Compared to the need of a well-arranged mesh in finite element analysis, robustness against irregular node distribution and freedom from element connectivity in meshfree methods is one of the most important advantages, especially for difficult-to-solve nonlinear problems. However, meshfree methods have various disadvantages: the imposition of essential boundary conditions (EBC), high cost for domain integration and calculation of shape functions and their derivatives, complexity of the treatment of non-convex domain, and so on.
In general, the shape functions in most meshfree methods based on moving least-squares or reproducing kernel method do not satisfy the Kronecker delta property at nodal points. Furthermore, the shape functions of internal nodes do not vanish at boundary where EBC are prescribed. This makes the imposition of EBC in meshfree methods very complex. In this thesis, a new scheme, named LU-decomposition method with full pivoting (LUDMP), for imposing the EBC represented as a linear system in meshfree methods is presented. The LUDMP eliminates redundant or inconsistent constraints from the constraint equations and partition unknowns automatically into a unique set of constrained and independent variables. So, the LUDMP can be easily generalized to apply to any kinds of linear constraints between the unknown variables. The LUDMP results in a symmetric, positive definite and banded stiffness matrix with a reduced degree of freedoms in the Galerkin method.
The most important property of being free from element division and operation possible in meshfree methods is obtained only with very high cost. That is, domain integration in common Galerkin meshfree methods cannot be exactly calculated even by using high order Gauss quadrature because of very complex non-polynomial shape functions. The integration error causes inaccuracy of solution and failure of patch test although the mesh-free shape functions can reproduce linear fields. To overcome the weakness, a new solution approximation scheme, embodied in a combination of finite element and moving least-square method frequently used in mesh-free methods is proposed. This scheme is named "the node-based least-square element method (NLEM)." In this method, the approximaton of a variable u is represented by $u^h = {\boldmath p(x)}^T {\boldmath a(ξ)}$ where ${\boldmath p(x)}$ is a basis function vector, and the coefficient vector ${\boldmath a(ξ)}$ is assumed as ${\boldmath a(ξ)} = {\sum \chooseI} {\boldmath a}_I N_I({\boldmath ξ})$ where ${\boldmath a$_I$ are nodal coefficient vectors and $N_I ({\boldmath ξ})$ the shape functions of simplex linear finite elements. The ${\boldmath a}_I$ are calculated at each node by using the conventional MLSM (moving least-square method). In the NLEM, the shape functions can reproduce all basis polynomials ${\boldmath p(x)}$ and not only the shape functions but also its derivatives with respect to global coordinates ${\boldmath x}$ are piecewise simple polynomials even under an irregular node distribution, and the supports of the shape functions coincide with the finite element mesh used in the solution approximation. These mean that the domain integration in the Galerkin method can be exactly calculated by a suitable domain integration scheme in a linear problem. So, the NLEM passes not only the first order but also high order patch tests for the all basis polynomials ${\boldmath p(x)}$. The NLEM is more expensive then FEM but less than EFGM (element-free Galerkin method) because of the following reasons: 1) It uses the MLSM to calculate the ${\boldmath a}$ only at nodes instead of integration points. 2) It does not require the derivatives of weight functions. 3) And it uses the lower order Gauss quadrature than EFGM. The numerical results show the property and consequently the convergent rate of the NLEM is demonstrated to be robust to the irregularity of node distributions.
Finally, the proposed NLEM is applied to a hyperelastic structural analysis. The Yeoh`s energy density function is employed to describe the hyperelastic structural behavior. For the numerical study, compression of a rectangular rubber band, tension of a tapered rubber bar and compression of the rubber pad of a keyboard are employed to demonstrate the feasibility of the proposed method. And, a continuum-based shape design sensitivity analysis (SDSA) for a hyperelastic structure is also presented. Both the adjoint variable and direct differentiation methods are applied for SDSA. The tapered rubber bar problem is studied as a numerical example of SDSA. The shape of tapered part is selected as design boundary and the averaged strain energy over a selected region is considered as performance functional. The numerical results of SDSA show good agreement with those obtained by finite difference method.
