Richard-von-Mises Prize Lectures

Isogeometric analysis (IGA) is a new approach in computational mechanics where functions used to describe geometries in CAD are adopted for analysis, aiming at a better integration of design and analysis. Non-Uniform Rational B-Splines (NURBS), the standard technology in CAD modeling tools, are used as basis functions in IGA. In various studies, IGA has proven superior approximation properties compared to standard finite element analysis for many different applications, which is attributed to the higher order and continuity of the NURBS basis functions.

In structural mechanics, the high inter-element continuity provides a significant advantage for the implementation of the classical theories for thin structures, such as the Kirchhoff-Love shell theory. Such “thin shell” theories typically require smooth basis functions, which has always been a major obstacle for an efficient implementation with finite elements and promoted the development of shell formulations based on the Reissner-Mindlin theory. Within IGA, the necessary continuity is provided naturally and efficient Kirchhoff-Love element formulations can be developed, avoiding various problems typically encountered in Reissner-Mindlin type finite elements, such as shear locking or finite rotations (no rotational degrees of freedom are needed). In this presentation, an isogeometric framework for nonlinear Kirchhoff-Love shell analysis is presented, as well as its application in different fields of engineering, including fluid-structure-interaction (FSI), fracture mechanics, biomechanics, and shape optimization, which demonstrate the potential of this approach. Furthermore, it is shown that the continuity properties of IGA can also be exploited in the context of shear deformable, or “thick”, structural theories, and innovative formulations are developed with only one unknown variable, similar to the “thin” theories, but accounting for shear deformation. Besides being rotation-free, these formulations also completely avoid the classical shear locking problem and could constitute a new paradigm for the development of shear deformable structural elements.

Models based on partial differential equations (PDE) play an important role throughout mathematics, engineering, and life sciences. The solution of forward models has been a core problem in numerical analysis for the past decades. With mathematical and algorithmic advances more challenging tasks could be tackled and much research has been devoted to compute ‘optimal setups’ of the governing equations. In many applications it is of paramount importance to compute the parameters, forcing terms, etc. of the PDE that best match measured data or describe a desired behaviour. Such problems are often formulated as optimization problems with PDE-constraints.

This field has seen much progress over the last years in terms of a better understanding of the mathematical structures and correspondingly the development of efficient numerical techniques to solve these problems. Our field of interest is the solution of the linear systems that are often the computational bottleneck of the optimization procedure. This can be either representing the first order conditions or as part of a nonlinear outer solver such as Newton’s method. Our approach follows a strategy where we solve the system in a monolithic manner, i.e., discretizing in space and time and then solving a large space-time linear system. It is in many cases possible to derive efficient preconditioners for these systems in saddle point form.

Nevertheless, the problems suffer from the curse of dimensionality, which means that refining spatial or temporal dimensions will result in an exponential increase of the matrix dimensions and thus the computational effort. To break this curse we introduce a low-rank in time approach that utilizes the structure of the linear system and allows the solution of a parabolic problem at only a small multiple of the cost of the steady state problem. With recent progress made in the field of tensor solvers, we further show how this approach extends to tasks with an even higher dimensionality such as problems with uncertain coefficients.