T网格上的高光滑阶样条与异度样条
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摘要
T网格上的样条是一种具有局部加细功能的函数。已有的关于T网格上样条空间的理论和应用主要是基于样条次数与光滑度有较大差别的情形,如通常见到的PHT样条是T网格上的双三次C1函数。本文主要研究T网格上高光滑阶的样条空间以及T网格上的异度样条空间。
论文在第一章中概述了T网格上的样条空间的产生背景、已有工作,并且介绍了T网格上的样条空间和同调代数中的相关概念及结论。然后在第二章中回顾了研究T网格上样条空间的一些经典方法以及与本文相关的工作。在第三章中利用同调代数中可分的短正合列理论具体给出了直和分解理论的两种表现形式—基于内大边的满射条件以及基于内边的满射条件。在第四章中,我们利用基于内大边的满射条件研究T网格上高光滑阶的样条空间的维数问题。在第五章中则利用基于内边的满射条件研究T网格上高光滑阶样条空间的基函数构造。我们根据不同的应用领域构造了不同性质的基函数并且把这些基函数做了一些具体的应用。另外,近来等几何分析概念的产生推动了对T网格上各种样条的研究。在这个过程中产生了一些在T网格上新定义的样条函数。从等几何分析理论分析的角度,我们在第六章中分析了直和分解理论在研究这些新定义的样条线性生成空间和对应的T网格上高光滑阶的样条空间之间差别的可行性。
在第七章中我们提出了T网格上异度样条空间的概念并进行了初步的理论研究和应用。特别地,在这一章中我们对用于表达广泛存在的间断数据的-类T网格上的异度样条进行了具体的研究。给出了这类T网格上的异度样条空间的维数以及基函数构造。通过这类T网格上的异度样条在有问断特征的图片数据上的应用,我们可以看出这种样条可以更好的保持原来数据的间断特征。
最后在第八章中我们总结本文,同时从T网格上高光滑阶的样条空间和T网格上异度样条空间这两个方面对未来的工作进行展望。
Splines over T-meshes are a kind of function spaces with ability of local refine-ment. Up to now, the theory of spline spaces over T-meshes mainly focuses on the space with a considerable difference between the degree of the spline space and its smoothness order. For example, PHT-splines are C1piecewise bicubic functions. In this thesis, we work on spline spaces with highest order smoothness and spline spaces with mixed orders of continuity over T-meshes.
In Chapter1, we summarize the research background of spline spaces over T-meshes and the existing work of these spaces. Related concepts and results about spline spaces over T-meshes and Homology theory are also presented in this chap-ter. We review some classical methods for studying spline spaces over T-meshes and the related works in Chapter2."Direct sum decomposition theory" is proposed in Chapter3. This is the main contribution of the thesis. There are two manifestations of this theory based on the theory of short exact sequence in Homology Algebra. One is the surjection condition relying on the interior1-edges, and the other one is the surjection condition relying on the interior edges. In Chapter4, the dimension of a spline space with highest order smoothness over a T-mesh is studied by the sur-jection condition relying on the interior1-edges. The methods of constructing basis functions of this spline space are presented by the surjection condition relying on the interior edges in Chapter5. Moreover, due to interest triggered by the introduction of isogeometric analysis, we also defined new spline functions over T-meshes for iso-geometric analysis. In Chapter6, we analyze the feasibility of comparing the spline spaces spanned by these splines with spline spaces over corresponding T-meshes.
In Chapter7, we make a preliminary study of spline spaces with mixed orders of continuity over T-meshes. In particular, we work on the dimension and the method of constructing a set of basis functions of a type of spline space with mixed orders of continuity over a hierarchical T-mesh which used to process discontinuous data. By the examples of application of this type of spline to process image data, these spline functions keep the features of discontinuity of the discontinuous data in a better way.
In Chapter8, we sum up our paper and present future research problems from the view of spline spaces with highest order smoothness over T-meshes and spline spaces with mixed orders of continuity over T-meshes.
论文在第一章中概述了T网格上的样条空间的产生背景、已有工作,并且介绍了T网格上的样条空间和同调代数中的相关概念及结论。然后在第二章中回顾了研究T网格上样条空间的一些经典方法以及与本文相关的工作。在第三章中利用同调代数中可分的短正合列理论具体给出了直和分解理论的两种表现形式—基于内大边的满射条件以及基于内边的满射条件。在第四章中,我们利用基于内大边的满射条件研究T网格上高光滑阶的样条空间的维数问题。在第五章中则利用基于内边的满射条件研究T网格上高光滑阶样条空间的基函数构造。我们根据不同的应用领域构造了不同性质的基函数并且把这些基函数做了一些具体的应用。另外,近来等几何分析概念的产生推动了对T网格上各种样条的研究。在这个过程中产生了一些在T网格上新定义的样条函数。从等几何分析理论分析的角度,我们在第六章中分析了直和分解理论在研究这些新定义的样条线性生成空间和对应的T网格上高光滑阶的样条空间之间差别的可行性。
在第七章中我们提出了T网格上异度样条空间的概念并进行了初步的理论研究和应用。特别地,在这一章中我们对用于表达广泛存在的间断数据的-类T网格上的异度样条进行了具体的研究。给出了这类T网格上的异度样条空间的维数以及基函数构造。通过这类T网格上的异度样条在有问断特征的图片数据上的应用,我们可以看出这种样条可以更好的保持原来数据的间断特征。
最后在第八章中我们总结本文,同时从T网格上高光滑阶的样条空间和T网格上异度样条空间这两个方面对未来的工作进行展望。
Splines over T-meshes are a kind of function spaces with ability of local refine-ment. Up to now, the theory of spline spaces over T-meshes mainly focuses on the space with a considerable difference between the degree of the spline space and its smoothness order. For example, PHT-splines are C1piecewise bicubic functions. In this thesis, we work on spline spaces with highest order smoothness and spline spaces with mixed orders of continuity over T-meshes.
In Chapter1, we summarize the research background of spline spaces over T-meshes and the existing work of these spaces. Related concepts and results about spline spaces over T-meshes and Homology theory are also presented in this chap-ter. We review some classical methods for studying spline spaces over T-meshes and the related works in Chapter2."Direct sum decomposition theory" is proposed in Chapter3. This is the main contribution of the thesis. There are two manifestations of this theory based on the theory of short exact sequence in Homology Algebra. One is the surjection condition relying on the interior1-edges, and the other one is the surjection condition relying on the interior edges. In Chapter4, the dimension of a spline space with highest order smoothness over a T-mesh is studied by the sur-jection condition relying on the interior1-edges. The methods of constructing basis functions of this spline space are presented by the surjection condition relying on the interior edges in Chapter5. Moreover, due to interest triggered by the introduction of isogeometric analysis, we also defined new spline functions over T-meshes for iso-geometric analysis. In Chapter6, we analyze the feasibility of comparing the spline spaces spanned by these splines with spline spaces over corresponding T-meshes.
In Chapter7, we make a preliminary study of spline spaces with mixed orders of continuity over T-meshes. In particular, we work on the dimension and the method of constructing a set of basis functions of a type of spline space with mixed orders of continuity over a hierarchical T-mesh which used to process discontinuous data. By the examples of application of this type of spline to process image data, these spline functions keep the features of discontinuity of the discontinuous data in a better way.
In Chapter8, we sum up our paper and present future research problems from the view of spline spaces with highest order smoothness over T-meshes and spline spaces with mixed orders of continuity over T-meshes.
引文
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[3]秦荣.1981.样条有限点法.数值计算与计算机应用,(2)：68-81.
[4]石钟慈.1979.样条有限元.计算数学,(1)：5072.
[5]王建国,沈鹏程.1984.样条函数法解圆柱壳问题.土木工程学报,17(2)：75-85.
[6]王仁宏.1979.任意剖分下的多元样条分析.中国科学,数学专辑(1),325-340.
[7]王仁宏等.1994.多元样条函数及其应用.中国科学,科学出版社.
[8]袁驷.1984.样条矩形元.计算结构力学及其应用,1(2)：41-47.
[9]朱述尧.1981.用样条有限点法解薄板动力问题.上海力学,2(4)：23-29.
[10]P. Alfeld and L. Schumaker.1987. The dimension of bivariate spline spaces of smooth-ness r for degree d≥4r+1. Constr. Approx.,3:189-197.
[11]H. Antes.1974. Bicubic fundamental splines in plate bending. Int. J. Num. Meth. Eng.,8:503-511.
[12]Y. Bazilevs, J. Calo and J. Evans.2010. Isogeometric analysis using T-splines. Corn-put. Methods Appl. Mech. Engrg.,199:229-264.
[13]P. Bezier.1972. Numerical control-mathematics and application (Translated by For-rest A R). John Wiley and Sons.
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[15]L. Billera.1988. Homology of smooth splines:generic triangulations and a conjecture of strang. Trans. A.M.S.,310:325-340.
[16]W. Boehm.1980. Inserting new knots into B-spline curves. CAD,12(4):199-201.
[17]L. Billera and L. Rose.1991. A Dimension Series for Multivariate Splines. Discrete Comput. Geom.,6:107-128.
[18]S. Brenner and L. Scott.2002. Finite Element Methods, NewYork, Springer.
[19]A. Buffa, D. Cho and G. Sangalli.2010. Linear independence of the T-spline blending functions associated with some particular T-meshes. Comput. Methods Appl. Mech. Engrg.,199:1437-1445.
[20]G. Cheri and P. Cella.1974. Spline-blended approximation of Hartmann's flow. Int. J. Num. Meth. Eng.,8:529-536.
[21]E. Cohen, T. Lyche and R. Riesenfeld.1980. Discrete B-splines and subdivision tech-niques in computer-aided geometric design and computer graphics. Computer Graph- ics and Image Processing,14:87-111.
[22]J. Cottrell, T. Hughes and etc.2009. Isogeometric Analysis:Toward Integration of CAD and FEA, John Wiley & Sons, Chichester, UK.
[23]M. Cox.1972. The numerical evaluation of B-splines. J. Inst. Math. App.,10:134-149.
[24]C. de Boor.1972. On calculating with B-splines. J. Approx. Theory,6(1):50-62.
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[26]P. de Casteljau.1963. Courbes et surfaces a poles. Technical Report, A. Citrie, Paris.
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[28]J. Deng, F. Chen, X. Li and etc.2008. Polynomial splines over hierarchical T-meshes. Graphical Models,74:76-86.
[29]J. Deng, F. Chen and L. Jin.2008. Dimensions of biquadratic spline spaces over T-meshes. http://arxiv.org/abs/0804.2533.
[30]T. Dokken, T. Lyche, K. Pettersen and etc.2010. Slide:Locally refined splines, http://sintef.net/upload/IKT/9011/ geometri/Dokken_Pavia_version_July_2.pdf.
[31]T. Dokken.2012. Slide:Linear independence and Locally Refined B-splines,34-37, http://www.sintef.no/upload/IKT/9011/geometri/ LR\%20B-splines/Linear_Independence_LR_B-splines_March_14.pdf.
[32]T. Dokken, T. Lyche and K. Pettersen.2012. Locally Refinable Splines over Box-Partitions. Preprint, http://www.sintef.no/home/Publications/ Publication/?pubid=SINTEF+A22403.
[33]M. Dorfel, B. Juttler and B. Simeon.2010. Adaptive isogeometric analysis by local h-refmement with T-splines. Comput. Methods Appl. Mech. Engrg.,199:264-275.
[34]M. Eck, T. DeRose and T. Duchamp.1995. Multiresolution Analysis of Arbitrary Meshes. SIGGRAPH'95 Proceedings of the 22nd annual conference on Computer graphics and interactive techniques 173-182.
[35]S. Fan and Y. Cheung.1983. Analysis of shallow shells by spline finite strip method. Engng. Struct., (5):255-263.
[36]S. Fan and M. Luah.1992. New spline finite element for analysis of shells of revolution. ASCE J Engineering Mechanics,118(6):1065-1081.
[37]G. Farin.1979. Subsplines iiber dreiecken. Diss. TU Braunschweig.
[38]G. Farin.2001. Curves and surfaces for CAGD, Morgan Kaufmann.
[39]J. Ferguson.1964. Multivariable Curve Interpolation. ACM, 11(2):221-228
[40]A. Forrest.1972. Interactive interpolation and approximation by Bezier polynomials. Computer J.,15(1):71-79.
[41]D. Forsey and R. Bartels.1988. Hierarchical B-spline refinement. Computer Graphics, 22(4):205-212.
[42]C. Giannelli and B. Jiittler.2011. Bases and dimensions of bivariate hierarchical tensor-product splines. DK-Report No.2011-11.
[43]W. Gordon and R. Riesenfled.1974. Bernstein-Bezier methods for the computer-aided design of free form curves and surfaces. J. ACM,21(2):293-310.
[44]W. Gordon and R. Riesenfled.1974. B-spline curves and surfaces. Computer Aided Geometric Design, Academic Press,95-126.
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