Meshless Collocation Research Articles

Meshless generalized finite difference method for two- and three-dimensional transient elastodynamic analysis

In this paper, a meshless collocation method is introduced for two-dimensional (2D) and three-dimensional (3D) transient elastodynamic problems by applying the generalized finite difference method (GFDM) in conjunction with the Houbolt scheme. Coupled equilibrium equations with a time-dependent loading are transformed into the static equations at time nodes by adopting the Houbolt method. After then, the solution of the static equations is achieved with the GFDM with second-order and fourth-order expansions. Several numerical examples involving complicated geometries and different initial and boundary conditions are simulated to validate the performance of the present approach.

Surface stress effect on nonlinear dynamical performance of nanobeam-type piezoelectric energy harvesters via meshless collocation technique

The chief point of the present study is to analyze numerically the surface stress effects on the nonlinear dynamic performance of nanosized lightweight piezoelectric bridge-type energy harvesters having agglomerated nanocomposite passive bulk. In this regard, the Gurtin-Murdoch continuum elasticity is formulated based upon the quasi-3D beam theory incorporating the relevant surface elastic constants as well as the surface residual stress. Afterwards, an effective numerical solving procedure employing the meshless collocation technique is developed to discretize the nonlinear governing equations via a combination of the polynomial as well as multiquadric basis functions to avoid any possible singularity. It is indicated that the surface stress effects result in to decrease the peaks of achieved voltage from the nonlinear dynamical response of nanobeam-type energy harvesters, but the associated frequency increases. Accordingly, for simply supported nanosized energy harvesters having the thickness of 50 nm, 20 nm, and 10 nm, the reductions in the average achieved voltage in order are about 7.79%, 25.57%, and 37.98%, but the required time decreases from 40.41 μs to 39.78 μs, from 35.17 μs to 33.57 μs, and from 28.37 μs to 26.46 μs, respectively. On the other hand, for clamped nanosized energy harvesters having the thickness of 50 nm, 20 nm, and 10 nm, the reductions in the average achieved voltage due to the surface stress effects are, respectively, about 8.33%, 27.35%, and 48.85%, but the required time decreases from 38.92 μs to 38.43 μs, from 33.42 μs to 32.04 μs, and from 26.58 μs to 24.62 μs, respectively.

Nonlinear dynamical performance of microsize piezoelectric bridge-type energy harvesters based upon strain gradient-based meshless collocation approach

In the current exploration, the size-dependent nonlinear dynamical performance of piezoelectric bridge-type energy harvesters at microscale is analyzed. The proposed microbeam-type energy harvesters are made of a lightweight agglomerated nanocomposite polymeric passive bulk reinforced with carbon nanotubes (CNTs) and integrated with piezoelectric facesheets. In order to perform a highly accurate analysis, the modified strain gradient continuum elasticity is adopted within the framework of the quasi-3D beam theory incorporating various microstructural-dependent strain gradient tensors. Thereafter, the size-dependent nonlinear coupled electromechanical differential equations are solved numerically via employing the meshless collocation technique using a combination of the radial basis and polynomial basis functions to avoid any possible singularity. It is demonstrated that by increasing the amount of CNTs inside clusters, the effect of strain gradient tensors on the reduction in the value of induced deflection in microsized energy harvester decreases from 25.57% to 22.65% for simply supported boundary conditions, and from 33.96% to 31.56% for clamped boundary conditions. Also, the average of achieved voltage by the microsized energy harvesters increases from 245.1 mv to 262.8 mv for simply supported boundary conditions, and from 79.5 mv to 96.7 mv for clamped boundary conditions.

A meshless collocation method based on Pascal polynomial approximation and implicit closest point method for solving reaction–diffusion systems on surfaces

A meshless collocation method based on Pascal polynomial approximation and implicit closest point method for solving reaction–diffusion systems on surfaces

Experimental and numerical analysis of hyperelastic plates using Mooney-Rivlin strain energy function and meshless collocation method

In this research, the flexural behavior of a hyperelastic plate made of silicone under uniformly distributed loading was investigated. The displacements and strains were formulated using the first-order shear deformation plate theory (FSDPT). The right Cauchy-Green tensor and Mooney-Rivlin strain energy function were employed to formulate hyperelastic silicone plate governing equations. The strong-form of the governing equations of the silicone plate using Mooney-Rivlin strain energy function and Euler-Lagrange relations were derived for the first time. In the governing equations, the effects of geometrical and material nonlinearity effects were considered simultaneously. The meshless collocation method (MCM) and thin-plate spline radial basis function were utilized to discretize the governing equations. The arc-length continuation algorithm was also used for analyzing the system of non-linear equations. MCM results were compared to those of the experimental test to validate the results of the meshless method. For this purpose, a square silicone plate with simply supported boundary conditions under uniformly distributed loading was investigated via the digital image correlation (DIC) technique. According to the obtained results, the MCM based on radial basis function is an adequate method for non-linear analysis of hyperelastic plates with Mooney-Rivlin strain energy function under uniformly distributed loading.

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.

A localized Fourier collocation method for solving high-order partial differential equations

This paper is concerned with the application of the localized Fourier collocation method (LFCM), a newly-developed meshless collocation method, for the numerical solution of high-order partial differential equations (PDEs). In the present method, the entire computational domain is divided into a set of overlapping subdomains in which the Fourier series expansion and moving-least square approximation are applied to construct the local systems of linear equations. By satisfying the governing equations and the corresponding boundary conditions, a sparse and banded stiffness matrix can be established which makes the method very attractive for large-scale engineering simulations. Preliminary numerical experiments for fourth-order PDEs in both two- and three-dimensions are presented to demonstrate the accuracy and efficiency of the present method.

Pharmacokinetic parameter identification using a meshless method approach for transdermal drug delivery

The Localized Collocation Meshless Method (LCMM) is applied to the solution of a set of partial differential equations governing the diffusive transport of a dermally injected compound as described by the Compartmental Pharmacokinetic (CPK) model. This efficient and accurate numerical technique then provides the framework for the estimation of pharmacokinetic (PK) parameters by the comparison of two models, both containing information derived from experimental results, using the iterative Golden Section search algorithm. This comparison is quantified using a norm between the two solution spaces. While this method does provide an estimate of the effective-diffusion coefficient, it ultimately demonstrates that a simple, free-diffusion model is inadequate for quantifying the spatial and temporal distribution of a dermally-injected compound if elimination effects are not accounted for properly.

Modified refinement algorithm to construct Lyapunov functions using meshless collocation

<p style='text-indent:20px;'>Lyapunov functions are functions with negative derivative along solutions of a given ordinary differential equation. Moreover, sublevel sets of a Lyapunov function are subsets of the domain of attraction of the equilibrium. One of the numerical construction methods for Lyapunov functions uses meshless collocation with radial basis functions.</p><p style='text-indent:20px;'>Recently, this method was combined with a grid refinement algorithm (GRA) to reduce the number of collocation points needed to construct Lyapunov functions. However, depending on the choice of the initial set of collocation point, the algorithm can terminate, failing to compute a Lyapunov function. In this paper, we propose a modified grid refinement algorithm (MGRA), which overcomes these shortcomings by adding appropriate collocation points using a clustering algorithm. The modified algorithm is applied to two- and three-dimensional examples.</p>

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.

Low-rank kernel approximation of Lyapunov functions using neural networks

<p style='text-indent:20px;'>We study the use of single hidden layer neural networks for the approximation of Lyapunov functions in autonomous ordinary differential equations. In particular, we focus on the connection between this approach and that of the meshless collocation method using reproducing kernel Hilbert spaces. It is shown that under certain conditions, an optimised neural network is functionally equivalent to the RKHS generalised interpolant solution corresponding to a kernel function that is implicitly defined by the neural network. We demonstrate convergence of the neural network approximation using several numerical examples, and compare with approximations obtained by the meshless collocation method. Finally, motivated by our theoretical and numerical findings, we propose a new iterative algorithm for the approximation of Lyapunov functions using single hidden layer neural networks.</p>

A meshless method for numerical solutions of linear and nonlinear time-fractional Black-Scholes models

<abstract><p>The numerical solution of the time-fractional Black-Scholes model for European and American options is presented using a local meshless collocation approach based on hybrid Gaussian-cubic radial basis functions with polynomials is presented. The approach is then expanded to a nonlinear time-fractional model for an option with transaction costs in a market with low liquidity. The spatial derivatives of the models are discretized using the proposed meshless technique. Numerical experiments are carried out for the American option, European option, and nonlinear transaction cost option models. In order to evaluate the effectiveness and precision of the suggested meshless approach, $ L_{\infty} $ and $ L_{rel} $ error norms are utilized. Both call and put option volatility is explored. A non-uniform grid customized around the strike price region is also used to determine the prices of European call and American put options. The methods described in literature are compared with the numerical results.</p></abstract>

Transient analysis of functionally graded graphene oxide powders-reinforced porous composite beams resting on elastic foundations using the reproducing kernel particle meshless method

In this paper, we propose a numerical algorithm, combining the reproducing kernel particle collocation (RKPC) meshless method with the implicit Newmark time integration scheme to examine the transient response of porous composite beams (PCB) strengthened with functionally graded (FG) graphene oxide powders (GOP), and resting on elastic foundations (EF). The PCB are subjected to dynamic loads (DL) with different boundary conditions (BC), and rested on EF modeled using the Winkler–Pasternak foundation model. The first-order shear deformation theory (FSDT) is employed to model the studied PCB reinforced with FG-GOP. The Halpin–Tsai homogenization technique and the rule of mixture are used to determine the effective material properties of the PCB taking the porosity into consideration. The obtained equations of motion using the principle of virtual displacement, are rewritten in a matrix–vector strong form and solved using the adopted numerical strategy. The performance and efficiency of the present approach are validated by comparison with numerical results of the finite element method (FEM). Parametric studies are performed to investigate the effect of GOP weight fraction, GOP distribution patterns, boundary conditions, porosity volume fraction, size of GOP, and elastic foundation stiffness on the transient behavior of FG-GOP reinforced PCB.

A novel spatial-temporal radial Trefftz collocation method for the backward heat conduction analysis with time-dependent source term

In this paper, a novel spatial-temporal radial Trefftz collocation method (STRTCM) is proposed to solve 2D and 3D backward heat conduction equations with time-dependent source term. Unlike the traditional time-domain discretization strategies (Laplace/Fourier transformation and time-stepping methods), the proposed spatial-temporal meshless collocation method employs the derived semi-analytical spatial-temporal radial Trefftz basis function as basis function, which naturally satisfies the governing equation with zero source term in advance. Due to this property, only the final conditions and boundary conditions need to be discretized. For non-zero source term, the extended multiple reciprocity method (E-MRM) is introduced to transform the original governing equation with several specific source terms to the high-order governing equation without the source term. By deriving the high-order semi-analytical spatial-temporal radial Trefftz basis function, the STRTCM can also avoid the node discretization of governing equation. For solving the backward heat conduction equations, the present numerical scheme uses the final temperature distribution and the related boundary conditions to obtain the unknown initial temperature field. Finally, the accuracy and efficiency of the proposed method is numerically verified by four benchmark examples about backward heat conduction equations with time-dependent source term.

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.

Multi-domain physics-informed neural network for solving forward and inverse problems of steady-state heat conduction in multilayer media

In this paper, a novel deep learning technique, called multi-domain physics-informed neural network (M-PINN), is presented to solve forward and inverse problems of steady-state heat conduction in multilayer media. By adopting the domain decomposition technique, the multilayer media is first divided into several sub-domains. Then, the fully connected neural network is employed to approximate the temperature field on each sub-domain. Finally, a large total network framework is formed by combining subnetworks of all the mediums and using continuity conditions on interfaces. By training the total network, we can obtain the temperature distribution over the whole computational domain, including the interface between every two mediums. In the proposed method, the boundary conditions are introduced into the loss function, and the governing equation is used as a constrain item, which ensures the accuracy and stability of numerical approximation. As a meshless collocation technology, the M-PINN does not require tedious procedures such as meshing and numerical integration, and can freely address forward and inverse problems of thin body and coating structure. Several numerical examples are given to illustrate the efficiency and performance of the new method. Results indicate that the Swish and the Sigmoid functions are two better activation functions for such problems. As the number of nodes increases, the number of hidden layers does not need to be increased. Even for the thin film at nanoscale, the M-PINN still obtains accurate results. Moreover, the proposed scheme shows better performance than the traditional boundary element method in solving nonlinear heat conduction problems.

Nonlinear free vibration and flexural analysis of hyperelastic beam utilizing a meshless method based on radial basis function

This paper investigates nonlinear free and flexural analysis of hyperelastic beams. The constitutive relations of the hyperelastic beam were derived using the neo-Hookean strain energy function and Timoshenko beam theory. Also, the nonlinear governing equations and nonlinear natural boundary conditions were derived using Hamilton’s principle. The meshless collocation method based on the multiquadric radial basis function (MQ-RBF) was utilized to discretize the nonlinear governing equations. Also, the arc-length algorithm was used to solve the nonlinear system of equations. To validate the results of the meshless method, different boundary conditions (clamped–clamped, simply supported–simply supported, clamped–simply supported, and clamped-free) were examined, and the results obtained from the meshless method were compared with those of the finite element method in Abaqus finite element software. The results show that the maximum and minimum differences between meshless and finite element methods occur in clamped and free-boundary conditions, respectively. The results also show that the meshless method based on the MQ-RBF has good accuracy compared to the finite element method for bending and free vibration analysis of hyperelastic beams.

Development of Three-Dimensional LES Based Meshless Model of Continuous Casting of Steel

A large-eddy simulation (LES) based meshless model is developed for the three-dimensional (3D) problem of continuous casting (CC) of steel billet. The local collocation meshless method based on radial basis functions (RBF) is applied in 3D. The method applies scaled multiquadric (MQ) RBF with a shape parameter on seven nodded local sub-domains. The incompressible turbulent fluid flow is described using mass, energy, and momentum conservation equations and the LES turbulence model. The solidification system is solved with the mixture continuum model. The Boussinesq approximation for buoyancy and the Darcy approximation for porous media are used. Chorin’s fractional step method is used to couple velocity and pressure. The microscopic model is closed with the lever rule model. The LES model is compared to the two-equation Low Re k−ε turbulence Reynolds Averaged Navier–Stokes (RANS) model in terms of temperature, velocity and computational times. The LES model resolves transient character of vortices which RANS-type turbulence models are unable to tackle. The computational cost of LES models is considerably higher than in RANS. On the other hand, it results in a much lower computational cost than the direct numerical simulation (DNS). The paper demonstrates the ability of the method to solve realistic industrial 3D examples. Trivial adjustment of nodal densities, high accuracy, and low numerical diffusivity are the main advantages of this meshless method.

A meshless method to solve the variable-order fractional diffusion problems with fourth-order derivative term

Using the meshless collocation method, a scheme for solving nonlinear variable-order fractional diffusion equations with a fourth-order derivative term is presented. Here the fractional derivative term is approximated by weighted and shifted Grünwald difference (WSGD) approximation formula. The difficulty caused by the nonlinear terms is carefully handled by quasilinearization technique. Using the radial basis functions, the solution of the problem is written in terms of the primary approximation, and the related correcting functions at each time step. Then the approximation is substituted back to the governing equations where the unknown parameters can be determined. Finally, the method is supported by several numerical experiments on irregular domains.

A localized meshless technique for solving 2D nonlinear integro-differential equation with multi-term kernels

This paper studies an accurate localized meshless collocation approach for solving two-dimensional nonlinear integro-differential equation (2D-NIDE) with multi-term kernels. The proposed strategy discretizes the unknown solution in two phases. First, the semi-discrete scheme is obtained by using backward Euler finite difference (FD) approach and the first-order convolution quadrature rule for the first order temporal derivative and the Riemann-Liouville (R-L) fractional integral, respectively. Second, the spatial discretization is established by means of the local radial basis function based on partition of unity (LRBF-PU) in the space variable and its partial derivatives. Furthermore, the unconditionally stable result and first-order convergence of the time semi-discrete scheme in L2-norm are proved by the energy method. It is shown that the proposed method is accurate and that the numerical results support the theoretical analysis.