The MLPG Method in Multiphysics and Scale Dependent Problems

Two multiphysical and scale dependent problems in flexoelectricity and thermoelectricity are analysed by the meshless Petrov-Galerkin (MLPG) method. The size-effect is considered in constitutive equations by the strain- and electric field-gradients in the flexoelectricity. The thermal conductivity can be reduced significantly in nano-sized structures and higher-grade heat flux theory is developed in the thermoelectricity. The governing equations are derived for both problems by the variational principle. The coupled governing partial differential equations (PDE) are satisfied in a local weak-form on small fictitious subdomains. A simple test function is applied there. The moving least-squares (MLS) approximation is used for all physical fields. It is convenient approach for problems with higher order derivatives occurred in gradient theories. Applying the spatial approximations in local integral equations a system of algebraic equations is obtained for the nodal values.