CAE-Companion-2018-2019

Theory WISSEN CAE

Meshless Methods: Smoothed Particle Hydrodynamics Method Introduction

but of the range around that point that the Kernel in question is spanning. Therefore, a “smoothing” of the domain has taken place, hence the term “smoothed” particle hydrodynamics. Although the SPH concept may differ completely from that of Finite Elements, there are similarities such the continuity of some basic variables within a limited region in space and that the SPHmethod may also be derived from a Galerkin formulation The Particle approximation is the next step after the Kernel approximation and it says that the domain around the point in question where we seek to define the value of a function, is NOT continuous. Instead it consists of a number of “topologically unconnected finite elements” which we will call from now on PARTICLES in order to distinguish them from the classical finite elements which have a pre-defined and rigid topological connection (the “connectivity” defined at the input level). The consequence of this approximation is in replacing the integral by a sum and modify the algebra to ac- count for the “number density” of the domain (ie. howmany particles can be found within a given domain volume defined by the Kernel we use). This is expressed as below : The above equation reads like : the contribution of each parti- cle within the Kernel range (taking into account its number density) is summed over all the particles in order to produce the smoothed value of a function at a point. Hence the above approximation has also its roots close to those of the classical FE method. In order for the Kernel and Particle approximations to be pragmatic, the choice of Kernel should be such that the following is satisfied : „ „ Compact form ie. acting over a finite range, zero outside that range „ „ Positive within this range „ „ Respecting the “delta function properties” „ „ Monotonically decreasing „ „ Degenerating in the limit to a delta function The reader should be reminded that indeed the first two requirements listed above are the same for the classical in- terpolation functions of the FE method. Therefore the Kernel should be seen as a form of an interpolation function. Figure 1 illustrates graphically the similarity between the FE and the SPH approximations. A patch of 9 elements is shown in both the FE and the equivalent SPH approximation. The interpolation functions have been overlaid upon the central

Advanced engineering applications have traditionally relied upon numerical methods like the Finite Element Method (FEM) in order to achieve the accuracy that closed-form mathematical solutions could not possibly offer. However, when the applications moved from the linear elastic domain to the non-linear large displacement / large strain domain, the classical FEM suffered strong limitations due to the loss of accuracy within highly distorted meshes. The practical restrictions imposed upon the usage of FEM in scenarios in- volving strong topological changes (like fracture) or involving Fluid-Structure Interaction (FSI) meant that a new class of methods had to be adopted, which would not suffer from the topological non-uniqueness problems that highly distorted FEMmeshes suffer. This was achieved by adopting “parti- cle-type” methods that do not maintain a strict connectivity in the domain, hence they do not follow a “mesh” (meshless methods). The most fundamental of these methods is the Smoothed Particle Hydrodynamics method, or SPH, which is a cornerstone in the Virtual Performance Solution (VPS) suite of codes of ESI Group. Overview of SPHMethod in VPS SPH is a gridless Lagrangian method whose corner stones are two approximations, namely : „ „ The Kernel approximation, and „ „ The Particle approximation The Kernel approximation is derived from the following identity :

which says nothing else than that the “value at a point” of a continuous function over a continuous domain could be ex- tracted from its integral by using a delta function as a “filter”. Assuming now that the delta function is replaced by another function which spans a certain “range” but still obeys the basic delta function property

then equation 1 will yield the following form

which is similar in appearance as before except for the functionWwhich will be called the KERNEL function and the range of influence it spans is controlled by the “smoothing length”. What that equation says is that the value of a func- tion at a point contains information about not just that point

90