# Project B7

## Contents

# A High-End Toolbox for Simulation and Optimisation of Multi-Physics PDE Models

One of the most fascinating perspectives of numerical simulation is the possibility to use the numerical model to optimise the underlying system with respect to certain parameters (control inputs, shape, material properties, model parameters, e.g.). Such optimisation tasks arise in several of the DC, with an important example being the inversion of a model with respect to quantities of interest (initial conditions, e.g.).

Especially, complex systems that are modelled by systems of PDE and involve multi-physics are at MAC's focus. They require efficient new techniques for coupling single-physics components or for integrating them into an all-at-once approach. The successful optimisation, control, or inversion of such systems is impossible without a close and well-designed interplay between state-of-the-art derivative-based optimisation methods and cutting-edge PDE solvers. Unfortunately, the majority of today's simulation codes are not at all well-prepared for optimisation, and the same holds for multi-physics modelling and simulation.

# Objectives

The project aims at bundling forces to overcome conceptional drawbacks of current simulation software and to make a big step towards a future generation of simulation and optimisation tools for complex systems. The goal is to develop a rapid prototyping HPC software platform for both simulation and optimisation. The design will be hierarchical, with high performance components on all levels, ranging from problem formulation via discretisation to numerics and parallelisation. Work will be interwoven with theoretical investigations of innovative numerical algorithms.

# Research Focus

- Identification of a best possible representation of complex systems with respect to both rapid multi-physics modelling and suitedness for optimisation,
- automated generation of the FEM-discretisation (stiffness matrices, etc.) from this description, in particular supporting high order ansatz spaces,
- coupling to efficient parallel scientific computing packages, in particular Trilinos and PETSc,
- embedding of existing numerical simulation codes,
- development of tailored structure-exploiting adjoint-based optimisation and inversion methods,
- integration of environments for multi-physics coupling,
- formulation and implementation of benchmark problems.

The general design and development tasks 1–3 will be done by all participating researchers and will profit from already existing software developments. Working packages 4 and 7 will adapt and integrate existing numerical models from computational mechanics (Adams, Manhart, Rank), earth science (Bunge), and civil and environmental engineering (Manhart, Rank). WP 5 and 6 will rely on the team's expertise in optimisation (Brokate, Ulbrich), inversion (Brokate, Bunge, Ulbrich), and fluid-structure interaction (Bungartz, Rank).

# Research Highlights

The subproject Rank focuses on the further development of immersed boundary methods for solid mechanics problems, in particular the finite cell method (FCM). In a first step, we extended the finite cell method to geometrically nonlinear problems of solid mechanics, with a special focus on large deformation analysis of foam-like structures. We also introduced a possibility for local refinement by the hp-d approach, which combines a high-order p-version basis for the resolution of the global problem with a local hierarchical linear basis for the resolution of local features. The research of the first part has been conducted in strong collaboration with the group of Prof. A. Düster at the Technische Universität Hamburg-Harburg.

In a second step, we combined the FCM concept with higher order and higher continuity B-spline approximations. For the B-spline version of the finite cell method, we demonstrated similar benefits as provided by the p-version of the FCM, such as simple mesh generation irrespective of the geometric complexity involved and exponential rates of convergence for smooth problems.

In a third step, we explored hierarchical refinement of NURBS for adaptive isogeometric and immersed boundary analysis. We used the principle of B-spline subdivision and ideas of the hp-d adaptive approach to derive a local refinement procedure, which combines full analysis suitability of the basis with straightforward implementation in tree data structures and simple generalization to higher dimensions. We successfully tested hierarchical refinement of NURBS for some elementary fluid and structural analysis problems in two and three dimensions. Using the B-spline version of the finite cell method, we demonstrated a proof of concept for immersed boundary methods as a seamless IGA design-through-analysis procedure for complex engineering parts, for example a ship propeller. We showed that hierarchical refinement considerably increases the flexibility of this approach by adaptively resolving local features. The research of the third part has been conducted in strong collaboration with the group of Prof. T.J.R. Hughes at the Institute for Computational Engineering and Sciences (ICES) of the University of Texas at Austin.

# Participants

**Principal Investigators:**

Prof. Dr. Michael Ulbrich (coordination) | Mathematical Optimisation |

Prof. Dr.-Ing. Nikolaus A. Adams | Aerodynamics |

Prof. Dr. Martin Brokate | Mathematical Modelling |

Prof. Dr. Hans-Joachim Bungartz | Scientific Computing in Computer Science |

Prof. Dr. Hans-Peter Bunge | Geophysics |

Prof. Dr.-Ing. habil. Michael Manhart | Hydromechanics |

Prof. Dr. Ernst Rank | Computing in Engineering |

**Project Members:**

Tuure Y. Döring |

Irene Hofmann |

Dr. Janos Benk |

Dr. Dominik Schillinger |

**Partners:** Universität Augsburg (Hoppe)

# Publications

- D. Schillinger, A. Düster, E. Rank:
. International Journal for Numerical Methods in Engineering.**The hp-d adaptive Finite Cell Method for geometrically nonlinear problems of solid mechanics** - D. Schillinger, E. Rank:
. Computer Methods in Applied Mechanics and Engineering 200(47-48), pp. 3358-3380, 2011.**An unfitted hp-adaptive finite element method based on hierarchical B-splines for interface problems of complex geometry** - D. Schillinger, S. Kollmannsberger, R.-P. Mundani, E. Rank:
IOP Conference Series: Materials Science and Engineering 10(1), 012170, 2010.**The finite cell method for geometrically nonlinear problems of solid mechanics.** - D. Schillinger, R.-P. Mundani:
In: Tagungsband 22. Forum Bauinformatik 2010, Berlin, Germany, 2010.**Finite Element Code Design with Sandia’s Library Package Trilinos: Efficient data structures, solver interfaces and parallelization.** - D. Schillinger, S. Kollmannsberger, R.-P. Mundani, E. Rank:
In: Proc. of the 4th European Conference on Computational Mechanics (ECCM 2010), Paris, France, 2010.**The Hierarchical B-Spline Version of the Finite Cell Method for Geometrically Nonlinear Problems of Solid Mechanics** - D. Schillinger,
, Dissertation, 2012**The p- and B-spline versions of the geometrically nonlinear finite cell method and hierarchical refinement strategies for adaptive isogeometric and embedded domain analysis**

- J. Benk,
, Dissertation, 2012**Immersed Boundary Methods within a PDE Toolbox on Distributed Memory Systems**