Meshless Methods for Simulations of Solid Mechanics
Finite element (FE) has been a staple of engineering analysis for quite a few decades due to its powerful accuracy when solving a wide range of problems. The quality of a FE analysis is highly dependent on the structure of the mesh that maps the physical system. For solid mechanics problems, if the mesh deforms too much, the solution will likely be inaccurate or even diverge. Typical fixes for this problem include increasing the mesh density or function order around problem areas, or adaptive meshing as the solution progresses. These fixes, however, can be incredibly computationally taxing, especially for time-dependent problems. For problems in which the object of interest physically breaks, either by cutting or fracturing, they might be impossible to solve with traditional FE methods.
To circumvent this issue, some computational methods have arose that do not require a traditional mesh. This research reviews the literature on meshless computational methods for solid mechanics, specifically looking at applications for cutting and fracturing. The three main meshless formulations in use today are smoothed particle hydrodynamics (SPH), moving least squares (MLS), and the material point method (MPM). There are also extensions of the traditional finite element method called XFEM that use discontinuous functions in between mesh nodes. These methods have also found some success in the field.
SPH treats the material as a flow of particles rather than a distinctly continuous body. The advantages of SPH are that there is no remeshing, and that there is natural material separation through the movement of particles around the cutting tool. SPH also seems to have nice convergence properties and speed, however it is much less researched than MLS.
MLS formulations are numerous and quite similar to finite element. Instead of a node connecting to a few specific nodes around it, it is given a radius of influence. Any other node within the radius of influence is included into a weighting function. Special attention needs to be given to nodes that are close to discontinuities. This boundary problem is a topic of great research and many different methods have arisen to deal with it because it adds significant computational time to the problem. Further more, MLS methods still require a background mesh, therefore is not truly meshfree. There has been some success in creating cutting and fracturing simulations for arbitrary objects and cutting patterns using MLS methods.
The MPM uses a finite volume style formulation for solid mechanics. A bucket sorting mechanism to group particles into uniform cell volumes which are weighted based on the number of particles and their function orders. Though there is still a lack of accuracy to this method for problems with discontinuities, it did perform faster than its FE counterpart. An interesting study coupled MPM with molecular dynamics to study nanoscale cutting and achieved favorable results.
In conclusion, MLS methods seem to have the greater research focus with reasonable accuracy. But due to its computational complexity, it is best used in conjunction with FE, using MLS for regions closer to the discontinuity and FE everywhere else. It may also be the case that advancements in remeshing schemes for FE or XFEM will overtake meshless methods. There is still no clear for the field, except for further research.