The T-meshes are local modification of rectangular meshes which allow T-junctions. The splines over T-meshes are involved in many fields, such as finite element methods, CAGD etc. The dimension of a spline space is a ...The T-meshes are local modification of rectangular meshes which allow T-junctions. The splines over T-meshes are involved in many fields, such as finite element methods, CAGD etc. The dimension of a spline space is a basic problem for the theories and applications of splines. However, the problem of determining the dimension of a spline space is difficult since it heavily depends on the geometric properties of the partition. In many cases, the dimension is unstable. In this paper, we study the instability in the dimensions of spline spaces over T-meshes by using the smoothing cofactor-conformality method. The modified dimension formulas of spline spaces over T-meshes with T-cycles are also presented. Moreover, some examples are given to illustrate the instability in the dimensions of the spline spaces over some special meshes.展开更多
In this paper,we construct a bijective mapping between a biquadratic spline space over the hierarchical T-mesh and the piecewise constant space over the corresponding crossing-vertex-relationship graph(CVR graph).We p...In this paper,we construct a bijective mapping between a biquadratic spline space over the hierarchical T-mesh and the piecewise constant space over the corresponding crossing-vertex-relationship graph(CVR graph).We propose a novel structure,by which we offer an effective and easy operative method for constructing the basis functions of the biquadratic spline space.Themapping we construct is an isomorphism.The basis functions of the biquadratic spline space hold the properties such as linear independence,completeness and the property of partition of unity,which are the same as the properties for the basis functions of piecewise constant space over the CVR graph.To demonstrate that the new basis functions are efficient,we apply the basis functions to fit some open surfaces.展开更多
The truncated hierarchical B-spline basis has been proposed for adaptive data fitting and has already drawn a lot of attention in theory and applications. However the stability with respect to the Lp-norm, 1 ≤ p 〈 ...The truncated hierarchical B-spline basis has been proposed for adaptive data fitting and has already drawn a lot of attention in theory and applications. However the stability with respect to the Lp-norm, 1 ≤ p 〈 ∞, is not clear. In this paper, we consider the Lp stability of the truncated hierarchical B-spline basis, since the Lp stability is useful for curve and surface fitting, especially for least squares fitting. We prove that this basis is weakly Lp stable. This means that the associated constants to be considered in the stability analysis are at most of polynomial growth in the number of the hierarchy depth.展开更多
针对三角面元目标提出了一种高效率的空间分割算法.该方法以一种空间点与单位立方体位置关系的判断法则为基础,并逐渐延拓到参数直线、三角形的空间分割上,给出了一种新的三角形面元目标快速分割的解决方法.介绍了该方法在参数曲线、NUR...针对三角面元目标提出了一种高效率的空间分割算法.该方法以一种空间点与单位立方体位置关系的判断法则为基础,并逐渐延拓到参数直线、三角形的空间分割上,给出了一种新的三角形面元目标快速分割的解决方法.介绍了该方法在参数曲线、NURBS(Non-Uniform Rational B-Spline)曲面目标的空间均匀分割上的应用,并给出了非均匀分割的处理方法.与计算机图形技术中最常用的BSP(Binary Space Partitioning)技术的比较中发现,对于特定情形,该算法的执行效率优于BSP法.通过实例证明了该算法的有效性和可靠性.展开更多
The characteristics of urban space expansion reflect the changes of urban spatial layout and structure, as well as the orientation of urban development in the future. This paper uses the regional sector division metho...The characteristics of urban space expansion reflect the changes of urban spatial layout and structure, as well as the orientation of urban development in the future. This paper uses the regional sector division method to divide the urban land into 8 orientations, based on the urban land space compaction index, and designs a sector partition compaction index. Based on the remote sensing image data of 1990, 2000, 2010, and 2020, the spatial characteristics of urban land expansion of Luoyang are analyzed by using the partition compaction index, expansion intensity index, fractal dimension, and standard deviation ellipse. The results show that: from 1990 to 2020, the urban overall planning of Luoyang has effectively guided the urban development, the urban land expands rapidly, the urban land compaction has been maintained at a low level, and the urban form has been tending to be reasonable;the urban land centroid gradually shifts to the southwest, and the distribution axis rotates clockwise from southwest-northeast to northwest-southeast, and the directionality of distribution gradually disappears;the urban land has gone through the development process of land filling-enlarging-refilling. The urban land expansion is relatively active in the region with an azimuth of 90°- 225°, and the urban expansion in the north of Luo River is relatively stable and is always filling mode.展开更多
为解决轮盘形状优化过程中变量多且高度非线性的问题,建立了基于核主成分分析(Kernelized Principal Component Analysis,KPCA)技术的航空发动机轮盘优化方法。利用控制点形式的非均匀有理B样条(NURBS)曲线构造涡轮盘截面形状,以控制点...为解决轮盘形状优化过程中变量多且高度非线性的问题,建立了基于核主成分分析(Kernelized Principal Component Analysis,KPCA)技术的航空发动机轮盘优化方法。利用控制点形式的非均匀有理B样条(NURBS)曲线构造涡轮盘截面形状,以控制点的坐标作为设计变量;利用实验设计(DOE)进行采样,从中选取60个较优样本,计算核矩阵K,应用核矩阵K的特征向量计算投影向量的系数矩阵,采用投影后的变量作为新的设计变量,重构设计空间;最后,在降维后的设计空间中优化涡轮盘。结果表明:优化后轮心当量应力降低4.21%,周向应力降低0.26%,轮背径向应力降低1.58%,子午截面平均周向应力降低2.57%,圆柱截面平均径向应力降低6.51%。展开更多
基金Acknowledgments. This work is partly supported by the National Natural Science Foundation of China （Nos. 11290143, Ul135003, 11471066, 11271060, 11301052）, Fundamental Research of Civil Aircraft （No. MJ-F-2012-04）, and the Fundamental Research Funds for the Central Universities （Nos. DUT13LK07, DUT13LK45, DUT14YQ111）.
文摘The T-meshes are local modification of rectangular meshes which allow T-junctions. The splines over T-meshes are involved in many fields, such as finite element methods, CAGD etc. The dimension of a spline space is a basic problem for the theories and applications of splines. However, the problem of determining the dimension of a spline space is difficult since it heavily depends on the geometric properties of the partition. In many cases, the dimension is unstable. In this paper, we study the instability in the dimensions of spline spaces over T-meshes by using the smoothing cofactor-conformality method. The modified dimension formulas of spline spaces over T-meshes with T-cycles are also presented. Moreover, some examples are given to illustrate the instability in the dimensions of the spline spaces over some special meshes.
基金supported by the NSF of China(No.11771420,No.12001197).
文摘In this paper,we construct a bijective mapping between a biquadratic spline space over the hierarchical T-mesh and the piecewise constant space over the corresponding crossing-vertex-relationship graph(CVR graph).We propose a novel structure,by which we offer an effective and easy operative method for constructing the basis functions of the biquadratic spline space.Themapping we construct is an isomorphism.The basis functions of the biquadratic spline space hold the properties such as linear independence,completeness and the property of partition of unity,which are the same as the properties for the basis functions of piecewise constant space over the CVR graph.To demonstrate that the new basis functions are efficient,we apply the basis functions to fit some open surfaces.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1129014311471066)+1 种基金Fundamental Research of Civil Aircraft(Grant No.MJ-F-2012-04)the Fundamental Research Funds for the Central Universities(Grant No.DUT15LK44)
文摘The truncated hierarchical B-spline basis has been proposed for adaptive data fitting and has already drawn a lot of attention in theory and applications. However the stability with respect to the Lp-norm, 1 ≤ p 〈 ∞, is not clear. In this paper, we consider the Lp stability of the truncated hierarchical B-spline basis, since the Lp stability is useful for curve and surface fitting, especially for least squares fitting. We prove that this basis is weakly Lp stable. This means that the associated constants to be considered in the stability analysis are at most of polynomial growth in the number of the hierarchy depth.
文摘针对三角面元目标提出了一种高效率的空间分割算法.该方法以一种空间点与单位立方体位置关系的判断法则为基础,并逐渐延拓到参数直线、三角形的空间分割上,给出了一种新的三角形面元目标快速分割的解决方法.介绍了该方法在参数曲线、NURBS(Non-Uniform Rational B-Spline)曲面目标的空间均匀分割上的应用,并给出了非均匀分割的处理方法.与计算机图形技术中最常用的BSP(Binary Space Partitioning)技术的比较中发现,对于特定情形,该算法的执行效率优于BSP法.通过实例证明了该算法的有效性和可靠性.
文摘The characteristics of urban space expansion reflect the changes of urban spatial layout and structure, as well as the orientation of urban development in the future. This paper uses the regional sector division method to divide the urban land into 8 orientations, based on the urban land space compaction index, and designs a sector partition compaction index. Based on the remote sensing image data of 1990, 2000, 2010, and 2020, the spatial characteristics of urban land expansion of Luoyang are analyzed by using the partition compaction index, expansion intensity index, fractal dimension, and standard deviation ellipse. The results show that: from 1990 to 2020, the urban overall planning of Luoyang has effectively guided the urban development, the urban land expands rapidly, the urban land compaction has been maintained at a low level, and the urban form has been tending to be reasonable;the urban land centroid gradually shifts to the southwest, and the distribution axis rotates clockwise from southwest-northeast to northwest-southeast, and the directionality of distribution gradually disappears;the urban land has gone through the development process of land filling-enlarging-refilling. The urban land expansion is relatively active in the region with an azimuth of 90°- 225°, and the urban expansion in the north of Luo River is relatively stable and is always filling mode.
文摘为解决轮盘形状优化过程中变量多且高度非线性的问题,建立了基于核主成分分析(Kernelized Principal Component Analysis,KPCA)技术的航空发动机轮盘优化方法。利用控制点形式的非均匀有理B样条(NURBS)曲线构造涡轮盘截面形状,以控制点的坐标作为设计变量;利用实验设计(DOE)进行采样,从中选取60个较优样本,计算核矩阵K,应用核矩阵K的特征向量计算投影向量的系数矩阵,采用投影后的变量作为新的设计变量,重构设计空间;最后,在降维后的设计空间中优化涡轮盘。结果表明:优化后轮心当量应力降低4.21%,周向应力降低0.26%,轮背径向应力降低1.58%,子午截面平均周向应力降低2.57%,圆柱截面平均径向应力降低6.51%。
