Isogeometric multi-patch shells and multigrid solvers

In this research project we develop mathematical techniques to analyze how surfaces are deformed when certain loads are applied. We specifically focus on partial differential equations that describe the physical behavior of so-called thin plates and shells. These equations are particularly useful in understanding how thin objects, e.g. industrial parts made from sheet metal, delicate architectural structures such as domes or roofs, or biological formations like membranes or blood vessels, respond to different conditions. To understand how the object reacts in a specific situation it might deform in unusual ways or break when forces are applied the shape of the studied object needs to be represented accurately and the solution of the problem must be computed with high precision. This is of particular interest in many technical applications, where accurate simulations can be performed instead of costly testing (and possible destruction) of components. Consider, for example, a sheet metal object like a car part. The way it deforms in case of a crash depends on intricate details such as holes, specific corner shapes, creases, and reinforced sections. In this project, our aim is to investigate how such features of the shape of an object can be represented with high precision and whether certain details can be simplified for analysis. To find a good balance is crucial. While an extremely detailed model might be necessary for simulating certain physical processes, like deformation under loads, it can also make the problem more complex and slow down the computation or make them impossible. Our goal is to develop a shape representation together with a suitable solution algorithm that is both flexible in its geometric accuracy and efficient in terms of computational speed. We adopt the framework of Isogeometric Analysis, wherein a common mathematical representation describes both the shape of the object and the solution of the physical problem. We particularly focus on representations that are based on free-form splines, which are used in CAD systems. To achieve faster computation speed, we employ the so-called multigrid method, a technique from high-performance computing that numerically solves complex mathematical problems. This method requires the shape to be described by a structure that is easily refined, that is, by a hierarchy of grids with varying resolution, ranging from coarse to fine. The method then iteratively solves the problem, starting from a coarse solution, which is computationally simpler, and then progressing through the different levels of resolution, using on each level the solution of the previous level. This way it optimizes the computational process and ultimately converges to a highly accurate result after very few steps. The research project is led by Thomas Takacs of the Johann Radon Institute for Computational and Applied Mathematics (RICAM) of the Austrian Academy of Sciences and will be carried out by researchers from RICAM together with cooperation partners at JKU Linz and TU Delft in the Netherlands.