Introduction to Mesh Based Generated Lumped Parameter Models for Electromagnetic Problems using Triangular Elements

This paper is an introduction to mesh based generated reluctance network modeling using triangular elements.Many contributions on mesh based generated reluctance networks using rectangular shaped elements have been published,but very few on those generated from a mesh using triangular elements.The use of triangular elements is aimed at extending the application of the approach to any shape of modeled devices.Basic concepts of the approach are presented in the case of electromagnetic devices.The procedure for coding the approach in the case of a flat linear permanent magnet machine is presented.Codes developed under MATLAB environment are also included.

TRIANGULAR ELEMENTS FOR REISSNER-MINDLIN PLATE

A general method to construct locking free Reissner-Mindlin plate elements is presented. According to this method the shear strain is replaced by its proper interpolation polynomial, which corresponds to the Kirchoff conditions at the interpolation points as the thickness of plate tends to zero, so the element is locking free. We construct two triangular elements by this method - a 3-node element and a 6-node element. The numerical results are provided.

The Numerical Integration of Discrete Functions on a Triangular Element

With the application of Hammer integral formulas of a continuous function on a triangular element, the numerical integral formulas of some discrete functions on the element are derived by means of decomposition and recombination of base functions. Hammer integral formulas are the special examples of those of the paper.

Analysis of Linear Triangular Elements for Convection-diffusion Problems by Streamline Diffusion Finite Element Methods

这份报纸被奉献给学习 superconvergence 优化散开为传送对流散开问题的有限元素方法。在里面[8 ] ，在条件下面那 h2 最佳的有限元素错误估计在 L2 标准被获得。然而，在现在的纸，一样的错误估计结果在更弱的条件下面被获得那 h。

SYMMETRIC POINT STRUCTURE OF SUPERCONVERGENCE FOR CUBIC TRIANGULAR ELEMENTS -A CONSULTATION WITH ZHU

Superconvergence structures for rectangular and triangular finite elements are summarized. Two debatable issues in Zhu's paper [18] are discussed. A superclose polynomial to cubic triangular finite element is constructed by area coordinate.

Triangular element partition method with consideration of crack tip

In fracture simulation,how to model the pre-existing cracks and simulate their propagation without remeshing is an important topic.The newly developed triangular element partition method(TEPM)provides an efficient approach to this problem.It firstly meshes the cracked body regardless of the geometry integrity of the interesting object with triangular elements.After the meshing procedure is completed,some elements are intersected by cracks.For the element intersected by a crack,the TEPM takes the element partition technique to incorporate the discontinuity into the numerical model without any interpolation enrichment.By this approach,the TEPM can simulate fracture without mesh modification.In the TEPM,all the cracked elements are treated as the usual partitioned elements in which the crack runs through.The virtual node pairs(the intersection points of crack faces and elements)at the opposite faces of the crack move independently.Their displacements are respectively determined by their neighbor real nodes(nodes formatted in the original mesh scheme)at the same side of the crack.However,among these cracked elements,the element containing a crack tip,referred to as the crack tip element thereafter,behaves differently from those cut through by the crack.Its influence on the singular field at the vicinity of the fracture tip becomes increasingly significant with the element size increasing.In the crack tip element,the virtual node pair at the crack tip move consistently before fracture occurs while the virtual node pair separate and each virtual node moves independently after the fracture propagates.Accordingly,the crack tip element is automatically transformed into the usual partitioned element.In the present paper,the crack tip element is introduced into the TEPM to account for the effect of the crack tip.Validation examples indicate that the present method is almost free from the element size effect.It can reach the same precision as the conventional finite element method under the same meshing scheme.But the TEPM is much more efficient and convenient than the conventional finite element method because the TEPM avoids the troubles that the conventional finite element method suffers,e.g.,the meshing problem of cracked body,modification of mesh scheme,etc.Though the extended finite element method can also avoid these troubles,it introduces extra degrees of freedom due to node interpolation enrichment.Due to the simplicity of the present TEPM,it is believed that its perspective should be highly inspiring.

A 3-Node Co-Rotational Triangular Finite Element for Non-Smooth,Folded and Multi-Shell Laminated Composite Structures

Based on the first-order shear deformation theory,a 3-node co-rotational triangular finite element formulation is developed for large deformation modeling of non-smooth,folded and multi-shell laminated composite structures.The two smaller components of the mid-surface normal vector of shell at a node are defined as nodal rotational variables in the co-rotational local coordinate system.In the global coordinate system,two smaller components of one vector,together with the smallest or second smallest component of another vector,of an orthogonal triad at a node on a non-smooth intersection of plates and/or shells are defined as rotational variables,whereas the two smaller components of the mid-surface normal vector at a node on the smooth part of the plate or shell(away from non-smooth intersections)are defined as rotational variables.All these vectorial rotational variables can be updated in an additive manner during an incremental solution procedure,and thus improve the computational efficiency in the nonlinear solution of these composite shell structures.Due to the commutativity of all nodal variables in calculating of the second derivatives of the local nodal variables with respect to global nodal variables,and the second derivatives of the strain energy functional with respect to local nodal variables,symmetric tangent stiffness matrices in local and global coordinate systems are obtained.To overcome shear locking,the assumed transverse shear strains obtained from the line-integration approach are employed.The reliability and computational accuracy of the present 3-node triangular shell finite element are verified through modeling two patch tests,several smooth and non-smooth laminated composite shells undergoing large displacements and large rotations.

Finite layer and triangular prism element method to subsidence prediction and stress analysis in underground mining

The application of the finite layer & triangular prism element method to the 3D ground subsidence and stress analysis caused by mining is presented. The layer elements and the triangular prism elements have been alternatively used in the numerical simulation system, the displacement pattern, strain matrix, elastic matrix, stiffness matrix, load matrix and the stress matrix of the layer element and triangular prism element have been presented. By means of the Fortran90 programming language, a numerical simulation system based on finite layer & triangular prism element have been built up, and this system is suitable for subsidence prediction and stress analysis of all mining condition and mining methods. Comparing with the infinite element method, this approach dramatically reduces the size of the set of equations that need to be solved, and greatly reduces the amount of data preparation required. It not only saves the internal storage, and the computation time, but also decreases the cost.

The Triangular Hermite Finite Element Complementing the Bogner-Fox-Schmit Rectangle

The Bogner-Fox-Schmit rectangular element is one of the simplest elements that provide continuous differentiability of an approximate solution in the framework of the finite element method. However, it can be applied only on a simple domain composed of rectangles or parallelograms whose sides are parallel to two different straight lines. We propose a new triangular Hermite element with 13 degrees of freedom. It is used in combination with the Bogner-Fox-Schmit element near the boundary of an arbitrary polygonal domain and provides continuous differentiability of an approximate solution in the whole domain up to the boundary.

An explicit finite volume element method for solving characteristic level set equation on triangular grids

Level set methods are widely used for predicting evolutions of complex free surface topologies,such as the crystal and crack growth,bubbles and droplets deformation,spilling and breaking waves,and two-phase flow phenomena.This paper presents a characteristic level set equation which is derived from the two-dimensional level set equation by using the characteristic-based scheme.An explicit finite volume element method is developed to discretize the equation on triangular grids.Several examples are presented to demonstrate the performance of the proposed method for calculating interface evolutions in time.The proposed level set method is also coupled with the Navier-Stokes equations for two-phase immiscible incompressible flow analysis with surface tension.The Rayleigh-Taylor instability problem is used to test and evaluate the effectiveness of the proposed scheme.

Finite Element Analysis of Magnetohydrodynamic Mixed Convection in a Lid-Driven Trapezoidal Enclosure Having Heated Triangular Block

A numerical research on magnetohydrodynamic mixed convection flow in a lid-driven trapezoidal enclosure at non-uniform heating of bottom wall has been studied numerically. The enclosure consists of insulated top wall and cold side walls, too. It also contains a heated triangular block (Rot = 0° - 90°) located somewhere inside the enclosure. The boundary top wall of the enclosure is moving through uniform speed U0. The geometry of the model has been represented mathematically by coupled governing equations in accordance with proper boundary conditions and then a two-dimensional Galerkin finite element based numerical approach has been adopted to solve this paper. The numerical computations have been carried out for the wide range of parameters Prandtl number (0.5 ≤ Pr ≤ 2), Reynolds number (60 ≤ Re ≤ 120), Rayleigh number (Ra = 103) and Hartmann number (Ha = 20) taking with different rotations of heated triangular block. The results have been shown in the form of streamlines, temperature patterns or isotherms, average Nusselt number and average bulk temperature of the fluid in the enclosure at non-uniform heating of bottom wall. It is also indicated that both the streamlines, isotherm patterns strongly depend on the aforesaid governing parameters and location of the triangular block but the thermal conductivity of the triangular block has a noteworthy role on the isotherm pattern lines. Moreover, the variation of Nuav of hot bottom wall and θav in the enclosure is demonstrated here to show the characteristics of heat transfer in the enclosure.

Finite Element Analysis of Magnetohydrodynamic Natural Convection within Semi-Circular Top Enclosure with Triangular Obstacles

The phenomena of magneto-hydrodynamic natural convection in a two-dimensional semicircular top enclosure with triangular obstacle in the rectangular cavity were studied numerically. The governing differential equations are solved by using the most important method which is finite element method (weighted-residual method). The top wall is placed at cold Tc and bottom wall is heated Th. Here the sidewalls of the cavity assumed adiabatic. Also all the wall are occupied to be no-slip condition. A heated triangular obstacle is located at the center of the cavity. The study accomplished for Prandtl number Pr = 0.71;the Rayleigh number Ra = 103, 105, 5 × 105, 106 and for Hartmann number Ha = 0, 20, 50, 100. The results represent the streamlines, isotherms, velocity and temperature fields as well as local Nusselt number.

仿生平面三角形单元积分声场的波函数构造

波叠加法在求解声源外部辐射声场时,需要对所有离散单元进行数值积分计算导致计算效率较低,而等效源法由于过度简化单元始终存在较大的积分近似误差.针对以上缺陷,利用Helmholtz方程在球坐标系下的解构造了一种与单元积分等效且无需积分的波函数.受仿生复合材料三角形缝合结构启发,以适用范围最广的平面三角形单元为例,构造了波函数的一般形式和内推形式.最后,通过数值仿真对比了两种波函数与直接积分的计算声场的精度和效率.结果表明,两种波函数与直接积分的计算误差低于0.5%,且内推波函数的计算效率约为直接积分的6倍.

接触问题的三角形载荷离散FFT加速算法

接触问题控制方程的有效求解,往往涉及到复杂的数学理论知识,而在实际工程应用中,接触应力分布又具有高度随机性。为高效快速求解任意载荷分布下固体的接触响应,基于三角形载荷离散单元,嵌入离散卷积快速傅里叶变换(DC-FFT)算法,提供了一种高精度、高可靠度的计算方法。相比于通常采用的分段均布载荷离散方法,三角形单元的解析求解略显复杂,但能更好地模拟接触载荷任意分布的特性,对于接触边缘处载荷由零递增或递减为零的情况,也可以予以充分考虑。为优化三角形载荷离散单元的求解方法,根据接触影响系数矩阵的“激励-响应”特性,推导了三角形载荷单元和均布载荷单元作用下的应力分量解析解。通过构造包含影响系数矩阵的离散卷积形式应力解,将某一目标节点在所有载荷单元作用下,重复度极高的矩阵运算叠加过程,采用DC-FFT算法来简化加速计算。通过程序编程计算,分析验证了所提出算法的精确度和高效性。

Highly efficient H^1-Galerkin mixed finite element method (MFEM) for parabolic integro-differential equation

A highly efficient H1-Galerkin mixed finite element method （MFEM） is presented with linear triangular element for the parabolic integro-differential equation. Firstly, some new results about the integral estimation and asymptotic expansions are studied. Then, the superconvergence of order O（h^2） for both the original variable u in H1 （Ω） norm and the flux p = u in H（div, Ω） norm is derived through the interpolation post processing technique. Furthermore, with the help of the asymptotic expansions and a suitable auxiliary problem, the extrapolation solutions with accuracy O（h^3） are obtained for the above two variables. Finally, some numerical results are provided to confirm validity of the theoretical analysis and excellent performance of the proposed method.

静电成形薄膜反射面天线的裁剪设计

该文通过裁剪和拼接平面薄膜片来对静电成形薄膜反射面天线的表面进行设计制作。由于抛物面型薄膜反射面是一个不可展开表面,通常由许多平面薄膜片制作而成,将不可避免地降低反射面表面的精度。因此,需要详细地分析裁剪和拼接过程对反射面表面精度的影响,从而获得设计需要的表面精度。该文基于旋转抛物面的无矩理论建立了工作状态的反射面模型,采用常应变三角形单元离散反射面,建立了反射面的展平和拼接有限元模型。将同口径反射面沿经线划分若干组不同数量子曲面,通过子曲面进行展平和拼接数值仿真分析,对薄膜反射面进行裁剪设计。最后,制作了0.55 m口径的薄膜反射器,并进行了摄影测量实验。验证了该方法的有效性,并讨论了制造误差的影响。

THE GENERATION OF NON－LINEAR STIFFNESS MATRIX OF TRIANGLE ELEMENT WHENCONSIDERING BOTH THE BENDING AND IN－PLANEME MBRANE FORCES

Using Stricklin Melhod￣[5],we have this paper has derived the formulas for the ge-neration of non-linear element stiffness matrix of a triangle element when considering both the bending and the in-plane membrane forces. A computer programme for the calculation of large deflection and inner forces of shallow shells is designed on theseformulas. The central deflection curve computed by this programme is compared with other pertaining results.

采棉机三角履带行走装置设计与仿真

为了适应长江、黄河流域棉区的棉花机械化采收需求,设计了一款采棉机三角履带行走装置。分析棉田工况、整车接地比压和车速等参数,确定了三角履带行走装置的主要技术参数;通过有限元分析,对基架、驱动轮和摆动销轴等关键部件进行了强度校核计算。结果表明:所设计结构与材料选用均能满足使用要求,单轮承载最高可达8 t。该装置可与原车轮胎快速互换,对于推进长江、黄河流域棉花收获机械化具有一定意义。

基于板-梁理论的双轴对称三角形管翼缘工字形梁组合扭转的理论研究及有限元验证

基于板-梁理论推导双轴对称三角形管翼缘工字形悬臂梁的总应变能方程,得到梁的翘曲刚度和自由扭转刚度,然后通过外力势能和总应变能得到总势能方程。再依据能量变分模型、微分方程模型及边界条件得出悬臂梁端部最大扭转角公式,将求出的最大扭转角与用ANSYS建立的有限元模型计算结果进行对比。结果表明,悬臂梁在端扭矩作用下的扭转角理论值与有限元值较为接近,误差最小的绝对值为2.40%,最大的绝对值为4.44%,可为工程设计提供参考。

Development of quadrilateral spline thin plate elements using the B-net method

The quadrilateral discrete Kirchhoff thin plate bending element DKQ is based on the isoparametric element Q8, however, the accuracy of the isoparametric quadrilateral elements will drop significantly due to mesh distortions. In a previous work, we constructed an 8-node quadrilateral spline element L8 using the triangular area coordinates and the B- net method, which can be insensitive to mesh distortions and possess the second order completeness in the Cartesian co- ordinates. In this paper, a thin plate spline element is devel- oped based on the spline element L8 and the refined tech- nique. Numerical examples show that the present element indeed possesses higher accuracy than the DKQ element for distorted meshes.