PHT-splines are defined as polynomial splines over hierarchical T-meshes with very efficient local refinement properties.The original PHT-spline basis functions constructed by the truncation mechanism have a decay phe...PHT-splines are defined as polynomial splines over hierarchical T-meshes with very efficient local refinement properties.The original PHT-spline basis functions constructed by the truncation mechanism have a decay phenomenon,resulting in numerical instability.The non-decay basis functions are constructed as the B-splines that are defined on the 2×2 tensor product meshes associated with basis vertices in Kang et al.,but at the cost of losing the partition of unity.In the field of finite element analysis and topology optimization,forming the partition of unity is the default ingredient for constructing basis functions of approximate spaces.In this paper,we will show that the non-decay PHT-spline basis functions proposed by Kang et al.can be appropriately modified to form a partition of unity.Each non-decay basis function is multiplied by a positive weight to form the weighted basis.The weights are solved such that the sum of weighted bases is equal to 1 on the domain.We provide two methods for calculatingweights,based on geometric information of basis functions and the subdivision of PHT-splines.Weights are given in the form of explicit formulas and can be efficiently calculated.We also prove that the weights on the admissible hierarchical T-meshes are positive.展开更多
This paper presents an adaptive collocation method with weighted extended PHT-splines.The authors modify the classification rules for basis functions based on the relation between the basis vertices and the computatio...This paper presents an adaptive collocation method with weighted extended PHT-splines.The authors modify the classification rules for basis functions based on the relation between the basis vertices and the computational domain. The Gaussian points are chosen to be collocation points since PHT-splines are C1 continuous. The authors also provide relocation techniques to resolve the mismatch problem between the number of basis functions and the number of interpolation conditions. Compared to the traditional Greville collocation method, the new approach has improved accuracy with fewer oscillations. Several numerical examples are also provided to test our the proposed approach.展开更多
Recently,it was found that during the process of certain refinement of hier-archical T-meshes,some basis functions of PHT-splines decay severely,which is not expected in solving numerical PDEs and in least square data...Recently,it was found that during the process of certain refinement of hier-archical T-meshes,some basis functions of PHT-splines decay severely,which is not expected in solving numerical PDEs and in least square data fitting since the matrices assembled by these basis functions are likely to be ill-conditioned.In this paper,we present a method to modify the basis functions of PHT-splines in the case that the sup-ports of the original truncated basis functions are rectangular domains to overcome the decay problem.The modified basis functions preserve the same nice properties of the original PHT-spline basis functions such as partition of unity,local support,linear independency.Numerical examples show that the modified basis functions can greatly decrease the condition numbers of the stiffness matrices assembled in solving Poisson’s equation with Dirichlet boundary conditions.展开更多
In this paper, we present a new surface reconstruction algorithm for polynomial spline surfaces of S(3, 3, 1, 1, T) over arbitrary hierarchical T-mesh T. The surface is piecewisely constructed by Coons surface interpo...In this paper, we present a new surface reconstruction algorithm for polynomial spline surfaces of S(3, 3, 1, 1, T) over arbitrary hierarchical T-mesh T. The surface is piecewisely constructed by Coons surface interpolation of the 16 parameters at four vertices on each rectangular cell of hierarchical T-mesh. For a given hierarchical T-mesh T and geometric information(the function values, the two first order partial derivatives and the mixed partial derivatives) at corresponding basis vertices of the hierarchical T-mesh, the surface is the same as the polynomial spline surface of S(3, 3, 1, 1, T), but our algorithm avoids the complexity of PHT-spline basis functions. Moreover, we give an adaptively refined surface algorithm for fitting scattered data points based on piecewise Coons surface construction. The numerical results show that the proposed adaptive algorithm is efficient in fitting scattered data points.展开更多
In isogeometric analysis(IGA),parametrization is an important and difficultissue that greatly influences the numerical accuracy and efficiency of the numericalsolution.One of the problems facing the parametrization in...In isogeometric analysis(IGA),parametrization is an important and difficultissue that greatly influences the numerical accuracy and efficiency of the numericalsolution.One of the problems facing the parametrization in IGA is the existence ofthe singular points in the parametrization domain.To avoid producing singular points,boundary-mapping parametrization is given by mapping the computational domainto a polygon domain which may not be a square domain and mapping each segmentof the boundary in computational domain to a corresponding boundary edge of thepolygon.Two numerical examples in finite element analysis are presented to show thenovel parametrization is efficient.展开更多
We study the accuracy and performance of isogeometric analysis on implicit domains when solving time-independent Schrödinger equation.We construct weighted extended PHT-spline basis functions for analysis,and the...We study the accuracy and performance of isogeometric analysis on implicit domains when solving time-independent Schrödinger equation.We construct weighted extended PHT-spline basis functions for analysis,and the domain is presented with same basis functions in implicit form excluding the need for a parameterization step.Moreover,an adaptive refinement process is formulated and discussed with details.The constructed basis functions with cubic polynomials and only C^(1) continuity are enough to produce a higher continuous field approximation while maintaining the computational cost for the matrices as low as possible.A numerical implementation for the adaptivemethod is performed on Schrödinger eigenvalue problem with doublewell potential using 3 examples on different implicit domains.The convergence and performance results demonstrate the efficiency and accuracy of the approach.展开更多
This paper provides a survey of local refinable splines,including hierarchical B-splines,T-splines,polynomial splines over T-meshes,etc.,with a view to applications in geometric modeling and iso-geometric analysis.We ...This paper provides a survey of local refinable splines,including hierarchical B-splines,T-splines,polynomial splines over T-meshes,etc.,with a view to applications in geometric modeling and iso-geometric analysis.We will identify the strengths and weaknesses of these methods and also offer suggestions for their using in geometric modeling and iso-geometric analysis.展开更多
基金The work was supported by the NSF of China(No.11801393)the Natural Science Foundation of Jiangsu Province,China(No.BK20180831).
文摘PHT-splines are defined as polynomial splines over hierarchical T-meshes with very efficient local refinement properties.The original PHT-spline basis functions constructed by the truncation mechanism have a decay phenomenon,resulting in numerical instability.The non-decay basis functions are constructed as the B-splines that are defined on the 2×2 tensor product meshes associated with basis vertices in Kang et al.,but at the cost of losing the partition of unity.In the field of finite element analysis and topology optimization,forming the partition of unity is the default ingredient for constructing basis functions of approximate spaces.In this paper,we will show that the non-decay PHT-spline basis functions proposed by Kang et al.can be appropriately modified to form a partition of unity.Each non-decay basis function is multiplied by a positive weight to form the weighted basis.The weights are solved such that the sum of weighted bases is equal to 1 on the domain.We provide two methods for calculatingweights,based on geometric information of basis functions and the subdivision of PHT-splines.Weights are given in the form of explicit formulas and can be efficiently calculated.We also prove that the weights on the admissible hierarchical T-meshes are positive.
基金supported by the National Natural Science Fondation of China under Grant Nos.11601114,11771420,61772167。
文摘This paper presents an adaptive collocation method with weighted extended PHT-splines.The authors modify the classification rules for basis functions based on the relation between the basis vertices and the computational domain. The Gaussian points are chosen to be collocation points since PHT-splines are C1 continuous. The authors also provide relocation techniques to resolve the mismatch problem between the number of basis functions and the number of interpolation conditions. Compared to the traditional Greville collocation method, the new approach has improved accuracy with fewer oscillations. Several numerical examples are also provided to test our the proposed approach.
基金is supported by the National Natural Science Foundation of China(Nos.11571338,11626253)Postdoctoral Science Foundation of China(2015M571931).
文摘Recently,it was found that during the process of certain refinement of hier-archical T-meshes,some basis functions of PHT-splines decay severely,which is not expected in solving numerical PDEs and in least square data fitting since the matrices assembled by these basis functions are likely to be ill-conditioned.In this paper,we present a method to modify the basis functions of PHT-splines in the case that the sup-ports of the original truncated basis functions are rectangular domains to overcome the decay problem.The modified basis functions preserve the same nice properties of the original PHT-spline basis functions such as partition of unity,local support,linear independency.Numerical examples show that the modified basis functions can greatly decrease the condition numbers of the stiffness matrices assembled in solving Poisson’s equation with Dirichlet boundary conditions.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11572081 11871137)the Program for Liaoning Innovation Talents in University(Grant No.LCR2018001)
文摘In this paper, we present a new surface reconstruction algorithm for polynomial spline surfaces of S(3, 3, 1, 1, T) over arbitrary hierarchical T-mesh T. The surface is piecewisely constructed by Coons surface interpolation of the 16 parameters at four vertices on each rectangular cell of hierarchical T-mesh. For a given hierarchical T-mesh T and geometric information(the function values, the two first order partial derivatives and the mixed partial derivatives) at corresponding basis vertices of the hierarchical T-mesh, the surface is the same as the polynomial spline surface of S(3, 3, 1, 1, T), but our algorithm avoids the complexity of PHT-spline basis functions. Moreover, we give an adaptively refined surface algorithm for fitting scattered data points based on piecewise Coons surface construction. The numerical results show that the proposed adaptive algorithm is efficient in fitting scattered data points.
基金973 Program 2011CB302400the National Natural Sci-ence Foundation of China(Nos.11371341 and 11426236)the 111 Project(No.b07033).
文摘In isogeometric analysis(IGA),parametrization is an important and difficultissue that greatly influences the numerical accuracy and efficiency of the numericalsolution.One of the problems facing the parametrization in IGA is the existence ofthe singular points in the parametrization domain.To avoid producing singular points,boundary-mapping parametrization is given by mapping the computational domainto a polygon domain which may not be a square domain and mapping each segmentof the boundary in computational domain to a corresponding boundary edge of thepolygon.Two numerical examples in finite element analysis are presented to show thenovel parametrization is efficient.
基金The work is supported by the NSF of China(No.11771420).
文摘We study the accuracy and performance of isogeometric analysis on implicit domains when solving time-independent Schrödinger equation.We construct weighted extended PHT-spline basis functions for analysis,and the domain is presented with same basis functions in implicit form excluding the need for a parameterization step.Moreover,an adaptive refinement process is formulated and discussed with details.The constructed basis functions with cubic polynomials and only C^(1) continuity are enough to produce a higher continuous field approximation while maintaining the computational cost for the matrices as low as possible.A numerical implementation for the adaptivemethod is performed on Schrödinger eigenvalue problem with doublewell potential using 3 examples on different implicit domains.The convergence and performance results demonstrate the efficiency and accuracy of the approach.
基金supported by National Natural Science Foundation of China(Grant Nos.11031007 and 60903148)the Chinese Universities Scientific Fund+2 种基金Scientific Research Foundation for the Returned Overseas Chinese ScholarsState Education Ministry,the Chinese Academy of Sciences Startup Scientific Research Foundationthe State Key Development Program for Basic Research of China(973 Program)(Grant No.2011CB302400)
文摘This paper provides a survey of local refinable splines,including hierarchical B-splines,T-splines,polynomial splines over T-meshes,etc.,with a view to applications in geometric modeling and iso-geometric analysis.We will identify the strengths and weaknesses of these methods and also offer suggestions for their using in geometric modeling and iso-geometric analysis.
