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General Formulas and Program Design for Manifold Method Based on Independent Covers Ⅰ:General Formulas
SU Hai-dong
Journal of Changjiang River Scientific Research Institute ›› 2025, Vol. 42 ›› Issue (4) : 193-201.
PDF(6589 KB)
PDF(6589 KB)
General Formulas and Program Design for Manifold Method Based on Independent Covers Ⅰ:General Formulas
Manifold method based on independent covers is a novel approach for numerically solving partial differential equations. By constructing approximate functions, it generates a “partitioned series solution” for partial differential equations. This method not only achieves the main functions of the finite element method (FEM) and other numerical techniques but also outperforms them in certain aspects, such as mesh generation flexibility and computational stability. However this also means that its calculation formulas and program design are different from existing methods. This paper reviews the major research outcomes in solid computation in recent years, and summarizes a set of simple and general calculation formulas in which the shape function of the local approximation function is expressed as the product of the Partition of Unity (PU) function, coordinate transformation matrix, and series matrix. The shape function and its derivatives under various scenarios are discussed in details. Different matrices and the time integration method are also given. These formulas can be applied to solve the differential equations of motion in elasticity, conduction equations, and wave equations, covering one-to-three-dimensional steady-state and transient analyses, along with three types of boundary conditions. They offer features such as high-order series, arbitrary mesh shapes, accurate boundary geometric simulation, precise application of essential boundary conditions, and local analytical series near the crack tip. Utilizing these formulas, a general program for the new method can be developed.
partial differential equations / series solutions / mesh division / exact geometry / independent covers / numerical manifold method
| [1] |
|
| [2] |
|
| [3] |
苏海东, 颉志强, 龚亚琦, 等. 基于独立覆盖的流形法的收敛性及覆盖网格特性[J]. 长江科学院院报, 2016, 33(2):131-136.
针对前期提出的基于部分重叠覆盖的数值流形方法,将其内涵范围缩小,仅研究其中的一种情况——基于独立覆盖的数值流形方法。从完备性和协调性2个方面讨论该方法的收敛性,特别强调其收敛性是基于各个独立覆盖的逼近而建立起来的,独立覆盖之间条形连接区域的尺寸要取小,并由此推断及用实例说明,覆盖网格可以具备“3个任意”的优良特性——任意形状、任意连接以及由此而来的可任意加密的能力,从而有望使数值计算的前处理工作大为简化。
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The scope of a numerical manifold method (NMM) based on partially overlapping covers is narrowed to a special case based on independent covers. Convergence of the new method is discussed from two aspects: completeness and coordination. The convergence of the method is due to the convergence of each independent cover. Results show that the size of the strips between independent covers should be small. Moreover, the cover meshes have three excellent features: arbitrary shape, arbitrary connection, and arbitrary refinement. Finally, some illustrations are given to verify these “arbitrary” features, and the method can be used to greatly simplify the pre-processing of numerical analysis.
|
| [4] |
苏海东, 付志, 颉志强. 基于任意形状网格和精确几何边界的数值计算[J]. 长江科学院院报, 2020, 37(7):167-174.
有限元网格形状要尽可能规则,网格之间必须通过结点连接,这些要求给复杂形状求解域的数值计算带来很大的前处理工作负担,而且实际的曲线边界一般要离散成有限单元能够描述的形式,难以模拟CAD模型的精确几何。针对这些问题,基于独立覆盖流形法提出任意形状且任意连接的覆盖网格,在CAE分析中模拟CAD模型的精确几何边界及其边界条件:将求解域划分为可包含曲线边的任意形状的块体网格,可以采用单纯形解析积分和数值积分2种方式进行块体积分;仅需在积分过程中考虑块体之间的窄条形(包括曲线条)的覆盖重叠区域,而不必在计算模型中生成这些条形;通过边界条实现本质边界条件的严格施加,包括曲线上的边界条件;给出2个数值算例验证了方法的有效性。任意形状的覆盖网格将为实现基于精确几何模型的数值计算及其完全自动化的前处理开辟新的路径。
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Finite element meshes should keep regular shape as much as possible, and ensure correct connections through nodes. These requirements pose a great burden to the pre-processing procedure of numerical computations for solving domains with complex shapes. On the other hand, curve boundaries in practical situations are usually discretized into shapes which finite element meshes can describe, resulting in an imprecise simulation of exact geometry defined in CAD. In view of this, cover meshes with arbitrary shapes and arbitrary connections are implemented using Manifold Method based on independent covers. Exact geometric boundaries of CAD models and boundary conditions are simulated in CAE analyses. The solving domain is divided into block meshes with arbitrary shapes which can contain curve boundaries. And two approaches, including analytical integration method with simplexes and numerical integration method, can be used for the block integration. The thin strips for cover overlapping are considered only in the integration process, but are not necessarily involved in the generation of computation models. Essential boundary conditions are strictly applied through boundary strips, including the boundary conditions on curves. Moreover, two numerical examples are given to illustrate the validity of the method. Cover meshes with arbitrary shapes bring about a new path for numerical computations based on exact geometric models and automatic pre-processing procedures.
|
| [5] |
祁勇峰, 苏海东. 3D裂缝应力强度因子的新算法[R]. 武汉: 长江科学院, 2016.
|
| [6] |
苏海东, 周朝, 颉志强, 等. 采用独立覆盖流形法分析精确几何描述的曲壳[J]. 长江科学院院报, 2018, 35(4): 158-166.
在前期研究的直梁和曲梁分析新方法的基础上,提出了基于独立覆盖流形法的曲壳分析方法。采用实体分析模式,只需使多项式覆盖函数中的某些项不参与计算,就能准确模拟三维平板和曲壳的Reissner-Mindlin假设,从而避免了推导曲壳控制方程及相应数值计算公式的复杂性。借助随中面参数方程变化的局部坐标系,并计算该坐标系的局部坐标和方向余弦关于整体坐标的导数,就能实现精确几何描述下的曲壳分析。给出了具体计算过程,包括刚度矩阵积分方式,以及相关的曲壳几何计算公式。通过球面壳和平板算例,验证了方法的收敛性。最后,结合前期的二维直梁和曲梁研究,以及本文的三维曲壳和平板研究,总结了基于独立覆盖流形法的梁板壳分析新方法的特点和优势,特别是彻底解决了自锁问题。
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On the basis of the new methods for straight and curved beam analysis proposed in previous study, a Numerical Manifold Method for curved shell analysis based on independent covers is presented. In the mode of solid analysis, the Reissner-Mindlin assumption for 3D plate and curved shells is strictly simulated just by eliminating some terms of polynomial cover functions. And therefore the complexity of the derivation for the governing equation of curved shells and the corresponding numerical calculation formula is avoided. By means of the local coordinate system varying with the middle layer of the shell described by parametric equations, and also by calculating the derivatives of the local coordinates and the direction cosines with respect to the global coordinates, curved shell analysis based on exact geometric description can be realized. The detailed procedures including the integrating method and geometric formula are given. Examples of a spherical shell and a plate are used to verify the convergence of the method. In the end, the characteristics and advantages of the new method for beam, plate and shell analysis are summarized, including the complete solution for locking problem, via previous studies of 2D straight and curved beams as well as the study of 3D plates and curved shells in this paper.
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| [7] |
苏海东, 韩陆超, 颉志强. 精确几何薄曲梁曲壳分析的分区级数解[J]. 长江科学院院报, 2022, 39(9):144-151.
薄梁板壳的数值计算涉及关于挠度的4阶微分方程,其困难在于构造C<sub>1</sub>连续的近似函数;同时,由于薄曲梁和曲壳控制方程的复杂性,通常用直梁或平板单元近似地模拟曲梁或曲壳,容易产生几何误差进而带来力学分析上的误差。前期研究采用独立覆盖流形法实现了基于厚梁板壳假设的精确几何曲梁和曲壳分析,本文在此基础上讨论了这种新型流形法的分区级数解的C<sub>1</sub>连续性,完成了基于Euler-Bernoulli梁理论和Kirchhoff-Love板壳理论的精确几何薄曲梁和曲壳分析,并解决了几何公式推导复杂的问题。详细给出了薄曲梁的计算公式,简述了薄曲壳的计算过程,将前期文献中的算例在薄梁板壳假设下重新计算,验证了方法的有效性,相比厚梁板壳假设可节省约30%的自由度。研究成果同时展示了应用独立覆盖流形法求解4阶微分方程的潜力。
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The numerical calculation of thin beam, plate and shell involves the fourth-order differential equation about deflection whose difficulty lies in constructing approximation functions with <i>C</i><sub>1</sub> continuity. In the meantime, due to the complexity of the governing equation, the thin curved beam and curved shell are usually simulated approximately by using straight beam or flat plate elements, which is prone to generate geometric errors and then brings errors in mechanical analysis. In our previous study, manifold method based on independent covers is used to analyze curved beam and shell with exact geometry based on the assumption of thick beam and shell. On this basis, the <i>C</i><sub>1</sub> continuity of the piecewise-defined series solutions of the new manifold method is discussed. The thin curved beam and shell with exact geometry is analyzed based on Euler-Bernoulli beam theory and Kirchhoff-Love shell theory, and the complexity of derivation of geometric formula is overcome. The calculation formula of thin curved beam is given in detail, and the process of thin curved shell is briefly described. The examples in previous study are recalculated under the assumption of thin beam, plate and shell, which verifies the effectiveness of the proposed method. Compared with the assumption of thick beam, plate and shell, the method saves about 30% of the degree of freedom. Meanwhile, the research demonstrates the potential of solving the fourth-order differential equations by applying manifold method based on independent covers.
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| [8] |
苏海东, 韦玉霞, 韩陆超, 等. 基于独立覆盖流形法的板壳与实体单元刚性连接研究[J]. 长江科学院院报, 2022, 39(9): 152-158.
有限元计算中,板壳单元与实体单元之间的连接需要进行特殊处理,且两者在连接处的网格必须匹配。前期基于独立覆盖流形法提出了梁板壳数值分析的分区级数解。在此基础上,研究了板壳与实体单元的刚性连接。由于板壳也采用了实体计算模式,因此与实体之间通过覆盖重叠区域自然连接。基于覆盖任意连接的特性,将板壳插入到实体中形成覆盖重叠区域。实体单元和板壳单元可以各自划分网格,在连接处不必要求网格匹配,有利于前处理工作,在网格划分达到一定密度的情况下能得到高精度的计算结果。通过变截面的悬臂梁算例、球面壳与实体基座连接算例,验证了方法的有效性,并初步展示了曲壳与实体相交曲线的精确几何。此外,还修正了新方法的三维弹性矩阵。
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In Finite Element Method, the connection between plate shell elements and solid elements needs special treatment, and their meshes at the connection must be matched, which brings some inconveniences. In our previous research, the piecewise-defined series solutions for the numerical analysis of beam, plate and shell are proposed by using manifold method based on independent covers. On this basis, the rigid connection between plate shell elements and solid elements is studied in this paper. The solid calculation mode is also adopted in the analysis of plate and shell which are naturally connected with the solid element through the overlapping area of the covers. In view of the characteristic of arbitrary connection of independent covers, the plate or the shell is inserted into the solid to form the overlapping area of the covers. The solid and the shell can be meshed separately in no need of mesh matching at the connection, which is very conducive to the preprocessing work. Highly precise results can be obtained when the mesh division reaches a certain density. The effectiveness of the method is verified by examples of a cantilever beam with variable sections, and a spherical shell with a solid base. The accurate geometry of the intersection curve between the curved shell and the solid is also preliminarily demonstrated. The three-dimensional elastic matrix of the new method is also modified.
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| [9] |
苏海东, 付志, 颉志强. 基于任意网格划分的二维自动计算[J]. 长江科学院院报, 2020, 37(7): 160-166.
针对有限元法网格剖分和加密不方便、难以实现精确几何建模以及人工操作量大等问题,采用前期提出的独立覆盖流形法,利用其覆盖网格所具有的任意形状、任意连接和任意加密的特性,基于“凸剖分”的思路提出二维求解域的一种任意网格划分方法。在此基础上,结合前期研究的误差估计和h-p型混合自适应分析手段,尝试二维结构线弹性静力分析的自动计算,包括求解域的自动细分、多项式级数的自动升阶等过程。通过重力坝和带圆孔平板的2个算例验证了方法的可行性,其中第2个算例演示了从CAD的几何信息和计算参数输入到基于精确几何的CAE自动建模、自适应分析、成果自动输出的全过程,初步实现了CAE自动计算以及CAD与CAE的融合。
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Finite Element Method (FEM) is inconvenient in mesh division and subdivision, difficult in precise modeling of exact geometry, and costs large amount of labor operations. In view of this, we propose an approach of arbitrary mesh subdivision in the 2D solving domain based on convex decomposition idea using Manifold Method based on independent covers presented previously, in which cover meshes are of arbitrary shape, arbitrary connection and arbitrary subdivision. On this basis, with the help of error estimation and h-p version self-adaptive technology in previous studies, we attempt to implement the automatic static analysis of 2D linear-elastic structure, including the automatic subdivision of the solving domain, and the automatic elevation of polynomial orders. Two numerical examples, one of which is a gravity dam and the other is a plate with a small circular hole, are given to illustrate the validity of the present method. Especially in the latter, the whole procedure is exhibited, involving the input of geometry information and computational parameters in CAD, automatic CAE modeling with exact geometry, automatic self-adaptive analysis, as well as the automatic output of computational results. Hence, the automatic CAE computation and CAD/CAE integration are realized preliminarily.
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| [10] |
韩陆超. 基于任意网格的三维数值计算初步研究[D]. 武汉: 长江科学院, 2022.
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| [11] |
苏海东, 陈积瞻, 颉志强, 等. 基于独立覆盖流形法的CAD与CAE融合研究[J]. 长江科学院院报, 2017, 34(12):133-139,154.
采用前期笔者提出的独立覆盖流形法,尝试CAD和CAE融合的新途径。以二维结构的线弹性静力分析为例,实现从CAD到CAE的无缝连接,以及CAE的完全自动化分析。在基于前期CAD几何的流形法研究基础上,给出NURBS曲线(CAD中的通用图形标准)与直线的切割算法,实现CAD几何模型在CAE建模和网格细化中的保形性;通过AutoCAD的DXF图形格式,将CAD中的结构形状、荷载及约束信息直接输出到CAE;基于矩形独立覆盖的自适应分析技术,实现结构静力分析的自动化计算;自动生成有限元网格用于计算结果后处理的图形输出。综合以上研究,用一个二维结构静力分析算例演示了从CAD几何建模和输出,到CAE的自动前处理、自动分析、自动后处理的完整过程,所有的人工操作仅限于CAD中,而CAE分析过程无需人工参与,就可以获得满足设定精度的计算结果。
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Using Numerical Manifold Method (NMM) based on independent covers proposed in previous study, we attempted a new way of integrating Computer Aided Design (CAD) and Computer Aided Engineering (CAE). Taking 2D static analysis of linear-elastic structures for example, we accomplished the seamless link from CAD to CAE and the total automatic CAE analysis. On the basis of previous study of NMM based on CAD geometry, we put forward algorithms for cutting NURBS (Non-uniform Rational B-spines, regarded as the general graphic standard in CAD) curves with straight lines, hence preserving the shape of geometric model in procedures of CAE modeling and mesh refinement. Meanwhile, data of structural shape, loads and constraints defined in CAD are directly transferred to CAE by means of DXF graphic format of AutoCAD. Furthermore, by using adaptive analysis based on rectangular independent covers, we conducted automatic static analysis of structures and automatic finite elementmeshes for post-processing graphic output of computational results. In association with the above studies, we took a 2D static structure as example to illustrate the entire procedures, including CAD modeling and output, automatic pre-processing, automatic analysis and automatic post-processing in CAE, in which all procedures were automatically accomplished by computer, except for manual operation of CAD model. Finally, we obtained calculated results with given precision.
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| [12] |
董鹏. 基于独立覆盖流形法的材料非线性分析[D]. 武汉: 长江科学院, 2021.
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| [13] |
苏海东, 董鹏, 颉志强. 在固定的独立覆盖网格中求解几何非线性问题[J]. 长江科学院院报, 2022, 39(9):159-166.
采用独立覆盖流形法(基于流形思想的“分区级数解”),提出在空间固定的网格中求解几何非线性问题的新方法:在当前构形中关注经过各空间点的物质点,通过级数“逆向追踪”物质点在上一时步的位置及其应力、速度等物理量,并采用最小二乘法形成新级数作为当前时步的初值,就可以在固定网格中求解拉格朗日型的控制方程;每步计算后更新材料体构形,即更新固定网格(独立覆盖)中的积分区域,以得到准确的材料边界;以覆盖合并方式处理边界网格中的“小块”问题,并通过“小块”实现新网格的信息传递。给出弹性体大变形、刚体旋转算例验证方法的有效性。新方法集合了拉格朗日法的跟踪物质点、控制方程简单、边界描述准确以及欧拉法的网格无扭曲的优点,避免了2种方法各自的缺陷,为下一步在固定网格中进行几何非线性的自适应分析打下基础。
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Using Manifold Method based on independent covers, or piecewise-defined series solutions based on manifold idea, a novel method of solving geometric nonlinear problems in fixed meshes is proposed. In the current configuration, the material point passing through each space point is paid attention to. Through the series solutions, the position of the material point in the previous step is traced backwards, and its stress, velocity and other physical quantities are obtained. A new series is formed as the initial values of the current time step using the least square method, hence the Lagrangian governing equation can be solved in fixed meshes. After each step, the material configuration is updated, that is, the integral region in the fixed meshes (independent covers) is updated so as to obtain the accurate material boundaries. The “small block” in the boundary meshes is processed by cover merging, and the information to the new mesh is transmitted through the “small block”. Some examples such as the large deformation of elastic body and the rotation of rigid body are given to verify the effectiveness of the method. The proposed method combines the advantages of Lagrangian method in tracking material points, simplicity of the governing equation, accurate boundary description, and of Eulerian method of undistorted meshes, and in the meantime avoids the defects of the two methods. The research finding lays a foundation for the next step adaptive analysis of solving geometric nonlinearity in fixed meshes.
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| [14] |
刘亚军, 龚亚琦, 苏海东. 一维对流扩散方程的独立覆盖分析方法[J]. 长江科学院院报, 2020, 37(7): 175-182.
针对现有的各种数值方法在求解一维对流扩散方程时容易出现的数值振荡、假扩散等计算稳定性和计算精度不足问题,提出应用独立覆盖流形法进行数值求解的新思路,即分区的多项式级数逼近。基于标准的伽辽金法推导一维对流扩散方程的独立覆盖流形法求解公式。采用场变量的一阶导数在独立覆盖之间的窄条形覆盖重叠区域是否连续的后验误差估计方法,通过覆盖加密和级数升阶的h-p型混合自适应进行自动求解。给出的稳态和非稳态分析算例结果表明:分区级数的数值解稳定地逼近于精确解,最终两者很好地吻合;对于对流占优问题,自适应求解可以有效避免数值振荡。另外还尝试了将数值解代回微分方程计算残差作为误差指标,如果能使微分方程逐点满足,那么将是对数值解最严格的误差判断。
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In solving one-dimensional convection-diffusion equations, present numerical methods are prone to suffer from stability and accuracy problems caused by numerical oscillation and pseudo-diffusion. In view of this, an idea of applying Numerical Manifold Method (NMM) based on independent covers (the approximation using polynomial series piecewise-defined) to the numerical solution is proposed. The solution formula of the one-dimensional convection-diffusion equation is derived based on the standard Galerkin method. The posterior error estimation method about the continuity of the first-order derivative of field variable in the narrow overlapping area between independent covers is used for the automatic solving by h-p hybrid self-adaptive analysis with mesh refinement and ascending series order. The results of the steady-state and unsteady-state analysis examples show that the numerical solution of the piecewise-defined series steadily approximates and finally well fits the exact solution. For the convection-dominated problem, the adaptive solution effectively avoids numerical oscillation. In addition, the error index of the residual by substituting the numerical result back to the differential equation is successfully attempted. If the differential equation is solved point by point, the method would the most stringent error judgment for the numerical solution so far.
|
| [15] |
刘亚军. 基于独立覆盖流形法的流体计算研究[D]. 武汉: 长江科学院, 2020.
(
|
| [16] |
|
| [17] |
|
| [18] |
|
| [19] |
|
| [20] |
|
| [21] |
|
| [22] |
徐芝纶. 弹性力学[M]. 2版. 北京: 人民教育出版社, 1982.
(
|
| [23] |
王元明. 工程数学-数学物理方程与特殊函数[M]. 4版, 北京: 高等教育出版社, 2012.
(
|
| [24] |
(
|
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|
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