Mathematics

Abstract: Isogeometric analysis is a recent method of computational analysis where functions used to describe geometries in Computer Aided Design (CAD) are adopted as basis for analysis. Due to this unified geometric representation, the model transfer from design to analysis, called mesh generation, is omitted providing a better integration of  design and analysis. NURBS (Non-Uniform Rational B-Splines) are the most widespread technology in today’s CAD modeling tools and therefore are adopted as basis functions for analysis. Apart from the geometrical advantages, NURBS-based isogeometric analysis has proven superior approximation properties compared to standard finite element analysis for manydifferentapplications due to the higher order and higher regularity of the basis functions. Furthermore, the higher continuity between elements also allows the implementation of formulations of second (or higher) order, which is not possible with C0-continuous Lagrange elements. An isogeometric shell element [1] based on the Kirchhoff-Love shell theory is presented. The formulation is completely displacement-based, i.e., without rotational degrees of freedom. The element is formulated geometrically nonlinear and therefore applicable 
to problems with large deformations. Furthermore, a method, called the bending strip method [2], is presented which enhances the application of this shell formulation to arbitrary structures consisting of multiple patches. Different examples show the good performance and accuracy of the method, for geometrically linear and nonlinear problems. The application in a fully coupled FSI simulation of rotating wind turbine blades demonstrates the relevance for realistic industrial structures [3]. Furthermore, the fact that both CAD description and shell analysis are surface-based, allows an effective integration of design and analysis, which is demonstrated by integrating the presented method into a commercial CAD software. Moreover, the presented method is extended to shape optimization. In traditional shape optimization using FE analysis, there are two different approaches for the parametrization of the optimization model, namely the CAD-based and the FE-based approach. Isogeometric shape optimization is introduced as a combination of and enhancement to the existing approaches which provides more flexibility in choosing the design space.

[1] Isogeometric shell analysis with Kirchhoff–Love elements, J. Kiendl, K.-U. Bletzinger, J. Linhard, and R. Wüchner, Computer Methods in Applied Mechanics and Engineering, 198:3902-3914, 2009.

[2] The bending strip method for isogeometric analysis of Kirchhoff–Love shell structures comprised of multiple patches, J. Kiendl, Y. Bazilevs, M.-C. Hsu, R. Wüchner, K.-U. Bletzinger, Computer Methods in Applied Mechanics and Engineering, 199:2403–2416, 2010.

[3] 3D simulation of wind turbine rotors at full scale. Part II: Fluid-structure interaction modeling with composite blades, Y. Bazilevs, M.-C. Hsu, J. Kiendl, R. Wüchner, and K.-U. Bletzinger, International Journal for Numerical Methods in Fluids, 65:236–253, 2011.