The element-free Galerkin(EFG)method,which constructs shape functions via moving least squares(MLS)approximation,represents a fundamental and widely studied meshless method in numerical computation.Although it achieve...The element-free Galerkin(EFG)method,which constructs shape functions via moving least squares(MLS)approximation,represents a fundamental and widely studied meshless method in numerical computation.Although it achieves high computational accuracy,the shape functions are more complex than those in the conventional finite element method(FEM),resulting in great computational requirements.Therefore,improving the computational efficiency of the EFG method represents an important research direction.This paper systematically reviews significant contributions fromdomestic and international scholars in advancing the EFGmethod.Including the improved element-free Galerkin(IEFG)method,various interpolating EFG methods,four distinct complex variable EFG methods,and a series of dimension splitting meshless methods.In the numerical examples,the effectiveness and efficiency of the three methods are validated by analyzing the solutions of the IEFG method for 3D steadystate anisotropic heat conduction,3D elastoplasticity,and large deformation problems,as well as the performance of two-dimensional splitting meshless methods in solving the 3D Helmholtz equation.展开更多
基于对单元能量投影(element energy projection,EEP)法误差项的直接推导及分析,用EEP简约格式的解计算出略掉的误差项,反补后得到比简约格式高一阶精度的EEP超收敛计算的加强格式。该文以一维Galerkin有限元为例,给出EEP加强格式的算...基于对单元能量投影(element energy projection,EEP)法误差项的直接推导及分析,用EEP简约格式的解计算出略掉的误差项,反补后得到比简约格式高一阶精度的EEP超收敛计算的加强格式。该文以一维Galerkin有限元为例,给出EEP加强格式的算法公式和数学证明。理论分析和算例验证表明:对于m (≥1)次单元,采用EEP加强格式计算的内点位移和内点导数都具有h^(min(m+3,2m))阶的收敛精度,对系数特例问题二者甚至可以分别达到h^(min(m+5,2m))和h^(min(m+4,2m))阶的收敛精度。并对该法的进一步拓展作了讨论。展开更多
针对一维四阶线性方程,研究了一种隐显式Runge-Kutta全离散局部间断Galerkin方法的稳定性和最优误差估计。空间离散采用局部间断Galerkin方法,时间离散选用强稳定显式Runge-Kutta方法和具有L稳定对角隐式Runge-Kutta方法相结合的三阶隐...针对一维四阶线性方程,研究了一种隐显式Runge-Kutta全离散局部间断Galerkin方法的稳定性和最优误差估计。空间离散采用局部间断Galerkin方法,时间离散选用强稳定显式Runge-Kutta方法和具有L稳定对角隐式Runge-Kutta方法相结合的三阶隐显式Runge-Kutta方法,数值流通量采用广义交替数值流通量,从而得到全离散LDG格式,分析了该格式的稳定性,同时引入全局Gauss-Radau投影,证明该格式具有k+1阶收敛。最后通过数值实验验证理论结果的正确性。The stability and error estimation of an implicit-explicit Runge-Kutta fully discrete local discontinuous Galerkin method for one-dimensional fourth-order linear equations are studied. The local discontinuity Galerkin method is used for spatial discretization, and the third-order implicit-explicit Runge-Kutta method combining the strong-stability-preserving explicit Runge-Kutta method and the implicit Runge-Kutta method with L-stable diagonal implicit is used for time marching, and the numerical circulation adopts the generalized alternating numerical flux, so as to obtain the fully discrete LDG scheme, and the stability of the scheme is analyzed, and the generalized Gauss-Radau projection is introduced to prove that the scheme has k+1order convergence. Finally, the theoretical results are verified by numerical experiments.展开更多
本文主要研究一维扩展的Fisher-Kolmogorov方程的有效数值算法。通过结合BDF2时间离散格式与直接间断有限元算法对一维扩展的Fisher-Kolmogorov方程进行求解。首先,引入辅助变量,将四阶的扩展的Fisher-Kolmogorov方程转化为低阶耦合方程...本文主要研究一维扩展的Fisher-Kolmogorov方程的有效数值算法。通过结合BDF2时间离散格式与直接间断有限元算法对一维扩展的Fisher-Kolmogorov方程进行求解。首先,引入辅助变量,将四阶的扩展的Fisher-Kolmogorov方程转化为低阶耦合方程,然后利用直接间断有限元求解耦合方程,最后使用BDF2方法,对时间格式进行离散。本文给出了详细的数值算法,并通过一个一维算例进行数值试验,验证了算法的有效性和收敛性。This paper mainly studies the effective numerical algorithm for the one-dimensional extended Fisher-Kolmogorov equation. By combining the BDF2 time discretization format with the direct discontinuous finite element algorithm, the one-dimensional extended Fisher-Kolmogorov equation is solved. Firstly, an auxiliary variable is introduced to transform the fourth-order extended Fisher-Kolmogorov equation into a low-order coupled equation. Then, the coupled equation is solved by using the direct discontinuous finite element method. Finally, the BDF2 method is used to discretize the time scheme. The detailed numerical algorithm is presented in this paper, and a one-dimensional example is used for numerical experiments to verify the effectiveness and convergence of the algorithm.展开更多
In this paper,we design a new error estimator and give a posteriori error analysis for a poroelasticity model.To better overcome“locking phenomenon”on pressure and displacement,we proposed a new error estimators bas...In this paper,we design a new error estimator and give a posteriori error analysis for a poroelasticity model.To better overcome“locking phenomenon”on pressure and displacement,we proposed a new error estimators based on multiphysics discontinuous Galerkin method for the poroelasticity model.And we prove the upper and lower bound of the proposed error estimators,which are numerically demonstrated to be computationally very efficient.Finally,we present numerical examples to verify and validate the efficiency of the proposed error estimators,which show that the adaptive scheme can overcome“locking phenomenon”and greatly reduce the computation cost.展开更多
In this paper,we develop an advanced computational framework for the topology optimization of orthotropic materials using meshless methods.The approximation function is established based on the improved moving least s...In this paper,we develop an advanced computational framework for the topology optimization of orthotropic materials using meshless methods.The approximation function is established based on the improved moving least squares(IMLS)method,which enhances the efficiency and stability of the numerical solution.The numerical solution formulas are derived using the improved element-free Galerkin(IEFG)method.We introduce the solid isotropic microstructures with penalization(SIMP)model to formulate a mathematical model for topology opti-mization,which effectively penalizes intermediate densities.The optimization problem is defined with the numerical solution formula and volume fraction as constraints.The objective function,which is the minimum value of flexibility,is optimized iteratively using the optimization criterion method to update the design variables efficiently and converge to an optimal solution.Sensitivity analysis is performed using the adjoint method,which provides accurate and efficient gradient information for the optimization algorithm.We validate the proposed framework through a series of numerical examples,including clamped beam,cantilever beam,and simply supported beam made of orthotropic materials.The convergence of the objective function is demonstrated by increasing the number of iterations.Additionally,the stability of the iterative process is analyzed by examining the fluctuation law of the volume fraction.By adjusting the parameters to an appropriate range,we achieve the final optimization results of the IEFG method without the checkerboard phenomenon.Comparative studies between the Element-Free Galerkin(EFG)and IEFG methods reveal that both methods yield consistent optimization results under identical parameter settings.However,the IEFG method significantly reduces computational time,highlighting its efficiency and suitability for orthotropic materials.展开更多
本文基于直接间断有限元(DDG)方法数值求解非局部粘性水波模型。该算法结合L1近似公式与BDF2方法,系统构建了非线性时间分数阶偏微分方程的高效数值算法。首先,运用分部积分法对模型的弱形式进行降阶处理。其次,通过引入边界项和选用合...本文基于直接间断有限元(DDG)方法数值求解非局部粘性水波模型。该算法结合L1近似公式与BDF2方法,系统构建了非线性时间分数阶偏微分方程的高效数值算法。首先,运用分部积分法对模型的弱形式进行降阶处理。其次,通过引入边界项和选用合适的数值通量,确保离散格式的稳定性。最后,针对时间导数项应用L1近似公式与BDF2时间差分的离散方法,建立全离散DDG格式。文中详细给出数值格式的构造过程并严格证明该算法的稳定性。数值实验部分选取无已知解析解的水波模型,验证该算法在时空离散上的高精度特性。This paper numerically solves the nonlocal viscous water wave model based on the Direct Discontinuous Galerkin (DDG) method. This algorithm combines the L1 approximation formula with the BDF2 method to systematically construct an efficient numerical algorithm for nonlinear time-fractional partial differential equations. Firstly, the integration by parts method is used to reduce the order of the weak formula. Secondly, the stability of the discrete scheme is ensured by introducing the boundary term and constructing a stable numerical flux. Finally, for the time derivative term, the discrete methods of the L1 approximation formula and the BDF2 time difference are applied to construct the fully discrete DDG scheme. This paper gives a detailed description of the construction process of the numerical scheme and strictly proves the stability of this algorithm. In the numerical experiment part, a water wave model without a known analytical solution is selected to verify the high-precision characteristics of this algorithm in space-time discretization.展开更多
We present the approaches to implementing the k-√k L turbulence model within the framework of the high-order discontinuous Galerkin(DG)method.We use the DG discretization to solve the full Reynolds-averaged Navier-St...We present the approaches to implementing the k-√k L turbulence model within the framework of the high-order discontinuous Galerkin(DG)method.We use the DG discretization to solve the full Reynolds-averaged Navier-Stokes equations.In order to enhance the robustness of approaches,some effective techniques are designed.The HWENO(Hermite weighted essentially non-oscillatory)limiting strategy is adopted for stabilizing the turbulence model variable k.Modifications have been made to the model equation itself by using the auxiliary variable that is always positive.The 2nd-order derivatives of velocities required in computing the von Karman length scale are evaluated in a way to maintain the compactness of DG methods.Numerical results demonstrate that the approaches have achieved the desirable accuracy for both steady and unsteady turbulent simulations.展开更多
基金supported by the National Natural Science Foundation of China(Grant No.12271341).
文摘The element-free Galerkin(EFG)method,which constructs shape functions via moving least squares(MLS)approximation,represents a fundamental and widely studied meshless method in numerical computation.Although it achieves high computational accuracy,the shape functions are more complex than those in the conventional finite element method(FEM),resulting in great computational requirements.Therefore,improving the computational efficiency of the EFG method represents an important research direction.This paper systematically reviews significant contributions fromdomestic and international scholars in advancing the EFGmethod.Including the improved element-free Galerkin(IEFG)method,various interpolating EFG methods,four distinct complex variable EFG methods,and a series of dimension splitting meshless methods.In the numerical examples,the effectiveness and efficiency of the three methods are validated by analyzing the solutions of the IEFG method for 3D steadystate anisotropic heat conduction,3D elastoplasticity,and large deformation problems,as well as the performance of two-dimensional splitting meshless methods in solving the 3D Helmholtz equation.
文摘基于对单元能量投影(element energy projection,EEP)法误差项的直接推导及分析,用EEP简约格式的解计算出略掉的误差项,反补后得到比简约格式高一阶精度的EEP超收敛计算的加强格式。该文以一维Galerkin有限元为例,给出EEP加强格式的算法公式和数学证明。理论分析和算例验证表明:对于m (≥1)次单元,采用EEP加强格式计算的内点位移和内点导数都具有h^(min(m+3,2m))阶的收敛精度,对系数特例问题二者甚至可以分别达到h^(min(m+5,2m))和h^(min(m+4,2m))阶的收敛精度。并对该法的进一步拓展作了讨论。
文摘针对一维四阶线性方程,研究了一种隐显式Runge-Kutta全离散局部间断Galerkin方法的稳定性和最优误差估计。空间离散采用局部间断Galerkin方法,时间离散选用强稳定显式Runge-Kutta方法和具有L稳定对角隐式Runge-Kutta方法相结合的三阶隐显式Runge-Kutta方法,数值流通量采用广义交替数值流通量,从而得到全离散LDG格式,分析了该格式的稳定性,同时引入全局Gauss-Radau投影,证明该格式具有k+1阶收敛。最后通过数值实验验证理论结果的正确性。The stability and error estimation of an implicit-explicit Runge-Kutta fully discrete local discontinuous Galerkin method for one-dimensional fourth-order linear equations are studied. The local discontinuity Galerkin method is used for spatial discretization, and the third-order implicit-explicit Runge-Kutta method combining the strong-stability-preserving explicit Runge-Kutta method and the implicit Runge-Kutta method with L-stable diagonal implicit is used for time marching, and the numerical circulation adopts the generalized alternating numerical flux, so as to obtain the fully discrete LDG scheme, and the stability of the scheme is analyzed, and the generalized Gauss-Radau projection is introduced to prove that the scheme has k+1order convergence. Finally, the theoretical results are verified by numerical experiments.
文摘本文主要研究一维扩展的Fisher-Kolmogorov方程的有效数值算法。通过结合BDF2时间离散格式与直接间断有限元算法对一维扩展的Fisher-Kolmogorov方程进行求解。首先,引入辅助变量,将四阶的扩展的Fisher-Kolmogorov方程转化为低阶耦合方程,然后利用直接间断有限元求解耦合方程,最后使用BDF2方法,对时间格式进行离散。本文给出了详细的数值算法,并通过一个一维算例进行数值试验,验证了算法的有效性和收敛性。This paper mainly studies the effective numerical algorithm for the one-dimensional extended Fisher-Kolmogorov equation. By combining the BDF2 time discretization format with the direct discontinuous finite element algorithm, the one-dimensional extended Fisher-Kolmogorov equation is solved. Firstly, an auxiliary variable is introduced to transform the fourth-order extended Fisher-Kolmogorov equation into a low-order coupled equation. Then, the coupled equation is solved by using the direct discontinuous finite element method. Finally, the BDF2 method is used to discretize the time scheme. The detailed numerical algorithm is presented in this paper, and a one-dimensional example is used for numerical experiments to verify the effectiveness and convergence of the algorithm.
基金supported by the National Natural Science Foundation of China(Grant Nos.12371393 and 11971150)Natural Science Foundation of Henan(Grant No.242300421047).
文摘In this paper,we design a new error estimator and give a posteriori error analysis for a poroelasticity model.To better overcome“locking phenomenon”on pressure and displacement,we proposed a new error estimators based on multiphysics discontinuous Galerkin method for the poroelasticity model.And we prove the upper and lower bound of the proposed error estimators,which are numerically demonstrated to be computationally very efficient.Finally,we present numerical examples to verify and validate the efficiency of the proposed error estimators,which show that the adaptive scheme can overcome“locking phenomenon”and greatly reduce the computation cost.
基金supported by the Graduate Student Scientific Research Innovation Project through Research Innovation Fund for Graduate Students in Shanxi Province(Project No.2024KY648).
文摘In this paper,we develop an advanced computational framework for the topology optimization of orthotropic materials using meshless methods.The approximation function is established based on the improved moving least squares(IMLS)method,which enhances the efficiency and stability of the numerical solution.The numerical solution formulas are derived using the improved element-free Galerkin(IEFG)method.We introduce the solid isotropic microstructures with penalization(SIMP)model to formulate a mathematical model for topology opti-mization,which effectively penalizes intermediate densities.The optimization problem is defined with the numerical solution formula and volume fraction as constraints.The objective function,which is the minimum value of flexibility,is optimized iteratively using the optimization criterion method to update the design variables efficiently and converge to an optimal solution.Sensitivity analysis is performed using the adjoint method,which provides accurate and efficient gradient information for the optimization algorithm.We validate the proposed framework through a series of numerical examples,including clamped beam,cantilever beam,and simply supported beam made of orthotropic materials.The convergence of the objective function is demonstrated by increasing the number of iterations.Additionally,the stability of the iterative process is analyzed by examining the fluctuation law of the volume fraction.By adjusting the parameters to an appropriate range,we achieve the final optimization results of the IEFG method without the checkerboard phenomenon.Comparative studies between the Element-Free Galerkin(EFG)and IEFG methods reveal that both methods yield consistent optimization results under identical parameter settings.However,the IEFG method significantly reduces computational time,highlighting its efficiency and suitability for orthotropic materials.
文摘本文基于直接间断有限元(DDG)方法数值求解非局部粘性水波模型。该算法结合L1近似公式与BDF2方法,系统构建了非线性时间分数阶偏微分方程的高效数值算法。首先,运用分部积分法对模型的弱形式进行降阶处理。其次,通过引入边界项和选用合适的数值通量,确保离散格式的稳定性。最后,针对时间导数项应用L1近似公式与BDF2时间差分的离散方法,建立全离散DDG格式。文中详细给出数值格式的构造过程并严格证明该算法的稳定性。数值实验部分选取无已知解析解的水波模型,验证该算法在时空离散上的高精度特性。This paper numerically solves the nonlocal viscous water wave model based on the Direct Discontinuous Galerkin (DDG) method. This algorithm combines the L1 approximation formula with the BDF2 method to systematically construct an efficient numerical algorithm for nonlinear time-fractional partial differential equations. Firstly, the integration by parts method is used to reduce the order of the weak formula. Secondly, the stability of the discrete scheme is ensured by introducing the boundary term and constructing a stable numerical flux. Finally, for the time derivative term, the discrete methods of the L1 approximation formula and the BDF2 time difference are applied to construct the fully discrete DDG scheme. This paper gives a detailed description of the construction process of the numerical scheme and strictly proves the stability of this algorithm. In the numerical experiment part, a water wave model without a known analytical solution is selected to verify the high-precision characteristics of this algorithm in space-time discretization.
基金supported by the National Natural Science Foundation of China(Grant Nos.92252201 and 11721202)the Fundamental Research Funds for the Central Universities.
文摘We present the approaches to implementing the k-√k L turbulence model within the framework of the high-order discontinuous Galerkin(DG)method.We use the DG discretization to solve the full Reynolds-averaged Navier-Stokes equations.In order to enhance the robustness of approaches,some effective techniques are designed.The HWENO(Hermite weighted essentially non-oscillatory)limiting strategy is adopted for stabilizing the turbulence model variable k.Modifications have been made to the model equation itself by using the auxiliary variable that is always positive.The 2nd-order derivatives of velocities required in computing the von Karman length scale are evaluated in a way to maintain the compactness of DG methods.Numerical results demonstrate that the approaches have achieved the desirable accuracy for both steady and unsteady turbulent simulations.
