In this paper,the dimension of the nonuniform bivariate spline space S_(3)^(1,2)(Δ_(mn)^((2))is discussed based on the theory of multivariate spline space.Moreover,by means of the Conformality of Smoothing Cofactor M...In this paper,the dimension of the nonuniform bivariate spline space S_(3)^(1,2)(Δ_(mn)^((2))is discussed based on the theory of multivariate spline space.Moreover,by means of the Conformality of Smoothing Cofactor Method,the basis ofS_(3)^(1,2)(Δ_(mn)^((2))composed of two sets of splines are worked out in the form of the values at ten domain points in each triangular cell,both of which possess distinct local supports.Furthermore,the explicit coefficients in terms of B-net are obtained for the two sets of splines respectively.展开更多
The quadrilateral discrete Kirchhoff thin plate bending element DKQ is based on the isoparametric element Q8, however, the accuracy of the isoparametric quadrilateral elements will drop significantly due to mesh disto...The quadrilateral discrete Kirchhoff thin plate bending element DKQ is based on the isoparametric element Q8, however, the accuracy of the isoparametric quadrilateral elements will drop significantly due to mesh distortions. In a previous work, we constructed an 8-node quadrilateral spline element L8 using the triangular area coordinates and the B- net method, which can be insensitive to mesh distortions and possess the second order completeness in the Cartesian co- ordinates. In this paper, a thin plate spline element is devel- oped based on the spline element L8 and the refined tech- nique. Numerical examples show that the present element indeed possesses higher accuracy than the DKQ element for distorted meshes.展开更多
In this paper,a 13-node pyramid spline element is derived by using the tetrahedron volume coordinates and the B-net method,which achieves the second order completeness in Cartesian coordinates.Some appropriate example...In this paper,a 13-node pyramid spline element is derived by using the tetrahedron volume coordinates and the B-net method,which achieves the second order completeness in Cartesian coordinates.Some appropriate examples were employed to evaluate the performance of the proposed element.The numerical results show that the spline element has much better performance compared with the isoparametric serendipity element Q20 and its degenerate pyramid element P13 especially when mesh is distorted,and it is comparable to the Lagrange element Q27.It has been demonstrated that the spline finite element method is an efficient tool for developing high accuracy elements.展开更多
Isoparametric quadrilateral elements are widely used in the finite element method, but the accuracy of the isoparametric quadrilateral elements will drop obviously deteriorate due to mesh distortions. Spline functions...Isoparametric quadrilateral elements are widely used in the finite element method, but the accuracy of the isoparametric quadrilateral elements will drop obviously deteriorate due to mesh distortions. Spline functions have some properties of simplicity and conformality. Two 8-node quadrilateral elements have been developed using the trian- gular area coordinates and the B-net method, which can ex- actly model the quadratic field for both convex and concave quadrangles. Some appropriate examples are employed to evaluate the performance of the proposed elements. The nu- merical results show that the two spline elements can obtain solutions which are highly accurate and insensitive to mesh distortions.展开更多
1 Definition of Surface of Multivariate B-form and Method for Construction In this section we present a definition of surface of multivariate B-form and themethod for constructing the surface. Usually R^m denotes m-di...1 Definition of Surface of Multivariate B-form and Method for Construction In this section we present a definition of surface of multivariate B-form and themethod for constructing the surface. Usually R^m denotes m-dimensional real number space, and Z_+~m denotes the set of allm-multiple nonnegative integer.展开更多
In this paper we present a C-1 interpolation scheme on a triangle. The interpolant assumes given values and one order derivatives at the vertices of the triangle. It is made up of partial interpolants blended with cor...In this paper we present a C-1 interpolation scheme on a triangle. The interpolant assumes given values and one order derivatives at the vertices of the triangle. It is made up of partial interpolants blended with corresponding weight functions. Any partial interpolant is a piecewise cubics defined on a split of the triangle, while the weight function is just the respective barycentric coordinate. Hence the interpolant can be regarded as a piecewise quartic. We device a simple algorithm for the evaluation of the interpolant. It's easy to represent the interpolant with B-net method. We also depict the Franke's function and its interpolant, the illustration of which shows good visual effect of the scheme.展开更多
Isopaxametric quadrilateral elements are widely used in the finite element method. However, they have a disadvantage of accuracy loss when elements are distorted. Spline functions have properties of simpleness and con...Isopaxametric quadrilateral elements are widely used in the finite element method. However, they have a disadvantage of accuracy loss when elements are distorted. Spline functions have properties of simpleness and conformality. A 17onode quadrilateral element has been developed using the bivaxiate quaxtic spline interpolation basis and the triangular area coordinates, which can exactly model the quartic displacement fields. Some appropriate examples are employed to illustrate that the element possesses high precision and is insensitive to mesh distortions.展开更多
Basic requirement for applying isoparametric element is that the element has to be convex and no violent distortion is allowed. In this paper, a cubic quadrilateral spline element with 12 nodes has been developed usin...Basic requirement for applying isoparametric element is that the element has to be convex and no violent distortion is allowed. In this paper, a cubic quadrilateral spline element with 12 nodes has been developed using the triangular area coordinates and the B-net method, which can exactly model the cubic field for quadrilateral element with both convex and concave shapes. Neither mapping nor coordinate transformation is required and the spline element can obtain high accuracy solutions and insensitive to mesh distortions.展开更多
In this paper,matrix representations of the best spline quasi-interpolating operator over triangular sub-domains in S_(2)^(1)(△_(mn)^(2),and coefficients of splines in terms of B-net are calculated firstly.Moreover,b...In this paper,matrix representations of the best spline quasi-interpolating operator over triangular sub-domains in S_(2)^(1)(△_(mn)^(2),and coefficients of splines in terms of B-net are calculated firstly.Moreover,by means of coefficients in terms of B-net,computation of bivariate numerical cubature over triangular sub-domains with respect to variables x and y is transferred into summation of coefficients of splines in terms of B-net.Thus concise bivariate cubature formulas are constructed over rectangular sub-domain.Furthermore,by means of module of continuity and max-norms,error estimates for cubature formulas are derived over both sub-domains and the domain.展开更多
R-function is a widely used tool when considering objects obtained through the Boolean operations start from simple base primitives.However,there is square root operation in the representation.Considering that the use...R-function is a widely used tool when considering objects obtained through the Boolean operations start from simple base primitives.However,there is square root operation in the representation.Considering that the use of splines will facilitate the calculations within the CAD system,in this paper,we propose a system of R-functions represented in spline form called Spline R-function(SR).After trans-forming the function ranges of two base primitives to a new coordinate system,a series of sign constraints following a specific Boolean operation are derived and the spline R-function can be formulated as a piecewise function.Representation of SR in both B´ezier form and B-spline form have been given.Among which the B´ezier ordinates are determined with the help of the B-net method through setting up a series of relations according to the sign constraints and properties of R-functions.The construction processes for both Boolean intersection and union operations with different smoothness are discussed in detail.Numerical experiments are conducted to show the potential of the proposed spline R-function.展开更多
基金supported by the National Natural Science Foundation of China(Nos.U0935004,11071031,11001037,10801024)the Fundamental Research Funds for the Central Universities(DUT10ZD112,DUT10JS02).
文摘In this paper,the dimension of the nonuniform bivariate spline space S_(3)^(1,2)(Δ_(mn)^((2))is discussed based on the theory of multivariate spline space.Moreover,by means of the Conformality of Smoothing Cofactor Method,the basis ofS_(3)^(1,2)(Δ_(mn)^((2))composed of two sets of splines are worked out in the form of the values at ten domain points in each triangular cell,both of which possess distinct local supports.Furthermore,the explicit coefficients in terms of B-net are obtained for the two sets of splines respectively.
基金supported by the National Natural Science Foundation of China(11001037,11102037,11290143)the Fundamental Research Funds for the Central Universities(DUT13LK07)
文摘The quadrilateral discrete Kirchhoff thin plate bending element DKQ is based on the isoparametric element Q8, however, the accuracy of the isoparametric quadrilateral elements will drop significantly due to mesh distortions. In a previous work, we constructed an 8-node quadrilateral spline element L8 using the triangular area coordinates and the B- net method, which can be insensitive to mesh distortions and possess the second order completeness in the Cartesian co- ordinates. In this paper, a thin plate spline element is devel- oped based on the spline element L8 and the refined tech- nique. Numerical examples show that the present element indeed possesses higher accuracy than the DKQ element for distorted meshes.
基金The project was supported by the National Natural Science Foundation of China(11001037,11102037,11072156)the Fundamental Research Funds for the Central Universities of China(DUT10ZD112,DUT10JS02)
文摘In this paper,a 13-node pyramid spline element is derived by using the tetrahedron volume coordinates and the B-net method,which achieves the second order completeness in Cartesian coordinates.Some appropriate examples were employed to evaluate the performance of the proposed element.The numerical results show that the spline element has much better performance compared with the isoparametric serendipity element Q20 and its degenerate pyramid element P13 especially when mesh is distorted,and it is comparable to the Lagrange element Q27.It has been demonstrated that the spline finite element method is an efficient tool for developing high accuracy elements.
基金supported by the National Natural Science Foundation of China(11001037,11102037 and 11290143)the Fundamental Research Funds for the Central Universities
文摘Isoparametric quadrilateral elements are widely used in the finite element method, but the accuracy of the isoparametric quadrilateral elements will drop obviously deteriorate due to mesh distortions. Spline functions have some properties of simplicity and conformality. Two 8-node quadrilateral elements have been developed using the trian- gular area coordinates and the B-net method, which can ex- actly model the quadratic field for both convex and concave quadrangles. Some appropriate examples are employed to evaluate the performance of the proposed elements. The nu- merical results show that the two spline elements can obtain solutions which are highly accurate and insensitive to mesh distortions.
文摘1 Definition of Surface of Multivariate B-form and Method for Construction In this section we present a definition of surface of multivariate B-form and themethod for constructing the surface. Usually R^m denotes m-dimensional real number space, and Z_+~m denotes the set of allm-multiple nonnegative integer.
文摘In this paper we present a C-1 interpolation scheme on a triangle. The interpolant assumes given values and one order derivatives at the vertices of the triangle. It is made up of partial interpolants blended with corresponding weight functions. Any partial interpolant is a piecewise cubics defined on a split of the triangle, while the weight function is just the respective barycentric coordinate. Hence the interpolant can be regarded as a piecewise quartic. We device a simple algorithm for the evaluation of the interpolant. It's easy to represent the interpolant with B-net method. We also depict the Franke's function and its interpolant, the illustration of which shows good visual effect of the scheme.
基金supported by the Natural Science Foundation of China China (Nos. 60533060, 10672032, and 10726067)the Science Foundation of Dalian University of Technology (No. SFDUT07001)
文摘Isopaxametric quadrilateral elements are widely used in the finite element method. However, they have a disadvantage of accuracy loss when elements are distorted. Spline functions have properties of simpleness and conformality. A 17onode quadrilateral element has been developed using the bivaxiate quaxtic spline interpolation basis and the triangular area coordinates, which can exactly model the quartic displacement fields. Some appropriate examples are employed to illustrate that the element possesses high precision and is insensitive to mesh distortions.
基金supported by the National Natural Science Foundation of China (11001037,11102037)the Fundamental Research Funds for the Central Universities(DUT13LK07)
文摘Basic requirement for applying isoparametric element is that the element has to be convex and no violent distortion is allowed. In this paper, a cubic quadrilateral spline element with 12 nodes has been developed using the triangular area coordinates and the B-net method, which can exactly model the cubic field for quadrilateral element with both convex and concave shapes. Neither mapping nor coordinate transformation is required and the spline element can obtain high accuracy solutions and insensitive to mesh distortions.
基金This work was supported by the Fundamental Research Funds for the Central Universities of Hohai University(Grant No.2019B19414,2019B44914)the Natural Science Foundation of Jiangsu Province for the Youth(Grant No.BK20160853)+2 种基金Key Laboratory of Ministry of Education for Coastal Disaster and Protection,Hohai University(Grant No.202011)the National Natural Science Foundation of China(Grant No.11601151)the National Science Foundation of Zhejiang Province(Grant No.LY19A010003).
文摘In this paper,matrix representations of the best spline quasi-interpolating operator over triangular sub-domains in S_(2)^(1)(△_(mn)^(2),and coefficients of splines in terms of B-net are calculated firstly.Moreover,by means of coefficients in terms of B-net,computation of bivariate numerical cubature over triangular sub-domains with respect to variables x and y is transferred into summation of coefficients of splines in terms of B-net.Thus concise bivariate cubature formulas are constructed over rectangular sub-domain.Furthermore,by means of module of continuity and max-norms,error estimates for cubature formulas are derived over both sub-domains and the domain.
基金We would like to thank the anonymous reviewers and our labo-ratory group for helpful discussions and comments.The work is supported by the NSF of China(No.11771420).
文摘R-function is a widely used tool when considering objects obtained through the Boolean operations start from simple base primitives.However,there is square root operation in the representation.Considering that the use of splines will facilitate the calculations within the CAD system,in this paper,we propose a system of R-functions represented in spline form called Spline R-function(SR).After trans-forming the function ranges of two base primitives to a new coordinate system,a series of sign constraints following a specific Boolean operation are derived and the spline R-function can be formulated as a piecewise function.Representation of SR in both B´ezier form and B-spline form have been given.Among which the B´ezier ordinates are determined with the help of the B-net method through setting up a series of relations according to the sign constraints and properties of R-functions.The construction processes for both Boolean intersection and union operations with different smoothness are discussed in detail.Numerical experiments are conducted to show the potential of the proposed spline R-function.
