The generalized-strain mesh-free (GSMF) formulation is a local meshfree method in the field of numerical analysis, completely integration free, working as a weighted-residual weak-form collocation. This method was first presented by Oliveira and Portela (2016), in order to further improve the computational efficiency of meshfree methods in numerical analysis. Local meshfree methods are derived through a weighted-residual formulation which leads to a local weak form that is the well known work theorem of the theory of structures. In an arbitrary local region, the work theorem establishes an energy relationship between a statically-admissible stress field and an independent kinematically-admissible strain field. Based on the independence of these two fields, this formulation results in a local form of the work theorem that is reduced to regular boundary terms only, integration-free and free of volumetric locking.
Advantages over finite element methods are that GSMF doesn't rely on a grid, and is more precise and faster when solving bi-dimensional problems. When compared to other meshless methods, such as rigid-body displacement mesh-free (RBDMF) formulation, the element-free Galerkin (EFG) and the meshless local Petrov-Galerkin finite volume method (MLPG FVM); GSMF proved to be superior not only regarding the computational efficiency, but also regarding the accuracy.
The moving least squares (MLS) approximation of the elastic field is used on this local meshless formulation.
Formulation
In the local form of the work theorem, equation:
The displacement field
For the sake of the simplicity, in dealing with Heaviside and Dirac delta functions in a two-dimensional coordinate space, consider a scalar function
which represents the absolute-value function of the distance between a field point
For a scalar coordinate
in which the discontinuity is assumed at
and
in which
Since the result of this equation is not affected by any particular value of the constant
Consider that
in which
Having defined the displacement and the strain components of the kinematically-admissible field, the local work theorem can be written as
Taking into account the properties of the Heaviside step function and Dirac delta function, this equation simply leads to
Discretization of this equations can be carried out with the MLS approximation, for the local domain
or simply
This formulation states the equilibrium of tractions and body forces, pointwisely defined at collocation points, obviously, it is the pointwise version of the Euler-Cauchy stress principle. This is the equation used in the Generalized-Strain Mesh-Free (GSMF) formulation which, therefore, is free of integration. Since the work theorem is a weighted-residual weak form, it can be easily seen that this integration-free formulation is nothing else other than a weighted-residual weak-form collocation. The weighted-residual weak-form collocation readily overcomes the well-known difficulties posed by the weighted-residual strong-form collocation, regarding accuracy and stability of the solution.