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PROCEEDINGS OF THE SEVENTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY
Edited by: B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru and M.L. Romero
Implementation of T-spline based Isogeometric Analysis within an Object Oriented Environment
D. Rypl and B. Patzák
Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Czech Republic
D. Rypl, B. Patzák, "Implementation of T-spline based Isogeometric Analysis within an Object Oriented Environment", in B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru, M.L. Romero, (Editors), "Proceedings of the Seventh International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 70, 2010. doi:10.4203/ccp.94.70
Keywords: isogeometric analysis, B-splines, NURBS, T-splines, structural analysis, object oriented environment.
The concept of isogeometric analysis (IGA), initially motivated by the gap between the computer aided design (CAD) and the finite element method, builds upon the concept of isoparametric elements, in which the same shape functions are used to approximate the geometry and the solution of a single finite element. The IGA goes one step further as it employs the same functions for the description of the geometry and for the approximation of the solution space on that geometry. This implies that the isogeometric mesh of the CAD geometry encapsulates the exact geometry no matter how coarse the mesh actually is. As a consequence, the need to have a separate representation for the original CAD model and another one for the actual computational geometry is completely eliminated.
The isogeometric approach has been originally developed using NURBS which are the basic building blocks in most CAD systems. However, this concept possesses several drawbacks when handling NURBS patches of two or more dimensions. One of the most significant is the fact that the h-refinement applied in a particular direction is propagated through the entire control grid in the remaining direction(s). This increases considerably the number of control points and variables. Moreover, if the compatibility of adjacent NURBS patches has to be maintained on their interfaces, the refinement must often propagate from one patch to another. This makes the growth of the number of control points too rapid. An attractive solution to the above problem is to use the so-called T-splines, which are a generalization of NURBS, by allowing a row of control points to terminate before reaching the patch boundary. The final control point in the partial row is called a T-junction and the control grid a T-mesh. The advantage of T-splines consists in the fact that they allow true local refinement, without propagating the entire row of control points. Moreover, they enable efficient merging of several NURBS patches that have different knot vectors into a single gap free model.
The aim of this paper is to present how the T-spline based IGA concept may be implemented into an inherent object oriented finite element environment. The class hierarchy and corresponding methods are designed in such a way, that most of the existing functionality is reused. The missing data and algorithms, namely those providing the support for handling isogeometric basis functions, forming the isogeometric mesh, application of an appropriate numerical integration scheme, and assignment of boundary conditions are developed and implemented in such a way that the object oriented features, such as modularity, extensibility, maintainability, and robustness, are fully retained. The functionality of the implementation and the performance of the T-spline based IGA is presented using a simple two-dimensional example.
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