A new Fragile Points Method (FPM) in computational mechanics, based on the concepts of Point Stiffnesses and Numerical Flux Corrections

Abstract

In this paper, a new method, named the Fragile Points Method (FPM), is developed for computer modeling in engineering and sciences. In the FPM, simple, local, polynomial, discontinuous and Point-based trial and test functions are proposed based on randomly scattered points in the problem domain. The local discontinuous polynomial trial and test functions are postulated by using the Generalized Finite Difference method. These functions are only piece-wise continuous over the global domain. By implementing the Point-based trial and test functions into the Galerkin weak form, we define the concept of Point Stiffnesses as the contribution of each Point in the problem domain to the global stiffness matrix. However, due to the discontinuity of trial and test functions in the domain, directly using the Galerkin weak form leads to inconsistency. To resolve this, Numerical Flux Corrections, which are frequently used in Discontinuous Galerkin methods are further employed in the FPM. The resulting global stiffness matrix is symmetric and sparse, which is advantageous for large-scale engineering computations. Several numerical examples of 1D and 2D Poisson equations are given in this paper to demonstrate the high accuracy, robustness and convergence of the FPM. Because of the locality and discontinuity of the Point-basedtrial and test functions, this method can be easily extended to model extreme problems in mechanics, such as fragility, rupture, fracture, damage, and fragmentation. These 2 extreme problems will be discussed in our future studies.

A new meshless “fragile points method” and a local variational iteration method for general transient heat conduction in anisotropic nonhomogeneous media. Part II: Validation and discussion

“…Finally, substituting the relation into Eqn. 26 and rearranging the sequence of the collocation equations:…”

“…In contrast to the complex, global continuous MLS approximated shape function used in the EFG and MLPG, local, simple, polynomial, piecewise continuous trial functions are applied in generating the Fragile Points Method (FPM) in [26]. Nevertheless, the method would become inconsistent if the Galerkin weak-form is employed directly with these discontinuous trial and test functions.…”

Meshless fragile points methods based on Petrov‐Galerkin weak‐forms for transient heat conduction problems in complex anisotropic nonhomogeneous media

“…The method is so named owing to its benefits in simulating problems involving fragility, rupture, and fragmentation. 25,26 The domain integrations in the FPM are simple and polynomial. Gaussian quadrature scheme with only one integration point in each subdomain is sufficient most of the time.…”

“…Furthermore, the trial function based on MLS approximation also lacks Delta‐function property, hence the essential boundary conditions cannot be enforced directly. Dong et al 25 proposed a fragile points method (FPM) based on symmetric Galerkin weak‐form which avoids the complicated numerical integration using local, polynomial, piecewise‐continuous trial functions. The method is so named owing to its benefits in simulating problems involving fragility, rupture, and fragmentation 25,26 .…”

A simple Galerkin meshless method, the Fragile Points method using point stiffness matrices, for 2D linear elastic problems in complex domains with crack and rupture propagation

“…But with these requirements, it is difficult to keep the trial and test functions continuous over the entire domain. In our previous article, we have developed the Fragile Points method (FPM) 8 for the first time, for Poisson's equations. The FPM approach employs Point‐based and discontinuous trial and test functions (piecewise polynomials) instead of continuous ones.…”