An implicit corrected SPH formulation for thermal diffusion with linear free surface boundary conditions

Abstract

SUMMARYThe smoothed particle hydrodynamics (SPH) method has proven useful for modeling large deformation of fluids including fluids with stress-free surfaces. Because of the Lagrangian nature of the method, it is well suited to address the thermal evolution of these free surface flows. Boundary conditions at the interface of the fluid with a solid wall are usually enforced through the use of boundary particles. However, applying conditions at free surfaces, in particular gradient boundary conditions, can be problematic with traditional SPH formulations due to the degradation of the gradient approximation in these regions. Compounding this difficulty is that traditional approximations of the Laplacian operator suffer a similar degradation near free surfaces. A new SPH formulation of the Laplacian operator is presented, which improves the accuracy near free surface boundaries. This new form is based on a gradient approximation commonly used in thermal, viscous, and pressure projection problems, but includes higher-order terms in the appropriate Taylor series. Comparisons with other approximations of second-order derivatives are given. The discretization is tested by solving steady-state and transient problems of thermal diffusion using the Backward Euler method with a GMRES solver. Boundary conditions are imposed through an augmented matrix.

A review on approaches to solving Poisson’s equation in projection-based meshless methods for modelling strongly nonlinear water waves

“…In their derivation, Jubelgas et al (2004) expanded a function into a Taylor series ignoring all thirdor higher-order terms. Following the same line, Schwaiger (2008) showed that…”

“…(14) will be the same as that given by Jubelgas et al (2004) and its discretized form will be LP-SPH03 with η = 0. Corresponding to his formulation, Schwaiger (2008) gave the following discrete Laplacian: LP-SPH04:…”

“…Compared to others, this formulation requires inverse matrixes −1 ββ and C αβ at each particles and so may bear extra computational costs. It was employed for solving thermal diffusion problem without a free surface in Schwaiger (2008), but extended and tested by Lind et al (2012) to solve water wave problems. Hu and Adams (2007) suggested the following approximation by considering particle-averaged spatial derivative:…”

“…For threedimensional (3D) problems, it will be a 3 × 3 matrix (see, e.g. Schwaiger 2008). Khayyer et al (2008) suggested a similar correction to the kernel gradient and applying it to estimating internal viscous force calculation to preserve both linear and angular momentum, but not for solving Poisson's equation.…”

“…After solving for all the second derivatives of pressure, the discrete Laplacian can be formed by summing up of p ,αα . This Laplacian discretization is named as corrective smoothed particle method (CSPM) following Schwaiger (2008). Fatehi and Manzari (2011) derived a new scheme (they called it as Scheme 4) using error analysis.…”