A Moving Kriging Interpolation Meshfree Method Based on Naturally Stabilized Nodal Integration Scheme for Plate Analysis

2019 ◽  
Vol 16 (04) ◽  
pp. 1850100 ◽  
Author(s):  
Chien H. Thai ◽  
H. Nguyen-Xuan
Keyword(s):  
Computational Cost ◽  
Meshfree Method ◽  
Weak Form ◽  
Delta Function ◽  
Current Approach ◽  
Kriging Interpolation ◽  
Integration Scheme ◽  
Nodal Integration ◽  
Moving Kriging Interpolation ◽  
Quadrature Scheme

A moving Kriging interpolation (MKI) meshfree method based on naturally stabilized nodal integration (NSNI) scheme is presented to study static, free vibration and buckling behaviors of isotropic Reissner–Mindlin plates. Gradient strains are directly computed at nodes similar to the direct nodal integration (DNI). Outstanding features of the current approach are to alleviate instability solutions in the DNI and to decrease computational cost significantly when compared with the traditional high-order Gauss quadrature scheme. The NSNI is a naturally implicit gradient expansion and does not employ a divergence theorem for strain fields as addressed in the stabilized conforming nodal integration method. The present formulation is derived from the Galerkin weak form and avoids a naturally shear-locking phenomenon without using any other techniques. Thanks to satisfied Kronecker delta function property of MKI shape function, essential boundary conditions (BCs) are easily and directly enforced similar to the finite element method. A variety of numerical examples with various geometries, stiffness ratios and BCs are studied to verify the effectiveness of the present approach.

A MOVING KRIGING INTERPOLATION-BASED MESHLESS LOCAL PETROV–GALERKIN METHOD FOR ELASTODYNAMIC ANALYSIS

2013 ◽  
Vol 05 (01) ◽  
pp. 1350011 ◽  
Author(s):  
BAODONG DAI ◽  
JING CHENG ◽  
BAOJING ZHENG
Keyword(s):  
Galerkin Method ◽  
Present Method ◽  
Time Integration ◽  
Weak Form ◽  
Kriging Interpolation ◽  
Special Treatment ◽  
Integration Scheme ◽  
Shape Functions ◽  
Heaviside Step Function ◽  
Moving Kriging Interpolation

A meshless local Petrov–Galerkin method (MLPG) based on the moving Kriging interpolation for elastodynamic analysis is presented in this paper. The present method is developed based on the moving Kriging interpolation for constructing shape functions at scattered points, and the Heaviside step function is used as a test function in each subdomain to avoid the need for domain integral in symmetric weak form. Since the shape functions constructed by this moving Kriging interpolation have the delta function property, the essential boundary conditions can be implemented easily, and no special treatment techniques are required. The discrete equations of the governing elastodynamic equations for two-dimensional solids are obtained using the local weak-forms. The Newmark method is adopted for the time integration scheme. Some numerical results are compared to that obtained from the exact solutions of the problem and other (meshless) methods. This comparison illustrates the efficiency and accuracy of the present method for solving the static and dynamic problems.

EIGENVALUE ANALYSIS OF THIN PLATE WITH COMPLICATED SHAPES BY A NOVEL MESH-FREE METHOD

2011 ◽  
Vol 03 (01) ◽  
pp. 21-46 ◽  
Author(s):  
TINH QUOC BUI ◽  
MINH NGOC NGUYEN
Keyword(s):  
Boundary Conditions ◽  
Thin Plate ◽  
Plate Theory ◽  
Orthogonal Transformation ◽  
Weak Form ◽  
Eigenvalue Analysis ◽  
Kriging Interpolation ◽  
Moving Kriging Interpolation ◽  
Mesh Free ◽  
Mesh Free Method

Further development of a novel mesh-free method for eigenvalue analysis of thin plate structures with complicated shapes is presented in this paper. A mesh-free method used the moving Kriging interpolation technique for constructing the shape functions, which possess the Kronecker's delta property, is formulated. Thus, it makes the present method efficient in enforcing the essential boundary conditions and none of any special techniques are required. The present plate theory followed the classical Kirchhoff's assumption and the deflection is in general approximated through the moving Kriging interpolation. Also, the mesh-free formulations for the vibration problem are formed in a simple way as finite element methods. The orthogonal transformation technique is used to implement the essential boundary conditions in the eigenvalue equation. A standard weak form is adopted to discrete the governing partial differential equation of plates. Some numerical examples are attempted to demonstrate the applicability, the effectiveness, and the accuracy of the method.

Meshfree Analysis of Higher-Order Gradient Crystal Plasticity Using Nodal Integration

2019 ◽  
Vol 794 ◽  
pp. 214-219
Author(s):  
Tota Niiro ◽  
Yuichi Tadano
Keyword(s):  
Size Effect ◽  
Crystal Plasticity ◽  
Meshfree Method ◽  
Higher Order ◽  
Integration Scheme ◽  
Plasticity Model ◽  
The Finite Element Method ◽  
Nodal Integration ◽  
Crystal Plasticity Model ◽  
Meshfree Analysis

The size effect of metallic materials is one of the important factors for understanding characteristics of material. The higher-order gradient crystal plasticity is a powerful model for describing the size effect. However, it is known that the finite element method sometimes provides an improper solution. In this study, we analyze the higher-order gradient crystal plasticity model using a meshfree method, and a nodal integration scheme is introduced to improve the analysis accuracy. The effectiveness and stability of the meshfree method for the higher-order gradient crystal plasticity model are quantitatively discussed.

scholarly journals Numerical Integration of Multibody Dynamic Systems Involving Nonholonomic Equality Constraints

2021 ◽  
Author(s):  
Sotirios Natsiavas ◽  
Panagiotis Passas ◽  
Elias Paraskevopoulos
Keyword(s):  
Dynamic Systems ◽  
Equations Of Motion ◽  
Time Integration ◽  
Nonholonomic Constraints ◽  
Coupled System ◽  
Weak Form ◽  
Integration Scheme ◽  
Motion Constraints ◽  
Multibody Dynamic ◽  
Time Integration Scheme

Abstract This work considers a class of multibody dynamic systems involving bilateral nonholonomic constraints. An appropriate set of equations of motion is employed first. This set is derived by application of Newton’s second law and appears as a coupled system of strongly nonlinear second order ordinary differential equations in both the generalized coordinates and the Lagrange multipliers associated to the motion constraints. Next, these equations are manipulated properly and converted to a weak form. Furthermore, the position, velocity and momentum type quantities are subsequently treated as independent. This yields a three-field set of equations of motion, which is then used as a basis for performing a suitable temporal discretization, leading to a complete time integration scheme. In order to test and validate its accuracy and numerical efficiency, this scheme is applied next to challenging mechanical examples, exhibiting rich dynamics. In all cases, the emphasis is put on highlighting the advantages of the new method by direct comparison with existing analytical solutions as well as with results of current state of the art numerical methods. Finally, a comparison is also performed with results available for a benchmark problem.

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