomers

We propose a multi-field implicit finite element method for analyzing the electromechanical behavior of dielectric elastomers. This method is based on a four-field variational principle, which includes displacement and electric potential for the electromechanical coupling analysis, and additional independent fields to address the incompressible constraint of the hyperelastic material. Linearization of the variational form and finite element discretization are adopted for the numerical implementation. A general FEM program framework is developed using C ++ based on the open-source finite element library deal.II to implement this proposed algorithm. Numerical examples demonstrate the accuracy, convergence properties, mesh-independence properties, and scalability of this method. We also use the method for eigenvalue analysis of a dielectric elastomer actuator subject to electromechanical loadings. Our finite element implementation is available as an online supplementary material.     

An efficient 3D stochastic finite element method

Real life structural systems are characterized by their inherent or externally induced uncertainties in the design parameters. This study proposes a stochastic finite element tool efficient to take account of these uncertainties. Here uncertain structural parameter is modeled as homogeneous Gaussian stochastic field and commonly used two-dimensional (2D) local averaging technique is extended and generalized for 3D random field. This is followed by Cholesky decomposition of respective covariance matrix for digital simulation. By expanding uncertain stiffness matrix about its reference value, the Neumann expansion method is introduced blended with direct Monte Carlo simulation. This approach involves decomposition of stiffness matrix only once for the entire simulated structure. Thus substantial saving of CPU time and also the scope of tackling several stochastic fields simultaneously are the basic advantages of the proposed algorithm. Accuracy and efficiency of this method with reference to example problem is also studied here and numerical results validate its superiority over direct simulation method or first-order perturbation approach.     

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。     

An extended finite element method for the simulation of particulate viscoelastic flows

We present an extended finite element method (XFEM) for the direct numerical simulation of the flow of viscoelastic fluids with suspended particles. For moving particle problems, we devise a temporary arbitrary Lagrangian–Eulerian (ALE) scheme which defines the mapping of field variables at previous time levels onto the computational mesh at the current time level. In this method, a regular mesh is used for the whole computational domain including both fluid and particles. A temporary ALE mesh is constructed separately and the computational mesh is kept unchanged throughout the whole computations. Particles are moving on a fixed Eulerian mesh without any need of re-meshing. For mesh refinements around the interface, we combine XFEM with the grid deformation method, in which nodal points are redistributed close to the interface while preserving the mesh topology. Our method is verified by comparing with the results of boundary fitted mesh problems combined with the conventional ALE scheme. The proposed method shows similar accuracy compared with boundary fitted mesh problems and superior accuracy compared with the fictitious domain method. If the grid deformation method is combined with XFEM, the required computational time is reduced significantly compared to uniform mesh refinements, while providing mesh convergent solutions. We apply the proposed method to the particle migration in rotating Couette flow of a Giesekus fluid. We investigate the effect of initial particle positions, the Weissenberg number, the mobility parameter of the Giesekus model and the particle size on the particle migration. We also show two-particle interactions in confined shear flow of a viscoelastic fluid. We find three different regimes of particle motions according to initial separations of particles.     

An all-at-once algebraic multigrid method for finite element discretizations of Stokes problem

We consider numerical solution of finite element discretizations of the Stokes problem. We focus on the transform-then-solve approach, which amounts to first apply a specific algebraic transformation to the linear system of equations arising from the discretization, and then solve the transformed system with an algebraic multigrid method. The approach has recently been applied to finite difference discretizations of the Stokes problem with constant viscosity, and has recommended itself as a robust and competitive solution method. In this work, we examine the extension of the approach to standard finite element discretizations of the Stokes problem, including problems with variable viscosity. The extension relies, on one hand, on the use of the successive over-relaxation method as a multigrid smoother for some finite element schemes. On the other hand, we present strategies that allow us to limit the complexity increase induced by the transformation. Numerical experiments show that for stationary problems our method is competitive compared to a reference solver based on a block diagonal preconditioner and MINRES, and suggest that the transform-then-solve approach is also more robust. In particular, for problems with variable viscosity, the transform-then-solve approach demonstrates significant speed-up with respect to the block diagonal preconditioner. The method is also particularly robust for time-dependent problems whatever the time step size.     

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.     

黏弹-Perzyna黏塑性有限元法应力更新隐式算法

黏弹-黏塑性耦合模型的黏弹性部分由弹簧、黏壶和Kelvin链串联而成,黏塑性部分为双曲线型DruckerPrager屈服函数、各向同性硬化和Perzyna黏塑性流动模型。基于黏弹性蠕变柔度,通过定义与弹性问题相对应的与时间增量相关的黏弹性剪切模量和体积模量,导出增量递推形式的本构方程。为保证算法的收敛和稳定性,把Perzyna黏塑性流动方程转化为与弹塑性相似的一致性条件,建立黏塑性增量因子单侧逼近其收敛值的N-R迭代算法。最后,给出应力更新完全隐式算法和最终计算公式。分别采用黏弹性、黏弹-塑性和黏弹-黏塑性本构关系对一地基蠕变模型进行三维有限元分析和比较,结果表明,本文算法具有较高的计算效率和稳定性。     

Gauge finite element method for incompressible flows

A finite element method for computing viscous incompressible flows based on the gauge formulation introduced in [Weinan E, Liu J‐G. Gauge method for viscous incompressible flows. Journal of Computational Physics (submitted)] is presented. This formulation replaces the pressure by a gauge variable. This new gauge variable is a numerical tool and differs from the standard gauge variable that arises from decomposing a compressible velocity field. It has the advantage that an additional boundary condition can be assigned to the gauge variable, thus eliminating the issue of a pressure boundary condition associated with the original primitive variable formulation. The computational task is then reduced to solving standard heat and Poisson equations, which are approximated by straightforward, piecewise linear (or higher‐order) finite elements. This method can achieve high‐order accuracy at a cost comparable with that of solving standard heat and Poisson equations. It is naturally adapted to complex geometry and it is much simpler than traditional finite element methods for incompressible flows. Several numerical examples on both structured and unstructured grids are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Tools for simulating non-stationary incompressible flow via discretely divergence-free finite element models

We develop simulation tools for the non-stationary incompressible 2D Navier--Stokes equations. The most important components of the finite element code are: the fractional step ?-scheme, which is of second-order accuracy and strongly A-stable, for the time discretization; a fixed point defect correction method with adaptive step length control for the non-linear problems (stationary Navier-Stokes equations); a modified upwind discretization of higher-order accuracy for the convective terms. Finally, the resulting nonsymmetric linear subproblems are treated by a special multigrid algorithm which is adapted to the quadrilateral non-conforming discretely divergence-free finite elements. For the graphical postprocess we use a fully non-stationary and interactive particle-tracing method. With extensive test calculations we show that our method is a candidate for a ‘black box’ solver.