Simulation of laminar backward facing step flow under magnetic field with explicit local radial basis function collocation method

This paper explores the application of the mesh-free Local Radial Basis Function Collocation Method (LRBFCM) [1] in solution of coupled heat transfer and fluid flow problems associated with solidification of a binary alloy. The involved temperature, velocity, species and pressure fields are represented on overlapping sub-domains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBF’s. The energy and momentum equations are solved through explicit time stepping. The pressure-velocity coupling is calculated iteratively, with pressure correction, predicted from the local continuity equation violation [2,3]. The solution procedure is demonstrated in simulation of solidification of 8 wt % Pb-Sn alloy and 5 wt % Sn-Pb alloy in rectangular cavity at conditions, defined in [4]. The numerical examples include studies with different uniform discretization and its influence on final macrosegregation map. The solution is assessed by comparison with the reference results of the finite volume and finite element method. The advantages of the method are simplicity, accuracy and straightforward applicability in non-uniform node arrangements. The methodology will be use in complementing our heat transfer model [5] of the continuous casting of steel.

A meshless collocation method for band structure simulation of nanoscale phononic crystals based on nonlocal elasticity theory

Using a local radial basis function collocation method to approximate radiation boundary conditions

The local radial basis function (RBF) method is a promising solver for variable‐order time fractional diffusion equation (TFDE), as it overcomes the computational burden of the traditional global method. Application of the local RBF method is limited to Fickian diffusion, while real‐world diffusion is usually non‐Fickian in multiple dimensions. This article is the first to extend the application of the local RBF method to two‐dimensional, variable‐order, time fractional diffusion equation in complex shaped domains. One of the main advantages of the local RBF method is that only the nodes located in the subdomain, surrounding the local point, need to be considered when calculating the numerical solution at this point. This approach can perform well with large scale problems and can also mitigate otherwise ill‐conditioned problems. The proposed numerical approach is checked against two examples with curved boundaries and known analytical solutions. Shape parameter and subdomain node number are investigated for their influence on the accuracy of the local RBF solution. Furthermore, quantitative analysis, based on root‐mean‐square error, maximum absolute error, and maximum error of the partial derivative indicates that the local RBF method is accurate and effective in approximating the variable‐order TFDE in two‐dimensional irregular domains.

Meshless approach to the large-eddy simulation of the continuous casting process

This article presents a local knot method (LKM) to solve inverse Cauchy problems of Helmholtz equations in arbitrary 2D and 3D domains. The Moore–Penrose pseudoinverse using the truncated singular value decomposition is employed in the local approximation of supporting domain instead of the moving least squares method. The developed approach is a semi‐analytical and local radial basis function collocation method using the non‐singular general solution as the basis function. Like the traditional boundary knot method, the LKM is simple, accurate and easy‐to‐program in solving inverse Cauchy problems associated with Helmholtz equations. Unlike the boundary knot method, the new scheme can directly reconstruct the unknowns both inside the physical domain and along its boundary by solving a sparse linear system, and can achieve a satisfactory solution. Numerical experiments, involving the complicated geometry and the high noise level, confirm the accuracy and reliability of the proposed method for solving inverse Cauchy problems of Helmholtz equations.

In many references, both the ill-posed and inverse boundary value problems are solved iteratively. The iterative procedures are based on firstly converting the problem into a well-posed one by assuming the missing boundary values. Then, the problem is solved by using either a developed numerical algorithm or a conventional optimization scheme. The convergence of the technique is achieved when the approximated solution is well compared to the unused data. In the present paper, we present a different way to solve an ill-posed problem by applying the localized and space-time localized radial basis function collocation method depending on the problem considered and avoiding the iterative procedure. We demonstrate that the solution of certain ill-posed and inverse problems can be accomplished without iterations. Three different problems have been investigated: problems with missing boundary condition and internal data, problems with overspecified boundary condition, and backward heat conduction problem (BHCP). It has been demonstrated that the presented method is efficient and accurate and overcomes the stability analysis that is required in iterative techniques.

Simulation of electromagnetic wave propagations in negative index materials by the localized RBF-collocation method

Simulation of linear and nonlinear advection-diffusion problems by the direct radial basis function collocation method

In this work we present an overview of the RBF finite collocation approach. The RBF-FC method is an alternative decomposition for the localisation of the radial basis function (RBF) collocation method for the solution of PDEs. In contrast to the popular finite difference approach, in which the PDE governing operator is reconstructed from simple RBF interpolants, the finite collocation formulation instead enforces all governing and boundary PDEs within the local RBF collocation systems, and assembles the field variable directly into a sparse global assembly. In this way the solution of the PDE is enforced by collocation, rather than by differencing. By including the PDE governing operator within the local collocation systems any data extracted from the RBF interpolation field naturally respects the local physics of the PDE, including information about the local convective field. This gives rise to an “implicit upwinding” effect, which allows stable solutions to be obtained in convective-dominant scenarios when using centred stencils. The approach also allows high rates of spatial convergence to be obtained. We present here an overview of the RBF-FC formulation, demonstrating the performance of the method with benchmark examples for linear elasticity and convection-diffusion PDEs. The adaptation of the method for unsteady problems is described, including a Richardson extrapolation method to improve rates of temporal convergence. The application of the method to the capture and transport

Background and ObjectiveLarge, uniformly spaced, complex and time varying datasets derived from high resolution medical image velocimetry can provide a wealth of information regarding small-scale transient physiological flow phenomena and pulsation of anatomical boundaries. However, there remains a need for interpolation techniques to effectively reconstruct a fully 4-dimensional functional relationship from this data. This paper presents a preliminary evaluation of a 4-dimensional local radial basis function (RBF) algorithm as a means of addressing this problem for laminar flows. MethodsA 4D interpolation algorithm is proposed based on a Local Hermitian Interpolation (LHI) using a combination of multi-quadric RBF with a partition of unity scheme. The domain is divided into uniform sub-systems with size restricted to immediately neighbouring points. The validity of the algorithm is first established on a known 4D analytical dataset and a CFD based laminar flow phantom. Application is then demonstrated through characterisation of a large 4D laminar flow dataset obtained from magnetic resonance imaging (MRI) measurements of cerebrospinal fluid velocities in the brain. ResultsPerformance of the algorithm is compared to that of a quad-linear interpolation, demonstrating favourable improvement in accuracy. The technique is shown to be robust, computationally efficient and capable of refined interpolation in Euclidean space and time. Application to MR velocimetry data is shown to produce promising results for the 4D reconstruction of the transient flow field and movement of the fluid boundaries at spatial and temporal locations intermediate to the original data. ConclusionThis study has demonstrated feasibility of an accurate, stable and efficient 4-dimensional local RBF interpolation method for large, transient laminar flow velocimetry datasets. The proposed approach does not suffer from ill-conditioning or high computational cost due to domain decomposition into local stencils where the RBF is only ever applied to a limited number of points. This work offers a potential tool to assist medical diagnoses and drug delivery through better understanding of physiological flow fields such as cerebrospinal fluid. Further work will evaluate the technique on a wider range of flow fields and against CFD simulation.

This study introduces a deep neural network approach that utilizes radial basis functions (RBFs) to solve forward and inverse problems in the process of diffusion. The input layer incorporates multiquadric (MQ) RBFs, symbolizing the radial distance between the boundary points on the spacetime boundary and the source points positioned outside the spacetime boundary. The output layer is the initial and boundary data given by analytical solutions of the diffusion equation. Utilizing the concept of the spacetime coordinates, the approximations for forward and backward diffusion problems involve assigning initial data on the bottom or top spacetime boundaries, respectively. As the need for discretization of the governing equation is eliminated, our straightforward approach uses only the provided boundary data and MQ RBFs. To validate the proposed method, various diffusion scenarios, including forward, backward, and inverse problems with noise, are examined. Results indicate that the method can achieve high-precision numerical solutions for solving diffusion problems. Notably, only 1/4 of the initial and boundary conditions are known, yet the method still yields precise results.

Recently, the localized oscillatory radial basis functions collocation method (L-ORBFs) has been introduced to solve elliptic partial differential equations in 2D with a large number of computational nodes. The research clearly shows that the L-ORBFs is very convenient and useful for solving large-scale problems, but this method is numerically less accurate. In this paper, we propose a numerical scheme to improve the accuracy of the L-ORBFs by adding low-degree polynomials in the localized collocation process. The numerical results validate that the proposed numerical scheme is highly accurate and clearly outperforms the results of the L-ORBFs.

Radial basis functions method for solving the fractional diffusion equations

In the paper, a numerical technique based on the multiquadric radial basis function (MQ-RBF) collocation method is presented for the fractional delay differential equations. The convergence analysis and the unique fixed point of the collocation methods are propounded in the norm. Numerical results illustrate the credibility and applicability of the numerical technique. The obtained results are compared with those obtained by other methods.