Radial Basis Function Finite Difference Research Articles

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.

An Efficient Meshfree Numerical Technique to Solve Fractional Schamel–KdV Equation for Ion-Acoustic Solitary Waves in Dusty Plasma

The Schamel–KdV equation plays a vital role in studying the effect of electron trapping on the nonlinear interaction of ion-acoustic waves in plasma and dusty plasma. This work presents a numerical scheme-based radial basis function-finite difference (RBF-FD) method to solve the time-fractional Schamel–KdV equation. The advantages of using this method are their meshfree nature and flexibility in dealing with complex geometries. In the discretization process, a finite difference (FD) technique is used to discretize the temporal derivative, while the multiquadric (MQ) RBF is used to approximate the spatial derivatives. The theoretical convergence and stability analysis of the time-discrete scheme are also established. In addition, numerical experiments are executed through some illustrated problems, and the obtained results are compared with that acquired by the tanh method and Kudryashov method solutions to show the high accuracy and plausibility of the proposed technique. Also, the graphical representations are given to demonstrate the physical interpretation of the resulting wave structures.

The RBF-FD and RBF-FDTD Methods for Solving Time-Domain Electrical Transient Problems in Power Systems

In this paper, the development and application of the radial basis function-finite difference (RBF-FD) method and the RBF-finite difference time domain (RBF-FDTD) method for solving electrical transient problems in power systems that are defined by the time-dependent ordinary differential equations (ODEs) and the time-dependent partial differential equations (PDEs), respectively, are presented. RBFs such as Gaussian (GA), Multiquadric (MQ), Inverse Quadric (IQ), and Inverse Multiquadric (IMQ) are used in these numerical methods to formulate the central finite difference approximations of the first- and second-order derivatives of a function. The algorithm of selecting “optimal” shape parameters for our basis functions is also applied, specifically to increase the accuracy of the suggested methods with regard to high accuracy needs. Finally, the accuracy, effectiveness, and applicability of our new approaches are evaluated through simulations of the switching transient voltages on a typical electrical circuit and a 220 kV single-phase transmission line, lightning-induced voltages on a 110 kV single-phase overhead distribution line, and transient voltages along two horizontal grounding electrodes excited by lightning impulse sources. The obtained numerical results demonstrate that our proposed RBF-based numerical approaches compare favorably to the traditional numerical methods.

An efficient radial basis function generated finite difference meshfree scheme to price multi-dimensional PDEs in financial options

In this paper, the purpose is to focus on proposing a new solver for pricing financial option at the presence of several assets. Toward this goal, first we introduce the weighting coefficients of the radial basis function-finite difference (RBF-FD) procedure for approximating the derivatives of the functions under the inverse multi-quadric function (IMQ). The weights are employed on a nonuniform mesh of points when tackling multi-dimensional partial differential equations (PDEs). A new solver based on RBF-FD methodology is then constructed employing matrix notations for solving option pricing problems. The theoretical discussions will be confirmed via numerical experiments on several test problems consisting of four to five underlying assets.

Iterative optimization method for determining optimal shape parameter in RBF-FD method

In this paper, we propose a novel iterative optimization method 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 reduce the numerical error significantly, for any type of radial basis functions (RBFs). Three numerical examples are presented to show the effectiveness and generality of the proposed method over previous methods. The relevant data and code can be found at https://github.com/hsbhc/IOM-RBF-FD.

Radial Basis Function Finite Difference Method Based on Oseen Iteration for Solving Two-Dimensional Navier-Stokes Equations.

In this paper, the radial basis function finite difference method is used to solve two-dimensional steady incompressible Navier-Stokes equations. First, the radial basis function finite difference method with polynomial is used to discretize the spatial operator. Then, the Oseen iterative scheme is used to deal with the nonlinear term, constructing the discrete scheme for Navier-Stokes equation based on the finite difference method of the radial basis function. This method does not require complete matrix reorganization in each nonlinear iteration, which simplifies the calculation process and obtains high-precision numerical solutions. Finally, several numerical examples are obtained to verify the convergence and effectiveness of the radial basis function finite difference method based on Oseen Iteration.

Accurate RBF-FD meshless numerical simulation of thermo-fluid problems for generic 3D geometries

In contrast to traditional mesh-based methods for the numerical solution of boundary value problems, e.g., Finite Element (FEM) and Finite Volume (FVM), in the recent period many meshfree approaches have been proposed in order to avoid those typical issues due to the mesh. For example, the quality of the mesh greatly affects the reliability of the final solution in the case of CFD problems and the human intervention of a professional is often still needed when dealing with complex-shaped domains. This in turn increases both cost and time required for the reliable simulation of problems of engineering relevance. Meshless methods, on the other side, usually rely on a simpler distribution of nodes and do not require the storage of connectivity information. Among others, one of the most promising meshless methods in terms of accuracy and flexibility is the one based on the Radial Basis Function – Finite Difference (RBF-FD) scheme. RBF-FD methods, however, are usually affected by severe ill conditioning issues when Neumann boundary conditions are employed. This fact is the main responsible for the appearance of large discretization errors near the boundary and for the lack of stability of traditional time integration schemes. In order to address this issue, some new algorithms for the robust treatment of boundary conditions have been developed and successfully employed to solve fluid flow problems with heat transfer. Furthermore, it is well acknowledged that the efficient resolution of boundary layers arising in this class of problems requires an adequate spatial discretization in the neighbourhood of the boundary, i.e., increased node/mesh density along the direction of large gradients only. This result is achieved by employing anisotropic node distributions, which is a novelty in the context of the RBF-FD method to the best of the authors’ knowledge. The method described above is successfully employed for the accurate solution of a representative 3D heat transfer problem with incompressible fluid flow.