Computational Particle Mechanics https://doi.org/10.1007/s40571-019-00245-0 Lagrangian–Eulerian enforcement of non-homogeneous boundary conditions in the Particle Finite Element Method M. Cremonesi 1 · S. Meduri 1 · U. Perego 1 Received: 24 December 2018 / Revised: 15 April 2019 / Accepted: 11 May 2019 © OWZ 2019 Abstract The Particle Finite Element Method (PFEM) is a Lagrangian finite element method with frequent remeshing, particularly suited for the simulation of fluid motions with evolving free surfaces, e.g., in the case of breaking waves or fluid–structure interactions with large displacements of the interaction surface. While the method has been successfully employed in a number of different engineering applications, there are several circumstances of practical interest where the Lagrangian nature of the method makes it difficult to enforce non-homogeneous boundary conditions. A novel mixed Lagrangian–Eulerian technique is proposed to the purpose of simplifying the imposition of this type of conditions with the PFEM. The method is simple to implement and computationally convenient, since only nodes on the boundary are considered Eulerian, while nodes inside the fluid body maintain their Lagrangian nature. A number of 2D and 3D examples, with analytical and numerical validations, confirm the excellent performance of the method. Keywords PFEM · Slip · Simmetry · Inflow/outflow · Non-homogeneous boundary conditions 1 Introduction In fluid mechanics problems, Navier–Stokes equations for incompressible fluid flows are typically formulated with boundary conditions such as inflow, outflow, slip, free- surface and symmetry. Following the general definition introduced in [64], these conditions can be classified based on the real or fictitious nature of the boundaries to which they are applied. Real boundaries are physical limits of the fluid domain where velocities or tractions can be defined (e.g., boundary walls, free surfaces). Fictitious boundaries are related to the fact that very frequently in fluid mechanics there are problems involving open domains (e.g., the flow past an aircraft wing or a bridge pile) or closed channel flows (e.g., flows in pipes). In these problems, the boundaries are simply limits of the computational domain and they are there- fore fictitious. Relevant cases belonging to this category are the inflow and outflow boundaries, which are fictitious sec- tions of the computational domain where velocity or traction B M. Cremonesi massimiliano.cremonesi@polimi.it 1 Department of Civil and Environmental Engineering, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milan, Italy profiles can be imposed specifying the state of the fluid par- ticles entering/exiting the computational domain [64]. Among the family of real boundaries, a particular case is represented by slip boundary conditions. More in detail, while the no-slip condition between the fluid and a basal surface or a confining wall is macroscopically accepted in most cases of fluid dynamics [8,49], there are several appli- cations involving fluids or fluid-like flows on solid surfaces where this condition is not realistic [54,56]. Relative fluid- wall slip can be observed in many industrial applications such as polymer extrusion [16,27,39], or in applications involv- ing granular flows, such as debris flows or silos discharge [10,40,53]. Navier slip boundary conditions define a linear correlation between the slip velocity and the basal tangen- tial stress, through a parameter summarizing the interaction between the fluid and the wall surface. This parameter con- trols the amount of slip at the wall surface, which can range from the no-slip condition with maximum tangential stress, to free slip with zero tangential stress. More realistic slip laws take into account the fact that, in general, slip occurs only when a critical threshold on the tangential stress is exceeded (see, e.g., [11]). A notable case of fictitious boundaries is the case of symmetry boundary conditions. In three-dimensional engi- neering applications, the size of the numerical model often 123