Local Radial Basis Function Research Articles

Size dependent free vibration analysis of 2D-functionally graded curved nanobeam by meshless method

The free vibration of two-directional functionally graded (2D-FG) thick curved nanobeam with concentrated mass is investigated for various boundary conditions. Hamilton’s principle is employed to develop the governing equations of the 2D-FG curved nanobeam using the first-order shear deformation theory (FSDT). The small scale effect is captured by the nonlocal elasticity theory of Eringen. The nanobeam is functionally graded in the transverse and circumferential directions of the curved nanobeam. The governing equations of the system are obtained as three coupled partial differential equations with non-constant coefficients. An appropriate meshless formulation is developed to discretize the governing equations based on local weak formulation and radial basis function. The presented meshless method is employed to study the free vibration of the one-directional and two-directional functional graded curved nanobeams. Simple-simple, free-free, clamped-clamped, and clamped-free edge conditions are investigated. Moreover, an analytical solution is developed for simple-simple transversely 1D-FG curved nanobeam. In the numerical results, the natural frequencies and mode shapes of shadow and deep curved nanobeam are studied and the influence of effective parameters such as nonlocal parameter, FG power indexes, opening angle, edge conditions, and concentrated mass are investigated. It is seen that shear locking effect is eliminated in present meshless formulation.

The local radial basis function collocation method for elastic wave propagation analysis in 2D composite plate

In this paper, the local radial basis function collocation method (LRBFCM) is applied to simulate the in-plane elastic wave propagation in composite plate. In order to reduce the influence caused by large aspect ratio, an improved LRBFCM based on direct method is developed so that the governing equations can be numerically decoupled by using bilinear nodes in different directions. Several numerical examples are given to demonstrate the accuracy and efficiency of the proposed technique in simulating the elastic wave propagation with large aspect ratio.

Finding an Efficient Computational Solution for the Bates Partial Integro-Differential Equation Utilizing the RBF-FD Scheme

This paper proposes a computational solver via the localized radial basis function finite difference (RBF-FD) scheme and the use of graded meshes for solving the time-dependent Bates partial integro-differential equation (PIDE) arising in computational finance. In order to avoid facing a large system of discretization systems, we employ graded meshes along both of the spatial variables, which results in constructing a set of ordinary differential equations (ODEs) of lower sizes. Moreover, an explicit time integrator is used because it can bypass the need to solve the large discretized linear systems in each time level. The stability of the numerical method is discussed in detail based on the eigenvalues of the system matrix. Finally, numerical tests revealed the accuracy and reliability of the presented solver.

A meshless multiscale method for simulating hemodynamics

Simulating hemodynamic quantities such as pressure and velocity are of great interest to clinicians to aid in surgical planning. To accurately simulate modifications to a region of vasculature, the entire system must be modeled. To facilitate this, a localized Radial-Basis Function (RBF) collocation Meshless flow solver is developed and tightly coupled to a 0D Lumped-Parameter Model (LPM) for solution of the peripheral circulation. The Meshless solver uses localized RBF collocations at data points that are automatically generated according to the geometry. The incompressible flow equations are updated by an explicit time-marching scheme based on a pressure-velocity correction algorithm. The inlets and outlets of the domain are tightly coupled with the LPM which contains elements that draw from a fluid-electrical analogy such as resistors, capacitors, and inductors that represent the viscous resistance, vessel compliance, and flow inertia, respectively. The localized RBF Meshless approach is well-suited for modeling complicated non-Newtonian hemodynamics due to ease of spatial discretization, ease of addition of multi-physics interactions such as fluid-structure interaction of the vessel wall, and ease of parallelization for fast computations. This work introduces the tight coupling of Meshless methods and LPMs for fast and accurate hemodynamic simulations.

Surrogate-Assisted Hybrid Meta-Heuristic Algorithm with an Add-Point Strategy for a Wireless Sensor Network

Meta-heuristic algorithms are widely used in complex problems that cannot be solved by traditional computing methods due to their powerful optimization capabilities. However, for high-complexity problems, the fitness function evaluation may take hours or even days to complete. The surrogate-assisted meta-heuristic algorithm effectively solves this kind of long solution time for the fitness function. Therefore, this paper proposes an efficient surrogate-assisted hybrid meta-heuristic algorithm by combining the surrogate-assisted model with gannet optimization algorithm (GOA) and the differential evolution (DE) algorithm, abbreviated as SAGD. We explicitly propose a new add-point strategy based on information from historical surrogate models, using information from historical surrogate models to allow the selection of better candidates for the evaluation of true fitness values and the local radial basis function (RBF) surrogate to model the landscape of the objective function. The control strategy selects two efficient meta-heuristic algorithms to predict the training model samples and perform updates. A generation-based optimal restart strategy is also incorporated in SAGD to select suitable samples to restart the meta-heuristic algorithm. We tested the SAGD algorithm using seven commonly used benchmark functions and the wireless sensor network (WSN) coverage problem. The results show that the SAGD algorithm performs well in solving expensive optimization problems.

An Efficient Localized RBF-FD Method to Simulate the Heston–Hull–White PDE in Finance

The Heston–Hull–White three-dimensional time-dependent partial differential equation (PDE) is one of the important models in mathematical finance, at which not only the volatility is modeled based on a stochastic process but also the rate of interest is assumed to follow a stochastic dynamic. Hence, an efficient method is derived in this paper based on the methodology of the localized radial basis function generated finite difference (RBF-FD) scheme. The proposed solver uses the RBF-FD approximations on graded meshes along all three spatial variables and a high order time-stepping scheme. Stability is also studied in detail to show under what conditions the proposed method is stable. Computational simulations are given to support the theoretical discussions.

Fractional Dual-Phase-Lag Model for Nonlinear Viscoelastic Soft Tissues

The primary goal of this paper is to create a new fractional boundary element method (BEM) model for bio-thermomechanical problems in functionally graded anisotropic (FGA) nonlinear viscoelastic soft tissues. The governing equations of bio-thermomechanical problems are briefly presented, including the fractional dual-phase-lag (DPL) bioheat model and Biot’s model. The more complex shapes of nonlinear viscoelastic soft tissues can be handled by the boundary element method, which also avoids the need for the interior domain to be discretized. The fractional dual-phase-lag bioheat equation was solved using the general boundary element method (GBEM) based on the local radial basis function collocation method (LRBFCM). The poroelastic fields are then calculated using the convolution quadrature boundary element method (CQBEM) The numerical findings show that our proposed numerical model is valid, efficient, and accurate.

Local meshless collocation scheme for numerical simulation of space fractional PDE

In this work, numerical solution of multi term space fractional PDE is calculated by using radial basis functions. The fractional derivatives of radial basis functions are evaluated by Caputo and Riemann-Liouville definitions. Local radial basis functions are applied to get stable and accurate solution the problem. Accuracy of the method is assessed by using double mesh procedure. Numerical solutions are presented for different fractional orders to show the effect of introducing fractionality.

Numerical Assessment of Nanofluid Natural Convection Using Local RBF Method Coupled with an Artificial Compressibility Model

In this paper, natural heat convection inside square and equilateral triangular cavities was studied using a meshless method based on collocation local radial basis function (RBF). The nanofluids used were Cu-water or -water mixture with nanoparticle volume fractions range of . A system of continuity, momentum, and energy partial differential equations was used in modeling the flow and temperature behavior of the fluids. Partial derivatives in the governing equations were approximated using the RBF method. The artificial compressibility model was implemented to overcome the pressure velocity coupling problem that occurs in such equations. The main goal of this work was to present a simple and efficient method to deal with complex geometries for a variety of problem conditions. To assess the accuracy of the proposed method, several test cases of natural convection in square and triangular cavities were selected. For Rayleigh numbers ranging from to , a validation test of natural convection of Cu-water in a square cavity was used. The numerical investigation was then extended to Rayleigh number , as well as -water nanofluid with a volume fraction range of . In a second investigation, the same nanofluids were used in a triangular cavity with varying volume fractions to test the proposed meshless approach on non-rectangular geometries. The numerical results appear to be in agreement with those from earlier investigations. Furthermore, the suggested meshless method was found to be stable and accurate, demonstrating that it may be a viable alternative for solving natural heat transfer equations of nanofluids in enclosures with irregular geometries.

Local Radial Basis Function Methods: Comparison, Improvements, and Implementation

Local Radial Basis Function Methods: Comparison, Improvements, and Implementation

A Novel Localized Meshless Method for Solving Transient Heat Conduction Problems in Complicated Domains

This paper first attempts to solve the transient heat conduction problem by combining the recently proposed local knot method (LKM) with the dual reciprocity method (DRM). Firstly, the temporal derivative is discretized by a finite difference scheme, and thus the governing equation of transient heat transfer is transformed into a non-homogeneous modified Helmholtz equation. Secondly, the solution of the non-homogeneous modified Helmholtz equation is decomposed into a particular solution and a homogeneous solution. And then, the DRM and LKM are used to solve the particular solution of the non-homogeneous equation and the homogeneous solution of the modified Helmholtz equation, respectively. The LKM is a recently proposed local radial basis function collocation method with the merits of being simple, accurate, and free of mesh and integration. Compared with the traditional domain-type and boundary-type schemes, the present coupling algorithm could be treated as a really good alternative for the analysis of transient heat conduction on high-dimensional and complicated domains. Numerical experiments, including two- and three-dimensional heat transfer models, demonstrated the effectiveness and accuracy of the new methodology.

Localized meshless approaches based on theta method and BDF2 for nonlinear Sobolev equation arising from fluid dynamics

This paper formulates the second-order BDF and theta schemes based on an efficient localized meshless technique for computing the approximate solution of the Sobolev model. The model is one of the useful description for modeling the flow of fluids through fractured rocks, movement of moisture in soils. This proposed approach discretizes the nonlinear Sobolev equation in two main steps. In the first step, temporal discretization is accomplished with the help of the second-order BDF and theta schemes, respectively. In the second step, the localized radial basis function partition of unity collocation method (L-RBF-PUCM) is used for spatial discretization, thereby constructing a fully-discrete algorithm. For large linear problems, global techniques have the disadvantage of high computational cost. The proposed method overcomes this shortcoming well and reduces the computational burden by sparsifying the linear system. The convergence analysis and unconditional stability of the semi-discrete formulation are examined by means of the energy method. Numerical results are carried out to verify the validity and accuracy of the L-RBF-PUCM.

4-dimensional local radial basis function interpolation of large, uniformly spaced datasets

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.

Adaptive selection strategy of shape parameters for LRBF for solving partial differential equations

Radial basis function (RBF) is a basis function suitable for scattered data interpolation and high dimensional function interpolation, wherein the independent shape parameters have a direct impact on the accuracy of calculation results. Research in this field is mostly concerned with the shape parameter selection strategy based on the premise of global distribution or block regional distribution. In this paper, a shape parameter selection strategy is proposed, which is used for the local RBF collocation method (LRBF) for solving partial differential equations. It overcomes many limitations of the traditional methods applied to LRBF. In this strategy, a set of twin matrices similar to the interpolation matrix are constructed to evaluate the error of the model. In addition, the penalty term contained in the twin matrix is used to relax the influence of the far end region on the target node. Since the objective problem is nonlinear, a particle swarm optimization algorithm (PSO) is employed to minimize the training objective and adjust the shape parameters of the basis function at each iteration. Extensive numerical results showed the effectiveness of the error estimation strategy, by providing a good shape parameter and better solution accuracy. At the end of the paper, the generality of the shape parameter optimization framework based on this strategy is discussed through three examples.

An efficient localized meshless collocation method for the two-dimensional Burgers-type equation arising in fluid turbulent flows

This paper focusses on the numerical technique based on a localized meshless collocation method for approximating the Burgers-type equation in two dimensions. The method uses two main stages to approximate the unknown solution. First, the time derivative is discretized by the backward Euler method and a semi-discrete approach is obtained. Meanwhile, the unconditional stability and convergence of the semi-discrete scheme are proved by virtue of the discrete energy method in an appropriate Sobolev space. Second, a local radial basis function partition of unity (LRBF-PU) method is applied to approximate the spatial direction and a fully-discrete scheme is established. The LRBF-PU is based on subdividing the original domain to several subdomains and uses the radial basis function interpolation on each local subdomain. Compared with the global collocation technique, the main advantages of this technique are that it can reduce the computational cost and obtain a well linear system. Three illustrative examples are presented to demonstrate the accuracy and efficiency of the proposed method.

Conjugate MHD natural convection of hybrid nanofluids in a square enclosure containing a complex conductive cylinder

PurposeThe purpose of this study is to investigate the enhancement and suppression of heat transfer for hybrid nanofluids (Cu–Al2O3/water) in a square enclosure containing a thermal-conductive cylinder when the Lorentz force is applied to the hybrid nanofluids.Design/methodology/approachSince the inner conductive cylinder in present research has a complex geometry, an in-house meshless method, namely, the local radial basis function (LRBF) method, is applied to solve the 2 dimensional (2D) incompressible Navier–Stokes equation in the fluid domain and Fourier heat conduction equation in solid domain. The solid–fluid interface remains the physical continuity of temperature and heat flux. Only the Lorentz force is considered for the presence of the magnetic field. The conjugate natural convection is assumed to be steady, thus only fully developed heat exchange from the nanofluids to solid or vice versa is comprehensively investigated.FindingsIt can be concluded that Lorentz force plays a more significant role than hybrid nanofluids in enhancing/suppressing heat transfer when the orientation of magnetic field is the same to the x direction. The thermal conductivity ratio can dramatically change the isotherms and streamlines as well as the mean value of the Nusselt number, resulting in totally different heat transfer phenomena. The included angle of magnetic field also has a significant effect on the heat transfer rate when it changes from horizontal to vertical.Research limitations/implicationsThe constant thermo-physical properties of incompressible fluid and the 2D steady flow are considered in this study.Originality/valueThe conjugate MHD natural convection of hybrid nanofluids is numerically investigated by an in-house meshless LRBF method. The enhancement and suppression of heat transfer under the combined influence of the volume fraction of nanoparticles, Hartmann number and the thermal conductivity ratio are comprehensively investigated.

Numerical simulation of 3D double-nozzles printing by considering a stabilized localized radial basis function collocation method

In this work, a stabilized local radial basis function collocation method (LRBFCM) is proposed to simulate the electrochemical 3D double-nozzles printing. A forward Euler-method is proposed to discretize time in the fluid flow described by Navier-Stokes (NS) equations. The background fictitious grids or meshes are employed to enhance the stability of the LRBFCM in solving the fluid flow. The details of the LRBFCM for simulating the pressure and velocity fields of the electrolyte during the electrochemical 3D printing process are introduced. The accuracy of the proposed method is validated by corresponding experiments, and good agreement can be observed between the numerical results and the experimental measurements.

A hybrid LRBF-DQ method for solving nonlinear (2 + 1) dimensional initial-boundary value problems

In this paper, a combination of local radial basis functions (LRBFs) and step-by-step differential quadrature method (DQM) is proposed to solve nonlinear (2 + 1) dimensional differential equations. The LRBFs are employed to estimate spatial derivatives whereas DQM is utilized to discretize temporal derivatives. Then, the capability of the method is proved via solving five examples including Burgers’, Zakharov-Kuznetsov, sine-Gordon and nonlinear chemical reaction-diffusion equations through both a rectangular and a complex domain. Furthermore, an experimental convergence analysis is conducted for each example. Moreover, the accuracy of the solution method is studied by calculating the error norms. Finally, the results are compared with those obtained in literature to prove the superiority of this scheme over particular previously proposed methods.

Transferable species distribution modelling: Comparative performance of Generalised Functional Response models

Predictive species distribution models (SDMs) are becoming increasingly important in ecology, in the light of rapid environmental change. However, the predictions of most current SDMs are specific to the habitat composition of the environments in which they were fitted. This may limit SDM predictive power because species may respond differently to a given habitat depending on the availability of all habitats in their environment, a phenomenon known as a functional response in resource selection. The Generalised Functional Response (GFR) framework captures this dependence by formulating the SDM coefficients as functions of habitat availability. The original GFR implementation used global polynomial functions of habitat availability to describe the functional responses. In this study, we develop several refinements of this approach and compare their predictive performance using two simulated and two real datasets. We first use local radial basis functions (RBF), a more flexible approach than global polynomials, to represent the habitat selection coefficients, and balance bias with precision via regularization to prevent overfitting. Second, we use the RBF-GFR and GFR models in combination with the classification and regression tree CART, which has more flexibility and better predictive powers for non-linear modelling. As further extensions, we use random forests (RFs) and extreme gradient boosting (XGBoost), ensemble approaches that consistently lead to variance reduction in generalization error. We find that the different methods are ranked consistently across the datasets for out-of-data prediction. The traditional stationary approach to SDMs and the GFR model consistently perform at the bottom of the ranking (simple SDMs underfit, and polynomial GFRs overfit the data). The best methods in our list provide non-negligible improvements in predictive performance, in some cases taking the out-of-sample R2 from 0.3 up to 0.7 across datasets. At times of rapid environmental change and spatial non-stationarity ignoring the effects of functional responses on SDMs, results in two different types of prediction bias (under-prediction or mis-positioning of distribution hotspots). However, not all functional response models perform equally well. The more volatile polynomial GFR models can generate biases through over-prediction. Our results indicate that there are consistently robust GFR approaches that achieve impressive gains in transferability across very different datasets.

On the Numerical Approximation of Mobile-Immobile Advection-Dispersion Model of Fractional Order Arising from Solute Transport in Porous Media

The fractional mobile/immobile solute transport model has applications in a wide range of phenomena such as ocean acoustic propagation and heat diffusion. The local radial basis functions (RBFs) method have been applied to many physical and engineering problems because of its simplicity in implementation and its superiority in solving different real-world problems easily. In this article, we propose an efficient local RBFs method coupled with Laplace transform (LT) for approximating the solution of fractional mobile/immobile solute transport model in the sense of Caputo derivative. In our method, first, we employ the LT which reduces the problem to an equivalent time-independent problem. The solution of the transformed problem is then approximated via the local RBF method based on multiquadric kernels. Afterward, the desired solution is represented as a contour integral in the left half complex along a smooth curve. The contour integral is then approximated via the midpoint rule. The main advantage of the LT-RBFs method is the avoiding of time discretization technique due which overcomes the time instability issues, second is its local nature which overcomes the ill-conditioning of the differentiation matrices and the sensitivity of the shape parameter, since the local RBFs method only considers the discretization points in each local domain around the collocation point. Due to this, sparse and well-conditioned differentiation matrices are produced, and third is the low computational cost. The convergence and stability of the numerical scheme are discussed. Some test problems are performed in one and two dimensions to validate our numerical scheme. To check the efficiency, accuracy, and efficacy of the scheme the 2D problems are solved in complex domains. The numerical results confirm the stability and efficiency of the method.