Plenary Lectures 2018

Hierarchical energy based modeling and numerical simulation for coupled multi-physics and multi-scale systems

We discuss an energy based modeling approach to deal with coupled systems from different physical domains that act on widely different scales. Each physical system is modeled via a model hierarchy (ranging from detailed models for simulation to reduced models for control and optimization) of port-Hamiltonian systems. The systems are coupled  via a network of uni-physics nodes coupled via power conserving interconnections so that the full system stays port-Hamiltonian. Using this very flexible approach, it is possible to control the accuracy of  each component seperately and to the need of the application. Error controlled model reduction and Galerkin projection as in Finite Element Modeling work in an analogous way.  We will demonstrate the approach with real world examples from gas transport optimization, power grid modeling, and the analysis of disk brake squeal.

Selection and Adaptation of Computational Models in the Presence of Uncertainty: Predictive Models of Tumor Growth and Random Hetero geneous Materials

In thispresentation, we describe a general Bayesian framework for selecting, calibrating, validating, and optimizing computational models of the behavior of complex, physical systems in the presence of uncertainties. The selection and adaptive control of models is based on the calculation of model plausibilities as weighted values of model evidences, on estimates of sensitivity of outputs to the choice of model parameters, and on a posteriori estimates of modeling and discretization error in quantities of interest. As an added benefit, the estimation of moments in modeling error between high-fidelity models and sequences of surrogate models can be used to significantly improve the efficiency of solvers, such as Multi Level Monte Carlo methods.

To demonstrate the theory and predictive paradigm, the problems of designing and selecting predictive models of the growth or decline of ava scular heterogeneous tumors in living tissue are explored and example applications using in vitro laboratory data are discussed . In addition, the estimation and control of modeling error and adaptive multiscale modeling of random heterogeneous media are also discussed.

Onsager reciprocity, gradient flows, and large deviations

The second law states that in a thermodynamically consistent system the ‘entropy’ is a Lyapunov function, a function which is monotonic along solutions of the corresponding differential equations. When the system can be written as a gradient flow of the entropy, then this statement is strengthened: not only is this functional monotonic, but it _drives_ the dissipative part of the evolution in a precise way, mediated by a ‘friction operator’. 

In this lecture I will go one step further. Onsager already pointed out how symmetry properties of linear friction operators arise through an upscaling procedure from a microscopic-reversibility property of the underlying system. Fluctuations figure centrally in his argument, but at that time their theory was not well developed, and more could not be said.

However, recently we have found that the connection between microscopic reversibility and macroscopic ‘symmetry’ properties is not at all limited to the close-to-equilibrium, linear-friction-operator context of Onsager’s. I will describe how the large-deviation theory of fluctuations allows one to make a much more general statement, where microscopic reversibility is one-to-one coupled to ‘symmetry’ at the macroscopic level – provided one generalizes the concept of symmetry in an appropriate way. 

Image Compression with Partial Differential Equations

Partial differential equations (PDEs) are widely used to model processes in engineering. In this talk we will see that they also have a high potential for lossy compression of digital images. The idea sounds temptingly simple: We keep only a few pixels and reconstruct the remaining data with PDE-based interpolation. However, this gives rise to several difficult and interrelated problems, e.g.: – Which data should be kept? – What are the most useful PDEs? – How can the selected data be encoded efficiently? – What are efficient numerical algorithms? Solving these problems requires to combine ideas from different mathematical disciplines such as mathematical modelling, optimisation, interpolation and approximation, and numerical methods for PDEs. After careful engineering we will end up with compression methods that can beat current standards such as JPEG. Since the talk is intended for a broad audience, we focus on the main ideas, and no specific knowledge in image processing is required.

On the continuum mechanical modelling of polymerphysical phenomena

Amorphous and semicrystalline polymers are applied in nearly all disciplines of engineering and daily life. For short-term applications under usual temperature conditions they exhibit a certain number of advantages but during larger times their mechanical material properties undergo significant changes. Caused by their chemical structure polymeric materials behave completely different in comparison with metals. They exhibit moderate characteristic temperatures which characterise glass transitions, melting and crystallisation regions or the onset of chemical ageing and diffusion processes. If, during the production of a polymer part or in a certain application, such a temperature is exceeded their mechanical material properties can change enormously. In consequence, detailed experimental investigations, the physical understanding, the constitutive representation in continuum mechanics or thermodynamics and the numerical simulation of polymers and parts which are made of them are essential and define big challenges in scientific research. In this presentation, three important polymerphysical phenomena, namely the glass transition, crystallisation and melting as well as the diffusion of fluids in polymers are highlighted in detail. The formulation of the constitutive models is done in accordance with the basic laws of thermodynamics. To this end, the physical motivation and all fundamental ideas as well as the related assumptions and constitutive theories are presented and discussed. The main properties of the different approaches are visualised using simulations and are discussed in the context of experimental data.

Why do turbulent flows have an increased state of symmetry?

Physical meaning and its implications

It was A. Einstein in his seminal work on special relativity in which he contemplated the symmetry principle as the axiomatic foundation of physics in general that confines the admissible laws of motion. For classical mechanics this is Galilean invariance, which is also admitted by the elementary equations of fluid motion – the Navier-Stokes equations. Interesting enough, all complete statistical turbulence theories, though uniquely derived from Navier-Stokes equations, admit more invariance properties, i.e. symmetries, which go beyond the Galilean group. It was recently shown that two of these so-called statistical symmetries mirror key properties of turbulence, i.e. intermittency and non-gaussianity. The recent discovery of statistical symmetries has important consequences for our understanding and derivation of turbulent scaling laws. Beside the classical logarithmic law of the wall, which is now understood to be a symmetry induced invariant solution of the underlying statistical equation of turbulence based on the new statistical symmetries, this finding will be exemplarily revealed employing examples of wall-bounded and free turbulent shear flows. Most important, these scaling laws not only describe the mean velocity but also higher order moments. Finally, symmetries and statistical symmetries have important implications for turbulence models. Since the 70th of the last century, essentially all turbulence models, both RANS and SGS models for LES, did comply with Galilean invariance. Interestingly, for a minor part, turbulence models have already been implicitly equipped with statistical symmetries, as they comply with the log-law. Still, symmetry based turbulence modelling is still at its infancy.

Particles – an Option for Unusual Computations

Simulation became an indispensible tool in the modern design process and there are several well established methods available like the finite element method or grid-based computational fluid dynamics. However, there are situations where these well investigated methods have weaknesses or can even not be applied at all.

For these challenging problems meshless methods can be an interesting alternative enriching and completing the toolbox of the engineer. In this talk two methods, the Discrete Element Method (DEM) and Smoothed Particle Hydrodynamics (SPH), are introduced.

The methods are illustrated by a number of applications from different fields of engineering. Examples from fluid dynamics, from fluid-structure interaction, from material processing and others are presented and challenges and limitations are shown. Although the presented methods are quite young compared to some of the well established computation methods they clearly already show their high potential. However, also some numerical problems and stability issues exist and concepts to overcome these challenges must be investigated. The mechanics of particles became a very active area of research with own conferences, journals, and a lively research community and progress is fast.

Particle simulations are always time consuming and require a lot of resources and care. In order to reduce computation times, parallel computation is advisable and efficient approaches are required.

Phase-field modeling and computation of fracture and fatigue

The talk discusses recent research results obtained in the group of the speaker in the framework of the phase-field approach to fracture, and the recent extensions to fracture in partially saturated porous media and to fatigue. Applications include the study of desiccation phenomena in soils and shrinkage in cementitious materials, as well as the investigation of fatigue phenomena in brittle materials. The first steps towards mesoscale modeling of fracture in concrete are also discussed.