Radial Basis Function Finite Difference Research Articles

Convergence properties of the radial basis function-finite difference method on specific stencils with applications in solving partial differential equations

We consider the problem of approximating a linear differential operator on several specific stencils using the radial basis function method in the finite difference scheme. We prove a linear convergence order on a non-equispaced five-point stencil. Then, we discuss how the convergence rate can be boosted up to the second-order on an equispaced stencil. Moreover, we show that including additional nearby nodes (six to twelve) in the stencil does not improve the convergence rate, thus increasing the computational load without enhancing convergence. To overcome this limitation, we propose a stencil that accelerates the convergence up to four using a nine-point stencil, unlike existing approaches which are based on thirteen-point equispaced stencils to achieve such an order of convergence. To support our findings, we conduct numerical experiments by solving Poisson equations and a parabolic problem.

Numerical analysis of small-strain elasto-plastic deformation using local Radial Basis Function approximation with Picard iteration

In this paper, we discuss a von Mises plasticity model with nonlinear isotropic hardening assuming small strains in a plane strain example of internally pressurised thick-walled cylinder subjected to different loading conditions. The elastic deformation is modelled using the Navier-Cauchy equation. In regions where the von Mises stress exceeds the yield stress, corrections are made locally through a return mapping algorithm. We present a novel method that uses a Radial Basis Function-Finite Difference (RBF-FD) approach with Picard iteration to solve the system of nonlinear equations arising from plastic deformation. This technique eliminates the need to stabilise the divergence operator and avoids special positioning of the boundary nodes, while preserving the elegance of the meshless discretisation and avoiding the introduction of new parameters that would require tuning. The results of the proposed method are compared with analytical and Finite Element Method (FEM) solutions. The results show that the proposed method achieves comparable accuracy to FEM while offering significant advantages in the treatment of complex geometries without the need for conventional meshing or special treatment of boundary nodes or differential operators.

A radial basis function-finite difference method for solving Landau–Lifshitz–Gilbert equation including Dzyaloshinskii-Moriya interaction

This paper investigates a numerical method for solving the two-dimensional Landau–Lifshitz–Gilbert (LLG) equation, governing the dynamics of the magnetization in ferromagnetic materials. Specifically, we incorporate the Dzyaloshinskii–Moriya interaction into the LLG equation—a crucial factor for the creation and stabilization of magnetic skyrmions. We propose a local meshless method that utilizes radial basis function-finite difference (RBF-FD) for spatial discretization and the Crank–Nicolson scheme for temporal discretization, along with an extrapolation technique to handle the nonlinear terms. We demonstrate the method’s accuracy, efficiency, and adaptability through numerical tests on domains of various shapes, showcasing its practical utility in simulating real-world magnetic phenomena and advanced materials.

Stability analysis and error estimates of implicit-explicit Runge-Kutta least squares RBF-FD method for time-dependent parabolic equation

In this paper, for the time-dependent parabolic equations defined on complex geometries domain, we develop and analyze the least-squares radial basis function finite difference method (RBF-FD) coupled with the implicit-explicit Runge-Kutta (IMEX-RK) time discretization up to third order accuracy, which improves stability and accuracy. We derive the absolute stability region and the optimal time-step constraint for four kinds of IMEX-RK schemes. Compared to the traditional explicit or implicit time discretization, these are not trivial. Under a wide time-step constraint, the stability and the error estimates in l2-norm are established. Finally, several numerical experiments on the regular domain and non-convex domain are performed to validate the theoretical analysis.

A Meshless Method of Radial Basis Function-Finite Difference Approach to 3-Dimensional Numerical Simulation on Selective Laser Melting Process

Selective laser melting (SLM) is a rapidly evolving technology that requires extensive knowledge and management for broader industrial adoption due to the complexity of phenomena involved. The selection of parameters and numerical analysis for the SLM process are both costly and time-consuming. In this paper, a three-dimensional radial basis function-finite difference (RBF-FD) meshless model is introduced to accurately and efficiently simulate the molten pool size and temperature distribution during the SLM process for austenitic stainless steel (AISI 316L). Two different volumetric moving heat source models were presented, namely the ray-tracing method heat source model and the double-ellipsoidal shape heat source model. The temperature-dependent material properties and phase change process were also considered, based on experiments and effective models. Results of the model for the molten pool size were validated with those of the literature. The proposed approach can be used to predict the effect of different laser power and scan speed on the molten pool size and temperature gradient along the depth direction. The result reveals that the depth of the molten pool is more sensitive to laser power than scan speed. Under the same scan speed, a 22% change in laser power (45 ± 10 W) affects the maximum temperature proportionally by about 9%. The developed algorithm is computationally efficient and suitable for industrial applications.

Efficient Scheme for the Economic Heston–Hull–White Problem Using Novel RBF-FD Coefficients Derived from Multiquadric Function Integrals

This study presents an efficient method using the local radial basis function finite difference scheme (RBF-FD). The innovative coefficients are derived from the integrals of the multiquadric (MQ) function. Theoretical convergence rates for the coefficients used in function derivative approximation are provided. The proposed scheme utilizes RBF-FD estimations on three-point non-uniform stencils to construct the final approximation on a tensor grid for the 3D Heston–Hull–White (HHW) PDE, which is relevant in economics and mathematical finance. Numerical evidence and comparative analyses validate the results and the proposed scheme.

A new efficient parametric level set method based on radial basis function-finite difference for structural topology optimization

To overcome a significant challenge in traditional parameterized level set methods based on globally supported radial basis functions, we propose employing a local differentiation construction of radial basis functions using finite difference, a technique previously applied to solving partial differential equations but novel in the context of topology optimization. We present a novel parameterized level set method for structural topology optimization of compliance minimization and compliant mechanism, with the main aim of reducing computational costs associated with fully dense matrices when approximating systems with a large number of collocation points. The new scheme implemented with rectangular mesh elements and polygonal mesh generation accommodates both rectangular and complex design domains. Numerical results are provided to demonstrate the algorithm's effectiveness.

Radial Basis Function–Finite Difference Solution Combined with Level-Set Embedded Boundary Method for Improving a Diffusive Logistic Model with a Free Boundary

In this paper, we propose an efficient numerical method to solve the problems of diffusive logistic models with free boundaries, which are often used to simulate the spreading of new or invasive species. The boundary movement is tracked by the level-set method, where the Hamilton–Jacobi weighted essentially nonoscillatory (HJ-WENO) scheme is utilized to capture the boundary curve embedded by the Cartesian grids via the embedded boundary method. Then the radial basis function–finite difference (RBF-FD) method is adopted for spatial discretization and the implicit–explicit (IMEX) scheme is considered for time integration. A variety of numerical examples are utilized to demonstrate the evolution of the diffusive logistic model with different initial boundaries.

A model reduction method for parametric dynamical systems defined on complex geometries

Dynamic mode decomposition (DMD) describes the dynamical system in an equation-free manner and can be used for the prediction and control. It is an efficient data-driven method for the complex systems. In this paper, we extend DMD to the parameterized problems and propose a model reduction method based on DMD to improve the computation efficiency. This method is an offline-online mechanism. In the offline phase, we need to generate the snapshots data by solving the parameterized equations for each parameter in the training set and perform the singular value decomposition (SVD) to get the reduced operator matrices, which would lead to the substantial computation. The weighted & interpolated nearest-neighbors algorithm (wiNN) is adopted to construct the efficient surrogate models of the reduced operator matrices (including the reduced Koopman operator matrix and SVD-modes). In the online phase, for each parameter, we only need to perform the operations based on the low-dimensional matrices to get the parameter DMD solution. Moreover, we choose the least squares radial basis function finite difference method for the spatial discretization. This can make our method more applicable to the parameterized problems defined on complex geometries. At last, the reaction-diffusion, the incompressible miscible flooding and the incompressible Navier-Stokes models defined on the complex geometries are presented to illustrate the effectiveness of the proposed method.

Numerical investigation of high-dimensional option pricing PDEs by utilizing a hybrid radial basis function - finite difference procedure

The target of this research is to resolve high-dimensional partial differential equations (PDEs) for multi-asset options, modeled as parabolic time-dependent PDEs. We present a hybrid radial basis function - finite difference (RBF-FD) solver, which combines the advantages of Gaussian and multiquadric functions. Additionally, we employ the Krylov subspace method on the resulting system of ordinary differential equations, reducing the computation load for finding the numerical solution. Computational tests, up to seven dimensions, and comparisons support the superiority of our hybrid solver.

Solving spatiotemporal partial differential equations with Physics-informed Graph Neural Network

Physics-informed neural networks (PINNs) have recently gained considerable attention as a prominent deep learning technique for solving partial differential equations (PDEs). However, traditional fully connected PINNs often encounter slow convergence issues attributed to automatic differentiation in constructing loss functions. In addition, convolutional neural network (CNN)-based PINNs face challenges when dealing with irregular domains and unstructured meshes. To address these issues, we propose a novel framework based on graph neural networks (GNNs) and radial basis function finite difference (RBF-FD). We introduce GNNs into physics-informed learning to better handle irregular domains. RBF-FD is employed to construct a high-precision difference format of the PDE to guide model training. We perform numerical experiments on various PDEs, including heat, wave, and shallow water equations on irregular domains. The results demonstrate that our method is capable of accurate predictions and exhibits strong generalization, allowing for inference with different initial. We evaluate the generalizability, accuracy, and efficiency of our approach by considering different Gaussian noise, PDE parameters, numbers of collection points, and various types of radial basis functions.

A hybrid radial basis function—Finite difference method for modelling two-dimensional thermo-elasto-plasticity, Part 2. Application to cooling of hot-rolled steel bars on a cooling bed

This paper represents Part 2 of the parallel paper Part 1, where the strong form hybrid RBF-FD method was developed for solving thermo-elasto-plastic problems. It addresses the industrial application of this novel meshless method to steel bars cooling on a cooling bed (CB) where the formation of residual stress is of primary interest. The study investigates the impact of the distance between the bars and the distance to the heat shield above the CB on radiative heat fluxes and, consequently, on thermo-mechanical response. The thermal model is solved on bars cross-section with a RBF-FD method where augmented polyharmonic splines are used for the local approximation. View factors, computed with a Monte-Carlo method, are included in radiative heat fluxes. The thermal solution is incrementally applied on a mechanical model that assumes a generalised plane strain state and captures bars bending. The study employs a hybrid RBF-FD method to resolve a nonlinear discontinuous mechanical problem successfully. The simulation of the process shows how different process parameters influence the thermo-mechanical response of the bars.

Adaptive residual refinement in an RBF finite difference scheme for 2D time-dependent problems

Adaptive residual refinement in an RBF finite difference scheme for 2D time-dependent problems

A hybrid radial basis function-finite difference method for modelling two-dimensional thermo-elasto-plasticity, Part 1: Method formulation and testing

A hybrid version of the strong form meshless Radial Basis Function-Finite Difference (RBF-FD) method is introduced for solving thermo-mechanics. The thermal model is spatially discretised with RBF-FD, where trial functions are polyharmonic splines augmented with polynomials. For time discretisation, the explicit Euler method is employed. An extension of RBF-FD, the hybrid RBF-FD, is introduced for solving mechanical problems. The model is one-way coupled, where temperature affects displacements. The thermo-elastoplastic material response is considered where the stress field is generally non-smooth. The hybrid RBF-FD, where the finite difference method is used to discretise the divergence operator from the balance equation, is shown to be successful when dealing with such problems. The mechanical model is introduced in a plane strain and in a generalised plane strain (GPS) assumption. For the first time, this work presents a strong form RBF-FD for GPS problems subjected to integral form constraints. The proposed method is assessed regarding h-convergence and accuracy on the benchmark with heating an elastoplastic square. It is proven to be successful at solving one-way coupled thermo-elastoplastic problems. The proposed novel meshless approach is efficient, accurate, and robust. Its use in an industrial situation is provided in Part 2 of this paper.

Solution of MHD-stokes flow in an L-shaped cavity with a local RBF-supported finite difference

One of the popular meshless methods for solving governing equations in applied sciences is a local radial basis function-finite difference (RBF-FD). In this paper, we proposed a new idea for an L- shaped (or like T- and Z-shaped) domain based on the domain decomposition. RBF-FD formulation is used at the interface points to get a better solution, while the classical FD is applied to all sub-regions. We use the algorithm based on the Gaussian-RBF (RBF-GA) in the stable calculation of the weights to avoid choosing optimal shape parameters. Stencil size is considered the nearest n-points (9,12,15) and benchmark results are presented for divided-lid driven cavity. Further, Navier–Stokes equations adding the Lorentz force term with Stokes approximation for a single-lid L-shaped cavity exposed to inclined magnetic field are solved by the devised numerical method. The flow structure is analyzed in aspect of streamline topology under the various magnetic field rotation (0∘≤α≤90∘ ) and its strength (M=10,30,50,100 ).

Solitary Wave Propagation of the Generalized Rosenau–Kawahara–RLW Equation in Shallow Water Theory with Surface Tension

This paper addresses a numerical approach for computing the solitary wave solutions of the generalized Rosenau–Kawahara–RLW model established by coupling the generalized Rosenau–Kawahara and Rosenau–RLW equations. The solution of this model is accomplished by using the finite difference approach and the upwind local radial basis functions-finite difference. Firstly, the PDE is transformed into a nonlinear ODEs system by means of the radial kernels. Secondly, a high-order ODE solver is implemented for discretizing the system of nonlinear ODEs. The main advantage of this technique is its lack of need for linearization. The global collocation techniques impose a significant computational cost, which arises from calculating the dense system of algebraic equations. The proposed technique estimates differential operators on every stencil. As a result, it produces sparse differentiation matrices and reduces the computational burden. Numerical experiments indicate that the method is precise and efficient for long-time simulation.

Meshfree algorithms for analysis and computational modeling of multidimensional hyperbolic wave models

PurposeThis article offered two meshfree algorithms, namely the local radial basis functions-finite difference (LRBF-FD) approximation and local radial basis functions-differential quadrature method (LRBF-DQM) to simulate the multidimensional hyperbolic wave models and work is an extension of Jiwari (2015).Design/methodology/approachIn the evolvement of the first algorithm, the time derivative is discretized by the forward FD scheme and the Crank-Nicolson scheme is used for the rest of the terms. After that, the LRBF-FD approximation is used for spatial discretization and quasi-linearization process for linearization of the problem. Finally, the obtained linear system is solved by the LU decomposition method. In the development of the second algorithm, semi-discretization in space is done via LRBF-DQM and then an explicit RK4 is used for fully discretization in time.FindingsFor simulation purposes, some 1D and 2D wave models are pondered to instigate the chastity and competence of the developed algorithms.Originality/valueThe developed algorithms are novel for the multidimensional hyperbolic wave models. Also, the stability analysis of the second algorithm is a new work for these types of model.

Generalized moving least squares vs. radial basis function finite difference methods for approximating surface derivatives

Generalized moving least squares vs. radial basis function finite difference methods for approximating surface derivatives

An ALE meshfree method for surface PDEs coupling with forced mean curvature flow

In this paper, we propose an arbitrary Lagrangian Eulerian (ALE) meshfree method that utilizes the radial basis function-finite difference (RBF-FD) method for solving diffusion-reaction equations on evolving surfaces. The surface evolution law is determined by a forced mean curvature flow (FMCF). We develop a parametric RBF-FD method to solving the FMCF, which has not been done previously in the framework of meshfree methods. The advantage of this parametric method is that it utilizes a suitable tangential velocity in the equation to maintain a fairly uniform distribution of nodes during the surface evolution. The tangential velocity also allows us to construct an ALE RBF-FD scheme for the surface PDEs, which performs better than the purely Lagrangian method. Furthermore, we discuss how to deal with the boundary conditions for both the FMCF and the PDEs on surfaces, and present a ghost node approach to efficiently discretize the boundary conditions. Some numerical experiments are shown, which not only demonstrate the effectiveness of the tangential velocity, but also illustrate two applications.

A new variable shape parameter strategy for RBF approximation using neural networks

The choice of the shape parameter highly effects the behaviour of radial basis function (RBF) approximations, as it needs to be selected to balance between the ill-conditioning of the interpolation matrix and high accuracy. In this paper, we demonstrate how to use neural networks to determine the shape parameters in RBFs. In particular, we construct a multilayer perceptron (MLP) trained using an unsupervised learning strategy, and use it to predict shape parameters for inverse multiquadric and Gaussian kernels. We test the neural network approach in RBF interpolation tasks and in a RBF-finite difference method in one and two-space dimensions, demonstrating promising results.