The Meshless Analysis of Scale Dependent Problems for Coupled Fields

Abstract

The meshless Petrov-Galerkin (MLPG) method is developed to analyse 2-D problems for flexoelectricity and thermoelectricity. Both problems are multiphysical and scale dependent. The size-effect is considered by the strain- and electric field-gradients in the flexoelecricity and higher-grade heat flux in the thermoelectricity. The variational principle is applied to de-rive the governing equations considered constitutive equations. The order of derivatives in governing equations is higher than in equations obtained from classical theory. The coupled governing partial differential equations (PDE) are satisfied in a local weak-form on small fic-titious subdomains with a simple test function. Physical fields are approximated by the mov-ing least-squares (MLS) scheme. Applying the spatial approximations in local integral equa-tions a system of algebraic is obtained for the nodal unknowns.

A New Meshless Fragile Points Method (FPM) With Minimum Unknowns at Each Point, For Flexoelectric Analysis Under Two Theories with Crack Propagation. Part I: Theory and Implementation

“…Whereas the E-formulation can also be incorporated with multiple numerical discretization methods, including primal and mixed FEM [13,14], Element-Free-Galerkin (EFG) method [15] and Meshless Local Petrov-Galerkin (MLPG) method [16], etc. The two formulations are inherently equivalent and can be unified by a linear relation between the electric polarization and electric displacement (the conjugate component of the electric field).…”

“…can be applied and thus lead to an asymmetric formulation. The MLPG method is also easily applicable in analyzing strain gradient [30] and flexoelectric [16,31] problems. The main advantage of the MLS approximation used in the EFG and MLPG methods is with an appropriate weight function, we can generate smooth trial functions with C 1 or higher order continuity.…”

A New Meshless Fragile Points Method (FPM) With Minimum Unknowns at Each Point, For Flexoelectric Analysis Under Two Theories with Crack Propagation. Part II: Validation and discussion

“…Both primal and mixed FPM formulations were developed, based on two flexoelectric theories with or without the electric field gradient effect and electroelastic stress. In general, the FPM presents clear advantages in its algorithmic formulation as compared to previous numerical methods including primal and mixed FEM [6,7], Element-Free Galerkin (EFG) method [8] and primal and mixed Meshless Local Petrov-Galerkin (MLPG) method [9,10] in analyzing flexoelectric behavior, as it does not require high quality meshes, has minimum number of DoFs at each Point, needs only very simple weak form integration schemes, and has great advantages in simulating crack and rupture initiation and propagation.…”