Stable Isogeometric Analysis of Trimmed Geometries

The project Stable Isogeometric Analysis of Trimmed Geometries" is aimed at making a significant contribution towards the goal of a seamless integration of design and analysis. In general, the application of functions used in Computer Aided Design (CAD) to numerical simulation is denoted as isogeometric analysis. CAD models provide the best geometrical representation of the problem considered and their direct integration improves the efficiency of the overall analysis process. However, one main challenge is the analysis of so-called trimmed geometries which consist of visible and invisible components. First of all, only the visible part defines the domain of interest and has to be considered for the computation. Moreover, trimmed objects may induce instabilities to the numerical simulation. The goal of the project is to develop robust procedures that allow a stable and higher order isogeometric analysis of trimmed geometries. In the proposed approach the stabilization of the simulation is realized by reformulating the instable functions as combination of stable ones. This is done in a general manner so that it can be applied to complex cases. A requirement of the method is that a sufficient number of stable functions is available. This is accomplished by the application of local refinement techniques. In addition, a tailored integration scheme is employed to preserve integration accuracy. The final outcome of the project will provide a big step towards the analysis of real-world problems without the need of generating an additional analysis model. In particular, an isogeometric analysis is derived which enables a stable and higher order analysis of the most common CAD models used in engineering design.

A significant contribution to the interoperability of Computer Aided Design (CAD) and numerical simulations was achieved within the current project. Numerical procedures have been developed which allow a seamless integration of CAD models into a simulation such that a robust analysis can be performed. CAD and numerical simulations are fundamental tools for product development, as they facilitate virtual drafting and testing of physical phenomena (e.g., the deformation due to an external force) on these virtual prototypes. Unfortunately, these fields have emerged as two autonomous disciplines leading to the absurd situation that their interplay is currently the most time-consuming and error-prone task of the analysis of practical problems. Within the current project, this issue has been systematically examined, setting thereby the focus on the most common CAD models in engineering design, so-called trimmed geometries. This revealed the source of the interoperability problems: certain CAD operations particularly surface-to-surface intersections cannot be performed exactly and as a result, ambiguous CAD model data may arise. Numerical simulation of practical trimmed models addresses the core issue of the interaction between CAD and analysis, namely the appropriate treatment of these shortcomings of CAD models. In order to enable a direct integration of trimmed geometries into an analysis, a novel approach was developed which processes CAD data directly and enhances it in such a way that a simulation can be performed in an effective and stable manner. This represents a significant contribution and in addition, is highly beneficial for modern product development procedures.