| The reproducing kernel particle method (RKPM) is a meshless method based on kernel approximation. Through the convolution theorem, the reproducing equations are developed and the reproducing conditions are derived by using Taylor series expansion. The RKPM shape function and its derivatives are constructed by imposing the consistency conditions.; By using this groundwork of RKPM, a Hermite reproducing kernel method (HRKM) is presented. In this approach, the reproducing conditions are enforced such that the derivatives can be treated as independent functions. Two special cases are presented for HRKM approximations: slope sampling involves only the slope at the available sampling points to recover the function properly; for window tailoring, the HRKM is expanded to provide a special tool in designing filters.; The efficient RKPM formulation is an extension of the standard RKPM formulation which allows a set of non-uniformly particles to be used. An irregular particle distribution corresponding to the geometry of the problem is transformed onto a uniformly spaced background particle distribution via the standard reproducing equations. Based on a uniform distribution, the shape function can be written explicitly leading to an efficient integration.; In the application of acoustic problem, the frequency/wave number relationship of the continuum case is studied for both wave equation and the Timoshenko beam. By performing a dispersion analysis of the RKPM, the effects of dilation parameters to frequency equations are investigated. An approach based on the minimization of the total beam energy to determine the optimal dilation parameters is proposed.; The methodology of computational cell modeling is introduced to establish the constitutive law during ductile fracture. A new model parameter function is obtained based on a computational cell modeling technique. For comparison with the Gurson model, the technique is applied to a set of numerical examples including necking behaviour of a tensile bar, a cracked panel under tension and an edge notch panel under pure bending. The applications of RKPM with the proposed pressure projection method for the ductile fracture process involving damage evolution is also studied. |
