Isogeometric topology optimization of shell structures using trimmed NURBS surfaces

• The topological changes of a shell are represented by NURBS trimming curves.
• The complex topologies of a shell structure are effectively analyzed by adopting TSA.
• The isogeometric shape optimization is combined with the topological change step.
• The holes of a design domain are adaptively treated during the optimization process.
• The surface curvatures and the topology of a shell can be optimized effectively.

In the present research, the isogeometric topology optimization of shell structures is proposed. There have been lots of successful studies on the shape optimization using isogeometric analysis (IGA). However, it is not straightforward to apply the conventional IGA to topology optimization. Topological changes of a domain are hard to be represented due to the tensor-product form of a Non-Uniform Rational B-Spline (NURBS) surface. Since only quadrilateral domains can be handled in the conventional IGA, a topologically complex domain which frequently appears during the optimization process should be built by introducing multiple untrimmed NURBS patches for analysis. Trimmed surface analysis (TSA) is the IGA where trimming techniques are employed, and it can handle a topologically complex domain with a single NURBS patch effectively. In the present work, the basic concept of the two-dimensional TSA is appropriately adapted to shell structures in order to handle the complex topologies. The whole optimization process includes the topological change step and the shape optimization step. The criteria based on the topological derivatives are used for the judgement of new hole creation and the decision of the hole position. Holes are represented by introducing NURBS trimming curves. The trimming curve control points as well as the surface control points are set as design variables for the shape optimization. In the optimization process, updating of the trimming curves is performed in adaptive manners to maintain smooth boundary representation and robust convergence. With numerical examples, it is shown that the proposed method gives appropriate and acceptable shell structure designs. By the inner or outer boundaries of a domain which are described by trimming curves, the smooth material layout is derived without gray scales or checkerboard patterns. Since the same IGES standard is used throughout the process, the final design derived by the present method can be directly communicated with CAD systems without any additional treatments.