To mesh or not to mesh, that is the question\u2026<\/em><\/p>\n\n\n\n If Shakespeare\u2019s Hamlet was a 21st-century computational fluid dynamics (CFD) engineer, that would probably be the revised version of his existential question. Triggered by the murder of a close relative, the original Hamlet wonders and counts arguments and counterarguments about what is more desirable: \u201cTo be or not to be\u201d is the question about being alive or not being alive. And if you look at the development of CFD technology these days, the rise of meshless technology makes you wonder: <\/p>\n\n\n\n Will Smoothed Particle Hydrodynamics (SPH) one day kill the mesh-based finite volume CFD approach? And is one better than the other?<\/em><\/p>\n<\/div>\n\n\n\n The original Hamlet concludes that the \u201cto be or not to be\u201d question remains unanswerable. There is no guarantee that each side, dead or alive, is better than the other and therefore, he cannot choose to be or not be.<\/p>\n\n\n\n But all philosophy aside\u2026 and back to fluid dynamics: Numerical simulation has increasingly become prominent in solving various applications inherent to fluid dynamics. Two approaches evolved parallelly in recent years.<\/p>\n\n\n\n Eulerian methods <\/strong>emerged in the late 1960s. Those techniques are based on geometric, fixed, or adaptive grids. Flow properties like velocity or pressure are estimated at each mesh cell via numerical formulations such as finite differences, finite elements, or finite volumes. The finite volume methods are very successful for most of the fluid applications today. However, the use of mesh-based methods results in difficulties for specific applications, particularly for highly dynamic free surface flows, complex moving objects, or large deformations. And while those can be overcome with modern finite volume technologies, this typically comes at an increased computational cost.<\/p>\n\n\n\n To address and lift these restrictions, Gingold, Monaghan (1977), and Lucy (1977) introduced a Lagrangian approach: the so-called Smoothed-Particle Hydrodynamics (SPH) method.<\/strong> This technology is part of a new generation of numerical methods, developed to overcome meshing-related constraints.<\/p>\n\n\n\n Still based on the Navier-Stokes equations, the SPH meshless method represents the fluid flow by moving elementary bodies referred to as particles. Moving particles thus discretize the continuous space domain. This defines a particle-based method.<\/p>\n\n\n\n Those particles are individually tracked: they have their own position, velocity, and size. They carry \u2013 like mesh cells in standard finite-volume methods \u2013 the state variables of the fluid parcel (velocity, density, pressure, temperature\u2026). This evaluation considers the neighbor particles’ contribution by using a kernel function, giving the smoothing characteristic of the formulation.<\/p>\n\n\n\n If you look at the video below of an experimental setup of a simplified gearbox lubrication application, the SPH method discretizes the liquid oil by fluid particles.<\/p>\n\n\n\n![\"Shakespeare's](\"https:\/\/blogs.sw.siemens.com\/wp-content\/uploads\/sites\/6\/2024\/02\/GettyImages-118374264.jpg\")
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\u201cTo mesh, or not to mesh\u201d\u2026 does the revamped CFD question also remain unanswerable? Is there no guarantee that the mesh-based method or meshless technology is better than the other? That is the question we will try to answer in this article.<\/p>\n\n\n\nA little bit of history<\/strong><\/h2>\n\n\n\n