A Spline Finite Point Method for Nonlinear Bending Analysis of FG Plates in Thermal Environments Based on a Locking-free Thin/Thick Plate Theory

A spline finite point method (SFPM) based on a locking-free thin/thick plate theory, which is suitable for analysis of both thick and thin plates, is developed to study nonlinear bending behavior of functionally graded material (FGM) plates with different thickness in thermal environments. In the proposed method, one direction of the plate is discretized with a set of uniformly distributed spline nodes instead of meshes and the other direction is expressed with orthogonal functions determined by the boundary conditions. The displacements of the plate are constructed by the linear combination of orthogonal functions and cubic B-spline interpolation functions with high efficiency for modeling. The locking-free thin/thick plate theory used by the proposed method is based on the first-order shear deformation theory but takes the shear strains and displacements as basic unknowns. The material properties of the FG plate are assumed to vary along the thickness direction following the power function distribution. By comparing with several published research studies based on the finite element method (FEM), the correctness, efficiency, and generality of the new model are validated for rectangular plates. Moreover, uniform and nonlinear temperature rise conditions are discussed, respectively. The effect of the temperature distribution, in-plane temperature force, and elastic foundation on nonlinear bending under different parameters are discussed in detail.

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Free Transverse Vibration Analysis of Axially Functionally Graded Tapered Euler-Bernoulli Beams through Spline Finite Point Method

Shock and Vibration ◽

10.1155/2016/5891030 ◽

2016 ◽

Vol 2016 ◽

pp. 1-23 ◽

Cited By ~ 4

Author(s):

Peng Liu ◽

Kun Lin ◽

Hongjun Liu ◽

Rong Qin

Keyword(s):

Boundary Conditions ◽

Transverse Vibration ◽

Spline Interpolation ◽

Computational Cost ◽

Functionally Graded ◽

Point Method ◽

Finite Point ◽

Various Boundary Conditions ◽

Finite Point Method ◽

Free Transverse Vibration

A new model for the free transverse vibration of axially functionally graded (FG) tapered Euler-Bernoulli beams is developed through the spline finite point method (SFPM) by investigating the effects of the variation of cross-sectional and material properties along the longitudinal directions. In the proposed method, the beam is discretized with a set of uniformly scattered spline nodes along the beam axis instead of meshes, and the displacement field is approximated by the particularly constructed cubic B-spline interpolation functions with good adaptability for various boundary conditions. Unlike traditional discretization and modeling methods, the global structural stiffness and mass matrices for beams of the proposed model are directly generated after spline discretization without needing element meshes, generation, and assembling. The proposed method shows the distinguished features of high modeling efficiency, low computational cost, and convenience for boundary condition treatment. The performance of the proposed method is verified through numerical examples available in the published literature. All results demonstrate that the proposed method can analyze the free vibration of axially FG tapered Euler-Bernoulli beams with various boundary conditions. Moreover, high accuracy and efficiency can be achieved.

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Natural Vibration Analysis of Piezoelectric Smart FGM Plate by Spline Finite Point Method

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.444-445.66 ◽

2013 ◽

Vol 444-445 ◽

pp. 66-71 ◽

Cited By ~ 1

Author(s):

Shuang Bei Li ◽

Jun Huang ◽

Rong Qin

Keyword(s):

Electric Field ◽

Natural Vibration ◽

Plate Theory ◽

Computational Cost ◽

Basic Function ◽

Point Method ◽

Parameter Method ◽

Finite Point ◽

Finite Point Method ◽

Piezoelectric Layers

The spline finite point method (SFPM) that is based on the classical laminated plate theory is introduced to investigate the natural vibration of smart FGM plate, which is integrated with piezoelectric layers. The spline basic function that meets different boundary conditions is constructed by the generalized parameter method, and a new dynamic computational scheme for smart FGM plate is established to analysis the natural frequency that considering influence of the axial force, which is generated by the electric field. To demonstrate the less computational cost and higher accuracy of the SFPM, several numerical examples are calculated. The effects of the electric field and the influence of the thickness ratio of piezoelectric layer and substrate on the fundamental frequency are also discussed.

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Real-Time Computer Simulation of Three Dimensional Elastostatics Using the Finite Point Method

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.110-116.2740 ◽

2011 ◽

Vol 110-116 ◽

pp. 2740-2745

Author(s):

Kirana Kumara P. ◽

Ashitava Ghosal

Keyword(s):

Real Time ◽

Graphics Processing Unit ◽

Numerical Technique ◽

Three Dimensional ◽

Point Method ◽

Processing Unit ◽

Finite Point ◽

Deformable Solids ◽

Finite Point Method ◽

Linear Elastostatics

Real-time simulation of deformable solids is essential for some applications such as biological organ simulations for surgical simulators. In this work, deformable solids are approximated to be linear elastic, and an easy and straight forward numerical technique, the Finite Point Method (FPM), is used to model three dimensional linear elastostatics. Graphics Processing Unit (GPU) is used to accelerate computations. Results show that the Finite Point Method, together with GPU, can compute three dimensional linear elastostatic responses of solids at rates suitable for real-time graphics, for solids represented by reasonable number of points.

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Two Implicit Meshless Finite Point Schemes for the Two-Dimensional Distributed-Order Fractional Equation

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2018-0009 ◽

2019 ◽

Vol 19 (4) ◽

pp. 813-831

Author(s):

Rezvan Salehi

Keyword(s):

Reproducing Kernel ◽

Point Method ◽

Least Square ◽

Computationally Efficient ◽

Finite Point ◽

Moving Least Square ◽

Point Collocation ◽

Finite Point Method ◽

Distributed Order ◽

Fractional Equation

AbstractIn this paper, the distributed-order time fractional sub-diffusion equation on the bounded domains is studied by using the finite-point-type meshless method. The finite point method is a point collocation based method which is truly meshless and computationally efficient. To construct the shape functions of the finite point method, the moving least square reproducing kernel approximation is employed. Two implicit discretisation of order{O(\tau)}and{O(\tau^{1+\frac{1}{2}\sigma})}are derived, respectively. Stability and{L^{2}}norm convergence of the obtained difference schemes are proved. Numerical examples are provided to confirm the theoretical results.

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