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 引用本文: 苏海东. 固定网格的数值流形方法研究[J]. 力学学报, 2011, 43(1): 169-178.
Su Haidong. Study on numerical manifold method with fixed meshes[J]. Chinese Journal of Theoretical and Applied Mechanics, 2011, 43(1): 169-178.
 Citation: Su Haidong. Study on numerical manifold method with fixed meshes[J]. Chinese Journal of Theoretical and Applied Mechanics, 2011, 43(1): 169-178.

## Study on numerical manifold method with fixed meshes

• 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.

Abstract: The numerical simulations of large deformations of continuums lead to thechoice of an appropriate kinematical description. In classical viewpoints,Lagrangian and Eulerian description approaches are alternatives. Lagrangianapproach tracks material particles, allowing for a clear delineation ofboundaries of material. However, meshes that adhere to material are easy tobe distorted, inducing a poor accuracy or even computation failure. On theother hand, Eulerian approach is very attractive in the point that fixedmeshes will never be distorted, but it suffers from the complexities ofhandling moving boundaries and convective terms of Eulerian governingequations. Thus ALE (Arbitrary Lagrangian-Eulerian) method, which isreported to take advantages of Lagrangian and Eulerian approaches to acertain extent by allowing motions of meshes, is developed in recent years.Nevertheless, how to devise a good mesh motion algorithm is a great burdento the user, and convective terms are still involved.This paper presents a novel method, numerical manifold method (NMM) withfixed mathematical meshes, for short, fixed-mesh NMM, for analyzing puregeometric non-linear problems. Making well use of the fact that mathematicalmeshes are independent of material boundaries in NMM, this method is basedon the Lagrangian description approach, but using fixed meshes. It has thevirtues of both Lagrangian description approach and Eulerian descriptionapproach, avoiding mesh distortion of the former, and complexities ofhandling moving boundaries and convection items of the latter.Following the time steps, equations of NMM for large deformations areadopted in this paper, providing an easy way to implement fixed-mesh NMM.There are only two special factors to consider: after each time step iscompleted, deformed material boundaries are intersected with fixedmathematical meshes to generate new manifold elements; initial stress loadsare handled in a proper way, which is most important to fixed-mesh NMM.Based on fixed rectangular mathematical meshes and one order polynomialcover functions, two methods are presented to compute initial stresses.Given results of large deflection of a cantilever beam show the feasibilityof the fixed-mesh NMM, and indicate that more research should be furtherdone on computational stability due to initial stress loads.

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