Final Report Summary - SIGMACADE (Subdivision based IsoGeometric Modelling Applied to Computer Aided Design and Engineering) The mechanical behavior of structures subject to environmental impact is described by partial differential equations which are usually solved approximately using finite element analysis (FEA). Standard FEA today requires a simulation mesh which is derived from the computer aided design (CAD) mesh. This approach is time consuming and introduces approximation errors. Interfacing shape models between design and analysis is a serious limitation in today's geometry modeling industry. The approach in which surface geometry designed in a CAD environment is directly used in analysis, is referred to as isogeometric analysis (IGA). There are several geometric representation employed in CAD which may be used in IGA. Nonuniform rational B-splines (NURBS) are the current standard geometry representation in the CAD industry. While there are clear advantages using NURBS (arbitrary degree, multiple knots to interpolate boundaries, easy evaluation), one mayor disadvantage is the tensor product construction. This restricts to a rectangular parametrization and requires a complicated surface to be decomposed into a set of NURBS surface patches which have to be stitched together to share common boundaries. Continuity problems, or even cracks at the boundaries of different patches lead to problems in analysis in an IGA approach. A subdivision scheme is an iterative refinement which generates a sequence of finer and finer nets which converge to a continuous surface. The refinement level of the subdivision surface can be chosen according to the resolution required. Subdivision schemes provide a flexible and efficient tool for arbitrary topology free-form surface modeling using a single surface, avoiding many of the problems inherent in traditional spline patch based approach. The aim of this project was to explore IGA based on subdivision surfaces. Initial investigation of IGA based on subdivision schemes looked closely at two standard subdivision schemes, namely Loop subdivision based on triangle meshes and Catmull-Clark subdivision based on quadrilateral meshes. This investigation showed a superior performance of the Catmull-Clark subdivision scheme. However, the Catmull-Clark algorithm is a generalization of degree three, uniform, non-rational B-spline surfaces to arbitrary topology. Non-uniform knot placement are not supported. Therefore, interpolated boundaries cannot be evaluated correctly for analysis. Because rational expressions are not available it is not possible to exactly describe conical sections. Also, because the Catmull-Clark subdivision algorithm generalizes B-splines of degree three, higher degree analysis, which has shown to provide better convergence properties in analysis, cannot be explored.However, the project objectives were to explore mesh refinement of an isogeometric framework based on subdivision surfaces. Two types of refinements are important in FE calculations: Refining or coarsening a mesh to adapt the number of degrees of freedomsavailable for the simulation, referred to as h-refinement. This can easily explored by adjusting the number of subdivision steps before analysis. The second is p-refinement which requires being able to adjust the polynomial degree of the basis functions, which is not possible using Catmull-Clark subdivision.In this project we successfully extended the concept of isogeometric analysis to NURBS compatible subdivision surfaces developed by Cashman et al. 2009. This algorithm is compatible with arbitrary degree NURBS in regular parts of the surface but also incorporates extraordinary vertices to model arbitrary topology with a single surface. Therefore, Cashman’s NURBS compatible subdivision combines the advantages of NURBS and subdivision surfaces for analysis:(i) The arbitrary topology property of subdivision surfaces provides us with watertight meshes. Their multilevel resolution property enables us to easily perform the analysis at different resolution levels (h-refinement).(ii) Like with NURBS, arbitrary degrees (p-refinement) and rational representations can be explored in the isogeometric framework and boundaries can now be parametrized correctly.In order to apply the FE method to NURBS compatible subdivision surfaces a parameterization of the surface in terms of elementary domain elements is required. Also, efficient evaluation routines for limit surface quantities such as first and second derivatives need to be available. In regular regions all NURBS evaluation algorithms apply. To evaluate Cashman's NURBS compatible subdivision surface around irregular regions an extension of Stam's exact evaluation algorithm to higher degrees was developed as part of the project.We are now able to evaluate the surface efficiently everywhere on the surface up to degree 9 and up to valence 11, by employing the eigenstructure of the corresponding subdivision matrix.Thin-walled structures, like car bodies or ship-hulls, are omnipresent in CAD and are usually designed as surfaces with a certain thickness. This corresponds to the structural model in shell analysis and the geometry data provided by CAD can be used directly in analysis. The analysis of a thin shell involves higher derivatives, and even a weak solution must have continuous first derivatives. Also, thin shell analysis is particularly sensitive to geometric inaccuracies. The IGA approach carries great advantages compared to std FEA where analysis requires extra continuity and a highly accurate representation of the geometry is required.A comparison of our approach to thin shell simulation results for which either highly accurate approximations or analytical solutions exist, demonstrated the correctness of our treatment of thin shells. We were also able to confirm that convergence properties improve as expected for higher degrees (p-refinement) as well as for denser surface representations (h-refinement). For the latter it is of outermost importance that a rational expression is available, since otherwise the description of geometry involving conic sections is inaccurate and analysis results are not useful. Using NURBS compatible subdivision interpolated boundaries as well as arbitrary shapes pose no difficulty.Another objective of this project was to develop a prototype modeling system which combines CAD and CAE through an IGA approach. The prototype developed as part of this project built on blender, a free and open source cross-platform 3D design suite. Two plug-ins have been developed: one to enable modeling using NURBS compatible subdivision surfaces and a second enabling thin shell IGA. The successful combination of design and analysis in one modeling system provides the designer with feedback about the physical plausibility of the product throughout product development. This leads to time and cost savings for the development of new products, guarantees a fast adoption to shorter product life-cycle and allows for an experimental, flexible design approach.Adding simulation modalities to the design software also increases the modeling possibilities of the designer by offering a range of new physically based modeling modalities: a designer is now also able to modify a design by applying forces or changing constraints within a thin shell simulation, next to the common modeling interface of adjusting control points.Our findings have been published on the projects web-site throughout the project http://www.cgv.tugraz.at/CGV/Research/Projects/SIGMACADE.. Results have also been presented at the First International Conference on Subdivision, Geometric and Algebraic Methods, Isogeometric Analysis and Refinability and also have been written up and are about to be submitted to the Journal of Applied Mathematics and Computation and the Computer Graphics Forum. The results and ongoing work resulting from this project demonstrate that NURBS compatible subdivision is an ideal basis for a common framework for design and analysis, because it combines advantages from NURBS, namely arbitrary degree, rational representations, non-uniform knot spacing for interpolation with the advantages offered by subdivision surfaces, namely watertight arbitrary shape meshes. We are now able to harvest both sets of advantages in one IGA modeling system, increasing the integration of CAD and CAE in an efficient and natural way. We expect the product development industry to be highly interested in the outcome of this project.
