Mathematical Physics - Volume II - Numerical Methods

5.2 Basic Formulation

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particles are fixed to FE nodes. This allows for a continuous coupling of the SPH and the FE domains. Sauer [82] proposed an SPH–FE coupling by extending the SPH domain onto the FE mesh. Different possibilities for exchanging forces between FE nodes and particles were shown, and the approach was used for adaptive conversion of elements into particles. The main difference with most other coupling methods is the use of a strong-form coupling. This approach was successfully applied to a number of impact problems, see Sauer et al. [83][84] and Hiermaier et al. [31][32]. Using a variation of the contact algorithm they developed for SPH, De Vuyst and Vignjevic [21] coupled Cranfield University SPH code with Lawrence Livermore National Laboratory DYNA3D. Coupling algorithms developed for other meshless techniques can be applied for use with SPH. Among many recently proposed techniques a selected few are mentioned below. A mixed hierarchical approximation based on meshless methods and FE, proposed by Huerta et al. [33][34], remove the discontinuities in the derivative across the interior boundaries when coupling FE and the element-free Galerkin method (EFG). Belytschko and Xiao [15] proposed the ‘bridging domain coupling method’ which uses Lagrange multipliers over a domain where FE and particle discretisations overlap. They applied this approach to multi-scale simulations for coupling continua with molecular dynamics. Another method for atomic and continua scale bridging was proposed by Wagner and Liu [103] and Kadowaki and Liu [41]. By matching dynamic impedances of different discretisation domains spurious wave reflection is prevented in this approach. A comprehensive overview of techniques for coupling of a range of meshless methods with FE with examples is given in Li and Liu [49]. 5.2 Basic Formulation The spatial discretisation of the state variables is provided by a set of points. Instead of a grid, SPH uses kernel interpolation to approximate the field variables at any point in a domain. For instance, an estimate of the value of a function f ( x ) at the location x is given in a continuous form by an integral of the product of the function and a kernel (weighting) function W ( | x − x ′ , h ) : ⟨ f ( x ) ⟩ = Z Ω f ( x ′ ) W ( | x − x ′ , h ) d x ′ . (5.1) Where: the angle brackets ⟨· ⟩ denote a kernel approximation h is a parameter that defines size of the kernel support known as the smoothing length x ′ is the new independent variable. The kernel function usually has the following properties: - Compact support, which means that it’s zero everywhere but on a finite domain, in conventional SPH this domain is taken to be all points within twice the smoothing length, h, of the centre: - W ( | x − x ′ | , h ) = 0 for | x − x ′ | ≥ 2 h . (5.2) - Normalised Z Ω W ( | x − x ′ , h ) d x ′ = 1 . (5.3)