Meshless Collocation Research Articles

A space-time meshfree method for heat transfer analysis in porous material

Abstract Porous material batteries are a new type of battery technology that uses porous materials as electrode materials, with advantages such as long lifespan. In the present study, we manily propose a space-time meshfree method for heat transfer problems in porous material energy storage battery. The thermal behavior is demonstrated based on a numerical solution of the energy conservation equation. A simple space-time meshless collocation scheme, which is based on a new type space-time radial basis function, is presented to get the approximate solution of the energy conservation equation. The energy conservation equation was transferred into a system of linear algebraic equations. By introducing prescribed boundary conditions, the heat transfer analysis in porous material energy storage battery can be shown with numerical results. Numerical discussions show that the proposed meshless collocation method is appropriate to simulate thermal behavior in porous material energy storage battery, while the traditional finite element method failed to provide the reasonable results for the tested examples in this paper.

Numerical computational technique for solving Volterra integro-differential equations of the third kind using meshless collocation method

The primary goal of this study is to give an approximate algorithm for solving Volterra integro-differential equations (VIDEs) of the third kind using meshless collocation techniques. The basic framework of the novel approach is based on a collocation scheme and radial basis functions (RBFs) created on scattered points. This technique requires no background approximation cells, and the algorithm is powerful, has greater stability, and does not require much computer memory. This approach represents the solution of VIDEs of the third kind by interpolating the RBFs based on the Gauss–Legendre quadrature formula. The problem is reduced to a system of algebraic equations that can be easily solved. A description of the technique for the proposed equations is provided. Furthermore, the error analysis of this scheme is examined. A few numerical experiments are presented to prove the reliability and precision of the suggested approach for solving VIDEs of the third kind. Certain problems were compared with analytical solutions, the moving least squares method, and other methods to prove the effectiveness and applicability of the approach described.

Flexural Analysis of Elastically Supported Bidirectional Monel–Zirconia Skew FGM Plate Subjected to Line Load Using Meshless Collocation Technique

Flexural Analysis of Elastically Supported Bidirectional Monel–Zirconia Skew FGM Plate Subjected to Line Load Using Meshless Collocation Technique

Resolving an Inverse Cauchy Problem of Mixed Boundary Value of Bi-harmonic using a meshless collocation method

Resolving an Inverse Cauchy Problem of Mixed Boundary Value of Bi-harmonic using a meshless collocation method

The meshless backward substitution method for inverse Cauchy problems in electroelastic piezoelectric structures

The backward substitution method is a recently proposed semi-analytical meshless collocation method. The main idea of the back substitution method is to obtain the boundary approximation from the boundary data, and this approximation does not need to satisfy the governing equation. Then, the traditional linear combination of basis functions is used to construct a correcting approximation that satisfies the homogeneous boundary conditions, the numerical solution is given by a linear combination system of boundary approximation and modified approximation, which satisfies the governing equation. In this way, the backward substitution method can be easily used to solve general problems without fundamental solutions or particular solutions. This paper makes the first attempt to extend the backward substitution method to the simulation of inverse Cauchy problems in piezoelectric structures. For the inverse Cauchy problems, the absence of data on some boundaries often leads to extreme instability of numerical solutions. A simple method is applied in this paper. For the points on the boundary without boundary information, the governing equation is enforced. Numerical results demonstrate that it will greatly increase the stability of the solution process.

A novel local meshless collocation method with partial upwind scheme for solving convection-dominated diffusion problems

A novel local meshless collocation method with partial upwind scheme for solving convection-dominated diffusion problems

Nonlinear dynamical response of sinusoidal impulsive actuated piezoelectric/porous sandwich nanoharvesters via GM-based meshfree collocation formulations

In this paper a new effective computational approach based upon a meshless collocation formulation of the Gurtin-Murdoch (GM) continuum elasticity is described. The newly developed computational model for the third-order shear flexible elastic plates is then employed to analyze the nonlinear dynamical response of piezoelectric/porous sandwich nanoharvesters subjected to a sinusoidal impulsive actuation. The material properties relevant to the porous passive core having uniform or two graded through thickness porosity dispersions are estimated using the approach of Gaussian random field. The weak form of model conception is discretized and solved numerically via employing an incorporation of the polynomial and radial basis functions having the capability to remove any feasible singularity as well as taking more precise approximation into account within the GM-based meshfree collocation formulations. It is deduced that by contemplating the surface stress tensor via the established GM-based model, the achieved voltage from the sinusoidal impulsive actuated sandwich nanoharvesters reduces, especially for those possessing lower thickness. Also, it is deduced that for the fully simply supported impulsive actuated sandwich nanoharvester, by changing the porosity decoration of distribution from the uniform decoration to FGO and FGX graded ones, the significance of the role of surface stress tensor in the achieved voltage reduces from 12.45% to 12.28%, and enhances from 12.45% to 12.64%, respectively. For the fully clamped impulsive actuated sandwich nanoharvester, by changing the porosity decoration of distribution from the uniform decoration to FGO and FGX graded ones, the significance of the role of surface stress tensor in the achieved voltage reduces from 15.01% to 14.68%, and enhances from 15.01% to 15.37%, respectively.

On the roles of strain gradient tensors in quasi‐3D nonlinear dynamics of microplate‐type piezoelectric systems of energy harvesting under triangular mechanical actuation

AbstractThe small‐scale‐dependent nonlinear dynamical attributes of microplate‐type laminated piezoelectric systems of harvesting energy subjected to a triangular mechanical actuation are explored in the current exploration. The core of harvesting systems is supposed to be made of nanocomposites reinforced by randomly oriented carbon nanotubes (CNTs) having various degrees of agglomeration coated by piezoelectric layers. In this regard, the microplate‐type systems of harvesting energy are modeled via a quasi‐3D plate theory combined with the modified strain gradient elasticity in the presence of various microsize‐dependent strain gradient tensors. Thereafter, the solution of the developed coupled electromechanical size‐dependent nonlinear problem is obtained numerically by utilizing the meshless collocation approach employing incorporation of the polynomial and radial basis functions to remove any feasible singularity for the associated moment matrices. It comes to the conclusion that by elevating the quantity of CNTs disclosed within the clusters, the significance of the microsize dependency in the achieved voltage enhances from 22.06% to 22.73% toward the simply supported bridge‐type context, and from 33.00% to 33.92% toward the clamped bridge‐type context.Highlights Development of a microstructural‐dependent quasi‐3D shear flexible energy harvester model. Incorporating the roles of various strain gradient tensors in the nonlinear dynamics of microharvesters. Size‐dependent time‐histories of achieved voltage from triangular mechanical‐loaded microharvesters. Influence of the CNT reinforcement clustering on the nonlinear dynamic performance.

Construction of a Contraction Metric for Time-Periodic Systems Using Meshless Collocation

Construction of a Contraction Metric for Time-Periodic Systems Using Meshless Collocation

Solving Bi-harmonic Cauchy problem using a meshless collocation method

Solving Bi-harmonic Cauchy problem using a meshless collocation method

Nonlocal strain gradient‐based nonlinear dynamics of sinusoidal impulsive actuated porous/piezoelectric multilayer energy nanoharvesters

AbstractThe applications of nanoelectromechanical systems integrating simultaneously the mechanical and electrical functionalities at nanoscale have been gotten wider in the last decade. The main objective of this exploration is to analyze the small scale‐dependent nonlinear dynamical feedback of multilayer energy nanoharvesters containing a graded porous passive core coated by piezoelectric facesheets under a sinusoidal impulsive external actuation. For this purpose, the technique of meshless collocation is constructed at the same time the nonlocal stress and strain gradient tensors without any necessary background meshes and integration procedure. It is found that by taking the nonlocal stress tensor into account, the average of extremum points of maximum lateral deflection increases from to , and from to for simply supported and clamped multilayer nanoharvesters, respectively. On the contrary, through considering the strain gradient tensor, the average of extremum values of the maximum lateral deflection reduces from to for the simply supported nanoharvester, and from to for the clamped one.Highlights Development a third‐order shear flexible nonlocal strain gradient (NSG)‐based plate‐type nanoharvester model Incorporating the role of nonlocal stress and strain gradient tensors in nonlinear dynamics of nanoharvesters Time‐histories of achieved voltage of sinusoidal impulsive loaded plate‐type nanoharvesters Influence of porosity on the NSG‐based nonlinear dynamic performance.

Investigation of combustion model via the local collocation technique based on moving Taylor polynomial (MTP) approximation/domain decomposition method with error analysis

In this paper, we develop a new meshless numerical procedure for simulating the combustion model. To that end, we employ a local meshless collocation method according to the moving Taylor polynomial (MTP) approximation. The space derivative is approximated by using the local approach and then the Crank–Nicolson algorithm is utilized to approximate the time derivative. The stability and convergence of the time-discrete formulation are discussed, analytically and numerically. The Broyden method is applied to solve this nonlinear system. Since the size of the physical domain is large, we employ the non-overlapping domain decomposition method (DDM) to obtain a faster numerical algorithm. The local meshless approaches are efficient numerical techniques to simulate models in the fluid flow. The obtained results show that the proposed numerical formulation has efficient results for solving this mathematical model.

A peridynamics approach to flexible multibody dynamics for fracture analysis of mechanical systems

The classical theory of continuum mechanics is formulated using partial differential equations (PDEs) that fail to describe structural discontinuities, such as cracks. This limitation motivated the development of peridynamics, reformulating the classical PDEs into integral-differential equations. In this theory, each material point interacts with its neighbours inside a characteristic length-scale through bond-interaction forces. However, while peridynamics can simulate complex multi-physics phenomena, its integration in the study of mechanical systems is still limited. This work presents a methodology that incorporates a peridynamics formulation into a planar multibody dynamics (MBD) formulation to allow the integration of flexible structures described by peridynamics into mechanical systems. A flexible body is described by a collection of point masses, in analogy with the meshless collocation scheme commonly used for peridynamics discretisations. Each point mass interacts with other point masses through nonlinear forces governed by a bond-based peridynamics (BBPD) formulation. The virtual bodies methodology enables the definition of kinematic joints connecting the flexible body with the neighbouring bodies. The implementation of the methodology proposed is illustrated using various mechanisms with different levels of complexity. Notched plates subjected to different loading conditions are compared with the results presented in the literature of the peridynamics field. The deformations of a flexible slider-crank mechanism compare well with the results obtained using a classical flexible MBD formulation. Additionally, three scenarios involving a rotating pendulum illustrate how the methodology proposed allows simulating impact scenarios. The results demonstrate how this methodology is capable to successfully simulate highly nonlinear phenomena, including crack propagation, in a multibody framework.

A computational approach for solving third kind VIEs by collocation method based on radial basis functions

This work presents a meshless collocation method based on radial basis functions for solving third kind VIEs. This method is an interpolation approach where the radial basis functions are used as basis functions and there is no meshing. The formulation of the technique for suggesting equations is described. In addition, we also study the error analysis of the suggested approach. Finally, illustrative examples demonstrate the reliability and effectiveness of the new approach. Some of these problems have been exposed and compared to other methods, which shows the accuracy of the proposed method.

HSDT mesh-free method for free vibration analysis of sandwich structures

The current study presents a meshless collocation method based on radial basis functions (MC-RBF) to analyze the vibrational response associated with a sandwich circular cylindrical nanopipe in a supersonic airflow and conveying fluid flow. The quasi-2D hybrid type of nonlocal strain gradient theory (QHNSGT) as a higher order shear deformation theory (HSDT) has been presented to model the size-dependent nanopipe equations extracted using the energy method for a sandwich nanopipe. The sandwich nanostructure is made of metal, ceramic, and bi-directional functionally graded (Bi-FG) material, respectively, lower, topper, and middle surfaces. By taking into account that the fluid flow is infinite, incompressible, uniform flow, Newtonian, laminar, as well as viscous, and with the help of the Navier-Stokes equation, the fluid-structure interaction is obtained. Also, the displacement fields are defined via the Quasi-2D hybrid type of high-order shear deformation theory. The governing equations are discretized via the MC-RBF method to obtain the natural frequencies. The verification section shows that the results of the current research are very near to the results of the published article in the literature. By using COMSOL multi-physics software, the results are verified, and the outputs show that the velocity of fluid flow has a significant role in the vibrational responses of the current composite structure.

A localized Fourier collocation method for 2D and 3D elastic mechanics analysis: Theory and MATLAB code

This paper documents the first attempt to apply the localized Fourier collocation method (LFCM), a recently published meshless collocation method, for solving 2D and 3D elastostatic problems. The approach first divides the entire domain into a series of intersecting sub-domains. Within each sub-domain, the unknown functions are approximated by utilizing the pseudo-spectral Fourier collocation technique. The method integrates the merits of the rapid convergence of the pseudo-spectral method and the high sparsity of the localized discretization technique. This combination results in a novel framework that is highly suitable for large-scale and high-precision engineering simulations. Extensive numerical experiments, involving both 2D and 3D elastostatic problems, are presented to showcase the precision and efficiency of the current method. The source codes for numerical examples studied in this work are also provided.

On the role of surface stress tensor in the nonlinear response of time-dependent mechanical actuated nanoplate-type energy piezo-harvesters

In the present exploration, the role of surface stress tensor as one of the most important size effects at nanoscale in the nonlinear dynamical behavior of nanoplate-type energy piezo-harvesters under a time-dependent mechanical actuation is investigated. The general form of the energy functional associated with a quasi-3D nanosize energy piezo-harvester incorporating the surface Lame constants and the residual surface stress is derived. After that, by taking the Hamilton’s principle into consideration, the weak form of the surface continuum mechanics-based governing equations for a nanoplate-type energy piezo-harvester is achieved. Afterwards, the meshless collocation strategy is utilized to numerically solve the established coupled electromechanical surface elastic-based nonlinear problem with the aid of implementing a combined form of the polynomial-kind and radial-kind basis functions to eliminate any singularity which may be observed for the associated moment matrices. One can find that for the nanoplate-type energy piezo-harvesters under simply edge supports owning the thickness of and the average value of the achieved voltages are equal to and respectively. However, by taking the role of surface stress tensor into consideration, these values in order reduce to and Moreover, for the nanoplate-type energy piezo-harvesters under clamped edge supports owning the thickness of and the conventional average values of the attained voltages are, respectively, and However, by taking the role of surface stress tensor into consideration, these values in order decrease to and respectively. Therefore, it is deduced that the role of surface stress tensor in the nonlinear dynamical behavior of nanoplate-type energy piezo-harvesters owing lower thickness is more prominent.

Nonlocal couple stress-based meshless collocation model for nonlinear dynamic performance of microbeam-type piezoelectric energy harvesters

The prime aim of the current exploration is that to analyze the size-dependent nonlinear dynamics of microsize laminated bridge-type piezoelectric energy harvesters containing a passive core made of an agglomerated nanocomposite and piezoelectric surface layers under a time-dependent mechanical load. For this purpose, the both nonlocal and couple stress tensors are incorporated to the classical quasi-3D shear flexible beam theory to model microsize beam-type laminated energy harvesters. After that, a numerical solution methodology based on the meshless collocation method is employed to discretize the obtained size-dependent nonlinear equations via using a coalesce basis function including polynomial and multiquadric parts on the basis of the Chebyshev node distribution scheme to remove any possible singularity. The nonlinear time histories relevant to the induced lateral deflection and extracted voltage are plotted in the presence and absence of the nonlocal and couple stress tensors as well as various values of the agglomeration constants relevant to the nanocomposite passive core of microsize energy harvesters. One can find that by increasing the cluster volume fraction, the effect of the nonlocal stress tensor on the extracted voltage from the microsize energy harvester increases from +16.50% to +17.53% in the case of simply supported end conditions, and from +21.84% to +22.99% in the case of clamped end conditions. In addition, the effect of the couple stress tensor enhances from −22.21% to −23.09% in the case of simply supported end conditions, and from −27.15% to −27.94% in the case of clamped end conditions.

Nonlocal strain gradient-based meshless collocation model for nonlinear dynamics of time-dependent actuated beam-type energy harvesters at nanoscale

There is a diverse and well-constructed application of energy harvesting systems due to their innovate technology to provide the necessary power for low-energy electronics. In this regard, the prime objective of the current study is to analyze the size-dependent nonlinear dynamic performance of piezoelectric beam-type energy harvesters at nanoscale subjected to a time-dependent mechanical uniform load. A laminated structure containing an agglomerated nanocomposite core integrated with top and bottom piezoelectric surface layers is considered for the nanoscale bridge-type energy harvesters. To take the size dependency into account, the nonlocal strain gradient continuum elasticity is formulated based upon a quasi-3D beam theory incorporating the both features of size effects. Thereafter, the size-dependent nonlinear problem is then solved numerically relevant to simply supported and clamped end conditions via employing the meshless collocation technique. Accordingly, the numerical solving procedure is established without any background meshes as well as eliminating the integration and singularity by using proper multiquadric radial basis functions.

Large deformation analysis of a hyperplastic beam using experimental / FEM/ meshless collocation method

In the present study, nonlinear numerical and experimental static bending analyses of hyperelastic beams are surveyed. Timoshenko's beam theory is used to obtain the Cauchy-Green deformation tensor. The governing equations are extracted using the neo-Hookean strain energy and Euler–Lagrange relation. Afterward, the nonlinear governing equations are solved numerically using the meshless collocation method (MCM). To validate the present investigation, a comparison study is accomplished between the acquired results from MCM and the experimental test of digital image correlation (DIC) and finite element method (FEM). The hyperelastic beam is made of natural rubber. Also, loading and boundary conditions are considered as concentrated force and clamped–clamped, respectively. In addition, the displacement contour is extracted by the DIC. The uniaxial tensile test based on the ASTM D412 standard is performed to extract the mechanical properties of natural rubber. The hyperelastic theory and neo-Hookean strain energy function are utilized to model the nonlinear behavior of rubber in the FEM. The error for the maximum displacement of system obtained from both numerical methods is less than 12% compared to the DIC. Additionally, it can be observed that in large deformations, unlike the FEM, MCM is able to predict the system's response with high accuracy.