Finite element method for piezoelectric vibration

A finite element formulation which includes the piezoelectric or electroelastic effect is given. A strong analogy is exhibited between electric and elastic variables, and a ‘stiffness’ finite element method is deduced. The dynamical matrix equation of electroelasticity is formulated and found to be reducible in form to the well-known equation of structural dynamics, A tetrahedral finite element is presented, implementing the theorem for application to problems of three-dimensional electroelasticity.     

Kinematics and dynamics of rigid and flexible mechanisms using finite elements and quaternion algebra

A technique for representing large finite rotations in terms of only three independent parameters, the conformal rotation vector, is described and applied to the finite element formulation of 3-D mechanisms problems. A beam finite element that takes into account large finite rotations and various types of rigid joints have been developed. Some test examples which demonstrate the applicability of the proposed technique are presented.     

Stress recovery and error estimation for the scaled boundary finite‐element method

The scaled boundary finite‐element method is a novel semi‐analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. This paper develops a stress recovery procedure based on a modal interpretation of the scaled boundary finite‐element method solution process, using the superconvergent patch recovery technique. The recovered stresses are superconvergent, and are used to calculate a recovery‐type error estimator. A key feature of the procedure is the compatibility of the error estimator with the standard recovery‐type finite element estimator, allowing the scaled boundary finite‐element method to be compared directly with the finite element method for the first time. A plane strain problem for which an exact solution is available is presented, both to establish the accuracy of the proposed procedures, and to demonstrate the effectiveness of the scaled boundary finite‐element method. The scaled boundary finite‐element estimator is shown to predict the true error more closely than the equivalent finite element error estimator. Unlike their finite element counterparts, the stress recovery and error estimation techniques work well with unbounded domains and stress singularities. Copyright © 2002 John Wiley & Sons, Ltd.     

Development and validation of a 3D computational tool to describe concrete behaviour at mesoscale. Application to the alkali-silica reaction

A finite element analysis of the large deformation of three-dimensional polycrystals is presented using pixel-based finite elements as well as finite elements conforming with grain boundaries. The macroscopic response is obtained through volume-averaging laws. A constitutive framework for elasto-viscoplastic response of single crystals is utilized along with a fully-implicit Lagrangian finite element algorithm for modeling microstructure evolution. The effect of grain size is included by considering a physically motivated measure of lattice incompatibility which provides an updated shearing resistance within grains. A domain decomposition approach is adopted for parallel computation to allow efficient large scale simulations. Conforming grids are adopted to simulate flexible and complex shapes of grains. The computed mechanical properties of polycrystals are shown to be consistent with experimental results for different grain sizes.     

Finite element analysis of sinusoidal small-amplitude vibrations in deformed viscoelastic solids. Part I: Theoretical development

A general method is presented for the isothermal mechanical analysis of incompressible material solids in which a small-amplitude time harmonic oscillation is superposed on a static finite deformation field. Material behaviour is assumed to be of the ‘fading memory’ type describable by the finite linear viscoelasticity theory of Coleman and Noll.⁴ Existing finite element codes, that treat finite elasticity material behaviour, can be extended with minor modifications to treat the subject problem. A specialized form of the finite linear viscoelasticity constitutive equations proposed by Morman⁸ has been implemented in the MARC nonlinear finite element program for predicting the response of statically deformed elastomeric components to small amplitude vibrations. Numerical results obtained with MARC for the static force-deflection behaviour and dynamic (complex) stiffness for a viscoelastic cylinder subject to combined axial and twisting pre-loads, are in excellent agreement with corresponding analytical results.     

Application of finite element methods to thermohydrodynamic lubrication

A finite element formulation is presented for the equations governing the steady thermohydrodynamic behaviour of liquid lubricated bearings. This formulation permits application of the iterative solution scheme to bearings of arbitrary geometry. A generalized Reynolds equation resulting from the combination of the mass and momentum conservation equations is cast into variational form and used to derive general finite element equations. The method of weighted residuals with Galerkin's criterion is used to generate finite element matrix equations for the thermal energy equation. In addition to the finite element formulation, a discussion of appropriate finite difference techniques is also given for problems without complex geometry. As an example, the formulations are applied to obtain numerical solutions for a three-dimensional sector thrust bearing operating in the thermohydrodynamic regime. Pressure, velocity and temperature distributions are give, and the thermohydrodynamic solutions are compared with the results of classical isothermal theory.     

Some properties of linear strain triangles and optimal finite element models

A detailed study of the linear strain triangle (LST), which is very applicable in plane stress finite element analysis, is presented. Based on analytical stiffnesses, it is shown that the LST stiffnesses are easily obtainable from the stiffnesses of the corresponding simple constant strain triangle (CST). The optimization of finite element models is related to the configuration of the elements, and this analysis is also based upon the LST element. A procedure for changing to an improved finite element model is presented which does not require re-analysis. Several examples are discussed.     

A Posteriori Error Estimates of Lowest Order Raviart-Thomas Mixed Finite Element Methods for Bilinear Optimal Control Problems

A Raviart-Thomas mixed finite element discretization for general bilinear optimal control problems is discussed. The state and co-state are approximated by lowest order Raviart-Thomas mixed finite element spaces, and the control is discretized by piecewise constant functions. A posteriori error estimates are derived for both the coupled state and the control solutions, and the error estimators can be used to construct more efficient adaptive finite element approximations for bilinear optimal control problems. An adaptive algorithm to guide the mesh refinement is also provided. Finally, we present a numerical example to demonstrate our theoretical results.     

Application of the control volume mixed finite element method to a triangular discretization

A two‐dimensional control volume mixed finite element method is applied to the elliptic equation. Discretization of the computational domain is based in triangular elements. Shape functions and test functions are formulated on the basis of an equilateral reference triangle with unit edges. A pressure support based on the linear interpolation of elemental edge pressures is used in this formulation. Comparisons are made between results from the standard mixed finite element method and this control volume mixed finite element method. Published 2011. This article is a US Government work and is in the public domain in the USA.     

Smooth finite element methods: Convergence,accuracy and properties

A stabilized conforming nodal integration finite element method based on strain smoothing stabilization is presented. The integration of the stiffness matrix is performed on the boundaries of the finite elements. A rigorous variational framework based on the Hu–Washizu assumed strain variational form is developed. We prove that solutions yielded by the proposed method are in a space bounded by the standard, finite element solution (infinite number of subcells) and a quasi‐equilibrium finite element solution (a single subcell). We show elsewhere the equivalence of the one‐subcell element with a quasi‐equilibrium finite element, leading to a global a posteriori error estimate. We apply the method to compressible and incompressible linear elasticity problems. The method can always achieve higher accuracy and convergence rates than the standard finite element method, especially in the presence of incompressibility, singularities or distorted meshes, for a slightly smaller computational cost. It is shown numerically that the one‐cell smoothed four‐noded quadrilateral finite element has a convergence rate of 2.0 in the energy norm for problems with smooth solutions, which is remarkable. For problems with rough solutions, this element always converges faster than the standard finite element and is free of volumetric locking without any modification of integration scheme. Copyright © 2007 John Wiley & Sons, Ltd.     

A two-step approach to finite element ordering

A two-step approach to finite element ordering is introduced. The scheme involves ordering of the finite elements first, based on their adjacency, followed by a local numbering of the nodal variables. The ordering of the elements is performed by the Cuthill-McKee algorithm. This approach takes into consideration the underlying structure of the finite element mesh, and may be regarded as a ‘natural’ finite element ordering scheme. The experimental results show that this two-step scheme is more efficient than the reverse Cuthill-McKee algorithm applied directly to the nodes, in terms of both execution time and the number of fill-in entries, particularly when higher order finite elements are used. In addition to its efficiency, the two-step approach increases modularity and flexibility in finite element programs, and possesses potential application to a number of finite element solution methods.     

A Treatment of wall boundaries for turbulent flows by the use of a transmission finite element method

A method to connect momentum Navier-Stokes equations with the universal law of the wall using the finite element method is developed for turbulent wall flows. This method is based on a domain decomposition of the fluid into subdomains near a solid boundary where the law of the wall is valid. A transmission formulation is introduced to match these regions and a new class of boundary finite element is used. This finite element takes into account the near-wall profile of the velocity and the transmission conditions. Computational results are presented for Poiseuille flow and flow over a backward-facing step.     

Application of the finite element method to an optimal control problem

The finite element method is applied to a typical problem in the optimal control theory. A finite element formulation is first set up, and then applied, as an example, to a problem of minimizing the transfer time of a low-thrust rocket vehicle between the orbits of Earth and Mars. A procedure which combines the gradient projection technique with the Newton-Raphson method is introduced to obtain numerically the solution of the minimum time transfer problem from the discretized functional. Numerical results show that the application of the finite element method to optimal control problems seems promising as well as encouraging.     

Comparative study of different nonconserving time integrators for wave propagation in hyperelastic waveguides

This paper addresses the formulation and numerical efficiency of various numerical models of different nonconserving time integrators for studying wave propagation in nonlinear hyperelastic waveguides. The study includes different nonlinear finite element formulations based on standard Galerkin finite element model, time domain spectral finite element model, Taylor–Galerkin finite element model, generalized Galerkin finite element model and frequency domain spectral finite element model. A comparative study on the computational efficiency of these different models is made using a hyperelastic rod model, and the optimal computational scheme is identified. The identified scheme is then used to study the propagation of transverse and longitudinal waves in a Timoshenko beam with Murnaghan material nonlinearity.     

有限变形下固-液耦合渗透固结问题的数值分析

基于多相连续介质力学的理论,把土体抽象为叠合连续体,建立了有限变形下固-液耦合渗透固结问题的数学模型。基于IFEPG有限元自动生成系统,编制了有限变形和小变形的计算程序。通过实例对有限变形和小变形理论下的压力场、位移场和应力场等进行了分析。     

Optimal stress locations in finite element models

The existence of optimal points for calculating accurate stresses within finite element models is discussed. A method for locating such points is proposed and applied to several popular finite elements.     

非线性系统的有限时间扩张状态观测器的设计

针对非线性系统的不确定项和外部干扰问题,提出了一种有限时间扩张状态观测器的设计方案,实现了非线性系统不确定项和外部干扰的有限时间估计。与传统的线性扩张状态观测器相比,有限时间扩张状态观测器基于终端滑模的思想而设计,其非线性项的引入保障了扩张状态的有限时间估计,进而辅助设计控制器,可以提高系统的鲁棒性能。利用Lyapunov有限时间稳定性理论得到该观测误差系统收敛的充分条件。最后,实例仿真进一步验证了该策略的有效性。     

Radial point interpolation based finite difference method for mechanics problems

A radial point interpolation based finite difference method (RFDM) is proposed in this paper. In this novel method, radial point interpolation using local irregular nodes is used together with the conventional finite difference procedure to achieve both the adaptivity to irregular domain and the stability in the solution that is often encountered in the collocation methods. A least‐square technique is adopted, which leads to a system matrix with good properties such as symmetry and positive definiteness. Several numerical examples are presented to demonstrate the accuracy and stability of the RFDM for problems with complex shapes and regular and extremely irregular nodes. The results are examined in detail in comparison with other numerical approaches such as the radial point collocation method that uses local nodes, conventional finite difference and finite element methods. Copyright © 2006 John Wiley & Sons, Ltd.     

Static Analysis of Shear Actuated Piezo-Electric Beams via Hierarchical FEM Theories

Several one-dimensional finite elements for the static analysis of shear actuated piezo-electric three-dimensional beams are presented. A generic expression of stiffness and mass matrices is obtained through a Unified Formulation. The derivation is general regardless of the approximation order of the displacements and the electric potential over the cross-section and the number of nodes along the axial direction. A Lagrange’s polynomials based layer-wise approximation is used. Several mechanical boundary conditions and sensor and actuator configurations are investigated. Results are assessed towards three-dimensional finite element solutions. It is demonstrated that the proposed class of finite elements is able to yield very accurate results.