Phase field simulations of coupled microstructure solidification problems via the strong form particle difference method

Particle difference method for hydrogen permeation through tubular membranes

“…We formulate a particle difference method to approximate the above bulk equations (14) for the chemical potential (r, t) and the displacement u(r, t) fields, together with mechanical and chemical boundary conditions (15). In the PDM framework introduced in [7,10,14], the Eqs. (14) and (15) are discretized based on the pointwise particle difference approximation.…”

“…Thus, the Taylor polynomial vector is defined by For constructing the approximated fields, the PDM uses the moving least squares (MLS) approximation method based on pointwise computation. This approach, as Song et al [7] observe, resembles that of the derivative approximation of the point collocation method, cf. [15,16].…”

“…Figure 1 depicts the onedimensional non-smooth weight functions given in (30). According to Song et al [7], although smooth weight functions have been often used in the conventional weakly formulated meshfree method, it was recognized that nonsmooth functions are preferable to smooth ones in the strongly formulated particle method because of their resemblance to the Dirac delta function; therefore, the use of smooth weight functions is very feasible in the PDM.…”

“…The PDM is a meshless point collocation method equipped with the fast particle derivative approximation which consists of the Taylor polynomial expanded by the moving least squares method (MLS) based on the node model [6]. The method approximates derivatives terms that are then used to discretize the strong-form equations employing the precomputed derivative operators at each nodal collocation point [7]. According to [6][7][8], the PDM effectively circumvents shortcomings of the conventional meshless methods, which are mostly related to the necessity of mesh structure for the integration of weak form; it Technical Editor: Paulo de Tarso Rocha de Mendonça.…”

“…Notwithstanding, the drawbacks of meshless methods are still challenging in the PDM framework such as ill-conditioned matrices, the reduced sparsity of the resulting system of equations, and lack of an error analysis if compared to typical existing numerical approaches, for instance, the finite element method. The PDM has been recently used for modeling dynamic systems as Stokes flow problem [9], elastic crack growth [10], tracking interfaces evolution [8], and phasefield simulations of coupled microstructure solidification problems [7].…”