Non-uniform algebraic-trigonometric B-splines shares most of the properties as those of the usual polynomial B-splines. But they are not orthogonai. We construct an orthogonal basis for the n-order（n ≥ 3） algebraic...Non-uniform algebraic-trigonometric B-splines shares most of the properties as those of the usual polynomial B-splines. But they are not orthogonai. We construct an orthogonal basis for the n-order（n ≥ 3） algebraic-trigonometric spline space in order to resolve the theo- retical problem that there is not an explicit orthogonai basis in the space by now. Motivated by the Legendre polynomials, we present a novel approach to define a set of auxiliary functions, which have simple and explicit expressions. Then the proposed orthogonal splines are given as the derivatives of these auxiliary functions.展开更多
A harmonic condition that can distinguish whether the dimension of spline space S 1 3( △ ) depends on the geometrical character of triangulation is presented, then on a type of general triangulation the dimension...A harmonic condition that can distinguish whether the dimension of spline space S 1 3( △ ) depends on the geometrical character of triangulation is presented, then on a type of general triangulation the dimension is got.展开更多
For a subdivision △ of a region in d-dimensional Euclidean space. we consider computation of dimension and of basis function in spline space S(△)consisting of all C(?)piecewise Polynomial func- tions over△of degree...For a subdivision △ of a region in d-dimensional Euclidean space. we consider computation of dimension and of basis function in spline space S(△)consisting of all C(?)piecewise Polynomial func- tions over△of degree at most k. A computational scheme is presented for computing the dimension and bases of spline space S(?)(△).This scheme based On the Grobner basis algorithm and the smooth co-factor method for computing multivariate spline. For bivariate splines, explicit basis functions of S(△)age obtained for any integer k and r when△is a cross-cut partition.展开更多
A3DT-mesh is generally a cuboid gridwhich allows hanging vertices.Here,a hanging vertex is an interior vertex,but it is not a corner point of eight cells.Spline function spaces with high order smoothness over 3D T-mes...A3DT-mesh is generally a cuboid gridwhich allows hanging vertices.Here,a hanging vertex is an interior vertex,but it is not a corner point of eight cells.Spline function spaces with high order smoothness over 3D T-meshes have great application prospect due to their local refinement and relatively low degrees of freedom,for example,3D isogeometric analysis and implicit representation of surfaces.However,there are still no available dimension formulae of those kinds of spline spaces for application.In this paper,we explore the dimensions of trivariate quadratic spline spaces with C^(1)continuity over hierarchical 3D T-meshes.By using space embedding method,the problem is converted into a system of linear constraints,and then a lower bound on the dimension of the spline space over a hierarchical 3D T-mesh is provided.For a special type of hierarchical 3D T-meshes,the explicit dimension formula is obtained.In addition,a topological explanation of the dimension is given,which presents a way to construct basis functions.展开更多
A T-mesh is basically a rectangular grid that allows T-junctions. Recently, Deng etal introduced splines over T-meshes, which are generalizations of T-splines invented by Sederberg etal, and proposed a dimension formu...A T-mesh is basically a rectangular grid that allows T-junctions. Recently, Deng etal introduced splines over T-meshes, which are generalizations of T-splines invented by Sederberg etal, and proposed a dimension formula based on the B-net method. In this paper, we derive an equivalent dimension formula in a different form with the smoothing cofactor method.展开更多
Presents information on a study which outlined the blossom approach to the dimension count of bivariate spline space. Smoothness conditions in blossoming form; Application of the approach to the case of Morgan-Scott p...Presents information on a study which outlined the blossom approach to the dimension count of bivariate spline space. Smoothness conditions in blossoming form; Application of the approach to the case of Morgan-Scott partition.展开更多
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
Multivariate spline function is an important research object and tool in Computational Geometry. The singularity of multivariate spline spaces is a difficult problem that is ineritable in the research of the structure...Multivariate spline function is an important research object and tool in Computational Geometry. The singularity of multivariate spline spaces is a difficult problem that is ineritable in the research of the structure of multivariate spline spaces. The aim of this paper is to reveal the geometric significance of the singularity of bivariate spline space over Morgan-Scott type triangulation by using some new concepts proposed by the first author such as characteristic ratio, characteristic mapping of lines （or ponits）, and characteristic number of algebraic curve. With these concepts and the relevant results, a polished necessary and sufficient conditions for the singularity of spline space S u＋1^u （△MS^u） are geometrically given for any smoothness u by recursion. Moreover, the famous Pascal＇s theorem is generalized to algebraic plane curves of degree n≥3.展开更多
Basis functions of biquadratic polynomial spline spaces over hierarchical T-meshes are constructed. The basis functions are all tensor-product B-splines, which are linearly independent, nonnegative and complete. To ma...Basis functions of biquadratic polynomial spline spaces over hierarchical T-meshes are constructed. The basis functions are all tensor-product B-splines, which are linearly independent, nonnegative and complete. To make basis functions more efficient for geometric modeling, we also give out a new basis with the property of unit partition. Two preliminary applications are given to demonstrate that the new basis is efficient.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
In this paper, the dimension of the spaces of bivariate spline with degree less that 2r and smoothness order r on the Morgan-Scott triangulation is considered. The concept of the instability degree in the dimension of...In this paper, the dimension of the spaces of bivariate spline with degree less that 2r and smoothness order r on the Morgan-Scott triangulation is considered. The concept of the instability degree in the dimension of spaces of bivariate spline is presented. The results in the paper make us conjecture the instability degree in the dimension of spaces of bivariate spline is infinity.展开更多
In this paper,the forecasting equations of a 2nd-order space-time differential remainder are deduced from the Navier-Stokes primitive equations and Eulerian operator by Taylor-series expansion.Here we introduce a cubi...In this paper,the forecasting equations of a 2nd-order space-time differential remainder are deduced from the Navier-Stokes primitive equations and Eulerian operator by Taylor-series expansion.Here we introduce a cubic spline numerical model(Spline Model for short),which is with a quasi-Lagrangian time-split integration scheme of fitting cubic spline/bicubic surface to all physical variable fields in the atmospheric equations on spherical discrete latitude-longitude mesh.A new algorithm of"fitting cubic spline—time step integration—fitting cubic spline—……"is developed to determine their first-and2nd-order derivatives and their upstream points for time discrete integral to the governing equations in Spline Model.And the cubic spline function and its mathematical polarities are also discussed to understand the Spline Model’s mathematical foundation of numerical analysis.It is pointed out that the Spline Model has mathematical laws of"convergence"of the cubic spline functions contracting to the original functions as well as its 1st-order and 2nd-order derivatives.The"optimality"of the 2nd-order derivative of the cubic spline functions is optimal approximation to that of the original functions.In addition,a Hermite bicubic patch is equivalent to operate on a grid for a 2nd-order derivative variable field.Besides,the slopes and curvatures of a central difference are identified respectively,with a smoothing coefficient of 1/3,three-point smoothing of that of a cubic spline.Then the slopes and curvatures of a central difference are calculated from the smoothing coefficient 1/3 and three-point smoothing of that of a cubic spline,respectively.Furthermore,a global simulation case of adiabatic,non-frictional and"incompressible"model atmosphere is shown with the quasi-Lagrangian time integration by using a global Spline Model,whose initial condition comes from the NCEP reanalysis data,along with quasi-uniform latitude-longitude grids and the so-called"shallow atmosphere"Navier-Stokes primitive equations in the spherical coordinates.The Spline Model,which adopted the Navier-Stokes primitive equations and quasi-Lagrangian time-split integration scheme,provides an initial ideal case of global atmospheric circulation.In addition,considering the essentially non-linear atmospheric motions,the Spline Model could judge reasonably well simple points of any smoothed variable field according to its fitting spline curvatures that must conform to its physical interpretation.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
Application of spline element and state space method for analysis of dynamic response of elastic rectangular plates is presented. The spline element method is used for space domain and the state space method in contro...Application of spline element and state space method for analysis of dynamic response of elastic rectangular plates is presented. The spline element method is used for space domain and the state space method in control theory of system is used for time domain. A state variable recursive scheme is developed, then the dynamic response of structure can he calculated directly. Several numerical examples are given. The results which are presented to demonstrate the accuracy and efficiency of the present method are quite satisfactory.展开更多
A one-dimensional monotone interpolation method based on interface reconstruction with partial volumes in the slope-space utilizing the Hermite cubic-spline, is proposed. The new method is only quartic, however is C2 ...A one-dimensional monotone interpolation method based on interface reconstruction with partial volumes in the slope-space utilizing the Hermite cubic-spline, is proposed. The new method is only quartic, however is C2 and unconditionally monotone. A set of control points is employed to constrain the curvature of the interpolation function and to eliminate possible nonphysical oscillations in the slope space. An extension of this method in two-dimensions is also discussed.展开更多
Over the last few years, there has been a significant increase in attention paid to fractional differential equations, given their wide array of applications in the fields of physics and engineering. The recent develo...Over the last few years, there has been a significant increase in attention paid to fractional differential equations, given their wide array of applications in the fields of physics and engineering. The recent development of using fractional telegraph equations as models in some fields (e.g., the thermal diffusion in fractal media) has heightened the importance of examining the method of solutions for such equations (both approximate and analytic). The present work is designed to serve as a valuable contribution to work in this field. The key objective of this work is to propose a general framework that can be used to guide quadratic spline functions in order to create a numerical method for obtaining an approximation solution using the linear space-fractional telegraph equation. Additionally, the Von Neumann method was employed to measure the stability of the analytical scheme, which showed that the proposed method is conditionally stable. What’s more, the proposal contains a numerical example that illustrates how the proposed method can be implemented practically, whilst the error estimates and numerical stability results are discussed in depth. The findings indicate that the proposed model is highly effective, convenient and accurate for solving the relevant problems and is suitable for use with approximate solutions acquired through the two-dimensional differential transform method that has been developed for linear partial differential equations with space- and time-fractional derivatives.展开更多
By using the method of space mapping,basis functions of biquadratic polynomial spline spaces over the hierarchical T-meshes without limitation for level difference can be constructed.In this paper,the basis functions ...By using the method of space mapping,basis functions of biquadratic polynomial spline spaces over the hierarchical T-meshes without limitation for level difference can be constructed.In this paper,the basis functions defined over hierarchical T-meshes with high level differences are adopted for the application in the isogeometric analysis problems with rapidly changing local features.Without subdividing redundant cells to ensure the level difference of the adjacent cells,the refinement becomes more local,and fewer cells are subdivided for each refinement of the hierarchical T-mesh.Therefore,the dimension of the biquadratic polynomial spline space over the hierarchical Tmesh can be reduced,the superfluous control points or coefficients can be avoided,and the quantity of calculations can be decreased.Numerical examples show that these basis functions can work well on physical domains with different boundaries for the application in IGA.展开更多
Tool path generated by space-filling curve always turns frequently causing trembling to machine,reducing toollife and affecting workpiece quality. Length and generation time of tool paths are both relatively long. In ...Tool path generated by space-filling curve always turns frequently causing trembling to machine,reducing toollife and affecting workpiece quality. Length and generation time of tool paths are both relatively long. In order to solve these problems,a toolpath generation method of NC milling based on space-filling curve is proposed. First,T-spline surface is regarded as the modeling surface,the grid,which is based on the limited scallop-height,can be got in the parameter space,and the influence value of grid node is determined. Second,a box is defined and planned,and the tool paths are got preliminarily,which is based on minimal spanning tree; Finally,based on an improved chamfering algorithm,the whole tool paths are got. A simulation system is developed for computer simulation,and an experiment is carried out to verify the method. The results of simulation and experiment show that the method is effective and feasible,and length and time of the tool paths are reduced.展开更多
基金Supported by the National Natural Science Foundation of China(60933008,61272300 and 11226327)the Science&Technology Program of Shanghai Maritime University(20120099)
文摘Non-uniform algebraic-trigonometric B-splines shares most of the properties as those of the usual polynomial B-splines. But they are not orthogonai. We construct an orthogonal basis for the n-order（n ≥ 3） algebraic-trigonometric spline space in order to resolve the theo- retical problem that there is not an explicit orthogonai basis in the space by now. Motivated by the Legendre polynomials, we present a novel approach to define a set of auxiliary functions, which have simple and explicit expressions. Then the proposed orthogonal splines are given as the derivatives of these auxiliary functions.
文摘A harmonic condition that can distinguish whether the dimension of spline space S 1 3( △ ) depends on the geometrical character of triangulation is presented, then on a type of general triangulation the dimension is got.
基金The Project is partly supported by the Science Technology New Star Plan of Beijing Education Committee of Beijing
文摘For a subdivision △ of a region in d-dimensional Euclidean space. we consider computation of dimension and of basis function in spline space S(△)consisting of all C(?)piecewise Polynomial func- tions over△of degree at most k. A computational scheme is presented for computing the dimension and bases of spline space S(?)(△).This scheme based On the Grobner basis algorithm and the smooth co-factor method for computing multivariate spline. For bivariate splines, explicit basis functions of S(△)age obtained for any integer k and r when△is a cross-cut partition.
基金supported by the NSF of China(No.12171453,No.12001197).
文摘A3DT-mesh is generally a cuboid gridwhich allows hanging vertices.Here,a hanging vertex is an interior vertex,but it is not a corner point of eight cells.Spline function spaces with high order smoothness over 3D T-meshes have great application prospect due to their local refinement and relatively low degrees of freedom,for example,3D isogeometric analysis and implicit representation of surfaces.However,there are still no available dimension formulae of those kinds of spline spaces for application.In this paper,we explore the dimensions of trivariate quadratic spline spaces with C^(1)continuity over hierarchical 3D T-meshes.By using space embedding method,the problem is converted into a system of linear constraints,and then a lower bound on the dimension of the spline space over a hierarchical 3D T-mesh is provided.For a special type of hierarchical 3D T-meshes,the explicit dimension formula is obtained.In addition,a topological explanation of the dimension is given,which presents a way to construct basis functions.
文摘A T-mesh is basically a rectangular grid that allows T-junctions. Recently, Deng etal introduced splines over T-meshes, which are generalizations of T-splines invented by Sederberg etal, and proposed a dimension formula based on the B-net method. In this paper, we derive an equivalent dimension formula in a different form with the smoothing cofactor method.
基金the 973 Project on Mathematical Mechanics!G1998030600NSF and SF of National Educational Committee of China
文摘Presents information on a study which outlined the blossom approach to the dimension count of bivariate spline space. Smoothness conditions in blossoming form; Application of the approach to the case of Morgan-Scott partition.
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 National Natural Science Foundation of China (Grant Nos.10771028 60533060)+1 种基金the programof New Century Excellent Fellowship of NECCfunded by a DoD fund (Grant No.DAAD19-03-1-0375)
文摘Multivariate spline function is an important research object and tool in Computational Geometry. The singularity of multivariate spline spaces is a difficult problem that is ineritable in the research of the structure of multivariate spline spaces. The aim of this paper is to reveal the geometric significance of the singularity of bivariate spline space over Morgan-Scott type triangulation by using some new concepts proposed by the first author such as characteristic ratio, characteristic mapping of lines （or ponits）, and characteristic number of algebraic curve. With these concepts and the relevant results, a polished necessary and sufficient conditions for the singularity of spline space S u＋1^u （△MS^u） are geometrically given for any smoothness u by recursion. Moreover, the famous Pascal＇s theorem is generalized to algebraic plane curves of degree n≥3.
基金Acknowledgement. The authors would like to thank Dr. F. Chen of School of Mathematical Sciences, University of Science and Technology of China for his constructive suggestions in the construction of the basis. The authors are grateful to the referees who address some questions about the basis that we did not consider. The authors are supported by the NSF of China （No. 11371341, No.11626253, No. 11601114, No. 11526069）, the Anhui Provincial Natural Science Foundation （No. 1608085QA14）.
文摘Basis functions of biquadratic polynomial spline spaces over hierarchical T-meshes are constructed. The basis functions are all tensor-product B-splines, which are linearly independent, nonnegative and complete. To make basis functions more efficient for geometric modeling, we also give out a new basis with the property of unit partition. Two preliminary applications are given to demonstrate that the new basis is efficient.
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Project Supported by the national Natural Science Foundation of China(No.19871010,No.69973010).
文摘In this paper, the dimension of the spaces of bivariate spline with degree less that 2r and smoothness order r on the Morgan-Scott triangulation is considered. The concept of the instability degree in the dimension of spaces of bivariate spline is presented. The results in the paper make us conjecture the instability degree in the dimension of spaces of bivariate spline is infinity.
文摘In this paper,the forecasting equations of a 2nd-order space-time differential remainder are deduced from the Navier-Stokes primitive equations and Eulerian operator by Taylor-series expansion.Here we introduce a cubic spline numerical model(Spline Model for short),which is with a quasi-Lagrangian time-split integration scheme of fitting cubic spline/bicubic surface to all physical variable fields in the atmospheric equations on spherical discrete latitude-longitude mesh.A new algorithm of"fitting cubic spline—time step integration—fitting cubic spline—……"is developed to determine their first-and2nd-order derivatives and their upstream points for time discrete integral to the governing equations in Spline Model.And the cubic spline function and its mathematical polarities are also discussed to understand the Spline Model’s mathematical foundation of numerical analysis.It is pointed out that the Spline Model has mathematical laws of"convergence"of the cubic spline functions contracting to the original functions as well as its 1st-order and 2nd-order derivatives.The"optimality"of the 2nd-order derivative of the cubic spline functions is optimal approximation to that of the original functions.In addition,a Hermite bicubic patch is equivalent to operate on a grid for a 2nd-order derivative variable field.Besides,the slopes and curvatures of a central difference are identified respectively,with a smoothing coefficient of 1/3,three-point smoothing of that of a cubic spline.Then the slopes and curvatures of a central difference are calculated from the smoothing coefficient 1/3 and three-point smoothing of that of a cubic spline,respectively.Furthermore,a global simulation case of adiabatic,non-frictional and"incompressible"model atmosphere is shown with the quasi-Lagrangian time integration by using a global Spline Model,whose initial condition comes from the NCEP reanalysis data,along with quasi-uniform latitude-longitude grids and the so-called"shallow atmosphere"Navier-Stokes primitive equations in the spherical coordinates.The Spline Model,which adopted the Navier-Stokes primitive equations and quasi-Lagrangian time-split integration scheme,provides an initial ideal case of global atmospheric circulation.In addition,considering the essentially non-linear atmospheric motions,the Spline Model could judge reasonably well simple points of any smoothed variable field according to its fitting spline curvatures that must conform to its physical interpretation.
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Application of spline element and state space method for analysis of dynamic response of elastic rectangular plates is presented. The spline element method is used for space domain and the state space method in control theory of system is used for time domain. A state variable recursive scheme is developed, then the dynamic response of structure can he calculated directly. Several numerical examples are given. The results which are presented to demonstrate the accuracy and efficiency of the present method are quite satisfactory.
文摘A one-dimensional monotone interpolation method based on interface reconstruction with partial volumes in the slope-space utilizing the Hermite cubic-spline, is proposed. The new method is only quartic, however is C2 and unconditionally monotone. A set of control points is employed to constrain the curvature of the interpolation function and to eliminate possible nonphysical oscillations in the slope space. An extension of this method in two-dimensions is also discussed.
文摘Over the last few years, there has been a significant increase in attention paid to fractional differential equations, given their wide array of applications in the fields of physics and engineering. The recent development of using fractional telegraph equations as models in some fields (e.g., the thermal diffusion in fractal media) has heightened the importance of examining the method of solutions for such equations (both approximate and analytic). The present work is designed to serve as a valuable contribution to work in this field. The key objective of this work is to propose a general framework that can be used to guide quadratic spline functions in order to create a numerical method for obtaining an approximation solution using the linear space-fractional telegraph equation. Additionally, the Von Neumann method was employed to measure the stability of the analytical scheme, which showed that the proposed method is conditionally stable. What’s more, the proposal contains a numerical example that illustrates how the proposed method can be implemented practically, whilst the error estimates and numerical stability results are discussed in depth. The findings indicate that the proposed model is highly effective, convenient and accurate for solving the relevant problems and is suitable for use with approximate solutions acquired through the two-dimensional differential transform method that has been developed for linear partial differential equations with space- and time-fractional derivatives.
文摘By using the method of space mapping,basis functions of biquadratic polynomial spline spaces over the hierarchical T-meshes without limitation for level difference can be constructed.In this paper,the basis functions defined over hierarchical T-meshes with high level differences are adopted for the application in the isogeometric analysis problems with rapidly changing local features.Without subdividing redundant cells to ensure the level difference of the adjacent cells,the refinement becomes more local,and fewer cells are subdivided for each refinement of the hierarchical T-mesh.Therefore,the dimension of the biquadratic polynomial spline space over the hierarchical Tmesh can be reduced,the superfluous control points or coefficients can be avoided,and the quantity of calculations can be decreased.Numerical examples show that these basis functions can work well on physical domains with different boundaries for the application in IGA.
基金Supported by the National Natural Science Foundation of China(No.51575143)
文摘Tool path generated by space-filling curve always turns frequently causing trembling to machine,reducing toollife and affecting workpiece quality. Length and generation time of tool paths are both relatively long. In order to solve these problems,a toolpath generation method of NC milling based on space-filling curve is proposed. First,T-spline surface is regarded as the modeling surface,the grid,which is based on the limited scallop-height,can be got in the parameter space,and the influence value of grid node is determined. Second,a box is defined and planned,and the tool paths are got preliminarily,which is based on minimal spanning tree; Finally,based on an improved chamfering algorithm,the whole tool paths are got. A simulation system is developed for computer simulation,and an experiment is carried out to verify the method. The results of simulation and experiment show that the method is effective and feasible,and length and time of the tool paths are reduced.
