Study on numerical manifold method with fixed meshes

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.