Radial Basis Collocation Method Research Articles

Weighted radial basis collocation method for large deformation analysis of rubber-like materials

A nonlinear formulation, based on the total Lagrange description of the weighted radial basis collocation method (WRBCM), is proposed for the large deformation analysis of rubber-like materials where the materials are hyperelastic and nearly incompressible. The WRBCM based on the strong form collocation is a genuinely meshfree method that eliminates the need for meshing. As a result, it effectively circumvents challenges associated with mesh distortion during large deformation analysis. The proper weights that should be imposed on the boundary collocation equations are first derived to achieve the optimal convergence in hyperelasticity. In the WRBCM, the support of the shape function remains unchanged throughout material deformation, thereby guaranteeing the absence of tension instability during large deformation analysis. Additionally, by combining WRBCM with a least-squares solution, volumetric locking in nearly incompressible hyperelastic problems can be suppressed. This is due to the infinite continuity possessed by the radial basis approximation, which ensures the divergence-free condition. Several numerical examples are examined, demonstrating the high accuracy and exponential convergence of WRBCM in hyperelastic large deformation analysis. Moreover, no volumetric locking can be observed which further substantiates the effectiveness of applying nonlinear WRBCM to nearly incompressible hyperelastic problems.

Weighted Radial Basis Collocation Method for the Nonlinear Inverse Helmholtz Problems

In this paper, a meshfree weighted radial basis collocation method associated with the Newton’s iteration method is introduced to solve the nonlinear inverse Helmholtz problems for identifying the parameter. All the measurement data can be included in the least-squares solution, which can avoid the iteration calculations for comparing the solutions with part of the measurement data in the Galerkin-based methods. Appropriate weights are imposed on the boundary conditions and measurement conditions to balance the errors, which leads to the high accuracy and optimal convergence for solving the inverse problems. Moreover, it is quite easy to extend the solution process of the one-dimensional inverse problem to high-dimensional inverse problem. Nonlinear numerical examples include one-, two- and three-dimensional inverse Helmholtz problems of constant and varying parameter identification in regular and irregular domains and show the high accuracy and exponential convergence of the presented method.

Meshfree approach for flexure analysis of bidirectional porous FG plate subjected to I, L, and T types of transverse loading

Various type of transverse loadings is generally used in practical applications, mainly in public transportation from buses, aeroplane, and railways to the places where ques are maintained in a desired way. These loads are applied on different types of structures which are made-up of composites/laminates and functionally graded materials. In present work, the flexure analysis of bi-directionally porous functionally graded material (BDPFG) rectangular plates subjected to L, I, and T types of transverse loading has been carried out for first time. Governing differential equations have been obtained using energy principle with five variable higher-order shear deformation theory (HSDT). The material properties of BDPFG rectangular plates are to be graded in the thickness and length directions according to modified power-law distribution with five porosity distributions. A code has been computed in MATLAB (2019) for obtaining the numerical results using meshfree approach based on radial basis collocation method (RBCM). The influence of the grading index, span to thickness ratio, aspect ratio, transverse loading type, porosity fraction, and porosity distribution on the normalized deflection and stresses of BDPFG rectangular plates is examined. New numerical results can provide benchmarks for future analyses of BDPFG plates and will be helpful in designing the structures.

Radial basis collocation method for free vibration analysis of elastically supported porous bi-directional FGM plate under various types of porosity distribution

ABSTRACT In this paper, the radial basis collocation method (RBCM) for the free vibration response of elastically supported porous bi-directional functionally graded (BDPFG) plate is investigated in a framework of higher-order shear deformation theory (HSDT). The material properties of BDPFG plates are assumed to be varying continuously in the length and thickness direction. The material properties are evaluated based on Voigt’s micro-mechanical model considering power law distribution method with arbitrary power index. Governing differential equations (GDEs) of the porous BFGM plate area achieved utilising Hamilton’s principle and multiquadric radial basis function (MQRBF) is used for discretizing the GDEs. To incorporate porosity in the BDPFG plate, five types of porosity distribution models are carried out and analysed. The present results are validated with the available published results in the literature to certify the efficacy and accuracy. The influence of grading index in the length and thickness direction, porosity distribution, porosity index, span to thickness ratio, and elastic foundation over natural frequency of bi-directional BDPFG plate is investigated which is the main contribution.

Radial basis collocation method with parameters optimized for estimating pollutant release history.

The identification of pollutant source release history in rivers is important for emergency response of pollution accidents and formulating remediation strategies. Space-time radial basis collocation method (RBCM), as a meshless method with strong applicability, can directly estimate the release history from the concentration data measured at downstream observation sites. However, the uncertainty of specific parameters in space-time RBCM is the main factor affecting the accuracy of estimation. Therefore, a way to solve the parameters efficiently and accurately is essential. For this purpose, a new model which combines space-time RBCM and differential evolution algorithm (DEA) is established to identify the source release history. First of all, efficient parameter optimizer DEA is introduced to search the parameters that affect the estimation accuracy of space-time RBCM. Then, a new loss function considering the imbalance configuration of RBCM nodes is designed to ensure the rationality of the parameters obtained by DEA. The results of numerical cases and real field case show that the proposed method can accurately estimate the real release history with low time consumption. It is also demonstrated that DEA is more efficient than k-fold cross-validation in searching the optimal parameters for space-time RBCM, and the parameters obtained from the new loss function can make the estimated release history more precise.

Gradient Enhanced Localized Radial Basis Collocation Method for Inverse Analysis of Cauchy Problems

This work proposes a gradient enhanced localized radial basis collocation method (GL-RBCM) for solving boundary value problems. In particular, the attention is paid to the solution of inverse Cauchy problems. It is known that the approximation by radial basis functions often leads to ill-conditioned systems due to the global nature. To this end, the reproducing kernel shape function and gradient reproducing kernel shape function are proposed to localize the radial basis function while the gradient approximation is aimed at reducing the computational intensity of carrying out the second derivatives of reproducing kernel shape function. In the proposed weighted collocation method, the weights on Neumann and Dirichlet boundary conditions are determined for both direct problems and inverse problems. From stability analysis, it is shown that the GL-RBCM can maintain high accuracy of approximating the first derivatives even under irregular perturbation added to boundary conditions. By comparing with the localized RBCM, the CPU saving of the GL-RBCM is manifested. The efficacy of the proposed method is therefore demonstrated.

Static and dynamic analysis of thin functionally graded shell with in-plane material inhomogeneity

The static and dynamic problems of thin functionally graded shells (FGSs) with in-plane material inhomogeneity are investigated in this work. Governing equations are derived based on the first-order shear deformation shell theory. A meshfree radial basis collocation method (RBCM) which employs infinitely continuous radial basis functions (RBFs) as the approximation and utilizes collocation method for the solution is introduced for the static and dynamic eigenvalue analysis. Comparison studies with the analytical solutions of the homogeneous shell problems demonstrate that this method can achieve high accuracy and spectral convergence. FGSs with three representative gradient distributions are studied. Numerical simulations reveal that material inhomogeneity can shift the dangerous positions of maximum displacements or stresses in statics, and has prominent influences on the natural frequencies and mode shapes in dynamics. These observations can be utilized for the design of material properties of shell structures in the engineering applications.

Solving fractional Laplacian visco-acoustic wave equations on complex-geometry domains using Grünwald-formula based radial basis collocation method

We propose a radial basis function collocation method (RBF method) to solve fractional Laplacian visco-acoustic wave equation for the Earth media having heterogeneous velocity model and complex geometry. Unlike the fractional Laplacian wave equation proposed in Zhu and Harris (2014), the wave equation we consider has a different definition for the fractional Laplacian. Specifically, spectral and Riesz fractional Laplacians are considered in Zhu and Harris (2014) and the present paper, respectively. Accordingly, the Fourier pseudospectral method (FPS method) and the RBF method are employed to solve the spectral and the Riesz fractional Laplacian wave equations. The two wave equations are observed to produce obviously different wavefields. We demonstrate the validity and flexibility of the proposed RBF method by considering five benchmarks of seismic forward modeling: (1) two-dimensional Earth media with four types of velocity models (homogeneous, two-layer, homogeneous but complex-geometry, and heterogeneous models) and (2) a spherical medium with homogeneous velocity model. We make a three-way comparison among numerical solutions to the Riesz fractional Laplacian, the spectral fractional Laplacian, and the integer-order visco-acoustic wave equations, and observe that when wave attenuation is weak the Riesz wave equation yields more similar wavefield to that of the integer-order wave equation than the spectral wave equation does. Furthermore, uniform and quasi-uniform layouts for collocation points of the RBF method are considered, and the latter layout turns out to be economical since it can preserve the solution accuracy with the minimum number of collocation points. The RBF method is truly mesh-free and dimension-free and can easily handle high-dimensional, irregular domains. Additionally, the method is easier to implement than element-based methods, such as finite element and spectral element methods, for discretizing the Riesz fractional Laplacian.

A weighted meshfree collocation method for incompressible flows using radial basis functions

A weighted strong form collocation method using radial basis functions and explicit time integration is proposed to solve the incompressible Navier-Stokes equations. The velocities and pressure are solved directly at the same time step and the continuity equation is satisfied at each time step which improve the solution accuracy and stability. No artificial compressibility coefficient needs to be introduced for modeling the incompressible flows and no pressure oscillation arises in the numerical solutions. Optimal convergence can be achieved by imposing the derived proper weights on the boundaries and the continuity equation. Radial basis collocation method in a Lagrangian form is quite easy to capture the moving boundary or free surface in flow problems. Moreover, solid boundary conditions can be enforced directly and no special treatments are required. Further, critical time step for the explicit time integration is predicted in the stability analysis and the influences on the stability are evaluated. Numerical studies validate the high accuracy as well as good stability of the presented method.

Hermite Radial Basis Collocation Method for Unsaturated Soil Water Movement Equation

Due to the nonlinear diffusion term, it is hard to use the collocation method to solve the unsaturated soil water movement equation directly. In this paper, a nonmesh Hermite collocation method with radial basis functions was proposed to solve the nonlinear unsaturated soil water movement equation with the Neumann boundary condition. By preprocessing the nonlinear diffusion term and using the Hermite radial basis function to deal with the Neumann boundary, the phenomenon that the collocation method cannot be used directly is avoided. The numerical results of unsaturated soil moisture movement with Neumann boundary conditions on the regular and nonregular regions show that the new method improved the accuracy significantly, which can be used to solve the low precision problem for the traditional collocation method when simulating the Neumann boundary condition problem. Moreover, the effectiveness and reliability of the algorithm are proved by the one-dimensional and two-dimensional engineering problem of soil water infiltration in arid area. It can be applied to engineering problems.

Strong-Form Formulated Generalized Displacement Control Method for Large Deformation Analysis

The traditional analysis of geometric nonlinearity is mostly based on the weak-formulated Galerkin method such as the finite element method. The element nature has limited its application as a result of numerical integration in the governing equation and quality control of deformed mesh. In the middle of 1990s, the meshfree methods have been developed and become one leading research topic in computational mechanics. Especially, the strong form collocation methods require no additional efforts to process numerical integration and impose Dirichlet boundary condition, thereby making the collocation methods computationally efficient. In the incremental–iterative process, how to accurately reflect the change in the slope of the load–deflection curve of the structure and remain numerically stable are of major concerns. Thus, we propose a strong-form formulated generalized displacement control method to analyze geometric nonlinear problems, where the radial basis collocation method is adopted. The numerical examples demonstrate the ability of the proposed method for large deformation analysis.

Radial basis functions methods for boundary value problems: Performance comparison

We present in this paper comparisons on the performances among five typical radial basis functions methods, namely radial basis collocation method (RBCM), radial basis Galerkin method (RBGM), compactly supported radial basis collocation method (CSRBCM), compactly supported radial basis Galerkin method (CSRBGM), and finite subdomain radial basis collocation method (FSRBCM), for solving problems arising from engineering industries and applied sciences. Numerical comparison results demonstrate that the RBCM and FSRBCM possess high accuracy and superior convergence rates in which the FSRBCM particularly attains higher accuracy for problems with large gradients. The FSRBCM, CSRBCM and RBCM are computationally efficient while the CSRBCM, CSRBGM and FSRBCM can greatly improve the ill-conditioning of the resultant matrix. In conclusion, its advantages on high accuracy; exponential convergence; well-conditioning; and effective computation make the FSRBCM a first-choice among the five radial basis functions methods.

Strong-form framework for solving boundary value problems with geometric nonlinearity

In this paper, we present a strong-form framework for solving the boundary value problems with geometric nonlinearity, in which an incremental theory is developed for the problem based on the Newton-Raphson scheme. Conventionally, the finite element methods (FEMs) or weak-form based meshfree methods have often been adopted to solve geometric nonlinear problems. However, issues, such as the mesh dependency, the numerical integration, and the boundary imposition, make these approaches computationally inefficient. Recently, strong-form collocation methods have been called on to solve the boundary value problems. The feasibility of the collocation method with the nodal discretization such as the radial basis collocation method (RBCM) motivates the present study. Due to the limited application to the nonlinear analysis in a strong form, we formulate the equation of equilibrium, along with the boundary conditions, in an incremental-iterative sense using the RBCM. The efficacy of the proposed framework is numerically demonstrated with the solution of two benchmark problems involving the geometric nonlinearity. Compared with the conventional weak-form formulation, the proposed framework is advantageous as no quadrature rule is needed in constructing the governing equation, and no mesh limitation exists with the deformed geometry in the incremental-iterative process.

Buckling analysis of functionally graded thin plate with in-plane material inhomogeneity

Buckling analysis of functionally graded material (FGM) thin plates with in-plane material inhomogeneity is investigated based on radial basis functions associated with collocation method. No background mesh is required in the discretization and solution which makes it a truly meshfree method. Two independent problems raised in the buckling analysis are studied according to the procedure. First, radial basis collocation method (RBCM) is employed to yield the non-uniform pre-buckling stresses by solving a 2D plane stress problem. Afterwards, based on Kirchhoff assumption and employing the predetermined non-uniform pre-buckling stresses, Hermite radial basis function collocation method (HRBCM) is proposed to study the buckling loads of FGM thin plates with in-plane material inhomogeneity. Compared to an over-determined system resulting from the conventional RBCM, HRBCM introducing more degrees of freedom on the boundary nodes can lead to a determined system for the eigenvalue problem. Convergence and comparisons studies with analytical solutions demonstrate that the proposed method possesses high accuracy and exponential convergence. Numerical examples illustrate that the material inhomogeneity has considerable effects on the buckling loads and mode shapes of thin plates. As a result, material inhomogeneity can be exploited to optimize the in-plane stress distribution and prevent the buckling of thin plates.

Investigation of Radial Basis Collocation Method for Incremental-Iterative Analysis

We propose an incremental-iterative algorithm by using the strong form collocation method for solving geometric nonlinear problems. As nonlinear analyses concerning large deformation have been relied on the weak form-based methods such as the finite element methods and the reproducing kernel particle methods, the recently developed strong form collocation methods could be new research directions in that the mesh control and quadrature rule are abandoned in the collocation methods. In this work, the radial basis collocation method is adopted to perform the nonlinear analysis. The corresponding parameters affecting the deformation paths such as the increment of applied traction and shape parameter of the radial basis function are discussed. We also investigate the possibility of using the weighted collocation methods in the nonlinear analysis.

Estimation of river pollution source using the space-time radial basis collocation method

River contaminant source identification problems can be formulated as an inverse model to estimate the missing source release history from the observed contaminant plume. In this study, the identification of pollution sources in rivers, where strong advection is dominant, is solved by the global space-time radial basis collocation method (RBCM). To search for the optimal shape parameter and scaling factor which strongly determine the accuracy of the RBCM method, a new cost function based on the residual errors of not only the observed data but also the specified governing equation, the initial and boundary conditions, was constructed for the k-fold cross-validation technique. The performance of three global radial basis functions, Hardy's multiquadric, inverse multiquadric and Gaussian, were also compared in the test cases. The numerical results illustrate that the new cost function is a good indicator to search for near-optimal solutions. Application to a real polluted river shows that the source release history is reasonably recovered, demonstrating that the RBCM with the k-fold cross-validation is a powerful tool for source identification problems in advection-dominated rivers.

A New Inverse Solution Assessment for the Recovery Test Using Radial Basis Function Collocation Method

Recovery pumping tests are still one of the most commonly preferred preliminary design steps in the assessment of aquifer hydraulics. The use of the existing methods would be insufficient under non-ideal aquifer conditions (i.e. heterogeneity, measurement errors, and boundary effect, etc.) which violate the Theis model assumptions developed for recovery test analysis. In this study, a new parameter estimation methodology based on the Radial Basis Collocation Method (RBFCM) was formulated to access the hydraulic parameters in a reliable, robust and accurate manner using recovery pump test. The suggested approach was established on the idea that the dimensionless time value at the pump-cessation which serves as a matching value is obtained by means of RFBCM. The proposed approach is straightforward to implement; which requires no data refinement, additional parameter, and visual match. The performance of the proposed method was tested with several aquifer conditions including homogeneous synthetic data, heterogeneous aquifer simulation and real field applications. The results confirm that the proposed method has a parameter estimation capacity as high as the available techniques in the literature and provides the practitioners to understand a more accurate portrayal of the effects of heterogeneity on the hydraulic parameters during the test process. In addition, the suggested methodology for the interpretation of aquifer recovery test can be evaluated to be employed as a diagnostic tool to identify non-ideal conditions. The potential use of RBFCM in this study was also presented as the supplement of aquifer test interpretation assessment.

Meshless analyses for time-fractional heat diffusion in functionally graded materials

The meshless local radial basis function method is applied to solve stationary and transient heat conduction problems in 2-D and 3-D bodies with functionally graded material properties. Time fractional derivative using Caputo definition is considered to describe anomalous diffusion phenomena. For temporal discretization, the Caputo time fractional derivative is approximated within each time interval 〈tk,tk+1〉 by series of derivatives of integer order. The spatial discretization is performed by using the local radial basis collocation method. Numerical analyses are given on square (2D) and cubic (3D) domains to show the influence of the temporal fractional derivative parameter and gradation material parameter on the temperature distribution and temperature evolution in transient heat conduction problem.

Radial Basis Collocation Method for the Dynamics of Rotating Flexible Tube Conveying Fluid

A Meshfree Radial Basis Collocation Method (RBCM) associated with explicit and implicit time integration scheme is formulated to study the coupling dynamics of a rotating flexible tube conveying fluid, which involves a partial differential equation (PDE) with variable coefficients. Dispersion studies are performed and they indicate that the proposed RBCM has a very small dispersion error compared with conventional FEM and Galerkin-based meshfree methods. Numerical examples are conducted for the influence of initial flow rate of the fluid, discretization and shape parameter on the dispersion error. The critical time step is obtained from a Von Neumann stability analysis. For the eigenproblem, Hermite-type RBCM is proposed in order to construct square matrices and eigenvalue analysis gives the frequencies of the system. Subsequently, the influence of angular velocity, flow rate of the fluid and the time variation on the fundamental frequencies is studied. Though proposed for studying the dynamics of a rotating flexible tube conveying fluid, this solution scheme is applicable to other dynamical problems which have similar PDEs with variable coefficients.

Hermite radial basis collocation method for vibration of functionally graded plates with in-plane material inhomogeneity

A meshfree Hermite-type collocation method with radial basis functions is presented to investigate the free vibration of thin FGM plates with in-plane material inhomogeneity. For the eigenvalue analysis of thin plate vibration, conventional radial basis collocation method (RBCM) leads to an over-determined system for the eigenproblems, while Hermite radial basis collocation method (HRBCM) introducing more degrees of freedom can result in a determined system. Convergence and comparison studies with analytical solutions demonstrate the effectiveness and accuracy of the proposed method. Effects on the natural frequencies and mode shapes originated from the material inhomogeneity are numerically investigated.