A Mixed Finite Element Method for the Biharmonic Problem Using Biorthogonal or Quasi-Biorthogonal Systems

We consider a finite element method based on biorthogonal or quasi-biorthogonal systems for the biharmonic problem. The method is based on the primal mixed finite element method due to Ciarlet and Raviart for the biharmonic equation. Using different finite element spaces for the stream function and vorticity, this approach leads to a formulation only based on the stream function. We prove optimal a priori estimates for both stream function and vorticity, and present numerical results to demonstrate the efficiency of the approach.     

Subdivision Schemes for Thin Plate Splines

Thin plate splines are a well known entity of geometric design. They are defined as the minimizer of a variational problem whose differential operators approximate a simple notion of bending energy. Therefore, thin plate splines approximate surfaces with minimal bending energy and they are widely considered as the standard "fair" surface model. Such surfaces are desired for many modeling and design applications.
Traditionally, the way to construct such surfaces is to solve the associated variational problem using finite elements or by using analytic solutions based on radial basis functions. This paper presents a novel approach for defining and computing thin plate splines using subdivision methods. We present two methods for the construction of thin plate splines based on subdivision: A globally supported subdivision scheme which exactly minimizes the energy functional as well as a family of strictly local subdivision schemes which only utilize a small, finite number of distinct subdivision rules and approximately solve the variational problem. A tradeoff between the accuracy of the approximation and the locality of the subdivision scheme is used to pick a particular member of this family of subdivision schemes.
Later, we show applications of these approximating subdivision schemes to scattered data interpolation and the design of fair surfaces. In particular we suggest an efficient methodology for finding control points for the local subdivision scheme that will lead to an interpolating limit surface and demonstrate how the schemes can be used for the effective and efficient design of fair surfaces.     

基于组合杂交法的四边形薄板弯曲有限元

在研究工程结构的问题中,为了建立简睁高效的四边形薄板弯曲有限元算法,一种基于组合杂交变分原理的新型四边形杂交有限元CH18P被设计出来。单元首先独立插值力矩和挠度形函数,然后在能量协调的概念下使用挠度对力矩进行约束从而得到优化的力矩模式。单元只要求挠度及其偏导数在单元顶点处连续,从而降低了挠度设计的难度。由于挠度弱协调,并引入组合系数和能量协调条件,和其它杂交元相比,计算花费小,无条件稳定,并且保证收敛性。此外,单元还有很高的计算精度,并且对网格形状不敏感．实验证明是一种实用的有限元算法。     

基于有限元分析的堵板厚度设计

传统工艺中对压力容器汽缸中圆堵板的厚度设计采用经验公式,而为了保证产品的安全性能预留余量过大,造成了严重的材料浪费.针对此问题,采用ANSYS软件对常用碳素结构钢Q235-A铸成的不同直径的圆堵板进行不同承压情况下的有限元分析,以工程应用中的许用应力为标准,不断进行有限元计算,直到优化出在满足强度条件下以最小值承受一定压强的堵板厚度值.在将最佳厚度值与经验值进行对比分析的基础上,对经验公式进行修正,并应用到工程实际中,结果表明修正后的厚度在满足安全性的同时节省了30％的材料,因此采用该方法设计圆堵板厚度具有科学性和可行性.     

一种四边形广义协调薄板弯曲单元

为了降低位移型任意四边形薄板弯曲有限元构造的难度,基于修正最小势能变分原理,提出了一种新的广义协调位移的设计方法,并且构造出一种任意四边形网格下的位移型薄板弯曲单元Q8P。这种单元能够通过常应力分片检查,当弯矩,扭矩和横向剪力沿着单元边界的分布为常值时,由于位移在单元边界非逐点协调而产生的附加能量为零。这种新型的广义协调单元保证了有限元解的收敛性,计算效率高,具有较高的数值精度,并且对网格畸变不敏感。     

面向异构架构的混合精度有限元算法及其CUDA实现

长期以来,单精度似乎与科学计算无缘,然而从体系结构看,混合精度计算可以充分发挥向量部件、GPGPU设备的单精度性能,提供更高的效能,如降低通讯带宽要求、提高数据传输和通讯效率等。混合精度显格式有限元算法,结合材料强非线性多尺度有限元程序msFEM,实现了GPGPU上的有效加速。实验结果表明:混合精度显格式有限元程序实现了90%以上的计算通过单精度完成,其计算结果与全部使用双精度的结果相一致。该算法可以使得在不支持双精度格式的加速卡上实现科学计算功能。在支持双精度浮点格式的GPU上,混合精度算法与全部采用双精度计算相比其加速效果提高了1.6～1.7倍。     

基于分布式系统的有限元并行计算的研究

有限单元法是现代工程设计和分析的重要数值方法之一,但要对这些大型或超大型复杂结构进行有效的结构分析,需要有高性能的计算资源、有效的算法和先进的软件编制手段。Java多线程技术是构建并行系统的有效手段,文章基于分布式系统用Java多线程技术实现了一个有限元并行应用系统。     

薄板系统振动主动控制器的一种设计

根据基于模型的模糊控制策略,提出一种薄板系统振动主动控制器的设计,与传统的方法不同之处在于作者首先采用递归的ＲＡＭＡＸ算法和ＰＥＭ进行模糊参数识别,从而建模;其次,通过在线调速系数矩阵Θ（ａ）的元素和目标函数优化,进行自适应控制器合成,数值仿真和实验结果验证了本文所提方法的有效性和鲁棒性。     

A Mixed Finite Element Method for Elasticity in Three Dimensions

We describe a stable mixed finite element method for linear elasticity in three dimensions.     

A New Mixed Finite Element Method for Elastodynamics with Weak Symmetry

We provide a new mixed finite element analysis for linear elastodynamics with reduced symmetry. The problem is formulated as a second order system in time by imposing only the Cauchy stress tensor and the rotation as primary and secondary variables, respectively. We prove that the resulting variational formulation is well-posed and provide a convergence analysis for a class of \({\mathrm {H}}(\mathop {{\mathrm {div}}}\nolimits )\)-conforming semi-discrete schemes. In addition, we use the Newmark trapezoidal rule to obtain a fully discrete version of the problem and carry out the corresponding convergence analysis. Finally, numerical tests illustrating the performance of the fully discrete scheme are presented.     

A Mixed Finite Element Method for the Stokes Equations Based on a Weakly Over-Penalized Symmetric Interior Penalty Approach

We present a mixed finite element method for the steady-state Stokes equations where the discrete bilinear form for the velocity is obtained by a weakly over-penalized symmetric interior penalty approach. We show that this mixed finite element method is inf-sup stable and has optimal convergence rates in both the energy norm and the $L_2$ norm on meshes that can contain hanging nodes. We present numerical experiments illustrating these results, explore a very simple adaptive algorithm that uses meshes with hanging nodes, and introduce a simple but scalable parallel solver for the method.     

Symbolic Finite Element Modeling of Structural Systems

The application of light materials to space structures, aircraft, robots and automobiles has increased the demand for effective algorithms to model and predict the response of structural multibody systems. The understanding of mechanics can assist in developing better design and control strategies. Formulation of mathematical models of a multibody system using manual approaches is a difficult task and prone to errors. For non-linear and/or time-varying systems, numerical formulation provides limited information about physical insight. In this study, a computer-aided symbolic method is used to generate the equations of motion from Lagrange's method. Equations are converted into FORTRAN form ready for simulations and control synthesis. The 4–5th order Runge–Kutta–Fehlberg method (RKF45) was used to numerically solve the system of equations. Two examples, namely a slider–crank mechanism and an aircraft model are presented.     

Axisymmetric Finite Element Method for Analysis of Plate on Layered Soil

To obtain the fundamental solution of soil has become the key problem for the semi-analytical and semi-numerical (SASN) method in analyzing plate on layered soil. By applying axisymmetric finite element method (FEM),an expression relating the surface settlement and the reaction of the layered soil can be obtained. Such a reaction can be treated as load acting on the applied external load. Having the plate modelled by four-node elements,the governing equation of the plate can be formed and solved. In this ca...     

Mixed Dimensional Coupling in Finite Element Stress Analysis

Many analysis models utilize finite elements of reduced dimension. However, to capture stress concentrations at local details, it would be desirable to combine the reduced dimensional element types with higher dimensional elements in a single finite element model. It is therefore important in such cases to integrate into the analyses some scheme for coupling the element types that conforms to the governing equations of the problem. In this paper, a novel method that can correctly couple beams to solids, beams to shells and shells to solids for elastic problems is presented. The approach adopted is to equate the work done on either side of the interface between dimensions, and this leads to multi-point constraint equations, thus providing a relationship among nodal degrees of freedom between the differing element types. Example results show that the proposed technique does not introduce any spurious stresses at the dimensional interfaces. ID="A1" Correspondence and offprint requests to: C. G. Armstrong, School of Mechanical and Manufacturing Engineering, The Queen's University of Belfast, Ashby Building, Stranmillis Road, Belfast BT9 5AH, Northern Ireland. E-mail: c.armstrong@qub.ac.uk     

中厚板粗轧机轧辊强度的有限元模拟

根据济钢中厚板厂的实际生产工艺,利用MSC Marc有限元软件,建立粗轧机轧辊的有限元模型,模拟计算其在最大力能参数下的应力分布规律. 模拟结果和解析计算结果非常接近,表明轧辊强度可以满足生产要求.     

A 3D Finite Element Method for Flexible Multibody Systems

An efficient finite element (FE) formulation for the simulation of multibody systems is derived from Hamilton's principle. According to the classical assumptions of multibody systems, a large rotation formulation has been chosen, where large rotations and large displacements, but only small deformations of the single bodies are taken into account. The strain tensor is linearized with respect to a co-rotated frame. The present approach uses absolute coordinates for the degrees of freedom and forms an alternative to the floating frame of reference formulation that is based on relative coordinates and describes deformation with respect to a co-rotated frame. Due to the modified strain tensor, the present formulation distinguishes significantly from standard nodal based nonlinear FE methods. Constraints are defined in integral form for every pair of surfaces of two bodies. This leads to a small number of constraint equations and avoids artificial stress singularities. The resulting mass and stiffness matrices are constant apart from a transformation based on a single rotation matrix for each body. The particular structure of this transformation allows to prevent from the usually expensive factorization of the system Jacobian within implicit time--integration methods. The present method has been implemented and tested with the FE-package NGSolve and specific 3D examples are verified with a standard beam formulation.     

有限元指纹图像配准

针对指纹图像中形变复杂多样的特点,对指纹图像配准算法进行研究,发现目前指纹配准算法存在复杂度高,精度低,速度慢,易受指纹形变等影响的缺陷,引入有限元分析理论,提出了有限元指纹图像配准算法。该算法将指纹图像配准中的复杂形变转换为有限个离散形变单元组成的弹性形变,以分叉点、转折点、指纹图像上两点的连线所穿越的脊线数量等特征作为特征指标,求出采样指纹与标准指纹的模糊贴近度,并创造性地以此为相似性测度,构造总体刚度方程,进而迭代求解最佳位移,完成指纹图像配准。实验表明,该算法位移量的误差最小,用时为8.894 s,其有效降低了算法复杂度,提高了精度,同时也有效避免了指纹复杂多样的形变等因素对指纹配准算法精度的影响。