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EFG method proposed by Prof. Ted Belytschko in mid. 1990 is an advanced numerical method to deal with large deformation, short waves and moving-boundary problems in solid and structural analyses. Thanks for the unique characteristics of mesh-independence, multi-scale (multi-resolution) approximations and naturally conforming, the method has been implemented in LS-DYAN with many other features for solving challenging industrial applications.

ME-FEM method was developed at LSTC in 2011 mainly for the simulation of nearly incompressible solids such as rubble-like materials and metals experiencing large deformation. This method has rigorous mathematical proofs and is an inf-sup stable formulation. The method can also incorporate with cohesive elements to model material failure in brittle, semi-brittle and ductile materials for solids


There are two types of EFG shell formulation available for the implicit and explicit structural analyses. EFG shell using local projection algorithm is suitable for problems that require accurately representing the local bending modes while avoiding numerical instability due to mesh distortions. Typical applications include the modeling of crashing tubes, metal covers in barrier, panels and eigenvalue analyses. EFG-shell using global projection algorithm is another formulation applicable to the sheet metal forming analysis. Since EFG approximations inherent a multi-scale property, this formulation is suitable for sheet metal forming problems involving fine gradients and sharper corner. Its extension from standard formulation to adaptive formulation is also available in LS-DYNA. In addition to standard meshfree approximation, several up-to-date approximations have also been developed at LTSC for the improvements of accuracy as well as efficiency in the original meshfree shell formulation.


Standard EFG solids formulation was first released in 2003 mainly for the modeling of deformable form materials. In order to improve efficiency, a stabilized EFG formulation was developed together with a user-defined formulation switch between standard and stabilized formulations.  An automatic Lagrangian-to-Eulerian kernel switch is also introduced in the case of severe deformation to further advance the simulation results in foam applications. Cohesive EFG formulation is a by-product of standard EFG-SOLID formulation developed for dynamic crack propagation in brittle materials based on Belytschko's meshfree visibility criterion. Similar to EFG-shells, there are several new meshfree approximations have been developed at LSTC for the simulation of solids with better accuracy. The coupling of EFG-shell, EFG-solid and finite element formulation is also straight forward and without introducing any special constraint keywords.


Adaptive EFG solid provides users an accurate numerical analysis tool in metal forming problems such as metal forging and extrusion simulations. The ingredients of the adaptive-EFG-solid formulation include the interactive adaptive procedure, internal valuables mapping using higher-order meshfree interpolations and pressure projection scheme. Those special numerical techniques make the adaptive-EFG-solid attractive in a large-scale simulation of manufacturing problems. The adaptive-EFG-solid formulation is able to couple with thermal solver for the implicit as well as explicit dynamic analyses.


Inf-sup stability determines the convergence of the numerical solution as mesh is continuously refined in the incompressible limit. ME-FEM is a newly developed inf-sup stable finite element formulation with meshfree enrichments in LS-DYAN to handle the large deformation analysis for near-incompressible problem. This method is suitable for modeling rubber-like materials such as seals, oiling, engine mount, bushing and several particle-reinforced or wire-reinforced rubber compounds. It is also suitable for modeling the metal components such as joints and engine blocks. The pressure field obtained from this new method is free of locking and free of oscillation in both implicit and explicit dynamic analysis.


Modeling dynamic crack propagation problems is always a main stream in computational mechanics. The inf-sup stable 5-noded ME-FEM-SOLID element allows a direct coupling of cohesive elements to model brittle, semi-brittle and ductile fractures in solids. Compared to the existing continuum mechanic approaches using stress or strain-based failure criteria such as element deletion or meshfree Eulerian kernel approaches to model dynamic crack propagation problems, the failure model in this new method is based on cohesive zone fracture model and thus is energy conserved and is independent on the mesh size. The method offers an extended applications in industrial and defense type of problems.


Two kinds of meshfree Galerkin methods, meshfree method and meshfree-enriched FEM method, are available in LS-DYNA. Compared to traditional FEM, these methods have advantage in accuracy and stability to be used in solving challenging numerical problems in solid and structural applications. For those of you who are interested in the theoretical parts of the methods including mathematical proofs, error estimates and benchmarks examples, several journal articles are available upon request. Also please feel free to contact us for more detail information on how to use those formulations in your analyses.