On the formulation of the planar ANCF triangular finite elements

In this paper, new planar isoparametric triangular finite elements (FE) based on the absolute nodal coordinate formulation (ANCF) are developed. The proposed ANCF elements have six coordinates per node: two position coordinates that define the absolute position vector of the node and four gradient coordinates that define vectors tangent to coordinate lines (parameters) at the same node. To shed light on the importance of the element geometry and to facilitate the development of some of the new elements presented in this paper, two different parametric definitions of the gradient vectors are used. The first parametrization, called area parameterization, is based on coordinate lines along the sides of the element in the reference configuration, while the second parameterization, called Cartesian parameterization, employs coordinate lines defined along the axes of the structure (body) coordinate system. The fundamental differences between the ANCF parameterizations used in this investigation and the parametrizations used for conventional finite elements are highlighted. The Cartesian parameterization serves as a unique standard for the triangular FE assembly. To this end, a transformation matrix that defines the relationship between the area and the Cartesian parameterizations is introduced for each element in order to allow for the use of standard FE assembly procedure and define the structure (body) inertia and elastic forces. Using Bezier geometry and a linear mapping, cubic displacement fields of the new ANCF triangular elements are systematically developed. Specifically, two new ANCF triangular finite elements are developed in this investigation, namely four-node mixed-coordinate and three-node ANCF triangles. The performance of the proposed new ANCF elements is evaluated by comparison with the conventional linear and quadratic triangular elements as well as previously developed ANCF rectangular and triangular elements. The results obtained in this investigation show that in the case of small and large deformations as well as finite rotations, all the elements considered can produce correct results, which are in a good agreement if appropriate mesh sizes are used.     

求解大柔度梁/绳索的ANCF单元及隐式迭代格式

为模拟大柔度梁/绳索结构的变形和大范围运动,基于绝对节点坐标方法ANCF(Absolute nodal coordi-nate formulation)和HHT(Hilber-Hughes-Taylor)积分方法,建立了ANCF单元的隐式动力学迭代格式.得到了简洁的节点等效力向量,且进一步导出了切线刚度矩阵的全部公式,...     

Development of ANCF tetrahedral finite elements for the nonlinear dynamics of flexible structures

Nonlinear Dynamics - In this paper, methods for developing isoparametric tetrahedral finite elements (FE) based on the absolute nodal coordinate formulation (ANCF) are presented. The proposed ANCF...     

Behavior of thin rectangular ANCF shell elements in various mesh configurations

The absolute nodal coordinate formulation (ANCF) is characterized by being developed specifically for dynamic analysis of large deformation problems. The objective of the study is to investigate how the shape of the initial mesh configuration influences the obtained numerical solution. After a thorough review of three available formulations, they are used in three different convergence studies. Initially a reference study is conducted to determine how the ANCF performs in an uniform and rectangular mesh. Subsequently, the ANCF methods sensitivity to irregular mesh is investigated and finally, the ability of the ANCF method to describe curved structures is evaluated. This study concludes that thin ANCF shell elements are sensitive to both the initial shape and their loading condition. Furthermore, both the initial configuration and the loading condition affect how the ANCF-based models converge. It is suggested that models containing thin ANCF shell elements are subjected to extensive validation studies, before they are used in a design process.     

Analytical modeling of sandwich beam for piezoelectric bender elements

Piezoelectric bender elements are widely used as electromechanical sensors and actuators.An analytical sandwich beam model for piezoelectric bender elements was developed based on the first-order shear deformation theory(FSDT),which assumes a single rotation angle for the whole cross-section and a quadratic distribution function for coupled electric potential in piezoelectric layers,and corrects the effect of transverse shear strain on the electric displacement integration.Free vibration analysis of simply- supported bender elements was carried out and the numerical results showed that,solu- tions of the present model for various thickness-to-length ratios are compared well with the exact two-dimensional solutions,which presents an efficient and accurate model for analyzing dynamic electromechanical responses of bender elements.     

Shear deformable finite beam elements for composite box beams

The shear deformable thin-walled composite beams with closed cross-sections have been developed for coupled flexural, torsional, and buckling analyses. A theoretical model applicable to the thin-walled laminated composite box beams is presented by taking into account all the structural couplings coming from the material anisotropy and the shear deformation effects. The current composite beam includes the transverse shear and the restrained warping induced shear deformation by using the first-order shear deformation beam theory. Seven governing equations are derived for the coupled axial-flexural-torsional-shearing buckling based on the principle of minimum total potential energy. Based on the present analytical model, three different types of finite composite beam elements, namely, linear, quadratic and cubic elements are developed to analyze the flexural, torsional, and buckling problems. In order to demonstrate the accuracy and superiority of the beam theory and the finite beam elements developed by this study,numerical solutions are presented and compared with the results obtained by other researchers and the detailed threedimensional analysis results using the shell elements of ABAQUS. Especially, the influences of the modulus ratio and the simplified assumptions in stress–strain relations on the deflection, twisting angle, and critical buckling loads of composite box beams are investigated.     

Automatic locking of a parametrically resonating,base-excited,nonlinear beam

Nonlinear Dynamics - Described is a closed-loop control scheme capable of stabilizing a parametrically excited nonlinear structure in several vibration modes. By setting the relative phase between...     

空间曲梁单元的扭转修正系数研究

三维空间曲梁有限单元模型是模拟曲梁结构的有效数值方法,可以考虑曲梁的弯扭耦合特性,最为符合曲梁的几何和受力特征.由于有限元法采用梁理论的平截面假定,空间曲梁单元上的扭转剪应力分布与实际曲梁截面上的扭转剪应力不同,从而会导致扭转刚度和扭转变形的计算失真.本文基于剪切应变能等效原理,推导了不同长宽比的矩形截面空间曲梁单元的...     

General finite element description for non-uniform and discontinuous beam elements

The theory of generalized functions is used to address the static equilibrium problem of Euler–Bernoulli non-uniform and discontinuous 2-D beams. It is shown that if simple integration rules are applied, the full set of response variables due to end nodal displacements and to in-span loads can be derived, in a closed form, for most common beam profiles and arbitrary discontinuity parameters. On this basis, for finite element analysis purposes, a non-uniform and discontinuous beam element is implemented, for which the exact stiffness matrix and the fixed-end load vector are derived. Upon computing the nodal response, no numerical integration is required to build the response variables along the beam element.     

Curved mixed beam elements for the analysis of thin-walled free-form arches

Summary The paper is concerned with the development and application of two curved beam elements for the numerical analysis of arbitrarily shaped arches. The first of these elements has been formulated according to the so-called shallow beam theory, whereas in the case of the second element deep-arch theory was employed. Both elements are based on modified versions of the mixed variational theorem due to Hellinger/ Reissner. Also, for all elements the number of degress-of-freedom equals six. In the final part of the paper, numerical results are presented in order to demonstrate the high accuracy of the solutions obtained herein and to evaluate the performance of the proposed elements in comparison with each other.

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Über gekrümmte Balkenelemente in gemischter Darstellung

Übersicht Die Arbeit beschäftigt sich mit dem Einsatz von zwei neuentwickelten gekrümmten Balkenelementen zur numerischen Berechnung des Trag- und Deformationsverhaltens beliebiger Bögen. Dem ersten dieser Elemente liegt die Marguerresche Theorie des schwach-gekrümmten Bogens zugrunde, während es sich beim zweiten Element um ein stark gekrümmtes Element handelt. Die Formulierung beider Elemente basiert auf modifizierten Funktionalen vom Typ Hellinger/Reissner. Die Anzahl der Freiheitsgrade für jedes Element beträgt sechs. Im numerischen Teil der Arbeit werden umfangreiche Resultate über durchgeführte Testrechnungen mitgeteilt. Die Genauigkeit und praktische Relevanz der hier vorgestellten Elemente wird dadurch klar herausgestellt.

     

Validation of flexible beam elements in dynamics programs

A spatial beam element for static and dynamic problems which involve large displacements and rotations is described. This beam element is applied to static linear buckling problems, the simulation of the motion of a slider-crank mechanism with a flexible connecting rod and a planar and spatial spin-up motion of a flexible beam. Results are compared with those from the open literature.     

Variational base and solution strategies for non-linear force-based beam finite elements

This paper presents the variational bases for the non-linear force-based beam elements. The element state determination of these elements is obtained exactly from a two-field functional with independent stress and strain fields. The variational base of the non-linear force-based beam elements implemented in a general purpose displacement-based finite element program requires the inclusion of independent displacement field in the formulation. For this purpose, a three-field functional is considered with independent displacement, stress, and strain fields. Various local and global solution strategies come out from the mixed formulation of the beam element, and these are shown to yield the algorithms presented for non-linear force formulation beam elements in literature; thus removing any doubts on their variational bases. The presented numerical examples demonstrate the accuracy and robustness of the solution algorithms adapted for mixed formulation elements over popularly used displacement-based beam finite elements even for large structural systems.     

埃尔米特梁单元的分块对角与高阶质量矩阵

埃尔米特梁单元常用的集中质量矩阵,是由挠度自由度对应的一致质量矩阵元素通过行求和或节点积分构造。然而,数值结果表明该集中质量矩阵在求解包含自由端的梁振动问题时,会出现频率精度掉阶现象。本文首先从保障质量矩阵最优收敛性的数值积分精度出发,分别针对三次和五次梁单元,发展了质量矩阵的梯度增强节点积分方案。利用梯度增强节点积分方案,可以得到具有分块对角形式的单元质量矩阵,而其组装的整体质量矩阵除边界节点外仍然呈现对角形式。对于两种单元,其分块对角质量矩阵分别具有4阶最优精度和6阶次优精度。再者,将标准一致质量矩阵和具有同阶精度的梯度增强节点积分质量矩阵进行优化组合,建立了具有超收敛特性的高阶质量矩阵。最后,通过数值算例系统验证了三次和五次单元的分块对角与高阶质量矩阵的频率计算精度。     

Progress of the unified wave-particle methods for non-equilibrium flows from continuum to rarefied regimes

We utilize the nonlinear acoustic solver (NLAS) and Ffowcs-Williams/Hawkings (FW-H) equation to investigate the noise generation and radiation due to shock (wave) and boundary layer interaction (SBLI) in the inlet duct. A classical benchmark for SBLI is chosen to validate the flow features and numerical results show good agreement with experimental results. In the simulation of the noise generated by SBLI, the inlet buzz phenomenon is successfully observed. The oscillation of the normal shock is a kind of little buzz and the oscillation of inner shocks is a kind of big buzz with a frequency around 100 Hz. In the far-field, frequency spectrums show a dominant frequency close to the frequency of inner shocks oscillation. This indicates that the oscillation of inner shocks determines the magnitude of the overall sound pressure level (OASPL) of the far-field noise.

     

Finite element methods for a class of continuum models for immiscible flows with moving contact lines

In this paper, we present a finite element method for two‐phase incompressible flows with moving contact lines. We use a sharp interface Navier–Stokes model for the bulk phase fluid dynamics. Surface tension forces, including Marangoni forces and viscous interfacial effects, are modeled. For describing the moving contact lines, we consider a class of continuum models that contains several special cases known from the literature. For the whole model, describing bulk fluid dynamics, surface tension forces, and contact line forces, we derive a variational formulation and a corresponding energy estimate. For handling the evolving interface numerically, the level‐set technique is applied. The discontinuous pressure is accurately approximated by using a stabilized extended finite element space. We apply a Nitsche technique to weakly impose the Navier slip conditions on the solid wall. A unified approach for discretization of the (different types of) surface tension forces and contact line forces is introduced. Results of numerical experiments are presented, which illustrate the performance of the solver. Copyright © 2016 John Wiley & Sons, Ltd.