可商业化光滑有限元分析软件的研发

The meshfree methods (championed by Belytchko, G.R. Liu and many others) have attracted attentions in the area of computational methods in the past 20 years, due to the excellent features of robustness. The meshfree methods have not been widely commercialized due to the bottleneck problem of computation speed, compared to the well-established finite element method (FEM). The recent S-FEM (Smoothed Finite Element Methods)developed by G.R. Liu at el. possesses features of both meshfree (robustness) and FEM (speed), and has been widely applied to solve various complicated problems in science and engineering. With the rapid development of S-FEM and the proven fact the FEM is in theory a special case of S-FEM, it is believed that the S-FEM will be soon in place of the widely used FEM. This project aims to develop a set of commercial S-FEM software package that consists of a number of modules including edge-based S-FEM (ES-FEM), node-based S-FEM (NS-FEM), cell-based S-FEM (CS-FEM), and the combined NS-FEM and FEM modules. It should be capable of solving mechanics problems of solids and structures including statics, dynamics, linear, nonlinear, large deformation and contact problems. It will be fully comparable to the existing commercial software packages in the market in the capability regards. Because the S-FEM is based on the Weakened Weak (W2) formulation, it should be superior to its FEM counterpart in the following aspects: 1) The S-FEM model is proven in theory always "softer" than the FEM model, and hence can produce upper bound and ultra-accurate solutions; 2) It works well with the T-mesh (Triangle for 2D, Tetrahedron and Triangular-prism for 3D), hence it is ideal for the full-automation in computation and adaptive analysis, and the volumetric locking problems can be conveniently overcome; 3) It does not demand for high quality mesh as its FEM counterpart, and hence facilitate easy local mesh refinements in automatic fashions; 4) The computational efficiency can be up to one order of magnitude higher, due to the ultra-accuracy, simplicity in the formulation and implementation, and much better conditioning in the discretized system equations. Upon successful, the developed S-FEM can strategically change the current under-developed situation in China in terms of engineering software development, and compete with the international major software packages.

近年来，无网格法已引起人们的广泛关注。但由于计算速度的瓶颈，它并没有像FEM那样大规模的商业化。因此将无网格和FEM相结合，刘桂荣提出了光滑有限元法S-FEM，它兼有无网格法的健壮性和FEM的高效性等优点。而且S-FEM能有效解决FEM不能解决的许多复杂问题，同时也证明了FEM是S-FEM的一个特例。由于S-FEM使用弱弱形式，较FEM等使用弱形式方法具有突出优势：①S-FEM模型比FEM 模型"软"，因此可给出问题的上界解和超精确解；②S-FEM采用能自动生成的T-mesh，从而可用于自动建模和自适应分析；③S-FEM对网格质量要求很低；④S-FEM的计算效率比FEM高得多。本项目拟将开发一套可商业化的S-FEM软件来解决科学和工程中的固体力学和结构力学问题，包括静动态、非线性、大变形和断裂问题等。希望通过该软件的开发改变中国在工程分析软件上的落后局面，占领下一代工程软件开发的至高点。

有限元作为一种高效的数值计算方法已经被广泛应用到各领域的力学分析中，而且已经涌现出了许多成熟的商业软件。随着人们对有限元的不断研究和实践应用，发现该方法存在的一些问题和不足，例如：应力解精度不高、无法解决体积锁定问题、模型过硬等问题。针对这些问题，G.R. Liu提出S-FEM方法。该方法能够取得非常高的应力解，对网格依赖很小，而且具有许多特殊的属性。因此，本项目对S-FEM求解不同力学问题进行了理论以及算法实现方面的深入研究，取得的成果主要包括：1）高效T-Mesh自适应及基于T-Mesh五种光滑域数据结构的快速建立；2）研究了S-FEM 的应变光滑技术及光滑域的点插值法（PIM），并给出了S-FEM理论基础的G空间证明；3）建立了动态及稳态、瞬态热传导问题的S-FEM 公式，及对应的求解器。同时研究了显式S-FEM稳定的条件和提出了求解时间步长的简化公式；4）建立了接触问题的S-FEM公式和求解器，并研究了复合材料中S-FEM算法以及具有的性质；5）建立了几何非线性和材料非线性的S-FEM公式和算法；6）研究了弱弱形式的刚度矩阵及S-FEM 模型的高速求解器，包括光滑域的快速构建、基于多重网格的线性方程组快速求解器的设计等；7）针对二维的MFree2D软件，替换了原来的面积平均S-FEM算法的求解器，实现了严格按照S-FEM理论的求解器，完善了前后处理功能；针对三维S-FEM 软件包：S-FEM®，开发了图形用户界面的前、后处理模块，实现了固体力学问题的S-FEM求解器，为研究人员利用S-FEM分析实际力学问题提供了可视化操作平台。在项目实施过程中，完成论文21篇，检索12篇，收录9篇；出版译著1本；培养博士生4名，硕士生7名；资助1名教师出国进修1年；参加学术会议和研讨会7人次/年；主办国际学术会议一次。