A Green's discrete transformation meshfree method for simulating transient diffusion problems

Abstract

SUMMARYThis manuscript presents the formulation and application of the Green's discrete transformation method (GDTM) for the meshfree simulation of transient diffusion problems, including those with moving boundaries. The GDTM implements a linear combination of time-dependent Green's basis functions defined on a set of source points to approximate the field in the form of a solution series. A discrete transformation is implemented to evaluate unknown coefficients of this series, which eliminates the need to use time integration schemes. We will study the optimal number and location of the GDTM source points that yield the highest level of accuracy, while maintaining a manageable condition number for the resulting linear system of equations. The optimal values of these parameters, which are inherently independent of the domain geometry, are determined such that the basis functions have appropriate features for approximating the field. A comprehensive convergence study is presented to show the precision and convergence rate of the GDTM for modeling various diffusion problems. We also demonstrate the application of this method for simulating three diffusion problems with complex and evolving morphologies: heat transfer in a turbine blade, thermal response of a porous material, and localized (pitting) corrosion in stainless steel.

Further Modeling of Chloride Concentration and Temperature Effects on 1D Pit Growth

“…31 A good approximation for D • M+ is 0.7 × 10 −5 cm 2 s −1 . Other studies also used a single value of D M+ for different metal ions to analyze diffusion-controlled metastable 32,33 and stable 4,19,[34][35][36][37][38] pit growth.…”

Phase Field Modeling of Crystallographic Corrosion Pits

“…33 However, such sharpinterface approaches require extensive remeshing during the evolution of the pit, which significantly increases the computational cost. Some advanced techniques such as the Arbitrary Lagrangian-Eulerian method, 34,35 moving mesh approach, 36 and mesh-free methods 37,38 have been proposed to alleviate the remeshing problem but these approaches increase the complexity of the numerical simulations. Hence, in our work we have adopted a diffuse interface approach, namely phase field method, which is very efficient in simulating moving boundary problems 39 such as pitting corrosion.…”