A Cosserat non-linear finite element analysis software for blocky structures

The non-linear analysis of bidimensional finite element models of blocky structures by using the Cosserat theory is described in this paper. The Cosserat theory is employed to describe the behaviour of blocky structures, showing that this finite element analysis is suitable to determine the response of 2D models of elastoplastic problems including nodal rotations and displacements. The elastoplastic analysis of such models is carried out by using a tilting-block criterion derived from classical analysis. Plastic surfaces as well as flow vectors are formulated in order to analyse blocky structures currently used in masonry engineering situations, discussing how input parameters affect the overall behaviour of these models. The software developed is also presented and discussed, showing that the algorithms proposed are robust and efficient. Finally, some numerical examples are presented and discussed in detail.     

Localisation in a Cosserat continuum under static and dynamic loading conditions

Localisation studies have been carried out for an unconstrained, elasto-plastic, strain-softening Cosserat continuum. Because of the presence of an internal length scale in this continuum model a perfect convergence is found upon mesh refinement. A finite, constant width of the localisation zone and a finite energy dissipation are computed under static as well as under transient loading conditions. Because of the existence of rotational degrees-of-freedom in a Cosserat continuum additional wave types arise and wave propagation becomes dispersive. This has been investigated analytically and numerically for an elastic Cosserat continuum and an excellent agreement has been found between both solutions.     

基于Cosserat理论的平面等参元

为解释材料在微尺度下的尺度效应,基于Cosserat理论,从势能泛函驻值条件出发提出构造8节点Serendipity平面等参元,并建立平面有限元法．每个节点拥有3个独立节点自由度,分别为2个方向的线位移和1个逆时针方向的角位移．用该方法分析含中心小孔的无限平板在单轴拉伸情况下的应力集中问题．数值计算结果与Cosserat理论的解析解非常符合,表明应力集中因数k受泊松比μ,常数c及a／l值的影响很大;由于偶应力的存在,小孔周围的应力分布明显小于经典弹性力学理论的预测．通过对材料常数c的调节可以将该方法推广应用于基于Mindlin偶应力理论的数值分析中．     

Interpolation of rotation and motion

In Cosserat solids such as shear deformable beams and shells, the displacement and rotation fields are independent. The finite element implementation of these structural components within the framework of flexible multibody dynamics requires the interpolation of rotation and motion fields. In general, the interpolation process does not preserve fundamental properties of the interpolated field. For instance, interpolation of an orthogonal rotation tensor does not yield an orthogonal tensor, and furthermore, does not preserve the tensorial nature of the rotation field. Consequently, many researchers have been reluctant to apply the classical interpolation tools used in finite element procedures to interpolate these fields. This paper presents a systematic study of interpolation algorithms for rotation and motion. All the algorithms presented here preserve the fundamental properties of the interpolated rotation and motion fields, and furthermore, preserve their tensorial nature. It is also shown that the interpolation of rotation and motion is as accurate as the interpolation of displacement, a widely accepted tool in the finite element method. The algorithms presented in this paper provide interpolation tools for rotation and motion that are accurate, easy to implement, and physically meaningful.     

A generalized Hill’s lemma and micromechanically based macroscopic constitutive model for heterogeneous granular materials

Based on the Hill’s lemma for classical Cauchy continuum, a generalized Hill’s lemma for micro–macro homogenization modeling of gradient-enhanced Cosserat continuum is presented in the frame of the average-field theory. In this context not only the strain and stress tensors defined in classical Cosserat continuum but also their gradients are attributed to assigned micro-structural representative volume element (RVE), that leads to a higher-order macroscopic Cosserat continuum modeling and enables to incorporate the micro-structural size effects. The enhanced Hill–Mandel condition for gradient-enhanced Cosserat continuum is extracted as a corollary of the presented generalized Hill’s lemma. The derived admissible boundary conditions for the modeling are deduced to direct the proper presentation of boundary conditions to be prescribed on the RVE in order to ensure the satisfaction of the Hill–Mandel energy condition.With the link between the discrete particle assembly and its effective Cosserat continuum in an individual RVE, the boundary conditions prescribed on the RVE modeled as Cosserat continuum are transformed into those prescribed to the peripheral particles of the RVE modeled as the discrete particle assembly. The micromechanically based macroscopic constitutive model and corresponding rate forms of the macroscopic stress–strain relations taking into account the local microstructure and its evolution are formulated with neither need of specifying the macroscopic constitutive relation nor need of providing macroscopic material parameters.     

Cosserat Rods with Projective Dynamics

We present a novel method to simulate Cosserat rods with Projective Dynamics (PD). The proposed method is both numerically robust and accurate with respect to the underlying physics, making it suitable for a variety of applications in computer graphics and related disciplines. Cosserat theory assigns an orientation frame to each point and is thus able to realistically simulate stretching and shearing effects, in addition to bending and twisting. Within the PD framework, it is possible to obtain accurate simulations given the implicit integration over time and its decoupling of the local‐global solve. In the proposed method, we start from the continuous formulation of the Cosserat theory and derive the constraints for the PD solver. We extend the standard definition of PD and add body orientations as system variables. Thus, we include the preservation of angular momentum, so that twisting and bending can be accurately simulated. Our formulation allows the simulation of different bending behaviors with respect to a user‐defined Young's modulus, the radius of the rod's cross‐section, and material density. We show how different material specifications in our simulations converge within a few iterations to a reference solution, generated with a high‐precision finite element method. Furthermore, we demonstrate mesh independence of our formulation: Refining the simulation mesh still results in the same characteristic motion, which is in contrast to previous position based methods.     

基于Cosserat理论的头发动态模拟*

针对基于弹性杆理论的超螺旋模型中对动力学方程的求解影响头发模拟实时性的问题,采用Cosserat弹性杆理论对头发建模,引入角速度变量,并加入头发运动平衡的固有约束,得到改进的拉格朗日动力学运动方程;然后,将头发单体离散化,用角速度和四元数作为状态变量简化动力学方程,用半显式的欧拉方程加速方程的执行,大大降低了系统的运行时间,提高了模拟的速度,在满足实时性要求的同时提高了头发模拟的真实度。     

STRANDS: Interactive Simulation of Thin Solids using Cosserat Models

Strands are thin elastic solids that are visually well approximated as smooth curves, and yet possess essential physical behaviors characteristic of solid objects such as twisting. Common examples in computer graphics include: sutures, catheters, and tendons in surgical simulation; hairs, ropes, and vegetation in animation. Physical models based on spring meshes or 3D finite elements for such thin solids are either inaccurate or inefficient for interactive simulation. In this paper we show that models based on the Cosserat theory of elastic rods are very well suited for interactive simulation of these objects. The physical model reduces to a system of spatial ordinary differential equations that can be solved efficiently for typical boundary conditions. The model handles the important geometric non‐linearity due to large changes in shape. We introduce Cosserat‐type physical models, describe efficient numerical methods for interactive simulation of these models, and implementation results.     

Some results of finite element applications in finite elasticity

This paper examines a number of problems connected with the finite element analysis of finite elastic deformations. A brief review of formulation of equations governing finite deformations of highly elastic elements is given. The convergence of finite element approximations for static problems in elasticity is studied. Incremental stiffness equations are derived in general form and various types of incremental loading techniques are examined. A number of representative solved problems in finite elasticity are given.     

Finite element analysis of composites using chebyshev polynomials

A finite quasi-prismatic (FQP) element is modified to analyze anisotropic materials. The finite quasi-prismatic element is a three-dimensional finite element which uses conventional interpolating functions in two directions and functions based on Chebyshev polynomials in the third direction. This element is used to solve different anisotropic problems and the results are compared with that of conventional finite elements and analytical solutions.     

Finite element modeling within an integrated geometric modeling environment: Part II—Attribute specification,domain differences,and indirect element types

This is the second of a two part paper that addresses the integration of finite element modeling and geometric modeling. Instead of considering the integration of currently available systems, this paper addresses both modeling techniques in general terms and identifies the functions that are needed to integrate them, taking full advantage of the capabilities of both. A set of geometric communication operators are identified and defined for use in carrying out this integration process. Part I [1] considered the integration of geometric modeling and finite element mesh generation. This part considers the remaining areas of the specification of finite element analysis attribute information, accounting for domain differences between the geometric and finite element models and the generation of finite element models using element types that are of a lesser dimension than the geometric entity they represent.     

Complementary error bounds for foolproof finite element mesh generation

The use of complementary variational principles in finite element analysis is examined. It is shown that complementary finite element solutions provide an element by element measure of the accuracy of the solution. By solving a problem repeatedly, beginning with a coarse mesh and refining those elements having the largest errors, an automatic, foolproof finite element mesh generation procedure is developed. Finite element solutions obtained by the new procedure have the property that the finest elements are concentrated in regions of greatest need while large elements are found in less important regions. A computer program which implements the new algorithm is described and examples of finite element solutions generated by the program are presented.     

Maximization of the fundamental eigenfrequency of micropolar solids through topology optimization

Aim of this work is the maximization of the fundamental eigenfrequency of 2D bodies made of micropolar (or Cosserat) materials using a topology optimization approach. A classical SIMP–like model is used to approximate the constitutive parameters of the micropolar medium. A suitable penalization is introduced for both the linear and the spin inertia of the material, to avoid the occurrence of undesired local modes. The robustness of the proposed procedure is investigated through numerical examples; the influence of the material parameters on the optimal material layouts is also discussed. The optimal layouts for Cosserat solids may differ significantly from the truss–like solutions typical of Cauchy solids, as the intrinsic flexural stiffness of the material can lead to curved beam-like material distributions. The numerical simulations show that the results are quite sensitive to the material characteristic length and the spin inertia.     

An assessment of discretizations for convection-dominated convection–diffusion equations

The performance of several numerical schemes for discretizing convection-dominated convection–diffusion equations will be investigated with respect to accuracy and efficiency. Accuracy is considered in measures which are of interest in applications. The study includes an exponentially fitted finite volume scheme, the Streamline-Upwind Petrov–Galerkin (SUPG) finite element method, a spurious oscillations at layers diminishing (SOLD) finite element method, a finite element method with continuous interior penalty (CIP) stabilization, a discontinuous Galerkin (DG) finite element method, and a total variation diminishing finite element method (FEMTVD). A detailed assessment of the schemes based on the Hemker example will be presented.     

Higher order stabilized finite element method for hyperelastic finite deformation

This paper presents a higher order stabilized finite element formulation for hyperelastic large deformation problems involving incompressible or nearly incompressible materials. A Lagrangian finite element formulation is presented where mesh dependent terms are added element-wise to enhance the stability of the mixed finite element formulation. A reconstruction method based on local projections is used to compute the higher order derivatives that arise in the stabilization terms, specifically derivatives of the stress tensor. Linearization of the weak form is derived to enable a Newton–Raphson solution procedure of the resulting non-linear equations. Numerical experiments using the stabilization method with equal order shape functions for the displacement and pressure fields in hyperelastic problems show that the stabilized method is effective for some non-linear finite deformation problems. Finally, conclusions are inferred and extensions of this work are discussed.     

Vibration of thin cylindrical shells based on a mixed finite element formulation

A Hellinger-Reissner functional for thin circular cylindrical shells is presented. A mixed finite element formulation is developed from this functional, which is free from line integrals and relaxed continuity terms. This element is applied to the problem of vibration of rectangular cylindrical shells. Bilinear trial functions are used for all field variables. The element satisfies the compatibility and completeness requirements. The numerical results obtained in this work show that convergence is quite rapid and monotonic for a much smaller number of degrees of freedom than other finite element formulations.     

A nine-node assumed-strain finite element for composite plates and shells

A new finite element for modeling fiber-reinforced composite plates and shells is developed and its performance for static linear problems is evaluated. The element is a nine-node degenerate solid shell element based on a modified Hellinger-Reissner principle with independent inplane and transverse shear strains. Several numerical examples are solved and the solutions are compared with other available finite solutions and with classical lamination theory. The results show that the present element yields accurate solutions for the test problems presented. Convergence characteristics are good, and the solution is relatively insensitive in element distortion. The element is also shown to be free of locking even for thin laminates.     

基于分层模型的复合材料耦合Lamb波频率—波数域特性研究

针对复合材料层合板中耦合Lamb波的传播问题,基于分层模型提出解析建模与有限元数值模拟相结合的方法对其进行预测和评估。利用Legendre正交多项式展开法推导多层各向异性复合材料层合板中耦合Lamb波的控制方程,并对频率-波数域频散特性曲线实现数值求解。基于平面壳单元构建复合材料层合板的有限元模型,采用波结构加载法生成单一Lamb波基本模态,设计复合材料层合板的不同纤维取向、边界和界面约束条件,并经二维傅里叶变换获得有限元模拟数据的频率-波数域频散特性曲线。通过对比验证,结果表明两种方法均有较好的吻合性。     

一阶非定常双曲问题的间断-差分流线扩散法及其误差估计

本文提出了求解一阶非定常双曲问题的一种新型有限元方法．间断-差分流线扩散法(DFDSD方法),建立了Euler型DFDSD格式,并对格式解的稳定性和收敛性进行了理论分析,最后给出了数值算例说明算法的有效性．