Uniform Nodes Research Articles

The localized RBF interpolation with its modifications for solving the incompressible two-phase fluid flows: A conservative Allen–Cahn–Navier–Stokes system

In this research work, we apply a numerical scheme based on the first-order time integration approach combined with the modifications of the meshless approximation for solving the conservative Allen–Cahn–Navier–Stokes equations. More precisely, we first utilize a first-order time discretization for the Navier–Stokes equations and the time-splitting technique of order one for the dynamics of the phase-field variable. Besides, we use the local interpolation based on the Matérn radial function for spatial discretization. We should solve a Poisson equation with the proper boundary conditions to have the divergence-free property during the numerical algorithm. Accordingly, the applied numerical procedure could not give a stable and accurate solution. Instead, we solve a regularization system in a discrete form. To prevent the instability of the numerical solution concerning the convection term, a biharmonic term with a small coefficient based on the high-order hyperviscosity formulation has been added, which has been approximated by a scalable interpolation based on the combination of polyharmonic spline with polynomials (known as the PHS+poly). The obtained full-discrete problem is solved using the biconjugate gradient stabilized method considering a proper preconditioner. We investigate the potency of the numerical scheme by presenting some simulations via uniform, hexagonal, and quasi-uniform nodes on rectangular and irregular domains. Besides, we have compared the proposed meshless method with the standard finite element method due to the used CPU time.

On the accuracy of interpolation based on single-layer artificial neural networks with a focus on defeating the Runge phenomenon

Artificial Neural Networks (ANNs) are a tool in approximation theory widely used to solve interpolation problems. In fact, ANNs can be assimilated to functions since they take an input and return an output. The structure of the specifically adopted network determines the underlying approximation space, while the form of the function is selected by fixing the parameters of the network. In the present paper, we consider one-hidden layer ANNs with a feedforward architecture, also referred to as shallow or two-layer networks, so that the structure is determined by the number and types of neurons. The determination of the parameters that define the function, called training, is done via the resolution of the approximation problem, so by imposing the interpolation through a set of specific nodes. We present the case where the parameters are trained using a procedure that is referred to as Extreme Learning Machine (ELM) that leads to a linear interpolation problem. In such hypotheses, the existence of an ANN interpolating function is guaranteed. Given that the ANN is interpolating, the error incurred occurs outside the sampling interpolation nodes provided by the user. In this study, various choices of nodes are analyzed: equispaced, Chebychev, and randomly selected ones. Then, the focus is on regular target functions, for which it is known that interpolation can lead to spurious oscillations, a phenomenon that in the ANN literature is referred to as overfitting. We obtain good accuracy of the ANN interpolating function in all tested cases using these different types of interpolating nodes and different types of neurons. The following study is conducted starting from the well-known bell-shaped Runge example, which makes it clear that the construction of a global interpolating polynomial is accurate only if trained on suitably chosen nodes, ad example the Chebychev ones. In order to evaluate the behavior when the number of interpolation nodes increases, we increase the number of neurons in our network and compare it with the interpolating polynomial. We test using Runge’s function and other well-known examples with different regularities. As expected, the accuracy of the approximation with a global polynomial increases only if the Chebychev nodes are considered. Instead, the error for the ANN interpolating function always decays, and in most cases we observe that the convergence follows what is observed in the polynomial case on Chebychev nodes, despite the set of nodes used for training. Then we can conclude that the use of such an ANN defeats the Runge phenomenon. Our results show the power of ANNs to achieve excellent approximations when interpolating regular functions also starting from uniform and random nodes, particularly for Runge’s function.

Error analysis of the element-free Galerkin method for a nonlinear plate problem

In this work, we derive a geometrically nonlinear plate model based on the Kirchhoff hypothesis and the large deflection hypothesis, and provide an error analysis of the corresponding element-free Galerkin method. The penalty method is employed to enforce boundary conditions. The error estimate explicitly depends on nodal spacing, number of monomial basis functions, continuity of weight functions, and penalty factors, which provides some practical choices among these key parameters in engineering applications. In addition, we offer guidance on selecting appropriate penalty factors to improve numerical accuracy. Numerical experiments, involving clamped square and circle plates with uniform and nonuniform nodes, as well as a plate model with a corner singularity, are presented to validate the theoretical results.

Free vibrations of variablesection beams taking rotational and frictional forces into account

Introduction. Beams are widely used in the construction industry as load-bearing structures of bridges, overpasses, coverings, slabs, stairs, equipment platforms, etc. In order to fully utilize the bearing capacity of such structures and to reduce the rate of material consumption, beams of variable cross-section along the length can be used. During operation, such structural elements are subjected to various types of vibrations, which determines the relevance of studying various aspects of vibrational motion.Aim. To apply numerical methods for studying the influence of rotational inertial forces in the presence of viscous frictional forces on free vibrations of variable-section beams. This area of research presents interest, since the calculations are directly related to the determination of frequencies and forms of natural vibrations of structures.Materials and methods. Free vibrations were described by a homogeneous partial differential equation of hyperbolic type. The methods of variable separation and finite differences were used. A discrete domain in the form of a set of uniform grid nodes and a homogeneous system of algebraic equations were introduced. The system of equations in the matrix-vector form was used.Results. The spectra of natural frequencies, damping coefficients, and eigenforms of beam vibrations are determined. It is shown that the coefficient matrix has a banded and pentadiagonal form. The matrix elements are functions of the characteristic index. The damping coefficient and the frequency of free vibrations are determined from a system of two nonlinear equations. The solution of the system of equations is found using the method of coordinate descent. An example of calculation of a welded I-beam is considered. Five elements of the spectra of damping coefficients and natural frequencies are calculated.Conclusions. State-of-the-art MATLAB solvers allows numerical and graphical methods to be combined. In the solved examples, the advantages of these methods were successfully applied to determine the eigenvalues of matrices and eigenfunctions. The high validity and accuracy of the results obtained confirm the simplicity and versatility of the methodology for determining the characteristics of free vibrations of beams of variable cross-section.

Solutions of a three-dimensional multi-term fractional anomalous solute transport model for contamination in groundwater

The study presents a meshless computational approach for simulating the three-dimensional multi-term time-fractional mobile-immobile diffusion equation in the Caputo sense. The methodology combines a stable Crank-Nicolson time-integration scheme with the definition of the Caputo derivative to discretize the problem in the temporal direction. The spatial function derivative is approximated using the inverse multiquadric radial basis function. The solution is approximated on a set of scattered or uniform nodes, resulting in a sparse and well-conditioned coefficient matrix. The study highlights the advantages of meshless method, particularly their simplicity of implementation in higher dimensions. To validate the accuracy and efficacy of the proposed method, we performed numerical simulations and compared them with analytical solutions for various test problems. These simulations were carried out on computational domains of both rectangular and non-rectangular shapes. The research highlights the potential of meshless techniques in solving complex diffusion problems and its successful applications in groundwater contamination and other relevant fields.

A shape parameter insensitive CRBFs‐collocation for solving nonlinear optimal control problems

AbstractIn this work, we introduce a Coupled Radial Basis Functions collocation (CRBFs‐Coll) approach for solving optimal control problems. CRBFs are real‐valued Radial Basis Functions (RBFs) augmented with a conical spline. They show insensitivity to the shape parameter resulting in robust behavior in function approximation. The method is applied to classical nonlinear optimal control problems: Zermelo's problem, a Duffing oscillator with various boundary conditions, and a nonlinear pendulum on a cart problem. The nonlinear optimal control problem is solved by deriving the necessary conditions for optimality followed by collocation using CRBFs as basis functions for approximating the two‐point boundary value problem (TPBVP). The resulting system of nonlinear algebraic equations (NAEs) is then solved using a standard solver. Unlike existing methods that rely on nodal distributions that are denser at the boundaries to obtain a solution, the present CRBFs‐Coll approach is capable of solving the optimal control problem on uniform nodes without the need for interpolation. The present CRBFs‐Coll approach is shown to be simple to implement and provides accurate results over an equidistant nodal distribution while maintaining a continuous representation of the control and states.

A Hybrid Local Radial Basis Function Method for the Numerical Modeling of Mixed Diffusion and Wave-Diffusion Equations of Fractional Order Using Caputo’s Derivatives

This article presents an efficient method for the numerical modeling of time fractional mixed diffusion and wave-diffusion equations with two Caputo derivatives of order 0<α<1, and 1<β<2. The numerical method is based on the Laplace transform technique combined with local radial basis functions. The method consists of three main steps: (i) first, the Laplace transform is used to transform the given time fractional model into an equivalent time-independent inhomogeneous problem in the frequency domain; (ii) in the second step, the local radial basis functions method is utilized to obtain an approximate solution for the reduced problem; (iii) finally, the Stehfest method is employed to convert the obtained solution from the frequency domain back to the time domain. The use of the Laplace transform eliminates the need for classical time-stepping techniques, which often require very small time steps to achieve accuracy. Additionally, the application of local radial basis functions helps overcome issues related to ill-conditioning and sensitivity to shape parameters typically encountered in global radial basis function methods. To validate the efficiency and accuracy of the proposed method, several test problems in regular and irregular domains with uniform and non-uniform nodes are considered.

Simulation of Elastic Wave Propagation Based on Meshless Generalized Finite Difference Method with Uniform Random Nodes and Damping Boundary Condition

When the grid-based finite difference (FD) method is used for elastic wavefield forward modeling, it is inevitable that the grid divisions will be inconsistent with the actual velocity interface, resulting in problems related to the stepped grid diffraction and inaccurate travel time of reflected waves. The generalized finite difference method (GFDM), which is based on the Taylor series expansion and weighted least square fitting, solves these problems. The partial derivative of the unknown parameters in the differential equation is represented by the linear combination of the function values of adjacent nodes. In this study, the Poisson disk node generation algorithm and the centroid Voronoi node adjustment algorithm were combined to obtain an even and random node distribution. The generated nodes fit the internal boundary more accurately for model discretization, without the presence of diffracted waves caused by the stepped grid. To avoid the instability caused by the introduction of boundary conditions, a Cerjan damping boundary condition was proposed for boundary reflection processing. The test results generated by the different models showed that the generalized finite difference method can effectively solve the problems related to inaccurate travel time of reflection waves and stepped grid diffraction.

Эффективные массово-параллельные алгоритмы восстановления данных в многомерном случае

A data recovery problem in multidimensional case by given values in the uniform nodes is considered. An efficient computational data recovery algorithm for a bulk parallelism model is proposed. The algorithm is based on transformations applied to summatory on V. A. Steklov’s kernels operator. The dependency between computational complexity and the problem parameters is shown. A software implementation of the algorithm is carried out in Open CL standard widely used for general-purpose computing on graphic processor units. Examples of surfaces reconstruction by the software are given.

Direct generation of finite element mesh using 3D laser point cloud

An approach for the finite element (FE) mesh generation of tunnel structures considering the lack of detail, low automation, and intermediate link transformation in the FE mesh generation of tunnels based on the 3D laser point cloud is presented in this paper. To this end, we acquired the tunnel point cloud model using terrestrial laser scanning or mobile laser scanning, and uniform nodes from the raw point cloud using a clock simulation extraction algorithm we developed. In addition, we assign topology information to these nodes. Further, we generated an FE model of the tunnel structure suitable for numerical simulation analysis using Abaqus. We applied the developed method to three types of tunnels; the mesh satisfied the evaluation standards of practical engineering applications. We examined the impact of mesh size and used an engineering example to confirm the method's accuracy. Finally, we provide recommendations for future research considering the future trend of FE mesh generation based on the 3D laser point cloud and its improvements.

A three-dimensional B-spline wavelet finite element method for structural vibration analysis

In this study, a three-dimensional (3-D) B-spline wavelet finite element method (WFEM) is proposed by combining the wavelet theory and 3-D finite element method (FEM). Owing to the semi-orthogonality and multi-resolution analysis of spline wavelet over the solution domain, a cuboid with uniform distribution nodes is established for the 3-D B-spline wavelet on the interval (BSWI) element. The governing equations of vibration for the elastic structures are derived by introducing the 3-D B-spline wavelet theory into the displacement components of each node. A procedure for assembling the BSWI elements is presented by redefining the index element node functions. Numerical studies including the vibration analysis of solid and hollow beams and plates under different boundary conditions are performed to show the performance of the present method. A comprehensive comparison of the predicted vibration characteristics obtained by the present method, traditional FEM, and reference data is presented to verify the accuracy and efficiency of the 3-D B-spline WFEM. The core of the present method is involving the 3-D B-spline wavelet to replace the polynomial interpolation functions in the traditional FEM, which enhances computational accuracy during the vibration analysis of the elastic structures.

Effect of Number of Nodes and Distance between Communicating Nodes on WSN Characteristics

Wireless Sensor Networks (WSNs) are increasingly used in many applications. However, issues like topology, routing, energy, nodes distribution and density, and connection reliability are affecting their deployment. In this work, investigation into such parameters using shortest path first routing technique is carried out using MATLAB simulation in free space with ransom uniform nodes distribution. The principle of simulation is to select the best route based on node’s location and available energy to carry out data transmission. The investigation proved that as the number of WSN nodes grows, the number of discovered routes grows in a power relationship with the overall time of route discovery, which is accompanied by a decrease in the number of hops per route. The research also showed that as the distance between the source and destination nodes increases, the number of routes identified increases in an exponential relationship, as does the overall route discovery time and the number of hops per route. Also, the simulation results shows that there is a relationship between path loss and both distance, and number of WSN nodes, with exponential relationship between path loss and distance and power relationship between path loss and number of nodes. The work also established a logarithmic relationship between distance and number of WSN nodes. By relating path loss to both distance and nodes, energy consideration is being covered in this work as an indirect parameter that is affected by routes, distance, communication medium and number of nodes. The mathematical modeling indicates that there is a need to dynamically correlate distance between nodes, communication medium, node’s energy, and topology to enhance routing of exchanged messages and to enable lower energy consumption.

Application of a local meshless modified characteristic method to incompressible fluid flows with heat transport problem

Radial basis function (RBF) has been accurately used for the spatial discretization to solve PDE problems. In this paper, a practical stabilization of the local formulation of the radial basis function (RBF) method is presented to solve the incompressible fluid flows with heat transport problems. To avoid the nonlinearity of the convective term, we include the modified method of characteristics to construct stable and efficient methods. The governing equations are the incompressible Navier–Stokes equations/Boussinesq approximation coupled with the heat transport equation. The spatial discretization carried out using the local radial basis function (LRBF) method on a uniform and non-uniform nodes distribution in a complex domain. The proposed method can be described as a fractional step splitting where the convective and generalized Stokes parts are treated separately. To solve the generalized Stokes problem, we used a projection/fractional step method that requires velocity–pressure decoupling. The proposed approach’s performance is tested on three benchmark problems and natural convection flow in a regular and irregular domains. We compare the results with different numerical solutions published in the literature. The obtained numerical results demonstrate the accuracy and stability of the proposed meshless method.

The Fragile Points Method (FPM) to solve two-dimensional hyperbolic telegraph equation using point stiffness matrices

In this paper, the Fragile Points Method (FPM) has been extended to solve the two-dimensional hyperbolic telegraph equation with specified initial and boundary conditions. Based on a naturally partitioned domain with scattered nodes and the Voronoi Diagram, the discretized FPM equations are derived by a Galerkin weak-form in the spatial domain and a finite difference scheme in the time domain. For the spatial discretization, discontinuous point-based polynomial trial and test functions are utilized, with numerical fluxes to ensure the consistency. For the time-domain discretization, theorems of unconditional stability and convergence for the telegraph equations are given. Numerical examples confirm the accuracy and robustness of the developed numerical method with uniform or random nodes and with different time increments. And as compared to several other meshless methods for the telegraph equation, it is shown that the developed FPM method has a better computational efficiency. This is because that a single-point quadrature rule is sufficient for evaluating the weak-form integral which gives a symmetric and sparse stiffness matrix, with the specifically-designed piecewise-linear trial and test functions.

A meshless technique based on the radial basis functions for solving systems of partial differential equations

The radial basis functions (RBFs) methods were first developed by Kansa to approximate partial differential equations (PDEs). The RBFs method is being truly meshfree becomes quite appealing, owing to the presence of distance function, straight-forward implementation, and ease of programming in higher dimensions. Another considerable advantage is the presence of a tunable free shape parameter, contained in most of the RBFs that control the accuracy of the RBFs method. Here, the solution of the two dimensional system of nonlinear partial differential equations is examined numerically by a Global Radial Basis Functions Collocation Method (GRBFCM). It can work on a set of random or uniform nodes with no need for element connectivity of input data. For the time-dependent partial differential equations, a system of ordinary differential equations (ODEs) is derived from this scheme. Like some other numerical methods, a comparison between numerical results with analytical solutions is implemented confirming the efficiency, accuracy, and simple performance of the suggested method.

Meshless Technique for the Solution of Time-Fractional Partial Differential Equations Having Real-World Applications

In this article, radial basis function collocation scheme is adopted for the numerical solution of fractional partial differential equations. This method is highly demanding because of its meshless nature and ease of implementation in high dimensions and complex geometries. Time derivative is approximated by Caputo derivative for the values of α ∈ 0 , 1 and α ∈ 1 , 2 . Forward difference scheme is applied to approximate the 1st order derivative appearing in the definition of Caputo derivative for α ∈ 0 , 1 , whereas central difference scheme is used for the 2nd order derivative in the definition of Caputo derivative for α ∈ 1 , 2 . Numerical problems are given to judge the behaviour of the proposed method for both the cases of α . Error norms are used to asses the accuracy of the method. Both uniform and nonuniform nodes are considered. Numerical simulation is carried out for irregular domain as well. Results are also compared with the existing methods in the literature.

Development of Kinematic Ephemeris Generator for Korea Pathfinder Lunar Orbiter (KPLO)

This paper presents a kinematic ephemeris generator for Korea Pathfinder Lunar Orbiter (KPLO) and its performance test results. The kinematic ephemeris generator consists of a ground ephemeris compressor and an onboard ephemeris calculator. The ground ephemeris compressor has to compress desired orbit propagation data by using an interpolation method in a ground system. The onboard ephemeris calculator can generate spacecraft ephemeris and the Sun/Moon ephemeris in onboard computer of the KPLO. Among many interpolation methods, polynomial interpolation with uniform node, Chebyshev interpolation, Hermite interpolation are tested for their performances. As a result of the test, it is shown that all the methods have some cases that meet requirements but there are some performance differences. It is also confirmed that, the Chebyshev interpolation shows better performance than other methods for spacecraft ephemeris generation, and the polynomial interpolation with uniform nodes yields good performance for the Sun/Moon ephemeris generation. Based on these results, a Kinematic ephemeris generator is developed for the KPLO mission. Then, the developed ephemeris generator can find an approximating function using interpolation method considering the size and accuracy of the data to be transmitted.

An efficient meshless method based on the moving Kriging interpolation for two‐dimensional variable‐order time fractional mobile/immobile advection‐diffusion model

In this work, we introduce an efficient meshless technique for solving the two‐dimensional variable‐order time‐fractional mobile/immobile advection‐diffusion model with Dirichlet boundary conditions. The main advantage of this scheme is to obtain a global approximation for this problem which reduces such problems to a system of algebraic equations. To approximate the first and fractional variable‐order against the time, we use the finite difference relations. The proposed method is based on the moving Kriging (MK) interpolation shape functions. To discretize this model in space variables, we use the MK interpolation. Duo to the fact that the shape functions of MK have Kronecker's delta property, boundary conditions are imposed directly and easily. To illustrate the capability of the proposed technique on regular and irregular domains, several examples are presented in different kinds of domains and with uniform and nonuniform nodes. Also, we use this scheme to simulating anomalous contaminant diffusion in underground reservoirs.

Experimental and numerical fracture analysis of the plain and polyvinyl alcohol fiber-reinforced ultra-high-performance concrete structures

In this paper, a new procedure for manufacturing the plain and polyvinyl alcohol (PVA) fiber-reinforced ultra-high-performance concrete (UHPC) is introduced to improve its workability and to reduce its shrinkage behavior. To determine its fracture characteristics, a series of three-point bending tests are performed with the notched beam structures made of the produced UHPCs. In addition, with the rising demand for time and cost saving design methodologies associated with R&D experimentation for many industrial materials, such as UHPC structures, numerical modelling and simulation have become an essential tool. Therefore, a relevant discrete-level numerical modelling approach based on bond-based peridynamics is proposed to predict the fracture behaviors of the UHPC and UHPC-PVA structures. In the proposed method, the structures are discretized by uniform meshless nodes linked by standard peridynamic bonds, and then the bonds are randomly associated with mechanical and fracture parameters of the UHPC and PVA materials according to their corresponding volume fractions. With this approach, the reinforcement of the PVA fibers on the UHPC materials is expressed by the bonds with the parameters of PVA materials, which greatly simplifies the modelling process of randomly distributed PVA fibers reinforced UHPC structures. The comparison between numerical and experimental results validates the effectiveness and accuracy of the present method.

Local meshless differential quadrature collocation method for time-fractional PDEs

This paper is concerned with the numerical solution of time- fractional partial differential equations (PDEs) via local meshless differential quadrature collocation method (LMM) using radial basis functions (RBFs). For the sake of comparison, global version of the meshless method is also considered. The meshless methods do not need mesh and approximate solution on scattered and uniform nodes in the domain. The local and global meshless procedures are used for spatial discretization. Caputo derivative is used in the temporal direction for both the values of $ \alpha \in (0,1) $ and $ \alpha\in(1,2) $. To circumvent spurious oscillation casued by convection, an upwind technique is coupled with the LMM. Numerical analysis is given to asses accuracy of the proposed meshless method for one- and two-dimensional problems on rectangular and non-rectangular domains.