摘要
针对单纯几何非线性的材料大变形问题,提出一种新的研究思路——固定数学网格的数值流形方法,简称固定网格流形法,可以看作是采用了固定网格的拉格朗日方法.它充分利用数值流形方法的数学网格与材料物理边界分离的特性,具备拉格朗日法和欧拉法各自的优势,避免了原始拉格朗日法的网格扭曲问题以及欧拉法对移动边界难以精确描述和迁移项较难处理的问题.采用数值流形方法的大变形分步计算格式,使得固定网格流形法实现起来并不复杂,仅需要每步切割网格形成新的流形单元,以及对初应力载荷进行适当的处理,而后者是固定网格流形法的关键.针对固定的矩形数学网格开展研究,采用一阶多项式覆盖函数的高阶流形法,给出了两种初应力计算方法,并用悬臂梁大变形算例验证了固定网格流形法的可行性,将来需要进一步解决初应力载荷所带来的计算稳定问题.
The numerical simulations of large deformations of continuums lead to the choice of an appropriate kinematical description.In classical viewpoints,Lagrangian and Eulerian description approaches are alternatives. Lagrangian approach tracks material particles,allowing for a clear delineation of boundaries of material. However,meshes that adhere to material are easy to be distorted,inducing a poor accuracy or even computation failure.On the other hand,Eulerian approach is very attractive in the point that fixed meshes will never be distorted,but it suffers from the complexities of handling moving boundaries and convective terms of Eulerian governing equations.Thus ALE（Arbitrary Lagrangian-Eulerian） method,which is reported to take advantages of Lagrangian and Eulerian approaches to a certain extent by allowing motions of meshes,is developed in recent years.Nevertheless,how to devise a good mesh motion algorithm is a great burden to the user,and convective terms are still involved. This paper presents a novel method,numerical manifold method（NMM） with fixed mathematical meshes, for short,fixed-mesh NMM,for analyzing pure geometric non-linear problems.Making well use of the fact that mathematical meshes are independent of material boundaries in NMM,this method is based on the Lagrangian description approach,but using fixed meshes.It has the virtues of both Lagrangian description approach and Eulerian description approach,avoiding mesh distortion of the former,and complexities of handling moving boundaries and convection items of the latter. Following the time steps,equations of NMM for large deformations are adopted in this paper,providing an easy way to implement fixed-mesh NMM.There are only two special factors to consider：after each time step is completed,deformed material boundaries are intersected with fixed mathematical meshes to generate new manifold elements;initial stress loads are handled in a proper way,which is most important to fixed-mesh NMM. Based on fixed rectangular mathematical meshes and o
出处
《力学学报》
EI
CSCD
北大核心
2011年第1期169-178,共10页
Chinese Journal of Theoretical and Applied Mechanics
基金
国家自然科学基金资助项目(10772034)~~
关键词
大变形分析
拉格朗日描述
固定网格
数值流形方法
矩形数学网格
large deformation problems
Lagrangian description approach
fixed meshes
numerical manifold method（NMM）
rectangular mathematical meshes
