With the improved moving least-squares (IMLS) approximation, an orthogonal function system with a weight function is used as the basis function. The combination of the element-free Galerkin (EFG) method and the IMLS a...With the improved moving least-squares (IMLS) approximation, an orthogonal function system with a weight function is used as the basis function. The combination of the element-free Galerkin (EFG) method and the IMLS approximation leads to the development of the improved element-free Galerkin (IEFG) method. In this paper, the IEFG method is applied to study the partial differential equations that control the heat flow in three-dimensional space. With the IEFG technique, the Galerkin weak form is employed to develop the discretized system equations, and the penalty method is applied to impose the essential boundary conditions. The traditional difference method for two-point boundary value problems is selected for the time discretization. As the transient heat conduction equations and the boundary and initial conditions are time dependent, the scaling parameter, number of nodes and time step length are considered in a convergence study.展开更多
The multiple patterns of internal solitary wave interactions(ISWI)are a complex oceanic phenomenon.Satellite remote sensing techniques indirectly detect these ISWI,but do not provide information on their detailed stru...The multiple patterns of internal solitary wave interactions(ISWI)are a complex oceanic phenomenon.Satellite remote sensing techniques indirectly detect these ISWI,but do not provide information on their detailed structure and dynamics.Recently,the authors considered a three-layer fluid with shear flow and developed a(2+1)Kadomtsev-Petviashvili(KP)model that is capable of describing five types of oceanic ISWI,including O-type,P-type,TO-type,TP-type,and Y-shaped.Deep learning models,particularly physics-informed neural networks(PINN),are widely used in the field of fluids and internal solitary waves.However,the authors find that the amplitude of internal solitary waves is much smaller than the wavelength and the ISWI occur at relatively large spatial scales,and these characteristics lead to an imbalance in the loss function of the PINN model.To solve this problem,the authors introduce two weighted loss function methods,the fixed weighing and the adaptive weighting methods,to improve the PINN model.This successfully simulated the detailed structure and dynamics of ISWI,with simulation results corresponding to the satellite images.In particular,the adaptive weighting method can automatically update the weights of different terms in the loss function and outperforms the fixed weighting method in terms of generalization ability.展开更多
In this paper, the solution to the structure consisting of a bead and a board is given as a result of the application of the subregion function method which was suggested in ref. [1]. The same problem is also computed...In this paper, the solution to the structure consisting of a bead and a board is given as a result of the application of the subregion function method which was suggested in ref. [1]. The same problem is also computed with finite element method. The comparison between the two results shows that the application of the subregion function in the method of weighted residuals is practical and effective, especially for solving compound structures.展开更多
An independent method for paper [10] is presented. Weighted lattice paths are enumerated by counting function which is a natural extension of Gaussian multinomial coefficient in the case of unrestricted paths. Convolu...An independent method for paper [10] is presented. Weighted lattice paths are enumerated by counting function which is a natural extension of Gaussian multinomial coefficient in the case of unrestricted paths. Convolutions for path counts are investigated, which yields some Vandcrmondc-type identities for multinomial and q-multinomial coefficients.展开更多
The quadratic penalty function is considered for finding the weighted analytic center ofa polytope.By an-alyzing the properties of the penalty function,an exterior central path is introduced.It is shown that the exter...The quadratic penalty function is considered for finding the weighted analytic center ofa polytope.By an-alyzing the properties of the penalty function,an exterior central path is introduced.It is shown that the exterior cen-tral path has some similar properties as the interior one in a line ar programming case,If the starting point is close to the path,by following the path with an appropriate step,a polynomia1 algorithm is derived.Usually,the starting point is not in the polytope,the method can be seen as an infeasible interior point one.It is also discussed how to get an appropriate starting point.展开更多
基金the National Natural Science Foundation of China (Grant No. 11171208)Shanghai Leading Academic Discipline Project (Grant No. S30106)
文摘With the improved moving least-squares (IMLS) approximation, an orthogonal function system with a weight function is used as the basis function. The combination of the element-free Galerkin (EFG) method and the IMLS approximation leads to the development of the improved element-free Galerkin (IEFG) method. In this paper, the IEFG method is applied to study the partial differential equations that control the heat flow in three-dimensional space. With the IEFG technique, the Galerkin weak form is employed to develop the discretized system equations, and the penalty method is applied to impose the essential boundary conditions. The traditional difference method for two-point boundary value problems is selected for the time discretization. As the transient heat conduction equations and the boundary and initial conditions are time dependent, the scaling parameter, number of nodes and time step length are considered in a convergence study.
基金supported by the National Natural Science Foundation of China under Grant Nos.12275085,12235007,and 12175069Science and Technology Commission of Shanghai Municipality under Grant Nos.21JC1402500 and 22DZ2229014.
文摘The multiple patterns of internal solitary wave interactions(ISWI)are a complex oceanic phenomenon.Satellite remote sensing techniques indirectly detect these ISWI,but do not provide information on their detailed structure and dynamics.Recently,the authors considered a three-layer fluid with shear flow and developed a(2+1)Kadomtsev-Petviashvili(KP)model that is capable of describing five types of oceanic ISWI,including O-type,P-type,TO-type,TP-type,and Y-shaped.Deep learning models,particularly physics-informed neural networks(PINN),are widely used in the field of fluids and internal solitary waves.However,the authors find that the amplitude of internal solitary waves is much smaller than the wavelength and the ISWI occur at relatively large spatial scales,and these characteristics lead to an imbalance in the loss function of the PINN model.To solve this problem,the authors introduce two weighted loss function methods,the fixed weighing and the adaptive weighting methods,to improve the PINN model.This successfully simulated the detailed structure and dynamics of ISWI,with simulation results corresponding to the satellite images.In particular,the adaptive weighting method can automatically update the weights of different terms in the loss function and outperforms the fixed weighting method in terms of generalization ability.
文摘In this paper, the solution to the structure consisting of a bead and a board is given as a result of the application of the subregion function method which was suggested in ref. [1]. The same problem is also computed with finite element method. The comparison between the two results shows that the application of the subregion function in the method of weighted residuals is practical and effective, especially for solving compound structures.
文摘An independent method for paper [10] is presented. Weighted lattice paths are enumerated by counting function which is a natural extension of Gaussian multinomial coefficient in the case of unrestricted paths. Convolutions for path counts are investigated, which yields some Vandcrmondc-type identities for multinomial and q-multinomial coefficients.
文摘The quadratic penalty function is considered for finding the weighted analytic center ofa polytope.By an-alyzing the properties of the penalty function,an exterior central path is introduced.It is shown that the exterior cen-tral path has some similar properties as the interior one in a line ar programming case,If the starting point is close to the path,by following the path with an appropriate step,a polynomia1 algorithm is derived.Usually,the starting point is not in the polytope,the method can be seen as an infeasible interior point one.It is also discussed how to get an appropriate starting point.
