Modified PHT-splines And Its Applications

PHT-splines(Polynomial splines over Hierarchical T-meshes)possess many ad-vantages,i.e.,PHT-spline surfaces have properties such as convex-hull property,affine invariant,local support,etc.and the basis functions have properties such as linear in-dependence,partition of unity,etc.Based on these good properties,PHT-splines have been successfully applied in fitting models and isogeometric analysis.Note that the lo-cal refinement for hierarchical T-meshes of PHT-splines is achieved by cross insertion,i.e.,splitting each candidate cell into four subcells by inserting a cross.This simple refinement rule may still introduce redundant control points or coefficients when the application of PHT-splines to problems is with anisotropic features,i.e.,the problems are directionally dependent.In addition,some basis functions of PHT-splines may de-cay severely during the process of certain refinement of hierarchical T-meshes.The decay may cause the isogeometric analysis matrices assembled by these basis functions to be ill-conditioned.It is also known that the results of parametrization not only affect the accuracy and convergence speed of solutions but also affect the computation cost during the process of isogeometric analysis.In addition,there are some advantages for the WEB method(Weight Extended B-splines method)in solving equations.The prob-lem that how to better apply PHT-splines to the WEB method is worthy to consider.In this paper,we try to address the above problems,which are consisted of the following three parts:This paper proposes MPHT-splines(Modified PHT-splines)that aim to improve the flexibility of PHT-splines,which can handle models or problems with anisotropic features efficiently.Instead of performing refinement only by cross insertion,modified hierarchical T-meshes are obtained by allowing for the splitting of cells into halves(hor-izontal or vertical).We propose a refinement algorithm based on the anisotropic infor-mation and the adjacent relationships of cells in the mesh.Numerical experiments show that our new splines have advantages when it is applied to problems with anisotropic features.As a generalization of PHT-splines,MPHT splines are also faced with the problem that some basis functions may decay severely as PHT-splines during the process of cer-tain refinement of hierarchical T-meshes.In order to overcome this defect,we present a method to modify the basis functions of MPHT-splines when the supports of the original truncated basis functions are rectangular domains.Hence,the truncation operations are less used,which alleviates the decay phenomenon of MPHT-splines.In addition,the modified basis functions preserve the nice properties of the original MPHT-spline basis functions,e.g.,the partition of unity,local support,and linear independence.Numerical examples show that the condition numbers of isogeometric analysis matrices decrease significantly when new basis functions are applied.This paper also presents an adaptive collocation method with weighted extended PHT-splines.We modify the classification rules for basis functions based on the rela-tion between the basis vertices and the computational domain.The Gaussian points are chosen to be collocation points since PHT-splines are CI continuous.We also provide relocation techniques to solve the mismatch problem between the number of basis func-tions and the number of interpolation conditions.Compared to the traditional Greville collocation method,our new approach has improved accuracy with fewer oscillations.In summary,we generalize PHT-splines to MPHT-splines.For MPHT-splines,the improvement strategy of basis function is further discussed.In addition,we also intro-duce a collocation method for weighted extended MPHT-splines to solve partial differ-ential equations defined over complex domains.