# A Co-Rotational Meshfree Method for the Geometrically Nonlinear Analysis of Structures
## 1 Introduction
To circumvent the numerical problem, some researchers have taken strain vectors [13] or stress resultants [16] as the primary variables. For instance, [PERSON] and [PERSON] [13] presented a formulation in terms of the strain vectors such that the additive interpolation has no restrictions. On the contrary, [PERSON] and [PERSON] [16] took the stress vectors as the only interpolated degree of freedom. In this way, the displacement can be obtained by integration of the kinematic equations. [PERSON] et al. [21] developed a large rotation finite element analysis of three-dimensional beams based on the Mixed Integration Point (MIP) Newton iterative scheme [22], in which the co-rotational nodal rotations defined by incremental nodal rotation vectors are interpolated for the evaluation of the nonlinear strains. In the above studies, whether the stress or strain vectors have been employed to be the interpolated variables in the finite element formulation, it still deals with the weak form based on the principle of virtual work or the principle of minimum potential energy. As opposed to FEM, adopted to solve numerical analysis, in this study a meshfree method based on the Differential Reproducing Kernel (DRK) Approximation collocation method combined with the Newton-Raphson method was adopted to analyze geometrically nonlinear problems. The kinematics of beam structures undergoing large deformation are derived with the concepts of the co-rotational framework. It is well-known that the primary characteristic of meshfree methods is that the domain is completely discretized by a scattered set of nodal points instead of elements. However, this naturally circumvents mesh problems may result in numerical problems in FEM. Therefore, in the present scheme the solution of nonlinear equilibrium equations such as displacements, rotations, and force resultants of a beam that has undergone large deformation will be directly solved toward the strong form of the governing equations. The DRK implementation of the three-dimensional co-rotational beam formulation, leading to differentiation of the assumed kinematic field, is not necessary while evaluating strains, internal forces, and moments. This also shows that the primary interpolated variables will not be the dominant factor in the present scheme. In the literature, meshfree methods have been used by numerous researchers [23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34]. Several meshfree methods, including collocation and finite point methods based on different approximations, have been proposed to solve the strong form problems. [PERSON] [28] implemented a point collocation method to solve the differential equations by reproducing kernel approximations. [PERSON] et al. [29] adopted weighted least squares interpolations to develop a finite point method to analyze elasticity problems. [PERSON] et al. [30] proposed a radial basis collocation method with a strong-form generalized displacement control method to solve geometrically nonlinear problems. Most point collocation methods are based on the idea of reproducing kernel (RK)approximations. However, conventional RK methods compute the derivatives of the shape functions by differentiating toward the shape functions directly. In particular, this may lead to complex expressions and computations while computing high-order derivatives. Several methods were developed to solve the complicated problems. [PERSON] et al. [31] employed a trivial numerical implementation of second-order smoothed gradients to develop a superconvergent gradient smoothing meshfree collocation method. [PERSON] et al. [32], [PERSON] et al. [33], and [PERSON] [34] developed the DRK method for one-dimensional and two-dimensional elasticity analysis. In comparison with the conventional RK approximations, the DRK approximations involve the calculation of the derivatives of shape functions in the conventional RK.