This paper presents the isogeometric least-squares collocatio method,which determines the numerical solution by making the approximate differential operator fit the real differential operator in a least-squares sense....This paper presents the isogeometric least-squares collocatio method,which determines the numerical solution by making the approximate differential operator fit the real differential operator in a least-squares sense.The number of collocation points employed in IGA-L can be larger than that of the unknowns.Theoretical analysis and numerical examples presented in this paper show the superiority of IGA-L over state-of-the-art collocation methods.First,a small increase in the number of collocation points in IGA-L leads to a large improvement in the accuracy of its numerical solution.Second,IGA-L method is more flexible and more stable,because the number of collocation points in IGA-L is variable.Third,IGA-L is convergent in some cases of singular parameterization.Moreover,the consistency and convergence analysis are also developed in this paper.展开更多
Collocation has become a standard tool for approximation of parameterized systems in the uncertainty quantification(UQ)community.Techniques for leastsquares regularization,compressive sampling recovery,and interpolato...Collocation has become a standard tool for approximation of parameterized systems in the uncertainty quantification(UQ)community.Techniques for leastsquares regularization,compressive sampling recovery,and interpolatory reconstruction are becoming standard tools used in a variety of applications.Selection of a collocation mesh is frequently a challenge,but methods that construct geometrically unstructured collocation meshes have shown great potential due to attractive theoretical properties and direct,simple generation and implementation.We investigate properties of these meshes,presenting stability and accuracy results that can be used as guides for generating stochastic collocation grids in multiple dimensions.展开更多
A boundary collocation method based on the least-square technique and a corresponding adaptive computation process have been developed for the plate bending problem. The trial functions are constructed using a series ...A boundary collocation method based on the least-square technique and a corresponding adaptive computation process have been developed for the plate bending problem. The trial functions are constructed using a series of the biharmonic polynomials, and the local error indicators are given by the residuals of the energy density on the boundary. In comparison with the conventional collocation methods, the solution accuracy in the present method can be improved in an economical and efficient way. In order to demonstrate the efficiency and advantages of the adaptive boundary collocation method proposed in this paper, two numerical examples are presented for circular plates subjected to uniform loads and restrained by mixed boundary conditions. The numerical results for the examples show good agreement with ones presented in the literature.展开更多
At present,there exist two methods used to recover the bathymetry from altimeter data,i.e.the deterministic method and the stochastic method.In this paper,the principles of the two methods are introduced first.Then ac...At present,there exist two methods used to recover the bathymetry from altimeter data,i.e.the deterministic method and the stochastic method.In this paper,the principles of the two methods are introduced first.Then according to the theory of least_square collocation,a modified statistical model for recovering bathymetry from altimeter data is proposed.The new model has been used for computing the ocean depth in the South China Sea from altimeter_derived gravity anomalies.Finally the predicted depths are compared with the ship_borne depth.It shows that they agree with each other very well.展开更多
One of major difficulties in the implementation of meshfree methods using the mov- ing least square （MLS） approximation, such as element-free Galerkin method （EFG）, is the im- position of essential boundary condit...One of major difficulties in the implementation of meshfree methods using the mov- ing least square （MLS） approximation, such as element-free Galerkin method （EFG）, is the im- position of essential boundary conditions as the approximations do not pass through the nodal parameter values. Another class of meshfree methods based on the radial basis point interpola- tion can satisfy the essential boundary conditions exactly since its approximation function passes through each node in an influence domain and thus its shape functions possess the properties of delta function. In this paper, a coupled element-free Galerkin（EFG）-radial point interpola- tion method （RPIM） is proposed to enhance their advantages and avoid their disadvantages. Discretized equations of equilibrium are obtained in the RPIM region and the EFG region, respectively. Then a collocation approach is introduced to couple the RPIM and the EFG method. This method satisfies the linear consistency exactly and can maintain the stiffness matrix symmetric. Numerical tests show that this method gives reasonably accurate results consistent with the theory.展开更多
基金supported by the National Natural Science Foundation of China under Grant No.61872316the Natural Science Foundation of Zhejiang Province under Grant No.LY19F020004。
文摘This paper presents the isogeometric least-squares collocatio method,which determines the numerical solution by making the approximate differential operator fit the real differential operator in a least-squares sense.The number of collocation points employed in IGA-L can be larger than that of the unknowns.Theoretical analysis and numerical examples presented in this paper show the superiority of IGA-L over state-of-the-art collocation methods.First,a small increase in the number of collocation points in IGA-L leads to a large improvement in the accuracy of its numerical solution.Second,IGA-L method is more flexible and more stable,because the number of collocation points in IGA-L is variable.Third,IGA-L is convergent in some cases of singular parameterization.Moreover,the consistency and convergence analysis are also developed in this paper.
基金T.Zhou is supported by the National Natural Science Foundation of China(Award Nos.91130003 and 11201461).
文摘Collocation has become a standard tool for approximation of parameterized systems in the uncertainty quantification(UQ)community.Techniques for leastsquares regularization,compressive sampling recovery,and interpolatory reconstruction are becoming standard tools used in a variety of applications.Selection of a collocation mesh is frequently a challenge,but methods that construct geometrically unstructured collocation meshes have shown great potential due to attractive theoretical properties and direct,simple generation and implementation.We investigate properties of these meshes,presenting stability and accuracy results that can be used as guides for generating stochastic collocation grids in multiple dimensions.
基金the National Natural Science Foundation of China (No. 10472051)
文摘A boundary collocation method based on the least-square technique and a corresponding adaptive computation process have been developed for the plate bending problem. The trial functions are constructed using a series of the biharmonic polynomials, and the local error indicators are given by the residuals of the energy density on the boundary. In comparison with the conventional collocation methods, the solution accuracy in the present method can be improved in an economical and efficient way. In order to demonstrate the efficiency and advantages of the adaptive boundary collocation method proposed in this paper, two numerical examples are presented for circular plates subjected to uniform loads and restrained by mixed boundary conditions. The numerical results for the examples show good agreement with ones presented in the literature.
文摘At present,there exist two methods used to recover the bathymetry from altimeter data,i.e.the deterministic method and the stochastic method.In this paper,the principles of the two methods are introduced first.Then according to the theory of least_square collocation,a modified statistical model for recovering bathymetry from altimeter data is proposed.The new model has been used for computing the ocean depth in the South China Sea from altimeter_derived gravity anomalies.Finally the predicted depths are compared with the ship_borne depth.It shows that they agree with each other very well.
基金supported by the National Natural Science Foundation of China (No. 11172192)the College Postgraduate Research and Innovation Project of Jiangsu Province (No. CX10B 029Z)the Nominated Excellent Thesis for PHD Candidates Program of Soochow University (No. 23320957)
文摘One of major difficulties in the implementation of meshfree methods using the mov- ing least square （MLS） approximation, such as element-free Galerkin method （EFG）, is the im- position of essential boundary conditions as the approximations do not pass through the nodal parameter values. Another class of meshfree methods based on the radial basis point interpola- tion can satisfy the essential boundary conditions exactly since its approximation function passes through each node in an influence domain and thus its shape functions possess the properties of delta function. In this paper, a coupled element-free Galerkin（EFG）-radial point interpola- tion method （RPIM） is proposed to enhance their advantages and avoid their disadvantages. Discretized equations of equilibrium are obtained in the RPIM region and the EFG region, respectively. Then a collocation approach is introduced to couple the RPIM and the EFG method. This method satisfies the linear consistency exactly and can maintain the stiffness matrix symmetric. Numerical tests show that this method gives reasonably accurate results consistent with the theory.
