meshfree

We are in the framework of small-strain two-dimensional linear elasticity without any body forces. Consider a domain that is discretized by a union of triangles and/or quadrilaterals (`patch of elements').  For C⁰ conforming approximations such as triangular/quadrilateral finite elements, the finite element approximation can exactly reproduce an arbitrary linear displacement field. Hence, if the exact solution is linear, then the finite element solution must match (within machine precision) the exact solution. In simple terms, passing the patch test for linear elasticity with standard conforming finite elements provides verification of one's implementation and is used to assess the same when new elements are proposed. For conforming elements, it is a sufficient condition for convergence (2nd order PDEs), and hence is the first problem that is solved when a new element/method is proposed. To carry out the patch test, the following steps are performed:

In meshfree (this is more in vogue than the term meshless) methods, two key steps need to be mentioned: (A) construction of the trial and test approximations; and (B) numerical evaluation of the weak form (Galerkin or Rayleigh-Ritz procedure) integrals, which lead to a linear system of equations (Kd = f). In meshfree Galerkin methods, the main departure from FEM is in (A): meshfree approximation schemes (linear combination of basis functions) are constructed independent of an underlying mesh (union of elements).

However, since a Galerkin method is typically used in solid mechanics applications, (B) arises and the weak form integrals need to be evaluated. Three main directions have been pursued to evaluate these integrals:

For someone with a background in solid mechanics and finite element methods, where should he go to read up on the elementary ideas of the meshfree methods?