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The improved element-free Galerkin (IEFG) method is proposed in this paper for solving 3D Helmholtz equations. The improved moving least-squares (IMLS) approximation is used to establish the trial function, and the penalty technique is used to enforce the essential boundary conditions. Thus, the final discretized equations of the IEFG method for 3D...
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... same node distribution and background integral grids are selected, α = 2.0 × 10 4 , and the cubic spline function is used. Figure 1 shows the relationship between dmax and relative errors. Because of the error of computer itself, the relative error become larger when dmax = 1.2. ...
Context 2
... the IEFG method is used, the relative error is 2.4229%. The numerical solutions and analytical ones are compared in Figure 16. It is shown that the numerical results are in good agreement with the analytical ones. ...
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... using the EFG method to solve it, the same parameters are selected, and the relative errors of both methods are equal to 0.8646%. Figures 17-19 show the comparison of the numerical solutions and the analytical ones. We can see that numerical solutions are in good agreement with the analytical ones. ...
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Citations
... The MLS approximate functions have been improved to accelerate computational speed, including the Improved Moving Least-Squares (IMLS) approximation [6], interpolating MLS approximation [7,8], and complex variable MLS approximation [9,10]. Using these methods to construct shape functions resulted in the presentation of the IEFG method [11][12][13][14], interpolating EFG method [15][16][17][18][19][20], and complex variable EFG method [10,21,22], respectively. ...
... where Γ (k) u and Γ (k) q are essential and natural boundaries. (14) and (15) are then analyzed using the IEFG method. The discretization of the second-order partial derivative in the splitting direction is performed using the FEM, then we obtain the discretized equations. ...
Due to the low computational efficiency of the Improved Element-Free Galerkin (IEFG) method, efficiently solving three-dimensional (3D) Laplace problems using meshless methods has been a longstanding research direction. In this study, we propose the Dimension Coupling Method (DCM) as a promising alternative approach to address this challenge. Based on the Dimensional Splitting Method (DSM), the DCM divides the 3D problem domain into a coupling of multiple two-dimensional (2D) problems which are handled via the IEFG method. We use the Finite Element Method (FEM) in the third direction to combine the 2D discretized equations, which has advantages over the Finite Difference Method (FDM) used in traditional methods. Our numerical verification demonstrates the DCM’s convergence and enhancement of computational speed without losing computational accuracy compared to the IEFG method. Therefore, this proposed method significantly reduces computational time and costs when solving 3D Laplace equations with natural or mixed boundary conditions in a dimensional splitting direction, and expands the applicability of the dimension splitting EFG method.
... It is found that the motion trajectory of the internal store will change at different time under the same launch conditions, which is caused by the unsteady flow field of the internal bay. Some scholars [7][8][9][10][11][12] have studied the hybrid complex variable element-free Galerkin (HCVEFG) method, which can improve computational efficiency. The research of Westmoreland [13] showed that applying appropriate ejection force to the store is conducive to the separation. ...
To understand the influence of the initial release conditions on the separation characteristics of the store and improve it under high Mach number (Ma = 4) flight conditions, the overset grid method and the Realizable k−ε turbulence model coupled with an equation with six degrees of freedom are used to simulate the store released from the internal bay. The motion trajectory and the attitude angle of the store separation under the conditions of different centroid, velocity, height and control measures are given by the calculated result. Through analysis, the position of the centroid will affect the separation of the store, which needs to be considered in the design. Increasing the launching height is conducive to the separation of the store. If the store has an initial velocity, it can leave the internal bay more quickly and reduce the probability of collision with the wall. Cylindrical rod and slanted aft wall control measures can improve the attitude of the store and make the store fall more smoothly.
... Mathematics 2023, 11, 770 2 of 19 orthogonal basis function. Thus, the IEFG method [26][27][28][29][30] was presented for some partial differential equations and mechanics problems, and the higher computational efficiency of the IEFG method was proved. ...
In order to obtain the numerical results of 3D convection-diffusion-reaction problems with variable coefficients efficiently, we select the improved element-free Galerkin (IEFG) method instead of the traditional element-free Galerkin (EFG) method by using the improved moving least-squares (MLS) approximation to obtain the shape function. For the governing equation of 3D convection-diffusion-reaction problems, we can derive the corresponding equivalent functional; then, the essential boundary conditions are imposed by applying the penalty method; thus, the equivalent integral weak form is obtained. By introducing the IMLS approximation, we can derive the final solved linear equations of the convection-diffusion-reaction problem. In numerical examples, the scale parameter and the penalty factor of the IEFG method for such problems are discussed, the convergence is proved numerically, and the calculation efficiency of the IEFG method are verified by four numerical examples.
... The improved element-free Galerkin (IEFG) method [1], which is based on the improved moving least-squares (IMLS) approximation [2], is an important meshless method. In the paper submitted to this Special Issue by Cheng et al. [3], the IEFG method for solving three-dimensional (3D) Helmholtz equations is proposed. The IMLS approximation is used to form the trial function, the penalty method is used to enforce the essential boundary conditions and Galerkin weak form of 3D Helmholtz equations are used to obtain the final discretized equations. ...
In recent years, mathematical models, numerical methods and data analysis have been paid more attention [...]
... Acoustic radiation and scattering problems arise in many real-life applications, such as radar, sonar, non-destructive testing, and noise barrier, just to mention a few. In most cases, however, the acoustic problems are not solvable analytically, thus numerical methods are of considerable interest [1][2][3]. For numerical calculation, the discretization techniques are usually employed to discretize the computational domain or the boundary to elements or nodes. ...
This paper presents a precorrected-FFT (pFFT) accelerated singular boundary method (SBM) for acoustic radiation and scattering in the high-frequency regime. The SBM is a boundary-type collocation method, which is truly free of mesh and integration and easy to program. However, due to the expensive CPU time and memory requirement in solving a fully-populated interpolation matrix equation, this method is usually limited to low-frequency acoustic problems. A new pFFT scheme is introduced to overcome this drawback. Since the models with lots of collocation points can be calculated by the new pFFT accelerated SBM (pFFT-SBM), high-frequency acoustic problems can be simulated. The results of numerical examples show that the new pFFT-SBM possesses an obvious advantage for high-frequency acoustic problems.
A Novel 2-D mathematical modeling to determine LHP to ... ZASTITA MATERIJALA 64 (2023) broj 3 ABSTRACT 2-dimensional mathematical model of axisymmetric transient industrial quenched low carbon steel bar, to examine the influence of process history on metallurgical and material characteristics, a water-cooled model based on the finite element technique was adopted. A 2-dimensional axisymmetric mathematical model was utilized to predict temperature history and, as a result, the hardness of the quenched steel bar at any node (point). The LHP (lowest hardness point) is evaluated. In this paper, specimen points' hardness was evaluated by the transformation of determined characteristic cooling time for phase conversion t8/5 to hardness. The model can be used as a guideline to design a cooling approach to attain the desired microstructure and mechanical properties, such as hardness. The mathematical model was verified and validated by comparing its hardness results to commercial finite element software results. The comparison demonstrates that the proposed model is reliable.
In this study, by introducing the finite element method (FEM) into the improved element-free Galerkin (IEFG) method, the dimension coupling method (DCM) is proposed for solving three-dimensional (3D) Helmholtz and Poisson’s equations efficiently. The dimensional splitting method is introduced into the corresponding governing equations, thus 3D equations can be split into a series of 2D ones. The IEFG method is selected to discretize those 2D forms, thus the discretized equations are derived easily by using the weak forms. In the third direction, the FEM is selected to couple those discretized equations, thus the final linear equation of 3D equation is derived. In numerical examples, the good convergence of the DCM for Helmholtz and Poisson’s equations is shown. The solutions of numerical examples show that the computational efficiency of the IEFG method is improved significantly without losing the computational accuracy when the DCM is used. In addition, the DCM can enhance the computational efficiency of the hybrid element-free Galerkin (HEFG) method significantly without too many layers when the natural boundary conditions exist in the splitting direction.
In this study, we present the hybrid complex variable element-free Galerkin (HCVEFG) method for solving 3D Helmholtz equations. The dimension splitting method (DSM) will be introduced into the corresponding governing equation, thus a series of 2D forms can be obtained by splitting the problem domain of 3D Helmholtz equation. For every 2D problem, the shape function can be obtained by using the improved complex variable moving least-squares (ICVMLS) approximation, and the essential boundary condition can be imposed by using the penalty method, thus the discretized equations of 2D problems can be derived by using the corresponding Galerkin weak form. These equations can be coupled by using the finite difference method (FDM) in the dimension splitting direction, thus final formulae of the numerical solution for 3D Helmholtz equation can be obtained. In Sec. 4, the relative errors are given, and the convergence is analyzed numerically. The numerical result of these examples illustrates that the calculation speed can be improved greatly when the HCVEFG method is used rather than the improved element-free Galerkin (IEFG) method.
In this study, the hybrid element-free Galerkin (HEFG) method is studied to solve the 3D Helmholtz equations. The idea of the dimension splitting method (DSM) is introduced into the improved element-free Galerkin (IEFG) method, thus a sequence of 2D forms can be obtained by splitting the problem domain of 3D Helmholtz equation, each 2D form can be discretized by using the IEFG method, thus the corresponding 2D discretized equations can be obtained, and these equations can be coupled by employing the finite difference method (FDM) in the dimension splitting direction, thus final formulae of numerical solution for 3D Helmholtz equation can be obtained. In Sec. 4, the relative errors and the convergence are analyzed, respectively, and the numerical results show that the calculation resources can be saved greatly without losing the computational accuracy when using the HEFG method.