A new 48 d.o.f. quadrilateral shell element with variable‐order polynomial and rational B‐spline geometries with rigid body modes

SUMMARYA T-spline surface is a nonuniform rational B-spline (NURBS) surface with T-junctions, and is defined by a control grid called T-mesh. The T-mesh is similar to a NURBS control mesh except that in a T-mesh, a row or column of control points is allowed to terminate in the inner parametric space. This property of T-splines makes local refinement possible. In the present study, shell formulation based on the T-spline finite element method (FEM) is presented. Shell formulation based on NURBS or T-splines has fundamental limitations because rotational DOFs, which are necessary in the shell formulation, cannot be defined on control points. In this study, the simple mapping scheme, in which every control point is mapped into one geometric point on the surface, is employed to eliminate the limitations. Using this mapping scheme, T-spline FEM can be easily extended to the analysis of shells. The proposed shell formulation is verified through various benchmarking problems. This study is a part of the efforts by the authors for the integration of CAD-CAE processes.

Section: T-splinesmentioning

confidence: 99%

T‐spline finite element method for the analysis of shell structures

Uhm

Youn

2009

“…Fan and Luah [6] have reported results of plate analysis as a special case of the B-spline finite element method restricted to cubic approximation. In the study of Moore and Yang's [7], a quadrilateral shell element was introduced, in which the element geometry is described by rational B-spline functions in curvilinear co-ordinate system, while the displacement field is approximated by bi-cubic Hermite polynomials in Cartesian co-ordinates. This study is limited in analysing the general structural behaviour because they do not use the same shape functions in geometrical modelling and finite element analysis.…”

Section: Introductionmentioning

confidence: 99%

Integration of geometric design and mechanical analysis using B‐spline functions on surface

Roh

Cho

2005

SUMMARYB-spline finite element method which integrates geometric design and mechanical analysis of shell structures is presented. To link geometric design and analysis modules completely, the non-periodic cubic B-spline functions are used for the description of geometry and for the displacement interpolation function in the formulation of an isoparametric B-spline finite element. Non-periodic B-spline functions satisfy Kronecker delta properties at the boundaries of domain intervals and allow the handling of the boundary conditions in a conventional finite element formulation. In addition, in this interpolation, interior supports such as nodes can be introduced in a conventional finite element formulation. In the formulation of the mechanical analysis of shells, a general tensor-based shell element with geometrically exact surface representation is employed. In addition, assumed natural strain fields are proposed to alleviate the locking problems. Various numerical examples are provided to assess the performance of the present B-spline finite element.

A new 48 d.o.f. quadrilateral shell element with variable‐order polynomial and rational B‐spline geometries with rigid body modes

Moore

Yang

Anderson

1984

SUMMARYA 48 degrees-of-freedom (d.0.f.) quadrilateral thin elastic shell finite element using variable-order polynomial functions, B-spline functions and rational B-spline functions to model the shell surface is developed. This development may allow the stiffness formulation of the shell element to be linked to the geometry data bases created by computer aided design systems. The displacement functions are that of bicubic Hermitian polynomials. The displacement functions and d.0.f. are expressed and investigated in both the curvilinear and Cartesian forms. The curvilinear form is simpler and can provide the proper solution for a certain class of shell problems. For certain highly curved shells such as bellows, however, the curvilinear form fails to properly model some rigid body modes even with either the explicit inclusion of rigid body terms or the high order displacement functions. It is suggested in this study that such difficulty can be circumvented and the rigid body modes can be properly included if a Cartesian form is used for displacement functions. The straindisplacement equations are expressed in curvilinear co-ordinates. Thus, the Cartesian displacement functions require a transformation to curvilinear displacement at each numerical integration point. Examples include a pinched cylinder, a translational shell under central load, a uniformly loaded hypar shell, a pressurized ovel shell, a semi-toroidal bellows and a U-shaped bellows. For the first four examples, geometric modellings consist of polynomials of second-order (subparametric), third-order (isoparametric), and fourth and fifth-order (both superparametric) as well as B-spline functions of fourth-and fifth-order. The geometries of the pinched cylinder, the semi-toroidal bellows, and the U-shaped bellows were modelled exactly using rational B-spline functions. All the results obtained are in good agreement with alternative existing solutions.