Local Radial Point Interpolation Method Research Articles

A new analytical approach for the local radial point interpolation discretisation in space and applications to high-order in time schemes for two-dimensional fractional PDEs

In weak-form formulations of the local radial point interpolation method (LRPIM) for the solution of partial differential equations, space discretisation matrices have most often been obtained entirely through numerical integration. This work introduces a novel approach which derives closed-form expressions for obtaining the entries of the discretisation matrix for the solution of two-dimensional time-fractional diffusion problems. This analytical approach also yields a closed-form formula for the approximation of the Laplacian. These techniques are then applied for developing LRPIM-based numerical algorithms. Since the exact solutions usually have unbounded first-order time derivatives at time zero, a graded mesh is employed for a high-order in time approximation of the Caputo derivative. The analytical shape functions are used to develop a weak-form algorithm and a strong-form algorithm is developed using the analytical approximation of the Laplacian. We demonstrate that computed solutions obtained using the weak-form and strong-form algorithms have the accuracy levels consistent with the theoretical accuracy in space. An appropriate choice of the mesh grading parameter yields a high-order convergence rate in time. The unconditional stability and convergence of a LRPIM strong form algorithm on a uniform temporal mesh is established under the assumption that the exact solution has sufficient regularity.

A local meshfree radial point interpolation method for Berger equation arising in modelling of thin plates

In this study, two-dimensional (2D) Berger equation arising in deflection of thin plates is solved numerically. A local meshfree collocation technique based on radial point interpolation is developed for the Berger equation. As radial basis function thin plate splines are used to avoid selecting optimal shape parameter. The method is used to solve both linear and non-linear Berger equation on regular and irregular geometries. Obtained results during numerical simulations, are compared with different methods exist in literatrue such as method of fundamental solutions, local Kansa’s method and Pascal polynomial based meshless method. The comparisons affirm the accuracy and reliability of the suggested method.

An efficient meshfree computational approach to the analyze of thermoelastic waves of functionally graded materials in a two-dimensional space

In this paper, a meshless local radial point interpolation method (RPIM) is proposed for the thermoelastic analysis of functionally graded materials (FGMs) under thermal shocks. The properties of such materials are temperature-dependent varying and change gradually through thickness based on the Mori–Tanaka scheme and the volume fraction power law distribution (Appendix A). The characterization and visualization of the FGMs under a prescribed temperature gradient or heat flux are then detailed. The proposed method does not need any background cell-based integration and is thus very suitable to deal with complex geometries and scattered data. Moreover, the shape functions of this method possess the properties of the Kronecker delta, wherein the essential boundary conditions can be analytically satisfied. The numerical results are fully discussed and validated by the corresponding outcomes of the moving least squares (MLS) method, demonstrating the high capability and efficiency of RPIM proposed here in simulating the dynamic thermoelastic analysis of FGMs.

Groundwater flow simulation in a confined aquifer using Local Radial Point Interpolation Meshless method (LRPIM)

Groundwater flow problems are generally solved using analytical or numerical methods. Though analytical solutions are exact and preferable, they are not available for complex field problems. Hence numerical methods such as Finite Element and Finite Difference methods are used to solve complex groundwater problems. These conventional mesh/ grid-based numerical methods need construction of a detailed mesh/ grid. On the other hand, the meshless approach creates a system of algebraic equations on a collection of distributed nodes in the problem area and the boundary. As a result, it is easy to incorporate any modifications to the model at a later time by simply adding nodes to the domain. In this study a weak form meshless method known as local radial point interpolation method (LRPIM) which uses radial basis functions for approximation or interpolation is developed to solve the groundwater flow problems in a confined aquifer. The results obtained from the LRPIM model has been compared with other numerical methods for benchmark and real field problems, and are found to be satisfactory. Implementation of the essential boundary conditions was relatively easier in LRPIM and gave good accuracy for the problems considered. LRPIM can potentially be used as an alternative to the other conventional methods, especially where the domain boundary is irregular or varying with time.

An improved meshless artificial viscosity technology combined with local radial point interpolation method for 2D shallow water equations

The two-dimensional shallow water equations (SWEs) are a hyperbolic system of first-order nonlinear partial differential equations which have a characteristic of strong gradient. In this study, a newly-developed numerical model, based on local radial point interpolation method (LRPIM), is adopted to simulate discontinuity in shallow water flows. In order to accurately capture the information of wave propagation, the LRPIM is combined with the split-coefficient matrix (SCM) method to transform the SWEs into a characteristic form and the selection of the direction of local support domain is introduced into the LRPIM. An improved meshless artificial viscosity (MAV) technique is developed to minimize the non-physical oscillations near the discontinuities. Then, the LRPIM and the second-order Runge–Kutta method are adopted for spatial and temporal discretization of the SWEs, respectively. The feasibility and validity of the proposed numerical model are verified by the classical dam-break problem and the mixed flow pattern problem. The comparison of the obtained results with the analytical solution and other numerical results showed that the MAV method combined with LRPIM can accurately capture the shocks and has high accuracy in dealing with discontinuous flow by adding appropriate viscosity to the equations in the discontinuous region.

A meshless artificial viscosity method for wet-dry moving interfaces problems of shallow water flow

In this paper, an improved meshless artificial viscosity (AV) method is proposed and combined with the local radial point interpolation method (LRPIM) and the MacCormack method to establish a newly-developed meshless numerical model, which can efficiently and accurately solve wet-dry moving interfaces problems of shallow water flow. As an important model describing shallow water flow, the two-dimensional (2D) shallow water equations (SWEs) are a hyperbolic system of 1st-order nonlinear partial differential equations which have a characteristic of strong gradient. The LRPIM and MacCormack method are adopted to discretize the 2D SWEs spatially and temporally, respectively. Meanwhile, the method of selecting the different shape of support domain in the LRPIM is employed for properly cooperating with the MacCormack method to accurately capture the direction of wave propagation. For eliminating the non-physical oscillation at the discontinuity of shallow water flow, the improved meshless AV form is first presented. Four challenging numerical examples were selected to verify the proposed model by comparing with the other solutions and experimental observations. The results indicate that the proposed method with improved meshless AV can accurately capture the shock waves and efficiently simulate the wet-dry front changes.

Modeling of Shear Crack Propagation in Rock Masses Using Mesh-Free LRPIM

The modeling of shear cracks in materials is critical in various engineering applications, such as the safety analysis of concrete structures and stability analysis of rock slopes. Based on the idea of Goodman element, the elastic-plastic constitutive model of the shear cracks is derived, and the elastic-plastic analysis of shear crack propagation is realized in the local radial basis point interpolation method (LRPIM). This method avoids the loss of accuracy caused by the mesh in the analysis of fracture propagation, and the crack propagation of rock brittle material is simulated. The investigation indicates that (1) the LRPIM results are close to the FDM results, which demonstrates that it is feasible to analyze shear cracks in rock masses. (2) Compared with the results of the built-in oblique crack model, when the LRPIM is used to analyze crack propagation, the results are close to the experimental results, showing that the LRPIM can model shear crack propagation in a rock mass. (3) The propagation path using the LRPIM is not sufficiently smooth, which can be explained as the crack tip stress and strain not being sufficiently accurate and still requiring further improvement.

Analysis of Aeroacoustic Properties of the Local Radial Point Interpolation Cumulant Lattice Boltzmann Method

The lattice Boltzmann method (LBM) has recently been used to simulate wave propagation, one of the challenging aspects of wind turbine modeling and simulation. However, standard LB methods suffer from the instability that occurs at low viscosities and from its characteristic lattice uniformity, which results in issues of accuracy and computational efficiency following mesh refinement. The local radial point interpolation cumulant lattice Boltzmann method (LRPIC-LBM) is proposed in this paper to overcome these shortcomings. The LB equation is divided into collision and streaming steps. The collision step is modeled by the cumulant method, one of the stable LB methods at low viscosities. In addition, the streaming step, which is naturally a pure advection equation, is discretized in time and space using the Lax–Wendroff scheme and the local radial point interpolation method (RPIM), a mesh free method. We describe the propagation of planar acoustic waves, including the temporal decay of a standing plane wave and the spatial decay of a planar acoustic pulse. The analysis of these specific benchmark problems has yielded qualitative and quantitative data on acoustic dispersion and dissipation, and their deviation from analytical results demonstrates the accuracy of the method. We found that the LRPIC-LBM replicates the analytical results for different viscosities, and the errors of the fundamental acoustic properties are negligible, even for quite low resolutions. Thus, this method may constitute a useful platform for effectively predicting complex engineering problems such as wind turbine simulations, without parameter dependencies such as the number of points per wavelength Nppw and resolution σ or the detrimental effect caused by the use of coarse grids found in other accurate and stable LB models.

An efficient and accurate hybrid weak-form meshless method for transient nonlinear heterogeneous heat conduction problems

Our purpose is to establish a numerical method meeting the requirements of efficiency, stability, accuracy and easy-using. Faced with these demands, in this paper, a hybrid method combining the advantages of 2 weak-form meshless methods is proposed for solving steady and transient heat conduction problems with temperature-dependent thermophysical properties in heterogeneous media. In the process of the calculation for an arbitrary model discretized by the hybrid method, two different weak-form methods are used for the nodes categorized as: (1) boundary nodes, (2) near boundary interior nodes and (3) interior nodes. A Galerkin free element method (GFREM) is proposed for the interior nodes of the Cartesian grid; and the local radial point interpolation method (LRPIM) is used for the other two types of nodes. The GFREM using the Cartesian grid can greatly improve the accuracy and efficiency of the hybrid scheme. Special decomposition technology to determine the irregular integral domain for LRPIM can increase the flexibility for complex models. In addition, the full implicit finite difference scheme and Newton–Raphson method are used for solving transient nonlinear problems. And non-crossing interface integral domain and support domain are employed for heterogeneous media. Compared with other methods, numerical results show better accuracy, stability and efficiency from the numerical experiments.

Thermo-elastoplastic analysis of functionally graded sandwich plates using a three-dimensional meshless model

In this paper, for the first time, three-dimensional (3D) thermo-elastoplastic bending analysis of functionally graded (FG) sandwich plates with FG face sheets and FG/homogeneous core subjected to combined thermal and mechanical loads is performed. It is assumed that the functionally graded material (FGM) is composed of two different constituents (metal-ceramic) whose volume fractions vary continuously in the thickness direction according to a power law. The von-Mises yield criterion and isotropic strain hardening rule are employed, while temperature-dependent thermal and mechanical properties of the materials are taken into account. A complete 3D meshless numerical model based on the local radial point interpolation method (LRPIM) is developed and used in all analyses. Several numerical examples for temperature, deflection and stress analysis of FG sandwich plates are presented considering different material gradients, layer thickness ratios, thickness-to-length ratios, and boundary conditions. The numerical results are compared with the existing results of 3D, quasi-3D, and 2D shear deformation theories and an excellent agreement is observed.

Comparative modelling of crack propagation in elastic-plastic materials using the meshfree local radial basis point interpolation method and eXtended finite-element method.

The modelling and understanding of crack propagation for elastic–plastic materials is critical in various engineering applications, such as safety analysis of concrete structures and stability analysis of rock slopes. In this paper, the local radial basis point interpolation method (LRPIM) combined with elastic–plastic theory and fracture mechanics is employed to analyse crack propagation in elastic–plastic materials. Crack propagation in elastic–plastic materials is compared using the LRPIM and eXtended finite-element method (XFEM). The comparative investigation indicates that: (i) the LRPIM results are close to the model test results, which indicates that it is feasible for analysing the crack growth of elastic–plastic materials; (ii) compared with the LRPIM, the XFEM results are closer to the experimental results, showing that the XFEM has higher accuracy and computational efficiency; and (iii) compared with the XFEM, when the LRPIM method is used to analyse crack propagation, the propagation path is not smooth enough, which can be explained as the crack tip stress and strain not being accurate enough and still needing further improvement.

Reliability assessment for system-level turbine disc structure using LRPIM-based surrogate model considering multi-failure modes correlation

This paper presents a probabilistic analysis framework for the reliability evaluation of turbine disc considering the correlation of multi-failure modes. A system-level zone division method is first applied to decompose the whole structure into different serial zones. Due to the same input random variables, there is correlation between failure modes. Thus, a mathematical copula function method is introduced to quantify the correlation between failure modes after reliability calculation of separate zones, during which process, dependent random variables are transformed to independent ones using Nataf transformation method. Meanwhile, to guarantee the accuracy and efficiency of calculation, adaptive surrogate model based on local radial point interpolation method (LRPIM) is established in each zone. Two main failure modes, i.e., low cycle fatigue and creep-fatigue are considered during the reliability analysis on a turbine disc. The results reveal that the reliability of the turbine disc changes with the correlation between failure modes. Also, sensitivity analysis shows that rotating speed and maximum temperature are two dominant factors affecting the turbined disc's reliability. Finally, the comparisons among three methods including the proposed method, the zone-based method without considering correlation and Monte Carlo (MC) method based on physics of failure (POF) of correlation are conducted. It is demonstrated that the proposed method in this study is more efficient and accurate for evaluating structural reliability with multi-failure modes coupling. Moreover, the proposed method provides an available prospect for reliability-based design optimization of multiple failure structure, contributing to enhance reliability in mechanical design.

New local radial point interpolation-FD methods for solving fractional diffusion and damped-wave problems

A new approach is proposed for developing numerical schemes for the solution of fractional diffusion and diffusion-wave equations. The novelty lies in the construction of analytical shape functions in a three-point local radial point interpolation method. The gain is that numerical computations of matrix entries are not required and as in the finite difference setting, the availability of matrix coefficients in closed-form facilitates stability and convergence analyses. For the fractional diffusion equation, a weak-form formulation is used to develop an implicit meshless finite difference method. The unconditional stability and convergence of the scheme is theoretically established and numerical examples are given to illustrate its second-order accuracy in space. The expected time convergence and the dependence of the convergence on the multiquadric shape parameter are also investigated. The technique is applied to develop numerical schemes for the fractional diffusion-wave equation. For this problem, the new scheme is shown to perform well in terms of both temporal and spatial accuracies when compared to some existing methods.

Analytical shape functions and derivatives approximation formulas in local radial point interpolation methods with applications to financial option pricing problems

The radial point interpolation method is increasingly being applied for the numerical solution of partial differential equations in different fields. Most implementations in the literature for obtaining the matrix coefficients make use of numerical approximations of the shape functions and their derivatives. To avoid the solution of linear systems required for computation of derivative approximations, this work derives analytical shape functions for three and five-node support domains in a local radial point interpolation method (LRPIM) with multiquadrics as basis functions. A weak form algorithm for the Black–Scholes equation using three-node analytical shape functions is developed and its unconditional stability and convergence are theoretically established. LRPIM finite-difference (FD) formulas are derived and applied to the solution of one and two-asset financial options. A five-node LRPIM-FD method in one-dimension is shown to yield fourth-order accuracy and applications to two-asset problems also yield accurate prices for options on a minimum of two risky assets and exchange options.

Numerical simulation of electromagnetic wave scattering from perfectly conducting cylinders using the local radial point interpolation technique

ABSTRACTThis paper is devoted to the investigation of the electromagnetic scattering problems from infinite perfectly conducting cylinders with arbitrary cross-sections. The problems can be mathematically modeled as surface integral equations. Both the electric field integral equation (EFIE), magnetic field integral equation (MFIE) and their combined form are studied. An efficient computational technique based on the local radial point interpolation method is performed to investigate the models. Some test problems with various cross-sections have been analyzed to validate the capability of the proposed technique in calculating the current distribution induced on the scatterer and measuring the Radar Cross Section.

A local weak form meshless method to simulate a variable order time-fractional mobile–immobile transport model

In this study, a simulation of an anomalous mobile–immobile transport process in complex systems is produced numerically. The process is mathematically described as a two-dimensional time-fractional mobile–immobile advection–diffusion equation in Coimbra variable order derivative sense. A local weak form meshless method combined with a time stepping approach is discussed and implemented to simulate the model. For this purpose, firstly, an implicit difference stepping method is used to semi-discretize the model in time direction. For full discretization, a set of regularly distributed nodes is created in the primary spatial domain and the local radial point interpolation method is used to construct the spatial shape functions on the distributed data-sites. Then an efficient meshless procedure based on combination of local Petrov–Galerkin method and collocation technique is formulated to treat in the interior and on the boundary of the primal spatial domain, respectively. Finally, some benchmark problems are presented on regular and irregular domains to verify the validity, efficiency and accuracy of the method.

A MESHLESS DEFORMATION SIMULATION METHOD FOR VIRTUAL SURGERY

An accurate and computation-effective model of soft tissues is very important for surgery simulation. When soft tissues are interacted by a surgical instrument, their external deformation and internal stress and strain distribution are closely related. These stress and strain data can not only be used to describe the soft tissue deformation but also can be used to calculate interaction force rendered to the force feedback device, making the user to have a realistic sense of touch during the surgical procedure. It is difficult, however, to solve partial differential equations (PDE) in conventional soft tissue models, and stress and strain data cannot be easily extracted. This paper presents a meshless local radial point interpolation method for soft tissue modelling and demonstrates its feasibility by numerical tests.

Meshfree analysis of structures modeled as extensible slender rods

The analysis of members that can be modeled as extensible elastic slender rods is investigated. A meshfree formulation using a Local Radial Point Interpolation Method (LRPIM) is developed that utilizes radial basis functions in curvilinear coordinates. This approach bypasses the need to utilize more conventional element meshes and significantly reduces the number of equations needed for the numerical solution. The slender rod formulation presented allows for tension variation, axial stretch, incremental loading and distributed load variation along the rod. It is well suited for nonlinear problems that involve large deflections and rigid rotations. The position and tangent vectors are expressed using Hermite-type approximations, and radial basis functions, while the interpolation of tension variation and distributed loads are described using polynomials. The solution procedure of weighted residuals Galerkin weak formulation combined with an incremental iterative numerical scheme is introduced to address the incremental loading and large deflection issues for static and quasi-static problems. The implementation of the analytical formulation and the numerical procedure are illustrated using three nonlinear problems. The first two examples provide insight into the validity, accuracy, and efficiency of the methodology. The third example presents the case of a moving boundary condition problem which models a cable entangled by fishing boat-trawling equipment.

Novel adaptive surrogate model based on LRPIM for probabilistic analysis of turbine disc

Probabilistic lifetime assessment for aero-engine turbine disc is required to ensure structural safety and reliability. For probabilistic analysis of aero-engine turbine disc, a large amount of random variables involving load, geometry, and material properties result in a high dimensional nonlinear state function for the fatigue lifetime, which can become prohibitively expensive. This paper presents a novel adaptive surrogate model for the probabilistic analysis of an aero-engine turbine disc by integrating the local radial point interpolation method (LRPIM) and directional sampling technique. The directional sampling technique includes initial sampling, limit state recognition and subsequent sampling. In order to implement the high-dimension-probabilistic analysis for the turbine disc, an adaptive scheme is proposed involving three parts, i.e. scale adjustment of local support domain, convergence test and repeated procedure of subsequent sampling. Applied to an aero-engine turbine disc probabilistic analysis problem with 11 dimensional random variables, it is demonstrated that the novel approach proposed improves the accuracy and computational efficiency with reduced sampling amount as compared to other models such as response surface method (RSM), Kriging model (KM) and artificial neural network model (ANNM). A leave-one-out (LOO) validation test is performed to verify the robustness of the prediction of the adaptive surrogate model in the probabilistic analysis process of aero-engine turbine discs.

Meshless local radial point interpolation (MLRPI) for generalized telegraph and heat diffusion equation with non-local boundary conditions

In this paper, the meshless local radial point interpolation (MLRPI) method is formulated to the generalized one-dimensional linear telegraph and heat diffusion equation with non-local boundary conditions. The MLRPI method is categorized under meshless methods in which any background integration cells are not required, so that all integrations are carried out locally over small quadrature domains of regular shapes, such as lines in one dimensions, circles or squares in two dimensions and spheres or cubes in three dimensions. A technique based on the radial point interpolation is adopted to construct shape functions, also called basis functions, using the radial basis functions. These shape functions have delta function property in the frame work of interpolation, therefore they convince us to impose boundary conditions directly. The time derivatives are approximated by the finite difference time- -stepping method. We also apply Simpson’s integration rule to treat the non-local boundary conditions. Convergency and stability of the MLRPI method are clarified by surveying some numerical experiments.