Global parameterization of trimmed NURBS based CAD geometries for mesh deformation

Kurzfassung

In this talk we want to present a novel global parameterization scheme for points on a CAD geometry. This algorithm can then be used to compute mesh deformations for changes of the underlying geometry. The creation of structured meshes is a time-consuming trial and error process, which is not suitable for e.g. automatic optimization. Particularly gradient based optimization often performs only small changes of the design variables, which should result in only slightly different meshes. Therefore, methods are required that deform an initial mesh based on the change of the initial geometry. Here, we present a projection method that computes a bijective mapping between a point in space and its global parameterization with respect to the trimmed NURBS based CAD geometry. After a geometry change, the parameterized points can be back-projected into 3D space which eventually yields the deformed mesh. Providing support for trimmed NURBS geometries is particularly challenging, as their surface parameters u;v mighty be valid only in a non-rectangular trimming region. This region however varies on geometry changes, which would lead to a loss of mesh points, if this is not properly handled. To overcome this issue, we reparametrize the trimming region such that the domain of some new parameters u0;v0 is rectangular. Our projection algorithm is separated into three different problems: first – finding the face a mesh point belongs; second – reparametrize the face to get a bijective mapping; third – project the point onto the reparametrized surface. The first and third problem are comparable simple and can be performed using standard CAD algorithms. For the reparameterization problem, we provide a method that converts the 2d trimming domain of the NURBS into a series of two-dimensional untrimmed patches. This is done by first subdividing the original NURBS face into multiple faces. Then, we identify or create four boundary curves for each of these sub-faces. The four boundary curves are finally used to create a reparameterization patch e.g. using the Coons method. The projection of a point leads only to a unique solution, if the reparameterization patch is invertible. We check invertibility of the patch, by separating the patch into rational Bezier spline surfaces and check that their Jacobian determinant is larger than 0. This strategy allows a large range of different face types, including faces with holes and faces with more or less than four boundary curves. The back-projection method is analogous, and also requires the creation of the reparameterization surfaces. This algorithm is implemented in a C++ based library, which utilizes the CAD functionality of the Open CASCADE framework. The library is designed to be used on computing clusters by providing functions for the serialization and deserialization of the geometry. This enables the parallelized projection and back-projection of large computation meshes with millions of points to reduce the computational runtime. Since this algorithm only works for small geometry changes - i.e. for geometries with the same topology - we also added functions to compare and store the topology of two CAD objects. Our method is currently used within a DLR-internal project to enable a large scale gradient based optimization of an aircraft. In our work flow, the software TiGL converts the parametric aircraft description into a CAD representation. The initial structured mesh is created with a commercial mesh generator. In the subsequent iterations of the optimization, the mesh is deformed using the presented method. Details to the robustness of this algorithm and its computational performance will be presented at the talk.