Multi-patch isogeometric analysis with C1-smooth functions

Spline surfaces and volumes are two useful tools for the geometric design and for the numerical analysis of components in various engineering disciplines such as automotive, mechanical and civil engineering. Both structures can be employed to design complex three-dimensional objects. Spline surfaces allow the modeling of thin structures called shells. Examples of shells are autobodies in automotive engineering or roofs and walls in civil engineering. The use of shells is based on the idea to describe the thin three-dimensional object by an easier just two-parametric representation given by the surface. In contrast, spline volumes can represent thicker structures by also describing the inner part of the three-dimensional shape but requires a more complex three-parametric representation. Examples of volumes are e.g. fluids in mechanical engineering. The goal of this project is to perform numerical analysis of complex shells and volumes used in real-world applications. These objects cannot be modeled in general by one single surface or volume patch. Instead, so- called multi-patch geometries are needed which consists of several surface or volume patches. For the numerical analysis of these structures, we use a new and innovative simulation technique called Isogeometric Analysis which allows the direct link of the geometric design and of the numerical analysis of the shell or volume. This is advantageous compared to classical simulation techniques, where the two steps have to be done separately from each other, and radically simplifies the engineering process of the components. In this project we will develop the theory and the methods for performing Isogeometric Analysis of complex multi-patch shells and volumes. This comprises amongst others the construction of suitable representations of the shells and volumes, the design of the required functions for the numerical analysis as well as the development of corresponding algorithms for the Isogeometric Analysis. Moreover, we will implement all methods within the open-source software library G+Smo (http://gs.jku.at).

A research focus of our project was the development of methods for the numerical analysis of complex, thin-walled components, the so-called shells, which are used in various engineering disciplines such as automotive, mechanical and civil engineering. Examples of shells are autobodies in automotive engineering or roofs and walls in civil engineering. The concept of shells is based on the idea to describe the thin, three-dimensional objects by simpler, just two-parametric representations, called surfaces. In particular, spline surfaces, which are a common and useful tool for the geometric design and for the numerical analysis of two-parametric objects, play an important role. However, complex shells generally cannot be modeled by a single spline surface patch and require instead so-called multi-patch spline constructions, which consist of numerous individual spline surface patches. For the numerical analysis of the multi-patch shells, we used the concept of isogeometric analysis, which allows the direct link of the geometric design and of the numerical analysis of shells, and thus allows a fast and efficient optimization of the thin-walled components. Thereby, the numerical analysis of the complex shells is based on solving a fourth-order partial differential equation. Due to the high order of the partial differential equation and the necessity of the multi-patch representation of the shells, it was essential to develop a novel, suitable mathematical representation for the shells, new functions for the numerical analysis as well as corresponding algorithms for the isogeometric analysis of the shells. Further research results included, among other things, the extension of the developed isogeometric method to an adaptive scheme, which enables a further increase in the efficiency in the numerical analysis of the shells through a significantly reduced number of required degrees of freedom, as well as the development of functions for solving partial differential equations of an even higher order. In addition to complex multi-patch shells, we also dealt with multi-patch spline volumes as part of the project. Spline volumes can represent thicker structures by also describing the inner part of the three-dimensional shape but requires a more complex three-parametric representation. Examples of volumes are e.g., fluids in mechanical engineering. In this area, we focused amongst others on the design of functions which are especially suited for solving fourth-order partial differential equations over the considered multi-patch spline volumes.