The Meshless Analysis of Scale-Dependent Problems for Coupled Fields

The meshless local Petrov–Galerkin (MLPG) method was developed to analyze 2D problems for flexoelectricity and higher-grade thermoelectricity. Both problems were multiphysical and scale-dependent. The size effect was considered by the strain and electric field gradients in the flexoelectricity, and higher-grade heat flux in the thermoelectricity. The variational principle was applied to derive the governing equations within the higher-grade theory of considered continuous media. The order of derivatives in the governing equations was higher than in their counterparts in classical theory. In the numerical treatment, the coupled governing partial differential equations (PDE) were satisfied in a local weak-form on small fictitious subdomains with a simple test function. Physical fields were approximated by the moving least-squares (MLS) scheme. Applying the spatial approximations in local integral equations and to boundary conditions, a system of algebraic equations was obtained for the nodal unknowns.