Modeling of coupling mechanisms and frequency separation in double disk resonators

We present modeling expressions for the coupling mechanisms in vertically stacked disk resonators. Using lumped modeling techniques, analytic expressions for coupling mechanisms in a stack of two disks are derived. The modeling results are compared to those of high-fidelity finite element analyses and show that the analytic expressions derived can be used to predict frequency separation between radial extensional modes. The established model is useful for gaining insight into parameter sensitivity when designing a stacked double disk resonator.     

Selecting the optimum engineering model for the frequency response of fcc nanowire resonators

The full potential of the nanoelectromechanical systems, NEMS, as one of the leading examples among the new-generation sensing technologies, is yet to be realized. One of the main challenges on the road is the mechanical modeling of their core elements, the tiny mechanical building blocks such as the nanowire resonators. The success of the engineering design of such miniaturized systems will depend heavily on the availability of accurate mechanistic models with the least possible computational cost. Although a variety of models are available for this purpose, the boundaries between their admissible domains remain rather vague. For example, analytical approaches including Euler–Bernoulli and Timoshenko beam theories provide closed-form solutions and work reasonably well for moderate nanowire geometries, and hence, they are frequently utilized in the literature. However, their validity in the case of extreme surface-to-volume ratios remains questionable. Classical finite element method can partially be used to address these deficiencies. On the other hand, molecular dynamics provide accurate results, while nanowire geometries studied with this computationally demanding technique usually remain confined to dimensions below those of practical interest. To address these issues, a benchmarking study among analytical and numerical techniques is carried out, where Surface Cauchy–Born theory serves as the reference. Using gold nanowires with different dimensions and boundary conditions, it is observed that analytical models are applicable within a length-to-thickness ratio range of 7–11 in the fixed–fixed configuration, whereas they can be used safely within a length-to-thickness ratio range of less than 25 in the fixed–free configuration. Deviations as high as 50% are encountered for length-to-thickness ratios exceeding 11 for both the analytical approach and the classical finite element method in the fixed–fixed structure. The deviations are quantitatively linked to the dominance of the surface effect through the use of the Surface Cauchy–Born model. For length-to-thickness ratios less than 7, the lack of cross-sectional deformations in analytical treatment is also observed to lead to high deviations for the fixed–fixed configuration through the comparison with higher-order beam theories. Results are verified with silver nanowires as well. The work provides a guideline for selecting the optimum mechanical model given the nanowire resonator dimensions and boundary conditions.     

A gradient model for finite strain elastoplasticity coupled with damage

This paper describes the formulation of an implicit gradient damage model for finite strain elastoplasticity problems including strain softening. The strain softening behavior is modeled through a variant of Lemaitre's damage evolution law. The resulting constitutive equations are intimately coupled with the finite element formulation, in contrast with standard local material models. A 3D finite element including enhanced strains is used with this material model and coupling peculiarities are fully described. The proposed formulation results in an element which possesses spatial position variables, nonlocal damage variables and also enhanced strain variables. Emphasis is put on the exact consistent linearization of the arising discretized equations.

A numerical set of examples comparing the results of local and the gradient formulations relative to the mesh size influence is presented and some examples comparing results from other authors are also presented, illustrating the capabilities of the present proposal.     

Automatic finite element formulation and assembly of hyperelastic higher order structural models

The formulation of higher order structural models and their discretization using the finite element method is difficult owing to their complexity, especially in the presence of nonlinearities. In this work a new algorithm for automating the formulation and assembly of hyperelastic higher-order structural finite elements is developed. A hierarchic series of kinematic models is proposed for modeling structures with special geometries and the algorithm is formulated to automate the study of this class of higher order structural models. The algorithm developed in this work sidesteps the need for an explicit derivation of the governing equations for the individual kinematic modes. Using a novel procedure involving a nodal degree-of-freedom based automatic assembly algorithm, automatic differentiation and higher dimensional quadrature, the relevant finite element matrices are directly computed from the variational statement of elasticity and the higher order kinematic model. Another significant feature of the proposed algorithm is that natural boundary conditions are implicitly handled for arbitrary higher order kinematic models. The validity algorithm is illustrated with examples involving linear elasticity and hyperelasticity.     

Finite element simulations of window Josephson junctions

This paper deals with the numerical simulation of the steady state two dimensional window Josephson junctions by finite element method. The model is represented by a sine-Gordon type composite PDE problem. Convergence and error analysis of the finite element approximation for this semilinear problem are presented. An efficient and reliable Newton-preconditioned conjugate gradient algorithm is proposed to solve the resulting nonlinear discrete system. Regular solution branches are computed using a simple continuation scheme. Numerical results associated with interesting physical phenomena are reported. Interface relaxation methods, which by taking advantage of special properties of the composite PDE, can further reduce the overall computational cost are proposed. The implementation and the associated numerical experiments of a particular interface relaxation scheme are also presented and discussed.     

LOCAL AND PARALLEL FINITE ELEMENT METHOD FOR THE MIXED NAVIER-STOKES/DARCY MODEL WITH BEAVERS-JOSEPH INTERFACE CONDITIONS

In this paper, we consider the mixed Navier-Stokes/Darcy model with BeaversJoseph interface conditions. Based on two-grid discretizations, a local and parallel finite element algorithm for this mixed model is proposed and analyzed. Optimal errors are obtained and numerical experiments are presented to show the efficiency and effectiveness of the local and parallel finite element algorithm.     

Formulation and evaluation of a finite element model for the linear analysis of stiffened composite cylindrical panels

A finite element model for linear static and free vibration analysis of composite cylindrical panels with composite stiffeners is presented. The proposed model is based on a cylindrical shell finite element, which uses a first-roder shear deformation theory. The stiffeners are curved beam elements based on Timoshenko and Saint-Venant assumptions for bending and torsion respectively. The two elements are developed in a cylindrical coordinate system and their stiffness matrices result from a hybrid-mixed formulation where the element assumed stress field is such that exact equilibrium equations are satisfied. The elements are free of membrane and shear locking with correct satisfaction of rigid body motions. Several examples dealing with stiffened isotropic and laminated plates and shells with eccentric as well as concentric stiffeners are analyzed showing the validity of the models.     

Finite element modelling of coupled diffusion and reaction problems in gas cooled nuclear reactors

The modelling of gaseous diffusion and reaction taking place within the open porosity of the graphite moderator in a carbon dioxide cooled nuclear reactor is described. The physical model leads to a set of coupled nonlinear partial differential equations which are solved by finite element techniques using a Gauss-Seidel iterative method. Test cases with typical pore geometries are analysed to demonstrate some important aspects of the system.     

The analyses of dynamic response and reliability for failure-dependent stochastic micro-resonator with thermoelastic coupling effects

A crucial measure for the design of high-performance micro-resonators is to consider the randomness of structural parameters when analyzing the structural system reliability. In this work, the stochastic dynamic response analysis and subsequently, a dynamic reliability assessment of the random micro-resonators are originally presented, where the thermoelastic coupling effects are freshly incorporated in the models proposed. The dynamic characteristics equation of the deterministic micro-resonator is firstly established based on the finite element method. The random dynamic characteristics of the resonator are then solved by implementing the left and right eigenvectors and the block Lanczos algorithm, and the random temperature field and structural random dynamic stress are also tackled. Afterwards, the overall structural reliability is investigated with a comprehensive consideration of the strength failure and frequency resonance failure, in which the Copula function is used for describing the dynamic correlation between two failure modes. Finally, the feasibility and rationality of the method put forward are demonstrated via a practically motivated example.     

An iterative elastic SBFE approach for collapse load analysis of inelastic structures

The paper proposes a novel numerical approach that incorporates the use of a modified elastic compensation method, within a polygon scaled boundary finite element framework, to determine the maximum load capacity of structures at plastic collapse. The distinctive feature of the proposed scheme is its effective computational ability in performing a series of successive elastic analyses by systematically adjusting elastic material properties of structures up to failure. The quadtree structural discretization within a polygon scaled boundary finite element platform enables model construction of sophisticated geometries at modest computing effort and thus the effective analysis of large-scale structures. The approach overcomes the challenges associated with stress singularity and locking phenomena under incompressibility conditions, even in the presence of high-order nonlinear yield loci. The robustness and accuracy of the proposed scheme are validated through a number of benchmarks and available practical engineering applications in 2D and 3D spaces. These illustrate the influences of some key algorithmic parameters, and the satisfaction of a lower-bound limit given by the present analysis method for a sufficiently fine discretization.     

Geometrically nonlinear analysis with a 4-node membrane element formulated by the quadrilateral area coordinate method

Recently, a 4-node quadrilateral membrane element AGQ6-I, has been successfully developed for analysis of linear plane problems. Since this model is formulated by the quadrilateral area coordinate method (QACM), a new natural coordinate system for developing quadrilateral finite element models, it is much less sensitive to mesh distortion than other 4-node isoparametric elements and free of various locking problems that arise from irregular mesh geometries. In order to extend these advantages of QACM to nonlinear applications, the total Lagrangian (TL) formulations of element AGQ6-I was established in this paper, which is also the first time that a plane QACM element being applied in the implicit geometrically nonlinear analysis. Numerical examples of geometrically nonlinear analysis show that the presented formulations can prevent loss of accuracy in severely distorted meshes, and therefore, are superior to those of other 4-node isoparametric elements. The efficiency of QACM for developing simple, effective and reliable serendipity plane membrane elements in geometrically nonlinear analysis is demonstrated clearly.     

Fundamental size dependent natural frequencies of non-uniform orthotropic nano scaled plates using nonlocal variational principle and finite element method

In the present study, a nonlocal continuum model based on the Eringen’s theory is developed for vibration analysis of orthotropic nano-plates with arbitrary variation in thickness. Variational principle and Ritz functions are employed to calculate the size dependent natural frequencies of non-uniform nano-plates on the basis of nonlocal classical plate theory (NCLPT). The Ritz functions eliminate the need for mesh generation and thus large degrees of freedom arising in discretization methods such as finite element (FE). Effect of thickness variation on natural frequencies is examined for different nonlocal parameters, mode numbers, geometries and boundary conditions. It is found that thickness variation accompanying small scale effect has a noticeable effect on natural frequencies of non-uniform plates at nano scale. Also a comparison with finite element solution is performed to show the ability of the Ritz functions in fast converging to the exact results. It is anticipated that presented results can be used as a helpful source in vibration design and frequency optimization of non-uniform small scaled plates.     

Modelling methane migration in soil

Based on Bear-Bachmat porous medium model, the governing equations for the migration of gases through soil from a buried source are presented. A finite element solution system for the equations is developed. The resulting model is capable of incorporating medium anistropy and inhomogeneity in an axisymmetric configuration. The model has facilities for including time-varying fluid properties and boundary conditions. Convergence of the solution is examined. Potential applications for the modelling of gas migration from waste burial sites and the evaluation of control mechanisms are discussed.     

Recent advances in the numerical modeling of constitutive relations

In this paper we present an overview of the recent developments in the area of numerical and finite element modeling of nonlinear constitutive relations. The paper discusses elastic, hyperelastic, elastoplastic and anisotropic plastic material models. In the hyperelastic model an emphasis is given to the method by which the incompressibility constraint is applied. A systematic and general procedure for the numerical treatment of hyperelastic model is presented. In the elastoplastic model both infinitesimal and large strain cases are discussed. Various concerns and implications in extending infinitesimal theories into large strain case are pointed out. In the anisotropic elastoplastic case, emphasis is given to the practicality of proposed theories and its feasible and economical use in the finite element environment.     

非均质气藏水平井三维渗流的产能预测方法

建立了非均质气藏水平井三维渗流产能预测的数学模型,采用有限元方法对其进行求解,求解时将地质模型区块中不同空间位置处的渗透率值以及其它气藏物性参数分别布置到有限元模型的相应位置处的网格中,从而体现了三维空间中气藏的非均质特性.分析了非均质性对气藏水平井产能的影响.结果表明,气藏的非均质性对水平井的产气量影响很大,在相同的生产压差下,存在高渗透带的气藏水平井产量明显高于均质气藏水平井的产量;高渗透带条数越多、渗透率越大,导致气藏内压力消耗越小,水平井产量越高.最后,结合松辽盆地徐深层气田的地质特点和储层特性,给出了该气田的水平井产能预测实例.研究方法符合气藏的实际情况,为气藏水平井,特别是非均质气藏水平井的产能预测提供了一种行之有效的方法.     

Mathematical modeling and analysis of a meta-plate for very low-frequency band gap

Aiming to attenuate very low-frequency flexural wave in a thin plate, this paper proposes a meta-plate model by periodically attaching high-static-low-dynamic-stiffness (HSLDS) resonators onto the thin plate. The HSLDS resonator consists of a linear spring with a negative stiffness mechanism (NSM) in parallel, and thus its stiffness can be adjusted to any low values within a range from zero to the stiffness of the linear spring. Using the plane-wave expansion (PWE) method, and considering the linearized stiffness of the resonator, the dispersion relation of the meta-plate can be derived and the band structure can be obtained. A dynamic model of the meta-plate with the original nonlinear stiffness under external excitations is also established. The numerical simulations are carried out by the Galerkin method to evaluate the band structure and study the propagation of the flexural wave along the meta-plate with different azimuth angles. The analytical results exhibit good agreement with the numerical ones, which indicates that the proposed meta-plate can create a very low-frequency band gap.     

An improved modeling approach for circular flexure hinges with application to geometry optimization

Within this paper, a modeling approach for flexure hinges based on the Euler-Bernoulli beam theory for beams of variable cross section is investigated in a static analysis. The proposed approach is implemented in a finite beam element routine, for which two different discretizations are discussed. The results are compared to a full scale three dimensional model. It is shown that a circular flexure hinge cannot be modeled accurately with one element. An improved model with three elements across the flexure hinge length is presented which shows excellent accordance with the reference model. A geometry optimization is realized based on the improved, low-DOF model. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Characteristic Finite Volume Element Method for the Air Pollution Model

In this article, a characteristic finite volume element method is presented for solving air pollution models. The convection term is discretized using the characteristic method and diffusion term is approximated by finite volume element method. Compared with standard finite volume element method, our proposed method is more accurate and efficient, especially suitable to solve convection-dominated problems. The proposed numerical schemes are analyzed for convergence in L ₂ norm. Some numerical results are presented to demonstrate the efficiency and accuracy of the method.     

固体中短波传播的单位分解有限元法

提出了固体中短波传播数值模拟的单位分解有限元法．有限元空间由形成单位分解的标准等参有限元形函数乘以定义为局部子空间基函数的特殊形函数构成．特殊形函数使试空间中包含了关于波动方程的已有知识,因而在单个单元内能近似地再现高度振荡性质．数值例题显示了所提出单位分解有限元在计算精度和效率上的良好性能．     

模糊运算和模糊有限元静力控制方程的求解

根据模糊数的区间形式表达和区间运算的性质,给出了模糊数和模糊变量的运算规则.据此并依据区间有限元理论,提出了结构模糊有限元静力控制方程的几种求解方法.方法可根据输入模糊数的隶属函数,给出结构响应量的可能性分布.且计算量小,易于实施.算例分析说明了方法是实用和可行的.