Reproducing kernel mesh-free collocation analysis of structural vibrations

2019 ◽  
Vol 36 (3) ◽  
pp. 734-764 ◽  
Author(s):  
Dongliang Qi ◽  
Dongdong Wang ◽  
Like Deng ◽  
Xiaolan Xu ◽  
Cheng-Tang Wu
Keyword(s):  
Collocation Method ◽  
Fourth Order ◽  
Reproducing Kernel ◽  
Basis Functions ◽  
Structural Vibration ◽  
Structural Vibrations ◽  
Content Type ◽  
Mesh Free ◽  
Odd Degree ◽  
Vibration Problems

PurposeAlthough high-order smooth reproducing kernel mesh-free approximation enables the analysis of structural vibrations in an efficient collocation formulation, there is still a lack of systematic theoretical accuracy assessment for such approach. The purpose of this paper is to present a detailed accuracy analysis for the reproducing kernel mesh-free collocation method regarding structural vibrations.Design/methodology/approachBoth second-order problems such as one-dimensional (1D) rod and two-dimensional (2D) membrane and fourth-order problems such as Euler–Bernoulli beam and Kirchhoff plate are considered. Staring from a generic equation of motion deduced from the reproducing kernel mesh-free collocation method, a frequency error measure is rationally attained through properly introducing the consistency conditions of reproducing kernel mesh-free shape functions.FindingsThis paper reveals that for the second-order structural vibration problems, the frequency accuracy orders arepand (p− 1) for even and odd degree basis functions; for the fourth-order structural vibration problems, the frequency accuracy orders are (p− 2) and (p− 3) for even and odd degree basis functions, respectively, wherepdenotes the degree of the basis function used in mesh-free approximation.Originality/valueA frequency accuracy estimation is achieved for the reproducing kernel mesh-free collocation analysis of structural vibrations, which can effectively underpin the practical applications of this method.

Multilevel RBF collocation method for the fourth-order thin plate problem

2020 ◽  
pp. 2050079
Author(s):  
Lanling Ding ◽  
Zhiyong Liu ◽  
Qiuyan Xu
Keyword(s):  
Radial Basis Function ◽  
Collocation Method ◽  
Radial Basis Functions ◽  
Thin Plate ◽  
Fourth Order ◽  
Basis Function ◽  
Research Method ◽  
Meshfree Method ◽  
Basis Functions ◽  
Radial Basis

The radial basis functions meshfree method is a research method for thin plate problem which has gradually developed into a more mature meshfree method. It includes finite element, radial basis functions meshfree collocation method, etc. In this paper, we introduce the multilevel radial basis function collocation method for the fourth-order thin plate problem. We use nonsymmetric Kansa multilevel radial basis function collocation method to solve the fourth-order thin plate problem. Two numerical examples based on Wendland’s [Formula: see text] and [Formula: see text] functions are given to examine that the convergence of the multilevel radial basis function collocation method which is good for solving the fourth-order thin plate problem.

scholarly journals A Novel Meshfree Strategy for a Viscous Wave Equation With Variable Coefficients

2021 ◽  
Vol 9 ◽  
Author(s):  
Fuzhang Wang ◽  
Juan Zhang ◽  
Imtiaz Ahmad ◽  
Aamir Farooq ◽  
Hijaz Ahmad
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Collocation Method ◽  
Basis Function ◽  
Variable Coefficients ◽  
Basis Functions ◽  
Time Dependent ◽  
Mesh Free ◽  
Radial Basis ◽  
One Step

A one-step new general mesh free scheme, which is based on radial basis functions, is presented for a viscous wave equation with variable coefficients. By constructing a simple extended radial basis function, it can be directly applied to wave propagation by using the strong form-based mesh free collocation method. There is no need to deal with the time-dependent variable particularly. Numerical results for a viscous wave equation with variable coefficients show that the proposed mesh free collocation method is simple with accurate solutions.

A localized Newton basis functions meshless method for the numerical solution of the 2D nonlinear coupled Burgers’ equations

2017 ◽  
Vol 27 (11) ◽  
pp. 2582-2602 ◽  
Author(s):  
Fahimeh Saberi Zafarghandi ◽  
Maryam Mohammadi ◽  
Esmail Babolian ◽  
Shahnam Javadi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solution ◽  
Collocation Method ◽  
Meshless Method ◽  
Basis Functions ◽  
Content Type ◽  
Burgers Equations ◽  
Convection Dominated ◽  
Newton Basis Functions

Purpose The purpose of this paper is to introduce a local Newton basis functions collocation method for solving the 2D nonlinear coupled Burgers’ equations. It needs less computer storage and flops than the usual global radial basis functions collocation method and also stabilizes the numerical solutions of the convection-dominated equations by using the Newton basis functions. Design/methodology/approach A meshless method based on spatial trial space spanned by the local Newton basis functions in the “native” Hilbert space of the reproducing kernel is presented. With the selected local sub-clusters of domain nodes, an approximation function is introduced as a sum of weighted local Newton basis functions. Then the collocation approach is used to determine weights. The method leads to a system of ordinary differential equations (ODEs) for the time-dependent partial differential equations (PDEs). Findings The method is successfully used for solving the 2D nonlinear coupled Burgers’ equations for reasonably high values of Reynolds number (Re). It is a well-known issue in the analysis of the convection-diffusion problems that the solution becomes oscillatory when the problem becomes convection-dominated if the standard methods are followed without special treatments. In the proposed method, the authors do not detect any instability near the front, hence no technique is needed. The numerical results show that the proposed method is efficient, accurate and stable for flow with reasonably high values of Re. Originality/value The authors used more stable basis functions than the standard basis of translated kernels for representing of kernel-based approximants for the numerical solution of partial differential equations (PDEs). The local character of the method, having a well-structured implementation including enforcing the Dirichlet and Neuman boundary conditions, and producing accurate and stable results for flow with reasonably high values of Re for the numerical solution of the 2D nonlinear coupled Burgers’ equations without any special technique are the main values of the paper.

scholarly journals Recovering Heat Source from Fourth-Order Inverse Problems by Weighted Gradient Collocation

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 241
Author(s):  
Judy P. Yang ◽  
Hsiang-Ming Li
Keyword(s):  
Heat Source ◽  
Collocation Method ◽  
Fourth Order ◽  
Reproducing Kernel ◽  
A Priori ◽  
High Order ◽  
Order Partial Differential Equation ◽  
Benchmark Analysis ◽  
The Mathematical Model ◽  
Inverse Heat Source

The weighted gradient reproducing kernel collocation method is introduced to recover the heat source described by Poisson’s equation. As it is commonly known that there is no unique solution to the inverse heat source problem, the weak solution based on a priori assumptions is considered herein. In view of the fourth-order partial differential equation (PDE) in the mathematical model, the high-order gradient reproducing kernel approximation is introduced to efficiently untangle the problem without calculating the high-order derivatives of reproducing kernel shape functions. The weights of the weighted collocation method for high-order inverse analysis are first determined. In the benchmark analysis, the unclear illustration in the literature is clarified, and the correct interpretation of numerical results is given particularly. Two mathematical formulations with four examples are provided to demonstrate the viability of the method, including the extreme cases of the limited accessible boundary.

scholarly journals Effect of Thrust on the Structural Vibrations of a Nonuniform Slender Rocket

2020 ◽  
Vol 25 (2) ◽  
pp. 29
Author(s):  
Desmond Adair ◽  
Aigul Nagimova ◽  
Martin Jaeger
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Structural Vibrations ◽  
Unknown Parameters ◽  
One Dimensional ◽  
Vibration Problems ◽  
Nonuniform Beam

The vibration characteristics of a nonuniform, flexible and free-flying slender rocket experiencing constant thrust is investigated. The rocket is idealized as a classic nonuniform beam with a constant one-dimensional follower force and with free-free boundary conditions. The equations of motion are derived by applying the extended Hamilton’s principle for non-conservative systems. Natural frequencies and associated mode shapes of the rocket are determined using the relatively efficient and accurate Adomian modified decomposition method (AMDM) with the solutions obtained by solving a set of algebraic equations with only three unknown parameters. The method can easily be extended to obtain approximate solutions to vibration problems for any type of nonuniform beam.

An efficient application of PML in fourth-order precise integration time domain method for the numerical solution of Maxwell's equations

2013 ◽  
Vol 33 (1/2) ◽  
pp. 116-125
Author(s):  
Zhongming Bai ◽  
Xikui Ma ◽  
Xu Zhuansun ◽  
Qi Liu
Keyword(s):  
Finite Difference ◽  
Numerical Solution ◽  
Fourth Order ◽  
Time Domain ◽  
Fdtd Method ◽  
Time Domain Method ◽  
Integration Time ◽  
Precise Integration ◽  
Content Type ◽  
Domain Method

Purpose – The purpose of the paper is to introduce a perfectly matched layer (PML) absorber, based on Berenger's split field PML, to the recently proposed low-dispersion precise integration time domain method using a fourth-order accurate finite difference scheme (PITD(4)). Design/methodology/approach – The validity and effectiveness of the PITD(4) method with the inclusion of the PML is investigated through a two-dimensional (2-D) point source radiating example. Findings – Numerical results indicate that the larger time steps remain unchanged in the procedure of the PITD(4) method with the PML, and meanwhile, the PITD(4) method employing the PML is of the same absorbability as that of the finite-difference time-domain (FDTD) method with the PML. In addition, it is also demonstrated that the later time reflection error of the PITD(4) method employing the PML is much lower than that of the FDTD method with the PML. Originality/value – An efficient application of PML in fourth-order precise integration time domain method for the numerical solution of Maxwell's equations.

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