A Meshless Local Petrov-Galerkin (MLPG) Approach for 3-Dimensional Elasto-dynamics

A Meshless Local Petrov-Galerkin (MLPG) method has been developed for solving 3D elasto-dynamic problems. It is derived from the local weak form of the equilibrium equations by using the general MLPG concept. By incorporating the moving least squares (MLS) approximations for trial and test functions, the local weak form is discretized, and is integrated over the local sub-domain for the transient structural analysis. The present numerical technique imposes a correction to the accelerations, to enforce the kinematic boundary conditions in the MLS approximation, while using an explicit time-integration algorithm. Numerical examples for solving the transient response of the elastic structures are included. The results demonstrate the efficiency and accuracy of the present method for solving the elasto-dynamic problems; and its superiority over the Galerkin Finite Element Method.     

Nonlinear Dynamic Analysis of Three-Dimensional Elasto-Plastic Solids by the Meshless Local Petrov-Galerkin (MLPG) Method

The meshless local Petrov-Galerkin approach is proposed for the nonlinear dynamic analysis of three-dimensional (3D) elasto-plastic problems. Galerkin weak-form formulation is applied to derive the discrete governing equations. A weak formulation for the set of governing equations is transformed into local integral equations on local sub-domains by using a unit test function and local weak-form formulation in three dimensional continua for the general dynamic problems is derived. Three dimensional Moving Least-Square (MLS) approximation is considered as shape function to approximate the field variable of scattered nodes in the problem domain. Normality hypothesis of plasticity is adopted to define the stress-strain relation in elasto-plastic analysis and the unknown plastic multiplier is obtained by the consistency condition. Von Mises yield criterion in three dimensional space is used as a yield function to determine whether the material has yielded. The Newmark time integration method in an incremental form is used to solve the final system of nonlinear second order Ordinary Differential Equations (ODEs). Several numerical examples are given to demonstrate the accuracy and effectiveness of the present numerical approach.     

How to Achieve Kronecker Delta Condition in Moving Least Squares Approximation along the Essential Boundary

A novel way is proposed to fulfill Kronecker delta condition in moving least squares (MLS) approximation along the essential boundary. In the proposed scheme, the original MLS weight is modified to boundary interpolatable (BI) weight based on the observation that the support of weight function is exactly the same as the support of MLS nodal shape function. The BI weight is zero along the boundary edges except the edges containing the nodal point associated with the concerned weight. In order to construct the BI weight from the original weight, concept of edge distance function is introduced, and the BI weight construction procedure is presented in detail. Furthermore, it is explained theoretically why the MLS nodal shape functions obtained by BI weights satisfy Kronecker delta condition along the boundary edges. To identify the validity and usefulness of the proposed BI MLS approximation scheme through numerical tests, the scheme is applied to the model problems with rectangular domain and complex shaped domain. Through the tests, theoretical prediction is identified numerically, and it is confirmed that one can handle the essential and natural boundary conditions through the proposed BI MLS scheme in exactly the same manner used in traditional finite element methods.