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。

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.

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。

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.

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。

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.

采用前期研究的基于矩形独立覆盖的新型数值流形法,提出结构线弹性静力分析的自动计算方法,包括自动的前处理、自适应分析等。根据独立覆盖的特点提出几个后验误差指标:独立覆盖之间条形连接区域的应变连续性指标;边界应力指标和独立覆盖的高阶误差指标。利用新方法的h型网格加密及p型升阶的方便性,选择一种路径尝试h-p型的混合自适应,其中,对于矩形独立覆盖采用简单的二分法实现覆盖加密。通过几个二维算例验证了新方法实现自动计算的可行性,只需人工输入结构外形、材料参数和边界条件,其它工作完全交由计算机完成,最终得到满足一定精度的计算结果。

By means of numerical manifold method (NMM) based on independent rectangular covers proposed in previous study, we present an automatic computation method for static analysis of linear-elastic structures, including automatic pre-processing, self-adaptive analyses and so on. According to the characteristics of independent covers, we give 3 indexes for posterior error such as index of strain continuity in strip area between two covers, stress index on boundary surfaces and high-order error in an independent cover. By using convenient h-version mesh refinement and p-version order increasing in the new method, we implement h-p version self-adaptivity in a selected way to realize h-version refinement of rectangular covers by using simple bisection method. Some 2D numerical examples are given to illustrate the feasibility of automatic computation, in which all the procedures are automatically accomplished by the computer, except for necessary manual input of structural outlines, material parameters, and boundary conditions. Finally, we obtain calculated data with certain precision.