Article
Mathematics, Applied
Bengt Fornberg
Summary: The algorithm provides FD weights of optimal accuracy for approximating any order derivative at a specified location with arbitrarily distributed node locations in one-dimension, and can now also be applied to first derivative values. The MATLAB code for the algorithm is provided, with two examples illustrating its application in solving partial differential equations.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Sara Arefian, Davoud Mirzaei
Summary: In this work, the standard Hermite interpolation based RBF-HFD method is developed into a new faster and more accurate technique based on the PU method. The new approach solves much fewer local linear systems for calculating stencil weights, reducing computational cost. The method also allows flexibility in using different types of PU weight functions and utilizes the scaling property of PHS kernels.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Engineering, Multidisciplinary
Kyle W. Beggs, Eduardo Divo, Alain J. Kassab
Summary: In this paper, a localized Radial-Basis Function collocation Meshless flow solver is developed and tightly coupled to a 0D Lumped-Parameter Model for accurate hemodynamic simulations. This approach is well-suited for modeling complex non-Newtonian hemodynamics and allows for fast computations.
ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS
(2023)
Article
Mathematics
Malik Zaka Ullah, Abdullah Khamis Alzahrani, Hashim Mohammed Alshehri, Stanford Shateyi
Summary: In this work, a numerical scheme is presented to solve the time-fractional Black-Scholes PDE under the generalized multiquadric radial basis function. The scheme uses spatial uniform meshes and stencils with five adjacent discretization nodes and estimates the time-fractional derivative using an L1-scheme. The results of numerical tests demonstrate the efficacy of the presented solver.
Article
Mathematics, Applied
Jie Hou, Ying Li, Shihui Ying
Summary: In this paper, a novel iterative optimization method is proposed to determine the best parameter c for the Radial basis function finite difference (RBF-FD) method based on the Double Operator Error (DOE). This method is a general iterative optimization approach that can rapidly determine the optimal c for a given problem and significantly reduce the numerical error for any type of radial basis functions (RBFs). Three numerical examples demonstrate the effectiveness and generality of the proposed method compared to previous methods. The relevant data and code can be found at https://github.com/hsbhc/IQM-RBF-FD.& COPY; 2023 Published by Elsevier Ltd.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Multidisciplinary Sciences
Ying-Ting Chen, Cheng Li, Lin-Quan Yao, Yang Cao
Summary: This paper proposes a new hybrid radial basis function collocation method (HRBF-CM) for solving two-dimensional elastostatic symmetric problems. The method combines infinitely smooth RBF and piecewise smooth RBF, with two parameters (the shape parameter and the weight parameter). Discretization schemes are presented in detail. Numerical results using MATLAB show that the proposed method has higher accuracy compared to traditional methods, especially with a larger number of nodes. The effectiveness of the new method compared to widely used traditional RBF is discussed, as well as the effect of parameters on the method's performance.
Article
Engineering, Multidisciplinary
Victor Bayona, Mario Sanchez-Sanz, Eduardo Fernandez-Tarrazo, Manuel Kindelan
Summary: This research focuses on developing a high-order meshfree method to model combustion inside complex geometries using radial basis functions-generated finite differences, aiming to identify different combustion regimes and improve conversion efficiency.
APPLIED MATHEMATICAL MODELLING
(2021)
Article
Mathematics, Applied
Davoud Mirzaei
Summary: This paper proposes a new localized radial basis function method based on partition of unity for solving boundary and initial-boundary value problems. The new method, called the direct RBF partition of unity (D-RBF-PU) method, is faster and simpler than the standard RBF-PU method by avoiding derivatives of PU weight functions and lower derivatives of local approximants. Additionally, the method is more efficient and less expensive with the use of discontinuous PU weight functions, and numerical experiments on irregular domains support the efficiency of the new method.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Engineering, Multidisciplinary
Faranak Gholampour, Esmail Hesameddini, Ameneh Taleei
Summary: This work presents a numerical solution for two-dimensional elasticity problems involving multiple material phases. The proposed method, based on radial basis functions, has been proven to be effective in solving partial differential equations on irregular domains. By utilizing polynomial augmented polyharmonic spline radial basis functions, the method provides highly accurate results while bypassing stability issues and stagnation errors.
ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS
(2022)
Review
Mathematics
Archna Kumari, Vijay K. Kukreja
Summary: This article provides an overview of the widely used Hermite interpolating polynomials and their application in solving various types of differential equations. The use of Hermite interpolation has become an established tool in applied science.
Article
Computer Science, Interdisciplinary Applications
Nat H. Mathews, Natasha Flyer, Sarah E. Gibson
Summary: The study presents a novel magnetohydrostatic numerical model for directly solving the force-balanced magnetic field in the solar corona. The model utilizes Radial Basis Function Finite Differences with 3D polyharmonic splines plus polynomials as the core discretization. It addresses the challenges posed by the ill-posed and numerically intractable nature of the static force-balance equations in the limit of zero forcing.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Varun Shankar, Grady B. Wright, Aaron L. Fogelson
Summary: The study introduces a high-order radial basis function finite difference method for solving advection-diffusion equations on time-varying domains. The framework eliminates overlap parameters, enables tuning-free assembly of differentiation matrices on moving domains, and demonstrates high performance with high convergence rates.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Youssef El Seblani, Elyas Shivanian
Summary: This paper introduces an effective technique called MRPHI for solving partial differential equations with Neumann boundary condition by utilizing radial point interpolation and Hermite-type interpolation techniques. The method is tested on various two-dimensional diffusion equations to demonstrate stability across different arbitrary domains over time.
ENGINEERING WITH COMPUTERS
(2021)
Review
Mathematics
Pravin Singh, Nabendra Parumasur, Shivani Singh
Summary: This paper focuses on the application of piecewise polynomial functions in numerical analysis, specifically on the use of spline functions and Hermite functions to solve problems with irregular features. By deriving the quintic Hermite basis and utilizing orthogonal collocation method, error analysis and numerical simulations were conducted to enhance theoretical results.
Article
Physics, Multidisciplinary
Haifa Bin Jebreen, Fairouz Tchier
Summary: This paper investigates and optimizes the order of approximation formulas on nonuniform grids using the RBF-HFD approach, resulting in new weighting coefficients with higher convergence rates. Theoretical discussions are supported with several tests to demonstrate the effectiveness of the method.
Article
Engineering, Electrical & Electronic
Santosh Pokhrel, Varun Shankar, Jamesina J. Simpson
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION
(2018)
Article
Computer Science, Interdisciplinary Applications
Varun Shankar, Grady B. Wright
JOURNAL OF COMPUTATIONAL PHYSICS
(2018)
Article
Computer Science, Interdisciplinary Applications
Varun Shankar, Akil Narayan, Robert M. Kirby
JOURNAL OF COMPUTATIONAL PHYSICS
(2018)
Article
Mathematics, Applied
Varun Shankar, Robert M. Kirby, Aaron L. Fogelson
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2018)
Article
Computer Science, Interdisciplinary Applications
Kathryn P. Drake, Grady B. Wright
JOURNAL OF COMPUTATIONAL PHYSICS
(2020)
Article
Computer Science, Interdisciplinary Applications
Kathryn P. Drake, Grady B. Wright
JOURNAL OF COMPUTATIONAL PHYSICS
(2020)
Article
Computer Science, Interdisciplinary Applications
Varun Shankar, Grady B. Wright, Aaron L. Fogelson
Summary: The study introduces a high-order radial basis function finite difference method for solving advection-diffusion equations on time-varying domains. The framework eliminates overlap parameters, enables tuning-free assembly of differentiation matrices on moving domains, and demonstrates high performance with high convergence rates.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Kathryn P. Drake, Edward J. Fuselier, Grady B. Wright
Summary: This paper presents a new method for surface reconstruction from a point cloud by utilizing the approximate normals to the surface. By using curl-free radial basis function interpolation of the normals, an implicit surface approximation for the point cloud can be obtained. The method is combined with a partition of unity technique to better represent local features and handle noise in both the normals and the point positions.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Grady B. Wright, Andrew Jones, Varun Shankar
Summary: We propose a new meshfree geometric multilevel (MGM) method for solving linear systems arising from discretizing elliptic PDEs on point cloud surfaces. The method utilizes Poisson disk sampling for coarsening point clouds and uses polyharmonic splines for transferring information. It is applicable to various localized meshfree methods and has been tested on different problems, showing efficient convergence rates and scalability. The effectiveness of MGM is further demonstrated on challenging applications involving complicated surfaces.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Andrew M. Jones, Peter A. Bosler, Paul A. Kuberry, Grady B. Wright
Summary: Approximating differential operators on two-dimensional surfaces is a crucial problem in various fields. Localized meshfree methods, such as generalized moving least squares (GMLS) and radial basis function finite differences (RBF-FD), have been proven effective and efficient in achieving high accuracy at low computational cost for this task. However, a direct comparison of these methods for approximating surface differential operators (SDOs) has not been conducted yet. This study aims to fill this gap and compare the performance of GMLS with an RBF-FD method based on polyharmonic spline kernels and polynomials (PHS+Poly). Furthermore, we investigate the relationship between the tangent plane formulation of SDOs and the local coordinate formulation used in GMLS, and propose a new RBF-FD method for approximating the tangent space of an unknown point cloud surface using ideas from the GMLS SDO formulation. Evaluation: 8/10.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Kathryn P. Drake, Edward J. Fuselier, Grady B. Wright
Summary: The paper introduces a technique for constructing global approximants of divergence-free or curl-free vector fields by combining div/curl-free radial basis functions in a partition of unity framework, applicable to vector fields in 2D space and on surfaces, and providing approximations for scalar potentials. The method effectively bypasses the computational expense issue caused by the global nature of the problem.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Varun Shankar, Grady B. Wright, Akil Narayan
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2020)
Article
Mathematics, Interdisciplinary Applications
Sean D. Lawley, Varun Shankar
MULTISCALE MODELING & SIMULATION
(2020)
Article
Computer Science, Interdisciplinary Applications
Ashish Bhole, Herve Guillard, Boniface Nkonga, Francesca Rapetti
Summary: Finite elements of class C-1 are used for computing magnetohydrodynamics instabilities in tokamak plasmas, and isoparametric approximations are employed to align the mesh with the magnetic field line. This numerical framework helps in understanding the operation of existing devices and predicting optimal strategies for the international ITER tokamak. However, a mesh-aligned isoparametric representation encounters issues near critical points of the magnetic field, which can be addressed by combining aligned and unaligned meshes.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2024)
Article
Computer Science, Interdisciplinary Applications
Federico Vismara, Tommaso Benacchio
Summary: This paper introduces a method for solving hyperbolic-parabolic problems on multidimensional semi-infinite domains. By dividing the computational domain into bounded and unbounded subdomains and coupling them using numerical fluxes at the interface, accurate numerical solutions are obtained. In addition, computational cost can be reduced by tuning the parameters of the basis functions.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2024)
Article
Computer Science, Interdisciplinary Applications
Keigo Enomoto, Takato Ishida, Yuya Doi, Takashi Uneyama, Yuichi Masubuchi
Summary: We have developed a novel Moving Particle Simulation (MPS) method to accurately reproduce the motion of fibers in sheared liquids. By introducing the micropolar fluid model, we address the issue of fibers being aligned with the flow direction in conventional MPS simulations. Our method is capable of accurately reproducing the fiber motion predicted by Jeffery's theory.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2024)
