振动杆有限差分模型的一类特征值反问题

1引言杆是重要的工程构件之一,具有分布质量的杆的纵向振动由下面的偏微分方程描述:     

Calculation of stresses in the finite elements of deformed constructions

Under study is the problem of deformation of a curved rod in the form of a circular arc. Using the previously developed version of the functions of the form of a curvilinear finite element, we construct a solution that differs slightly from the exact one with respect to displacements even for few elements; however, the bending moment is calculated with a greater error. As a result of the direct integration of the equations of the problem for this rod, there are constructed some modified functions of the form from which an “exact” stiffness matrix is calculated. These functions yield the construction of the functions of the form with a parameter and the reason is clarified why the calculation of the force factors by differentiation of such functions fails to be exact. Also, we demonstrate a possible nonuniqueness of the obtained results for the stresses under the same errors in the stiffness matrix.     

An evolution equation based approach to topology optimization

The objective of topology optimization is to find a mechanical structure with maximum stiffness and minimal amount of used material for given boundary conditions [2]. There are different approaches. Either the structure mass is held constant and the structure stiffness is increased or the amount of used material is constantly reduced while specific conditions are fulfilled. In contrast, we focus on the growth of a optimal structure from a void model space and solve this problem by introducing a variational problem considering the spatial distribution of structure mass (or density field) as variable [3]. By minimizing the Gibbs free energy according to Hamilton's principle in dynamics for dissipative processes, we are able to find an evolution equation for the internal variable describing the density field. Hence, our approach belongs to the growth strategies used for topology optimization. We introduce a Lagrange multiplier to control the total mass within the model space [1]. Thus, the numerical solution can be provided in a single finite element environment as known from material modeling. A regularization with a discontinuous Galerkin approach for the density field enables us to suppress the well-known checkerboarding phenomena while evaluating the evolution equation within each finite element separately [4]. Therefore, the density field is no additional field unknown but a Gauß-point quantity and the calculation effort is strongly reduced. Finally, we present solutions of optimized structures for different boundary problems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Development of size-dependent consistent couple stress theory of Timoshenko beams

In this paper, a linear size-dependent Timoshenko beam model based on the consistent couple stress theory is developed to capture the size effects. The extended Hamilton's principle is utilized to obtain the governing differential equations and boundary conditions. The general form of boundary conditions and the concentrated loading are employed to determine the exact static/dynamic solution of the beam. Utilizing this solution for the beam's deformation and rotation, the exact shape functions of the consistent couple stress theory (C-CST) is extracted, which leads to the stiffness and mass matrices of a two-node C-CST finite element beam. Due to the complexity and high computational cost of using the exact solution's shape functions, in addition to the Ritz approximate solution, a two primary variable finite element model of C-CST is proposed, and the corresponding general deformation and rotation fields, shape functions, mass and stiffness matrices are calculated. The C-CST is validated by comparing the prediction of different beam models for a benchmark problem. For the fully and partially clamped cantilever, and free-free beams, the size dependency of the formulations is investigated. The static solutions of the classical and consistent couple stress Timoshenko beam models are compared, and a criterion for selecting the proper model is proposed. For a wide range of material properties, the relation between the beam length and length scale parameter is derived. It is shown that the validity domain of the consistent couple stress Timoshenko model barely depends on the beam's constituent material.     

Discrete maximum principles for finite element solutions of some mixed nonlinear elliptic problems using quadratures

The discrete maximum principles are proved for finite element solutions of some nonlinear elliptic problems with mixed boundary conditions. The effect of quadrature rules, used for the construction of the stiffness matrices, is taken into account.     

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.     

Dynamics of a rod undergoing a longitudinal impact by a body

A longitudinal elastic impact caused by a body on a thin rod is considered. The results of theoretical, finite element, and experimental approaches to solving the problem are compared. The theoretical approach takes into account both the propagation of longitudinal waves in the rod and the local deformations described in the Hertz model. This approach leads to a differential equation with a delayed argument. The three-dimensional dynamic problem is considered in terms of the finite element approach in which the wave propagation and local deformation are automatically taken into account. A benchmark test of these two approaches showed a complete qualitative and satisfactory quantitative agreement of the results concerning the contact force and the impact time. In the experiments, only the impact time was determined. The comparison of the measured impact time with the theoretical and finite element method’s results was satisfactory. Owing to the fact that the tested rod was relatively short, the approximate model with two degrees of freedom was also developed to calculate the force and the impact time. The problem of excitation of transverse oscillation after the rebound of the impactor off the rod is solved. For the parametric resonance, the motion has a character of beats at which the energy of longitudinal oscillation is transferred into the energy of transverse oscillation and vice versa. The estimate for the maximum possible amplitude of transverse oscillation is obtained.     

Pre- and post-processing for the finite element method

The finite element method provides a powerful procedure to mathematically model physical phenomena. The technique is numerically formulated and is effectively used on a broad range of computers. The method has increased in both popularity and functionality with the development of user friendly pre- and post-processing software. Pre-processing software is used to create the model, generate an appropriate finite element grid, apply the appropriate boundary conditions, and view the total model. Post-processing provides visualization of the computed results. This paper addresses the pertinent issues of pre- and post-processing for finite element analysis. It reviews the capabilities that are provided by pre- and post-processors and suggests enhancements and new features that will likely be developed in the near future.     

Fast simplicial finite element algorithms using Bernstein polynomials

Fast algorithms for applying finite element mass and stiffness operators to the B-form of polynomials over d-dimensional simplices are derived. These rely on special properties of the Bernstein basis and lead to stiffness matrix algorithms with the same asymptotic complexity as tensor-product techniques in rectangular domains. First, special structure leading to fast application of mass matrices is developed. Then, by factoring stiffness matrices into products of sparse derivative matrices with mass matrices, fast algorithms are also obtained for stiffness matrices.     

A gradient based iterative algorithm for solving model updating problems of gyroscopic systems

Model updating should be correlated with experimental data to ensure that it models the dynamics of the real structure and the updated model predicts the dynamics of a structure more accurately. Considering that the iterative methods for model updating have aroused little public attention, this paper studies an iterative algorithm for quadratic model updating problems which can incorporate the measured modal data into the finite element model to produce an adjusted finite element model on the mass, gyroscopic and stiffness matrices that closely match the experimental modal data. By this method, the best approximation symmetric and skew-symmetric solutions can be obtained by choosing a convergence factor. Numerical example shows that the introduced iterative algorithm is quite efficient.     

Discretization by finite elements of a model parameter dependent problem

The discretization by finite elements of a model variational problem for a clamped loaded beam is studied with emphasis on the effect of the beam thickness, which appears as a parameter in the problem, on the accuracy. It is shown that the approximation achieved by a standard finite element method degenerates for thin beams. In contrast a large family of mixed finite element methods are shown to yield quasioptimal approximation independent of the thickness parameter. The most useful of these methods may be realized by replacing the integrals appearing in the stiffness matrix of the standard method by Gauss quadratures.     

Optimal balance between mass and smoothed stiffness in simulation of acoustic problems

The Finite Element Method (FEM) is known to behave overly-stiff, which leads to an imbalance between the mass and stiffness matrices within discretized systems. In this work, for the first time, a model is developed that provides optimal balance between discretized mass and smoothed stiffness—the mass-redistributed alpha finite element method (MR-αFEM). This new method improves on the computational efficiency of the FEM and Smoothed Finite Element Methods (S-FEM). The rigorous research conducted ensures that stiffness with the parameter, α, optimally matches the mass with a flexible integration point, q. The optimal balance system significantly reduces the dispersion error of acoustic problems, including those of single and multi-fluids in both time and frequency domains. The excellent properties of the proposed MR-αFEM are validated using theoretical analyses and numerical examples.     

Modelling of Wave Propagation using Spectral Finite Elements

For the analysis of wave propagation at high frequencies, the spectral finite element method (SFEM) is under investigation. In contrast to the conventional finite element method high-order shape functions are used. They are composed of Lagrange polynomials with nodes at the Gauß-Lobatto-Legendre points. The Gauß-Lobatto-Legendre integration scheme is applied in order to obtain a diagonal mass matrix. So, the resulting system equations can be solved efficiently. In the numerical examples, spectral finite elements with shape functions of different order are applied to a plane strain problem. The numerical examples cover structures without and with stiffness discontinuities. It is shown that the results agree well with analytical solutions. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

具有对角线化的一致质量矩阵的动力有限元和弹塑性撞击计算

在EPIC[1、2]、NONSAP[3]等弹塑性撞击计算的有限元程序中,都有一些共同的弱点.所有这些程序,都采用静力学问题中常用的简单线性形状函数来描写各位移分量.在这样的有限元法中,应变和应力分量在每一有限元中都是常量.但在运动方程中,应力分量都是以它们的空间导数的形式出现的.于是,在采用了线性形状函数来表达的位移分量以后,应力分量对运动方程的贡献必恒等于零.克服这种困难的一般方法是通过虚位移原理,把运动方程化为能量关系的变分形式,从而建立既作用在结点上而又在每一有限元内自相平衡的人为内力平衡系统.把施加在某一结点上的所有相邻有限元的人为内力的作用叠加在一起,就能计算这一结点的加速度.但是从虚位移原理化为能量关系的变分形式时,要求位移和应力在积分域内处处连续.也就是说,要求位移和应力有限元都是协调的.我们很易看到,线性形状函数所描述的位移有限元是连续协调的,但其有关的应力分量在有限元界面上,则并不连续.所以,这样的有限元处理,是否收敛并无把握,即使从近似角度看,也是难以令人满意的.而且,为了计算结点的加速度,我们还应该有建立质量矩阵的计算规则.目前有两种计算方法:一种是集总(lumped)质量法,另一种是一致(consistent)质量法[4].     

Finite-element calculation of a plate compliant in transverse shear

A finite-element calculation of a plate with a low transverse shear stiffness is presented. As the basic kinematic parameters, the angles of transverse shear at each of four nodes of the finite element of the plate are selected. The results found for the stress-strain state of an isotropic homogeneous composite and a three-layer plates confirm that the finite-element model elaborated is also efficient in the cases of nonclassical boundary conditions for plates, including conditions for the angles of transverse shear.     

Implementing and using high-order finite element methods

We present theoretical analyses of and detailed timings for two programs which use high-order finite element methods to solve the Navier- Strokes equations in two and three dimensions. The analyses show that algorithms popular in low-order finite element implementations are not always appropriate for high-order methods. The timings show that with the proper algorithms high-order finite element methods are viable for solving the Navier-Stokes equations. We show that it is more efficient, both in time and storage, not to precompute element matrices as the degree of approximating functions increases. We also study the cost of assembling the stiffness matrix versus directly evaluating bilinear forms in two and three dimensions. We show that it is more efficient not to assemble the full stiffness matrix for high-order methods in some cases. We consider the computational issues with regard to both Euclidean and isoparametric elements. We show that isoparametric elements may be used with higher-order elements without increasing the order of computational complexity.     

Finite element buckling analysis of multi-layered graphene sheets on elastic substrate based on nonlocal elasticity theory

Graphene-polymer nano-composites are one of the most applicable engineering nanostructures with superior mechanical properties. In the present study, a finite element (FE) approach based on the size dependent nonlocal elasticity theory is developed for buckling analysis of nano-scaled multi-layered graphene sheets (MLGSs) embedded in polymer matrix. The van der Waals (vdW) interactions between the graphene layers and graphene-polymer are simulated as a set of linear springs using the Lennard-Jones potential model. The governing stability equations for nonlocal classical orthotropic plates together with the weighted residual formulation are employed to explicitly obtain stiffness and buckling matrices for a multi-layered super element of MLGS. The accuracy of the current finite element analysis (FEA) is approved through a comparison with molecular dynamics (MD) and analytical solutions available in the literature. Effects of nonlocal parameter, dimensions, vdW interactions, elastic foundation, mode numbers and boundary conditions on critical in-plane loads are investigated for different types of MLGS. It is found that buckling loads of MLGS are generally of two types namely In-Phase (INPH) and Out-of-Phase (OPH) loads. The INPH loads are independent of interlayer vdW interactions while the OPH loads depend on vdW interactions. It is seen that the decreasing effect of nonlocal parameter on the OPH buckling loads dwindles as the interlayer vdW interactions become stronger. Also, it is found that the small scale and polymer substrate have noticeable effects on the buckling loads of embedded MLGS.     

A new element for analyzing large deformation of thin Naghdi shell model. Part II: Plastic

In this paper a new element is developed that is based on Cosserat theory. In the finite element implementation of Cosserat theory shear locking can occur, especially for very thin shells. In the present investigation the director vector is constrained to remain perpendicular to the mid surface during deformation. It will be shown that this constraint yields accurate results in very large deformation of thin shells also the rate of convergency is very good. For plastic formulation, the model introduced by Simo is used and it has been reduced for constrained director vector and the consistent elasto-plastic tangent moduli is extracted for finite element solution. This model includes both kinematic and isotropic hardening. For numerical investigations an isoparametric nine node element is employed then by linearization of the principle of virtual work, material and geometric stiffness matrices are extracted. The validity and the accuracy of the proposed element is illustrated by the numerical examples and the results are compared with those available in the literature.     

Evolutionary structural optimization for problems with stiffness constraints

This paper presents a simple evolutionary procedure based on finite element analysis to minimize the weight of structures while satisfying stiffness requirements. At the end of each finite element analysis, a sensitivity number, indicating the change in the stiffness due to removal of each element, is calculated and elements which make the least change in the stiffness; of a structure are subsequently removed from the structure. The final design of a structure may have its weight significantly reduced while the displacements at prescribed locations are kept within the given limits. The proposed method is capable of performing simultaneous shape and topology optimization. A wide range of problems including those with multiple displacement constraints, multiple load cases and moving loads are considered. It is shown that existing solutions of structural optimization with stiffness constraints can easily be reproduced by this proposed simple method. In addition some original shape and layout optimization results are presented.