Although the PFEM was originally designed for fluid dynamics and FSI applications, the method is a powerful tool also for non-linear solid mechanics. In this field, PFEM is particularly useful for those processes where solid bodies undergo large motions and deformations, as it occurs in several manufacturing processes and many geotechnical applications. In this field, the PFEM has shown the capability to track accurately the deforming shape of the solid material and to deal with the complex interactions between the different solid bodies.

Pressing a quarter of a metal cylinder [12]

Orthogonal cutting test of a steel workpiece. Continous chip formation considering a deformable tool [14]

The first PFEM formulation for solid mechanics [2] was applied to complex industrial processes, such as metal forging, machining, or powder filling. Different types of manufacturing processes are studied in [6], [5] analyzed tunneling and excavation applications, while bed erosion in river dynamics was tackled in [1]. Several PFEM formulations have been proposed in the field of soil mechanics and geotechnical engineering, especially for the modeling of frictional materials and granular flows [7] and for different types of geomechanics problems [9]. A more extended literature review of the PFEM formulations for solid mechanics can be found in [11].

3D machining done by a rigid tool generating a continous chip from a metal piece [12]

One of the most critical points of PFEM for solid mechanics is the management and conservation of the historical variables during the remeshing step. The solution accuracy depends on how well the historical information is preserved along with the transition from the previous mesh to the new one. This remapping procedure may lead to smooth the historical solution. This problem has been extensively studied in the PFEM literature. To limit the solution smoothing, Oliver et al. [2] proposed to remap not the whole historical variable but just its time step increment. For the same purpose, [8] proposed to transfer the information directly from the Gauss points of the previous mesh to those of the new mesh in those zones where the connectivity has not changed.  Recently, [10] proposed to use nodal integration rather than the Gaussian one to improve the preservation of historical variables. In nodal integration schemes, all variables (including stresses and strains) are stored at the nodes. Hence, this information is not erased during the remeshing and is not affected by the remapping procedure.

The capability of the PFEM for dealing with contact interaction between solids was proved since the first applications of the method to non-linear solid mechanics problems. The first study that showed its suitability for contact problems was [1]. In this work, the authors studied FSI problems where the solid bodies, dragged by the fluid motion, could impact each other and against the rigid contours. 

The PFEM contact algorithm uses a mesh of interlayer elements between the boundaries of the interacting solid bodies. This auxiliary mesh is created following the same steps of the standard PFEM remeshing procedure described before. The contact elements that pass the Alpha-Shape check are furtherly reduced by considering for computation only those elements having a size smaller than a fixed critical value. Over these active contact elements, the elastic and frictional forces are computed either with penalty methods or using Lagrangian multipliers [11].

Machining by a rigid tool generating a segmental chip from a metal piece [8]

The contact algorithm was improved in [2] by applying an artificial contraction to the solid boundaries to capture more accurately the contact time and to reduce the distance between the interacting solid bodies. The same methodology was also used in the so-called Contact Domain Method [3] and formalized in [4]. The first 3D application of the PFEM contact algorithm was presented in [3], where the capabilities of the method were proved against complex multi-body interactions, either in the presence of water or not.

Cylindrical steel bar impacting with a wall [13]

References

[1] Oñate E, Celigueta M, Idelsohn S (2006) Modeling bed erosion in free surface flows by the particle finite element method. Acta Geotech 1(4):237–252     
[2] Oliver J, Cante J, Weyler R, González C, Hernández J (2007) Particle finite element methods in solid mechanics problems. In: Oñate E, Owen R (eds) Computational plasticity. Springer, Berlin
[3] Oliver J, Hartmann S, Cante J, Weyler R, Hernández J (2009) A contact domain method for large deformation frictional contact problems. Part 1. Theoretical basis. Comput Methods Appl Mech Eng 198(33–36):2591–2606
[4] Hartmann S, Oliver J, Weyler R, Cante J, Hernández J (2009) A contact domain method for large deformation frictional contact problems. Part 2. Numerical aspects. Comput Methods Appl Mech Eng 198(33–36):2607–2631
[5] Carbonell JM, Oñate E, Suarez B (2010) Modeling of ground excavation with the particle finite-element method. J Eng Mech 136(4):455–463
[6] Oñate E, Franci A, Carbonell JM (2014) A particle finite element method for analysis of industrial forming processes. Comput Mech 54(1):85–107
[7] Cante J, Dávalos C, Hernández J, Oliver J, Jonsén P, Gustafsson G, Häggblad H (2014) PFEM-based modeling of industrial Granular flows. Comput Part Mech 1(1):47–70
[8] Rodriguez J, Carbonell JM, Cante J, Oliver J (2016) The particle finite element method (PFEM) in thermo-mechanical problems. Int J Numer Methods Eng 107(9):733–785
[9] Monforte L, Arroyo M, Carbonell JM, Gens A (2017) Numerical simulation of undrained insertion problems in geotechnical engineering with the particle finite element method (PFEM). Comput Geotech 82:144–156
[10] Zhang W, Yuan W, Dai B (2018) Smoothed particle finite-element method for large-deformation problems in geomechanics. Int J Geomech 18(4):04018010
[11] Cremonesi M, Franci A, Idelsohn SR, Oñate E (2020) A state of the art review of the Particle Finite Element Method (PFEM). Archives of Computational Methods in Engineering, 17, 1709-1735
[12] Carbonell JM, Rodriguez JM, Oñate E (2020) Modelling 3D metal cutting problems with the particle finite element method. Computational Mechanics. Springer. 66-3, pp.603-624
[13] Rodriguez JM, Carbonell JM, Jonsen P. (2020) Numerical methods for the modelling of chip formation processes. Archives of computational methods in engineering. Springer. 27, pp.387-412
[14] Rodriguez JM, Larson S, Carbonell JM, Jonsen P. (2021) Implicit or explicit time integration schemes in the PFEM modeling of metal cutting processes Computational Particle Mechanics. Springer.