This paper addresses the numerical solution of nonlinear time-fractional Fisher equations via local meshless method combined with explicit difference scheme. This procedure uses radial basis ...functions to compute space derivatives while Caputo derivative scheme utilizes for time-fractional integration to semi-discretize the model equations. To assess the accuracy, maximum error norm is used. In order to validate the proposed method, some non-rectangular domains are also considered.
One of the main problems of the conventional meshless methods in the time domain is their conditional stability. Up to now, some algorithms have been introduced to remove the stability condition, ...like alternating-direction-implicit (ADI) and locally one-dimensional (LOD) schemes. ADI meshless method has been investigated completely, but there is no report to exploit the full efficiency of the LOD meshless method. So, in this study, LOD scheme is employed to a meshless method for solving three-dimensional transient electromagnetic problems. The initial form of the proposed meshless method solves time-domain Maxwell's equations in three sub-steps, so it is called LOD3. LOD3 meshless method has first-order temporal accuracy. Moreover, LOD5 meshless method is introduced to upgrade temporal accuracy. Also, the results show that LOD5 meshless method is more accurate than ADI and LOD3 meshless methods. The accuracy of the proposed method is due to two main factors; Crank–Nicolson scheme and solving the equation in five sub-steps instead of one sub-step. Stability and accuracy of the proposed method are assessed through numerical experiment. Also, the unconditional stability of the proposed method is proved, analytically.
A strong-form boundary collocation method, the singular boundary method (SBM), is developed in this paper for the wave propagation analysis at low and moderate wavenumbers in periodic structures. The ...SBM is of several advantages including mathematically simple, easy-to-program, meshless with the application of the concept of origin intensity factors in order to eliminate the singularity of the fundamental solutions and avoid the numerical evaluation of the singular integrals in the boundary element method. Due to the periodic behaviors of the structures, the SBM coefficient matrix can be represented as a block Toeplitz matrix. By employing three different fast Toeplitz-matrix solvers, the computational time and storage requirements are significantly reduced in the proposed SBM analysis. To demonstrate the effectiveness of the proposed SBM formulation for wave propagation analysis in periodic structures, several benchmark examples are presented and discussed The proposed SBM results are compared with the analytical solutions, the reference results and the COMSOL software.
The localized method of fundamental solutions (LMFS) is an efficient meshless collocation method that combines the concept of localization and the method of fundamental solutions (MFS). The resultant ...system of linear algebraic equations in the LMFS is sparse and banded and thus, drastically reduces the storage and computational burden of the MFS. In the LMFS, the moving least square (MLS) approximation, based on fundamental solutions, is used to construct approximate solution at each node. In this paper, this fundamental solutions-based MLS approximation, named as an augmented MLS (AMLS) approximation, is generalized to any point in the computational domain. Computational formulas, theoretical properties and error estimates of the AMLS approximation are derived. Then, taking Laplace equation as an example, this paper sets up a framework for the theoretical error analysis of the LMFS. Finally, numerical results are presented to verify the efficiency and theoretical results of the AMLS approximation and the LMFS. Convergence and comparison researches are conducted to validate the accuracy, convergence and efficiency.
In this study, the generalized finite difference method (GFDM) was used to stably and accurately solve two-dimensional (2D) inverse Cauchy problems in linear elasticity by using the Navier equations. ...In Cauchy problems, overdetermined boundary conditions are imposed on parts of the boundary, whereas there are missing boundary conditions on some parts of the boundary. In Cauchy problems, conventional numerical methods generally generate highly ill-conditioned matrices and thus provide unstable numerical solutions. Moreover, even if a slight noise is added in the boundary conditions, numerical errors are evidently magnified. The GFDM, one of the most promising meshless methods and an extension of the classical finite difference method, can avoid time-consuming tasks of mesh generation and numerical quadrature. The GFDM was applied in this study to stably solve the 2D Cauchy problems in linear elasticity and four numerical examples are provided to illustrate the consistency and accuracy of the presented meshless numerical scheme. Moreover, the stability of the presented scheme for inverse Cauchy problems was proved by adding noise into the boundary conditions.
In this study, a new framework for the efficient and accurate solutions of three-dimensional (3D) dynamic coupled thermoelasticity problems is presented. In our computations, the Krylov deferred ...correction (KDC) method, a pseudo-spectral type collocation technique, is introduced to perform the large-scale and long-time temporal simulations. The generalized finite difference method (GFDM), a relatively new meshless method, is then adopted to solve the resulting boundary-value problems. The GFDM uses the Taylor series expansions and the moving least squares approximation to derive explicit formulae for the required partial derivatives of unknown variables. The method, thus, is truly meshless that can be applied for solving problems merely defined over irregular clouds of points. For problem with complicated geometries, this paper also examines a new distance criterion for adaptive selection of nodes in the GFDM simulations. Preliminary numerical experiments show that the KDC accelerated GFDM methods are very promising for accurate and efficient long-time and large-scale dynamic simulations.
•Apply the GFDM for the first time to 3D dynamic coupled thermoelasticity problems.•Developing a combined KDC-GFDM scheme for long-time dynamic simulations.•Proposing a new distance criterion for the adaptive selection of nodes in the GFDM.•Extremely large time step-sizes can be used in temporal discretization.
We present a scheme implementing an a posteriori refinement strategy in the context of a high-order meshless method for problems involving point singularities and fluid–solid interfaces. The ...generalized moving least squares (GMLS) discretization used in this work has been previously demonstrated to provide high-order compatible discretization of the Stokes and Darcy problems, offering a high-fidelity simulation tool for problems with moving boundaries. The meshless nature of the discretization is particularly attractive for adaptive h-refinement, especially when resolving the near-field aspects of variables and point singularities governing lubrication effects in fluid–structure interactions. We demonstrate that the resulting spatially adaptive GMLS method is able to achieve optimal convergence in the presence of singularities for both the div-grad and Stokes problems. Further, we present a series of simulations for flows of colloid suspensions, in which the refinement strategy efficiently achieved highly accurate solutions, particularly for colloids with complex geometries.
•An a posteriori adaptive refinement strategy is proposed in the context of generalized moving least squares (GMLS), a compatible high-order meshless method.•The resulting adaptive GMLS method can efficiently resolve point singularities and capture lubrication effects in fluid–structure interactions and achieve optimal convergence.•Simulations for flows of colloid suspensions are presented, in which the adaptive GMLS efficiently achieves highly accurate solutions, particularly for colloids with complex geometries.
•Proposing an enriched LMFS approach for fracture crack analysis.•Oscillatory near-tip displacement and stress fields can be approximated properly.•SIFs of in-plane cracks can be calculated with very ...high accuracy.
In this paper, the localized method of fundamental solutions (LMFS), a recently developed meshless collocation method, is applied to the numerical solution of problems with cracks in linear elastic fracture mechanics. The main idea of the LMFS is to divide the entire computational domain into a set of overlapping sub-domains, and in each sub-domain, the classical MFS formulation and the moving least squares (MLS) technique are applied to form the corresponding local system of equations. The LMFS will finally generate a banded and sparse matrix system which makes the method very attractive for large-scale engineering applications. To deal with in-plane crack problems, an enriched LMFS approach is proposed by combining the LMFS formulation for linear elasticity problems and a set of enrichment functions which take into account the asymptotic behavior of the near-tip displacement and stress fields. The enriched LMFS can significantly improve the computational accuracy of the calculation of stress intensity factor (SIF) of the cracked materials, even with a very coarse LMFS node distribution. Several benchmark numerical examples are presented to illustrate the accuracy and efficiency of the proposed method.
Fractional differential equations depict nature sufficiently in light of the symmetry properties which describe biological and physical processes. This article is concerned with the numerical ...treatment of three-term time fractional-order multi-dimensional diffusion equations by using an efficient local meshless method. The space derivative of the models is discretized by the proposed meshless procedure based on the multiquadric radial basis function though the time-fractional part is discretized by Liouville–Caputo fractional derivative. The numerical results are obtained for one-, two- and three-dimensional cases on rectangular and non-rectangular computational domains which verify the validity, efficiency and accuracy of the method.
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