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.

NODAL INTEGRATION THIN PLATE FORMULATION USING LINEAR INTERPOLATION AND TRIANGULAR CELLS

2011 ◽  
Vol 08 (04) ◽  
pp. 813-824 ◽  
Author(s):  
X. Y. CUI ◽  
S. LIN ◽  
G. Y. LI
Keyword(s):  
Boundary Conditions ◽  
Thin Plate ◽  
Plate Theory ◽  
Weak Form ◽  
Linear Interpolation ◽  
Integration Domain ◽  
Nodal Integration ◽  
Thin Plate Theory ◽  
System Equations ◽  
Gradient Smoothing

This paper presents a thin plate formulation with nodal integration for bending analysis using three-node triangular cells and linear interpolation functions. The formulation was based on the classic thin plate theory, in which only deflection field was required and dealt with as the field variables. They were assumed to be piecewisely linear and expressed using a set of three-node triangular cells. Based on each node, the integration domain has been further derived, where the curvature in the domain was computed using a gradient smoothing technique (GST). As a result, the curvature in each integration domain is a constant whereby the deflection is compatible in the whole problem domain. The generalized smoothed Galerkin weak form is then used to create the discretized system equations where the system stiffness is obtained using simple summation operation. The essential rotational boundary conditions are imposed in the process of constructing the curvature field in conjunction with imposing the translational boundary conditions in the same way as undertaken in the standard FEM. A number of numerical examples were studied using the present formulation, including both static and free vibration analyses. The numerical results were compared with the reference ones together with those shown in the state-of-art literatures published. Very good accuracy has been achieved using the present method.

scholarly journals Buckling analysis of simply supported composite laminates subjected to an in-plane compression load by a novel mesh-free method

2011 ◽  
Vol 33 (2) ◽  
pp. 65-78 ◽  
Author(s):  
Tinh Quoc Bui
Keyword(s):  
Composite Laminates ◽  
Plate Theory ◽  
Numerical Technique ◽  
Buckling Analysis ◽  
Kriging Interpolation ◽  
Shape Functions ◽  
Compression Load ◽  
Mesh Free ◽  
Mesh Free Method ◽  
Plane Compression

Buckling analysis of composite laminates under an in-plane compression load based on the mesh-free Galerkin Kriging method is presented. The moving Kriging interpolation (MK) technique possessing the delta property is employed to construct the shape functions, and thus no special techniques for imposing the essential boundary conditions are required. The present formulation is based on the Kirchhoff plate theory. The applicability, the accuracy and the effectiveness of the method are illustrated through a number of numerical examples. The results calculated by the proposed method are compared with those of existing reference solutions available in the literature and very good agreements are observed. It can be said that the proposed method can be considered as an alternative numerical technique in terms of meshfree methods.

scholarly journals 3D Global Weak-Form Mesh-Free Method for Acoustic Attenuation Analysis of Expansion Chambers

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Chun-Hui Fang ◽  
Cheng-Yang Liu ◽  
Zhi Fang
Keyword(s):  
Transmission Loss ◽  
Interpolation Method ◽  
Weak Form ◽  
Computational Accuracy ◽  
Acoustic Attenuation ◽  
Expansion Chamber ◽  
Acoustic Characteristics ◽  
Acoustic Modes ◽  
Mesh Free ◽  
Mesh Free Method

In order to avoid the dependence of mesh method on grids, a 3D global weak-form mesh-free method (MFM) is applied to study the three-dimensional acoustic characteristics of silencers. For the expansion chamber silencers, the 3D acoustic modes are extracted and the transmission loss results are computed by using the 3D global weak-form (MFM), which is based on the radial basis function point interpolation method (RPIM) for calculating the shape functions and Galerkin method for discretizing the system equation. The first 15 order 3D acoustic modes and TL results of a special expansion chamber silencer are presented to validate the computational accuracy of the proposed technique, and the relative errors are controlled within 0.5% by comparing with the 3D finite element method (FEF) calculations. Additionally, the effects of axial modes on the acoustic characteristics are investigated, and the pass through frequencies can be eliminated to enhance the acoustic attenuation performance by locating the side branch outlet on the nodal lines of axial modes.

Implicit Boundary Approach for Reissner-Mindlin Plates

2013 ◽  
Author(s):  
Hailong Chen ◽  
Ashok V. Kumar
Keyword(s):  
Boundary Conditions ◽  
Plate Theory ◽  
Weak Form ◽  
Mindlin Plate ◽  
Benchmark Problems ◽  
Element Analysis ◽  
Boundary Method ◽  
Solid Models ◽  
Plate Elements ◽  
Background Mesh

Implicit boundary method enables the use of background mesh to perform finite element analysis while using solid models to represent the geometry. This approach has been used in the past to model 2D and 3D structures. Thin plate or shell-like structures are more challenging to model. In this paper, the implicit boundary method is shown to be effective for plate elements modeled using Reissner-Mindlin plate theory. This plate element uses a mixed formulation and discrete collocation of shear stress field to avoid shear locking. The trial and test functions are constructed by utilizing approximate step functions such that the boundary conditions are guaranteed to be satisfied. The incompatibility of discrete collocation with implicit boundary approach is overcome by using irreducible weak form for computing the stiffness associated with essential boundary conditions. A family of Reissner-Mindlin plate elements is presented and evaluated in this paper using several benchmark problems to test their validity and robustness.

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.

BUCKLING OF SYMMETRICALLY LAMINATED COMPOSITE PLATES USING THE ELEMENT-FREE GALERKIN METHOD

2002 ◽  
Vol 02 (03) ◽  
pp. 281-294 ◽  
Author(s):  
G. R. LIU ◽  
X. L. CHEN ◽  
J. N. REDDY
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Present Method ◽  
Plate Theory ◽  
Orthogonal Transformation ◽  
Laminated Composite ◽  
Composite Plates ◽  
Laminated Composite Plates ◽  
Element Free Galerkin ◽  
Element Free

An element free Galerkin (EFG) method is presented for buckling analyses of isotropic and symmetrically laminated composite plates using the classical plate theory. The shape functions are constructed using the moving least squares (MLS) approximation, and no element connectivity among nodes is required. The deflection can be easily approximated with higher-order polynomials as desired. The discrete eigenvalue problem is derived using the principle of minimum total potential energy of the system. The essential boundary conditions are introduced into the formulation through the use of the Lagrange multiplier method and the orthogonal transformation techniques. Since the dimension of the eigenvalue problem obtained by the present method is only one third of that in the conventional finite element method (FEM), solving the eigenvalue problem in the EFG is computationally more efficient compared to the FEM. Buckling load param-eters of isotropic and symmetrically laminated composite plates for different boundary conditions are calculated to demonstrate the efficiency of the present method.

Solution Structures for Imposing Boundary Conditions in Mesh Free Analysis of Heat Conduction Problems

2005 ◽  
Author(s):  
Linxia Gu ◽  
Ashok V. Kumar
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Step Function ◽  
Poisson’S Equation ◽  
Poisson's Equation ◽  
Structured Grid ◽  
Solution Structures ◽  
Mesh Free ◽  
Implicit Equations ◽  
Mesh Free Method

One of the main advantages of meshless methods is that it eliminates the mesh generation, but it is still necessary to place nodes with controlled spacing variation on the boundary and within the domain. However, due to lack of connectivity between nodes it is more difficult to interpolate the field variables and impose boundary conditions. In this paper, a mesh free method is presented for analysis using a structured grid that does not conform to the geometry of the domain. The geometry of the domain is independent of the structured grid and is represented using implicit equations. The implicit equations of the boundaries can be used to construct solution structures that satisfy boundary conditions exactly even though the nodes of the grid are not on the boundaries of the domain. The solution structures are constructed using the implicit equations of the boundary together with a piece-wise interpolation over the structured grid. The implicit equations are also used to construct step function of solid such that its value is equal to unity inside the solid and zero outside. The step function of the solid is used for volume integrations needed for the analysis. The traditional weak form for Poisson’s equation is modified by using this solution structure to eliminate the surface integration terms. The accuracy and implementation of the present mesh free method is illustrated for two-dimensional heat conduction problems governed by Poisson’s equation. Satisfactory results are obtained when compared with analytical results and results from commercial finite element software.

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.

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