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Flexoelectricity refers to a phenomenon which involves a coupling of themechanical strain gradient and electric polarization. In this study, a meshlessFragile Points Method (FPM), is presented for analyzing flexoelectric effectsin dielectric solids. Local, simple, polynomial and discontinuous trial andtest functions are generated with the help of a local meshless DifferentialQuadrature approximation of derivatives. Both primal and mixed FPM aredeveloped, based on two alternate flexoelectric theories, with or without theelectric gradient effect and Maxwell stress. In the present primal as well asmixed FPM, only the displacements and electric potential are retained asexplicit unknown variables at each internal Fragile Point in the finalalgebraic equations. Thus the number of unknowns in the final system ofalgebraic equations is kept to be absolutely minimal. An algorithm forsimulating crack initiation and propagation using the present FPM is presented,with classic stress-based criterion as well as a Bonding-Energy-Rate(BER)-basedcriterion for crack development. The present primal and mixed FPM approachesrepresent clear advantages as compared to the current methods for computationalflexoelectric analyses, using primal as well as mixed Finite Element Methods,Element Free Galerkin (EFG) Methods, Meshless Local Petrov Galerkin (MLPG)Methods, and Isogeometric Analysis (IGA) Methods, because of the following newfeatures: they are simpler Galerkin meshless methods using polynomial trial andtest functions; minimal DoFs per Point make it very user-friendly; arbitrarypolygonal subdomains make it flexible for modeling complex geometries; thenumerical integration of the primal as well as mixed FPM weak forms istrivially simple; and FPM can be easily employed in crack developmentsimulations without remeshing or trial function enhancement.

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