Mathematical Physics - Volume II - Numerical Methods

Chapter 5. Review of Development of the Smooth Particle Hydrodynamics (SPH) Method

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into account the boundary conditions through residual terms in the integral by parts. Very interesting work on boundary conditions in SPH is due to Takeda et al. [89], who applied SPH to a variety of viscous flows. A similar approach was also used by Randles [74] with the ghost particles added to enforce reflected and symmetry surface boundary conditions. Belytschko, Lu and Gu [13] imposed the essential boundary conditions by the use of Lagrange multipliers leading to an awkward structure of the linear algebraic equations, which are not positive definite. Krongauz and Belytschko [43] proposed a simpler technique for the treatment of the essential boundary conditions in meshless methods, by employing a string of finite elements along the essential boundaries. This allowed for the boundary conditions to be treated accurately, but reintroduced the shortcomings inherent to structured meshes. Randles et al. [74][78] were first to propose a more general treatment of boundary conditions based on an extension of the ghost particle method. In their approach the boundary is considered to be a surface one half of the local smoothing length away from the so-called boundary particles. A boundary condition is applied to a field variable by assigning the same boundary value of the variable to all ghost particles. A constraint is imposed on the boundary by interpolating it smoothly between the specified boundary particle value and the calculated values on the interior particles. This serves to communicate to the interior particles the effect of the specific boundary condition. There are two main difficulties in this: • Definition of the boundary (surface normal at the vertices). • Communication of the boundary value of a dependent variable from the boundary to internal particles. A penalty contact algorithm for SPH was developed by Campbell and Vignjevic [18]. This algorithm was tested on normalised SPH in combination with the Randles’ approach for treatment of free surfaces. The contact algorithm considered only particle-particle interactions, and allowed contact and separation to be correctly simulated. However, tests showed that when this approach is used zero-energy modes are often excited. Further development of this contact algorithm for the treatment of contact problems involving frictionless sliding and separation under large deformations was achieved by the contact conditions through the use a contact potential for particles in contact, see Vignjevic et al. [99]. Inter-penetration is checked as a part of the neighbourhood search. In the case of conventional SPH contact conditions are enforced on the boundary layer 2h thick while in the case of the normalized SPH contact conditions are enforced for the particles lying on the contact surface. In a number of engineering applications it is beneficial to discretise only certain parts of the domain with particles and the rest with finite elements. The main reasons for this are to take advantage of the strengths of both methods, which include significantly better numerical efficiency of the finite element (FE) method, and in SPH modelling arbitrary crack propagation, large deformations and adaptive refinement of the discretisation. One of the first coupling procedures for FE-SPH coupling was proposed by Attaway et al. [6]. They developed a penalty-based approach for modelling of fluid–structure interactions where the fluid was discretised with particles and the structure was modelled with finite elements. A similar approach was proposed by Johnson [38] and Johnson et al. [39]. In addition to the contact algorithm, they developed a tied interface where SPH