Biharmonic navigation using radial basis functions

Robotica ◽  
2021 ◽  
pp. 1-12
Author(s):  
Xu-Qian Fan ◽  
Wenyong Gong
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Radial Basis Functions ◽  
Real World ◽  
Biharmonic Equation ◽  
Finite Difference Methods ◽  
Basis Functions ◽  
Large Size ◽  
Radial Basis ◽  
Difference Methods

Abstract Path planning has been widely investigated by many researchers and engineers for its extensive applications in the real world. In this paper, a biharmonic radial basis potential function (BRBPF) representation is proposed to construct navigation fields in 2D maps with obstacles, and it therefore can guide and design a path joining given start and goal positions with obstacle avoidance. We construct BRBPF by solving a biharmonic equation associated with distance-related boundary conditions using radial basis functions (RBFs). In this way, invalid gradients calculated by finite difference methods in large size grids can be preventable. Furthermore, paths constructed by BRBPF are smoother than paths constructed by harmonic potential functions and other methods, and plenty of experimental results demonstrate that the proposed method is valid and effective.

scholarly journals A Finite Difference Fictitious Domain Wavelet Method for Solving Dirichlet Boundary Value Problem

2021 ◽  
Vol 14 (3) ◽  
pp. 706-722
Author(s):  
Francis Ohene Boateng ◽  
Joseph Ackora-Prah ◽  
Benedict Barnes ◽  
John Amoah-Mensah
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Finite Difference Methods ◽  
Dirichlet Boundary Conditions ◽  
Fictitious Domain ◽  
Dirichlet Boundary Value Problem ◽  
Wavelet Method ◽  
Difference Methods

In this paper, we introduce a Finite Difference Fictitious Domain Wavelet Method (FDFDWM) for solving two dimensional (2D) linear elliptic  partial differential equations (PDEs) with Dirichlet boundary conditions on regular geometric domain. The method reduces the 2D PDE into a 1D system of ordinary differential equations and applies a compactly supported wavelet to approximate the solution. The problem is embedded in a fictitious domain to aid the enforcement of the Dirichlet boundary conditions. We present numerical analysis and show that our method yields better approximation to the solution of the Dirichlet problem than traditional methods like the finite element and finite difference methods.

scholarly journals Analysis of Laminated Shells by Murakami’s Zig-Zag Theory and Radial Basis Functions Collocation

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
D. A. Maturi ◽  
A. J. M. Ferreira ◽  
A. M. Zenkour ◽  
D. S. Mashat
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Radial Basis Functions ◽  
Vibration Analysis ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Basis Functions ◽  
Transverse Displacement ◽  
Laminated Shells ◽  
Radial Basis

The static and free vibration analysis of laminated shells is performed by radial basis functions collocation, according to Murakami’s zig-zag (ZZ) function (MZZF) theory . The MZZF theory accounts for through-the-thickness deformation, by considering a ZZ evolution of the transverse displacement with the thickness coordinate. The equations of motion and the boundary conditions are obtained by Carrera’s Unified Formulation and further interpolated by collocation with radial basis functions.

A NUMERICAL METHOD FOR 1D TIME-DEPENDENT SCHRÖDINGER EQUATION USING RADIAL BASIS FUNCTIONS

2013 ◽  
Vol 10 (02) ◽  
pp. 1341010 ◽  
Author(s):  
TONGSONG JIANG ◽  
ZHAOLIN JIANG ◽  
JOSEPH KOLIBAL
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Numerical Method ◽  
Radial Basis Functions ◽  
Finite Difference Scheme ◽  
Good Accuracy ◽  
Basis Functions ◽  
Time Dependent ◽  
Numerical Examples ◽  
Radial Basis

This paper proposes a new numerical method to solve the 1D time-dependent Schrödinger equations based on the finite difference scheme by means of multiquadrics (MQ) and inverse multiquadrics (IMQ) radial basis functions. The numerical examples are given to confirm the good accuracy of the proposed methods.

The Coupling of RBF and FDM for Solving Higher Order Fractional Partial Differential Equations

2014 ◽  
Vol 598 ◽  
pp. 409-413 ◽  
Author(s):  
Zakieh Avazzadeh ◽  
Wen Chen ◽  
Vahid Reza Hosseini
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Radial Basis Functions ◽  
Basis Functions ◽  
Discrete Solution ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Time Direction ◽  
Radial Basis

In this work, we describe the radial basis functions for solving the time fractional partial differential equations defined by Caputo sense. These problems can be discretized in the time direction based on finite difference scheme and is continuously approximated by using the radial basis functions in the space direction which achieves the semi-discrete solution. Numerical results accuracy the efficiency of the presented method.

COMPARATIVE STUDY OF THE MULTIQUADRIC AND THIN-PLATE SPLINE RADIAL BASIS FUNCTIONS FOR THE TRANSIENT-CONVECTIVE DIFFUSION PROBLEMS

2006 ◽  
Vol 17 (08) ◽  
pp. 1151-1169 ◽  
Author(s):  
A. DURMUS ◽  
I. BOZTOSUN ◽  
F. YASUK
Keyword(s):  
Radial Basis Functions ◽  
Thin Plate ◽  
Numerical Solutions ◽  
Convective Diffusion ◽  
Finite Difference Methods ◽  
Basis Functions ◽  
Thin Plate Spline ◽  
Collocation Methods ◽  
Radial Basis ◽  
Diffusion Problems

The numerical solutions of the unsteady transient-convective diffusion problems are investigated by using multiquadric (MQ) and thin-plate spline (TPS) radial basis functions (RBFs) based on mesh-free collocation methods with global basis functions. The results of radial basis functions are compared with the mesh-dependent boundary element and finite difference methods as well as the analytical solution for high Péclet numbers. It is reported that for low Péclet numbers, MQ-RBF provides excellent agreement, while for high Péclet numbers, TPS-RBF is better than MQ-RBF.

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