Optimized Radius of Influence Domain in Meshless Approach for Modeling of Large Deformation Problems

RBF collocation path-following approach: optimal choice for shape parameter based on genetic algorithm

“…The equations of equilibrium in strong formulation for a nonlinear elastic structure in large deformations [8,43] assume that the displacements and forces imposed are proportional to a single scalar parameter λ called the load parameter as follows:…”

“…Note that (u(a), λ(a)) depends smoothly on the parameter "a" and x = (u T , λ) T . Using pseudoarc-length parametrization [8,46] as shown in the previous paper [43], we add an equation to close our system and to obtain a nonsingular Jacobian for the our problem statement.…”

“…We inject the developments in Eq. ( 9) into the non-linear problem (8) and identify the terms having the same powers of the parameter "a" to obtain a sequence of linear problems: Problem at order 1 :…”

“…where [K T ] is the stiffness tangent matrix at the starting point ({T 0 }, {S 0 }, {γ 0 }, {U 0 }). The new starting point of the new solution branch is obtained by continuation procedure {T (a max )}, {S(a max )}, {γ(a max )}, {U (a max )}, where a max is the convergence radius [8,46]:…”

“…So there is no need for numerical integrations, which reduces the computational cost compared to FEM. The meshless methods can be classified in two classes: (i) the meshless methods of Galerkin type [4,5] and of collocation type [6][7][8]. The first class combines the meshless methods with the weak formulation of PDEs and often these methods require a domain partition.…”