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Theoretical Analysis of the Reproducing Kernel Gradient Smoothing Integration Technique in Galerkin Meshless Methods
... Some advanced integration techniques [26][27][28][29][30][31] have been devised for Galerkin meshless methods, while the reproducing kernel gradient smoothing integration (RKGSI) [30,31] technique proposes one of the most effective quadrature rules. In the RKGSI, integration constraints are inherent, and quadrature rules can then be determined in the parametric space by algebraic precision [32]. The total number of quadrature points in the RKGSI is much smaller than that in Gauss quadrature rules. ...
... As shown in Refs. [32,35,36], if k ≤ m − 1 , then the smoothed gradient ∇Φ * i (x) meets the integration constraint in Eq. (18) in a numerical form as ...
... In addition, if then ∇Φ * i (x) also meets the reproducing condition [32,35], which corresponds to the reproducing condition of the original gradient ∇Φ i (x) in Eq. (17). ...
A stabilized element-free Galerkin (EFG) method is proposed in this paper for numerical analysis of the generalized steady MHD duct flow problems at arbitrary and high Hartmann numbers up to 1016. Computational formulas of the EFG method for MHD duct flows are derived by using Nitsche’s technique to facilitate the implementation of Dirichlet boundary conditions. The reproducing kernel gradient smoothing integration technique is incorporated into the EFG method to accelerate the solution procedure impaired by Gauss quadrature rules. A stabilized Nitsche-type EFG weak formulation of MHD duct flows is devised to enhance the performance damaged by high Hartmann numbers. Several benchmark MHD duct flow problems are solved to testify the stability and the accuracy of the present EFG method. Numerical results show that the range of the Hartmann number Ha in the present EFG method is 1≤Ha≤1016, which is much larger than that in existing numerical methods.
... The RKGSI is one of the most effectual integration approaches for Galerkin meshless methods. The RKGSI has some outstanding merits: (1) by using reproducing kernel formula to reformulate meshless smoothed gradients, not only the heavy calculation of meshless gradients is reduced, but also the RKGSI is suitable for arbitrary order basis functions; (2) the integration constraint or integration consistency is automatically and intrinsically satisfied, which guarantees both the accuracy of the associated Galerkin meshless methods and the ability to pass patch tests; (3) quadrature rules are formed explicitly in the reference space to accurately integrate polynomials up to the least algebraic precision [33], which greatly facilitate the application; and (4) the total number of integration points is minimized from a global point of view, which significantly reduces the amount of calculations. ...
... The integration constraints are completely independent not only of the prescribed functions f,ū andḡ in problem (6) but also of the penalty parameter β in Nitsche's method. Besides, these constraints are the same as those for scalar-valued potential problems [10,33,44]. Moreover, with the operation of integration by parts, it can be verified that Eq. (44) holds. ...
... As proved in Ref. [33], we have ...
A stabilized element-free Galerkin (EFG) method is developed and analyzed in this paper for the meshless numerical solution of Stokes problems. To accelerate the solution procedure and recover the optimal convergence impaired by Gauss integration, integration constraints of Galerkin numerical methods for Stokes problems are derived, and then the inherently consistent reproducing kernel gradient smoothing integration is incorporated into the EFG method with explicit quadrature rules in the reference space. By using Nitsche’s method to satisfy the Dirichlet boundary condition, the inf-sup stability, the existence and uniqueness, and the error estimation of the EFG solution with numerical integration are derived rigorously. Theoretical results reveal that the EFG error essentially comes from not only the meshless approximations of velocity and pressure, but also numerical integration of the Galerkin weak forms. It turns out a procedure on how to choose quadrature rules to ensure that the optimal convergence is not affected by the integration error. Numerical results demonstrate the consistency, efficiency and optimal convergence of the method, and support the theoretical results.
... With the improvement of meshfree methods, various approaches have been developed. Point cloud-based meshfree methods such as the Element Free Galerkin (EFG) [8,9] and the Radial Basis Function (RBF) [10,11], and particle-based meshfree methods, such as Smooth Particle Hydrodynamics (SPH) [12,13] and Particle Finite Element Method (PFEM) [14,15], have demonstrated significant advantages over mesh-based methods in certain applications [16,17,18,19]. Particle-based method based on a Lagrangian framework in which particles adaptively follow the movement of the material. ...
In this paper, a Feature-preserving Particle Generation (FPPG) method for arbitrary complex geometry is proposed. Instead of basing on implicit geometries, such as level-set, FPPG employs an explicit geometric representation for the parallel and automatic generation of high-quality surface and volume particles, which enables the full preservation of geometric features, such as sharp edges, singularities and etc. Several new algorithms are proposed in this paper to achieve the aforementioned objectives. First, a particle mapping and feature line extraction algorithm is proposed to ensure the adequate representation of arbitrary complex geometry. An improved and efficient data structure is developed too to maximize the parallel efficiency and to optimize the memory footprint. Second, the physics-based particle relaxation procedure is tailored for the explicit geometric representation to achieve a uniform particle distribution. Third, in order to handle large-scale industrial models, the proposed FPPG method is entirely parallelized on shared memory systems and Boolean operations are allowed to tackle structures with multiple assemblies. Intensive numerical tests are carried out to demonstrate the capabilities of FPPG. The scalability tests show that a speedup of ~10X is achieved through multi-threading parallelization with various models. Comparative studies with other particle generation methods show that FPPG achieves better performance in both runtime and accuracy. Last, two industrial cases of vehicle wading and gearbox oiling are studied to illustrate that FPPG is applicable to complex geometries.
... The selected numerical method, namely the Galerkin finite element method (GFEM) is implemented to provide the numerical finding. 35,36 However, the highaccuracy meshless numerical methods 37,38 can become the range of selections as the future works for the Darcy-Forchheimer model. Table 1 is a list of the thermophysical characteristics of HNF. ...
In this study, a new cavity shape was filled with an extension multi-walled carbon nanotubes-Fe 2 O 3 /H 2 O nanofluid under a constant magnetic field. The Darcy–Forchheimer model is used to account for the inertial impact of advection in the porous layer while maintaining the laminar and incompressible nature of the nanofluid flow. The dimensionless version of the governing equations is used to describe the issue and the finite element approach is used to resolve it. Through this complex geometry, various thermophysical factors such as Rayleigh number [Formula: see text], Hartmann number [Formula: see text], and nanoparticle concentration are considered [Formula: see text]. The porous layer's numerous characteristics are also explored. For example, its porosity [Formula: see text] and Darcy number [Formula: see text], which indicates the permeability of the porous medium. The content of the hybrid nanofluid is considered to be Newtonian, stable, incompressible, and following a constant Prandtl number for the base fluid [Formula: see text]. Calculations are made according to the finite element method. The results of this work are presented in terms of rheology, isotherms, entropy generation, and mean Nusselt numbers. They have demonstrated that increasing the Rayleigh and Darcy numbers improve heat transfer in the enclosure.
... In [8], the FC equation involving two integro-differential operators was solved by semi-discrete finite difference approximation, and the scheme was proved unconditionally stable. In reference [9], numerical integration with the reproducing kernel gradient smoothing integration are constructed. In reference [10], recursive moving least squares (MLS) approximation was constructed. ...
A fractional cable (FC) equation is solved by the barycentric rational interpolation method (BRIM). As the fractional derivative is a nonlocal operator, we develop a spectral method to solve the FC equation to get the coefficient matrix as the full matrix. First, the fractional derivative of the FC equation is changed to a nonsingular integral from the singular kernel to the density function. Second, an efficient quadrature of a new Gauss formula is constructed to compute it simply. Third, a matrix equation of the discrete FC equation is obtained by the unknown function replaced by a barycentric rational interpolation basis function. Then, convergence rate for FC equation of the BRIM is derived. At last, a numerical example is given to illustrate our results.
... In Ref. [32], the approximation errors of the Galerkin meshless method for linear elliptic problem are analyzed based on the nonsymmetric Nitsche method and an inverse assumption, and the effect of the numerical integration are discussed. The error estimates combined with the effect of numerical integration are also developed in [33,34] based on the reproducing kernel gradient smoothing integration method. Using the Nitsche method, a fast time discrete EFG method is analyzed for the fractional diffusion-wave equation [35]. ...
... In [10], the fractional KPP equation was solved by q-homotopy analysis transform method (q-HATM), then uniqueness and convergence analysis of q-HATM the projected problem was also presented. In [11], numerical integration of the reproducing kernel gradient smoothing integration were constructed and the existence, uniqueness and error estimates of the solution of Galerkin meshless methods were established. In reference [12], recursive moving least squares (MLS) approximation was constructed in meshless methods. ...
In this paper, we seek to solve the Kolmogorov-Petrovskii-Piskunov (KPP) equation by the linear barycentric rational interpolation method (LBRIM). As there are non-linear parts in the KPP equation, three kinds of linearization schemes, direct linearization, partial linearization, Newton linearization, are presented to change the KPP equation into linear equations. With the help of barycentric rational interpolation basis function, matrix equations of three kinds of linearization schemes are obtained from the discrete KPP equation. Convergence rate of LBRIM for solving the KPP equation is also proved. At last, two examples are given to prove the theoretical analysis.
... The trial and test functions for the EFG method are generated by the moving least squares (MLS) approximation [14]. During the past several decades, many research works have been devoted to improving and extending the MLS approximation, see [4,[15][16][17][18] for various details. To offset the lack of interpolating properties of the MLS shape functions, several interpolation-type MLS methods have been developed. ...
The element-free Galerkin (EFG) method with penalty for Stokes problems is proposed and analyzed in this work. A priori error estimates of the penalty method, which is used to deal with Dirichlet boundary conditions, are derived to illustrate its validity in a continuous sense. Based on a feasible assumption, it is proved that there is a unique weak solution in the modified weak form of penalized Stokes problems. Then, the error bounds with the penalty factor for the EFG discretization are derived, which provide a rationale for choosing an efficient penalty factor. Numerical examples are given to confirm the theoretical results.
A novel coupling algorithm of the proper orthogonal decomposition (POD)–Galerkin projection method (PGM) and the finite volume method (FVM) is developed. The proposed coupling algorithm can be easily applied to complex heat transfer problems with a significant reduction in computation time. PGM is only adopted to calculate the sub-domain of heat conduction while FVM is adopted to calculate the sub-domain of the complex flow and heat transfer with openings and turbulence. Moreover, the most critical subject of the coupling of PGM and FVM at the interface based on the interfacial physical variables is comprehensively analyzed. Two coupling strategies are tested and analyzed: (1) D–D (Dirichlet–Dirichlet): the temperature profile at the interface is selected as the transferred information; and (2) D–N (Dirichlet–Neumann): the temperature and the heat flux at the interface are selected as the transferred information. A two-dimensional temperature simulation for an insulated gate bipolar transistor (IGBT) module is selected to assess the performance of the proposed coupling method. All the algorithm analyses in this study are performed via a homemade code based on Python. Results suggest that both the above coupling strategies can obtain a continuous temperature field within the maximum deviation of 0.05 compared with the pure FVM simulation results on the entire computational domain for the given two-dimensional problem. The coupling algorithm of D–N(a) (PGM provides heat flux boundary for FVM and FVM provides temperature boundary for PGM at the interface) with the additional source term method achieves the best convergence and its calculation time is only about 1/3–2/5 of that required by pure FVM. The proposed coupling algorithm achieves a high level of accuracy and significantly reduces the computational effort, which has important reference value for subsequent research.
With the assistance of the moving least‐squares (MLS) interpolation functions, a two‐dimensional finite element code is developed to consider the effects of a stationary or moving solid body in a flow domain. At the same time, the mesh or grid is independent of the shape of the solid body. We achieve this goal in two steps. In the first step, we use MLS interpolants to enhance the pressure (P) and velocity (V) shape functions. By this means, we capture different discontinuities in a flow domain. In our previous publications, we have named this technique the PVMLS method (pressure and velocity shape functions enhanced by the MLS interpolants) and described it thoroughly. In the second step, we modify the PVMLS method (the M‐PVMLS method) to consider the effect of a solid part(s) in a flow domain. To evaluate the new method's performance, we compare the results of the M‐PVMLS method with a finite element code that uses boundary‐fitted meshes.
A stabilized element-free Galerkin (EFG) method is presented and analyzed for the numerical solution of the advection–diffusion–reaction problem. Through deriving a stabilized Nitsche-type weak form for the problem, the presented EFG method is valid for small diffusivity and large reaction coefficient. Theoretical error of the method is analyzed. Numerical results verify the effectiveness of the method.
The reproducing kernel gradient smoothing integration (RKGSI) is an efficient technique to tackle the integration problem and optimal convergence in meshless methods. In this paper, the effect of the RKGSI on the element-free Galerkin (EFG) method is studied for elliptic boundary value problems with mixed boundary conditions. Theoretical results of smoothed gradients in the RKGSI are provided. Fundamental criteria on how to determine integration points and weights of quadrature rules are established according to necessary algebraic precision. By using the Nitsche's technique to impose Dirichlet boundary condition, the existence, uniqueness and error estimations of the solution of the EFG method with numerical integration are analyzed. Numerical results validate the theoretical analysis and the optimal convergence of the method.
The generalized finite difference method (GFDM) is a typical meshless collocation method based on the Taylor series expansion and the moving least squares technique. In this paper, we first provide theoretical results of the meshless function approximation in the GFDM. Properties, stability and error estimation of the approximation are studied theoretically, and a stabilized approximation is proposed by revising the computational formulas of the original approximation. Then, we provide theoretical results consisting of error bound and condition number of the GFDM. Numerical results are finally provided to confirm these theoretical results.
The recursive moving least squares (MLS) approximation is a superconvergent technique for constructing shape functions in meshless methods. Computational formulas, properties and theoretical error of the recursive MLS approximation are analyzed in this paper. Theoretical results reveal that high order derivatives of the approximation have the same convergence order as the first order derivative. Numerical results confirm the superconvergence of the recursive MLS approximation.
Smoothed gradients of meshless shape functions have been widely used in meshless methods to enhance the performance of solving partial differential equations. In this paper, properties and error estimation of the moving least squares approximation with smoothed gradients are analyzed theoretically. Numerical results are also presented to evince the extra reproduction properties and superconvergence of smoothed gradients and confirm the theoretical analysis.
An implicit Galerkin meshfree scheme based on the element-free Galerkin (EFG) and the backward Euler methods is analyzed for general second-order parabolic problems. The penalty method is used to impose Dirichlet boundary conditions, and a priori approximation error has been derived to confirm its validity. The existence and uniqueness of the weak solution for the penalized parabolic problems under homogeneous mixed boundary conditions are proved, offering an insight for numerical discretizations. With an orthogonal projection being defined, the continuous error bounds with the penalty factor for the semi-discrete approximation are obtained, and meanwhile, the discrete error bounds with the penalty factor for the fully discrete approximation are also achieved, which provide a theoretical explanation for the value of a reasonable penalty factor. Finally, numerical examples are conducted to verify the theoretical results.
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