ERC Consolidator Grant (2020-2025)
Computational Random Multiscale Problems
PI: Prof. Dr. Daniel Peterseim
Geometrically or statistically heterogeneous microstructures and high physical contrast are the key to astonishing physical phenomena such as invisibility cloaking with metamaterials or the localization of quantum waves in disordered media. Due to the complex experimental observation of such processes, numerical simulation has very high potential for their understanding and control. However, the underlying mathematical models of random partial differential equations are characterized by a complex interplay of effects on many non-separable or even a continuum of characteristic scales. The attempt to resolve them in a direct numerical simulation easily exceeds today's computing resources by multiple orders of magnitude. The simulation of physical phenomena from multiscale models, hence, requires a new generation of computational multiscale methods that accounts for randomness and disorder in a hierarchical and adaptive fashion.
This project concerns the design and numerical analysis of such methods. The main goals are connected to fundamental mathematical and algorithmic challenges at the intersection of multiscale modeling and simulation, uncertainty quantification and computational physics:
(A) Numerical stochastic homogenization beyond stationarity and ergodicity,
(B) Uncertainty quantification in truly high-dimensional parameter space,
(C) Computational multiscale scattering in random heterogeneous media,
(D) Numerical prediction of Anderson localization and quantum phase transitions.
These objectives base upon recent breakthroughs of deterministic numerical homogenization beyond periodicity and scale separation and its deep links to seemingly unrelated theories ranging all the way from domain decomposition to information games and their Bayesian interpretation. It is this surprising nexus of classical and probabilistic numerics that clears the way to the envisioned new computational paradigm for multiscale problems at randomness and disorder.
DFG Project 446856041 (2020-2023)
Decoupled computational methods for nonlinear parabolic problems with dynamic boundary conditions
PI: Dr. Robert Altmann
Problems with dynamic boundary conditions appear in applications where one needs to reflect the effective properties on the surface. A representative example is the model of a heat source on the boundary of the computational domain. The basis of the project is an alternative formulation of the problem as a partial differential-algebraic equation (PDAE), which considers the bulk and surface dynamics as a coupled system. For this, an additional Lagrange multiplier is inserted to the system equations. Given the PDAE formulation, we aim to construct new approximation schemes, which are robust in terms of the underlying saddle point structure. Thus, we need to consider mixed finite element schemes in combination with techniques known from the field of differential-algebraic equations. In contrast to the standard formulation of the problem, we do not include the bulk-surface coupling into the ansatz space but as an additional constraint on the dynamics. This formal decoupling offers a great flexibility for the spatial discretization, since finite element meshes in the bulk and on the boundary can be chosen independently. This is advantageous, especially if the system contains different characteristic scales or inhomogeneities. The second goal of the project is to construct efficient splitting methods for these PDAE systems in order to deal with appearing nonlinearities.
Project within DFG Priority Program 1748 (2015-2021)
Adaptive isogeometric modeling of discontinuities in complex-shaped heterogeneous solids
PI: Prof. Dr. Daniel Peterseim
The development of innovative products demands multi-material lightweight designs with complex heterogeneous local material structures. Their computer-aided engineering relies on the constitutive modeling and, in particular, the numerical simulation of propagating cracks. The underlying numerical techniques have to account for the failure of interfaces and bulk material as well as their interaction in the form of crack branching and coalescence. In order to provide realistic predictions by simulation, the true 3D nature of the problem has to be captured. For this purpose, this project develops new numerical models and methods that combine adaptive spline-based approximations from Isogeometric Analysis (IGA) with phase-field models for crack propagation. The main goals of this project are linked to fundamental challenges in the fields of Computational Mechanics, Numerical Analysis and Material Sciences, e.g., the representation and adaptive refinement of unstructured (water-tight) spline surfaces, the feasible coupling of spline surfaces with structured bulk meshes, the regularized modeling of heterogeneous materials, and the rigorous error analysis and control in pre-asymptotic regimes.