9 research outputs found
Multiscale Modeling and Simulation of Deformation Accumulation in Fault Networks
Strain accumulation and stress release along multiscale geological fault networks are fundamental mechanisms for earthquake and rupture processes in the lithosphere. Due to long periods of seismic quiescence, the scarcity of large earthquakes and incompleteness of paleoseismic, historical and instrumental record, there is a fundamental lack of insight into the multiscale, spatio-temporal nature of earthquake dynamics in fault networks. This thesis constitutes another step towards reliable earthquake prediction and quantitative hazard analysis. Its focus lies on developing a mathematical model for prototypical, layered fault networks on short time scales as well as their efficient numerical simulation.
This exposition begins by establishing a fault system consisting of layered bodies with viscoelastic Kelvin-Voigt rheology and non-intersecting faults featuring rate-and-state friction as proposed by Dieterich and Ruina. The individual bodies are assumed to experience small viscoelastic deformations, but possibly large relative tangential displacements. Thereafter, semi-discretization in time with the classical Newmark scheme of the variational formulation yields a sequence of continuous, nonsmooth, coupled, spatial minimization problems for the velocities and states in each time step, that are decoupled by means of a fixed point iteration. Subsequently, spatial discretization is based on linear and piecewise constant finite elements for the rate and state problems, respectively. A dual mortar discretization of the non-penetration constraints entails a hierarchical decomposition of the discrete solution space, that enables the localization of the non-penetration condition. Exploiting the resulting structure, an algebraic representation of the parametrized rate problem can be solved efficiently using a variant of the Truncated Nonsmooth Newton Multigrid (TNNMG) method. It is globally convergent due to nonlinear, block Gauß–Seidel type smoothing and employs nonsmooth Newton and multigrid ideas to enhance robustness and efficiency of the overall method. A key step in the TNNMG algorithm is the efficient computation of a correction obtained from a linearized, inexact Newton step.
The second part addresses the numerical homogenization of elliptic variational problems featuring fractal interface networks, that are structurally similar to the ones arising in the linearized correction step of the TNNMG method. Contrary to the previous setting, this model incorporates the full spatial complexity of geological fault networks in terms of truly multiscale fractal interface geometries. Here, the construction of projections from a fractal function space to finite element spaces with suitable approximation and stability properties constitutes the main contribution of this thesis. The existence of these projections enables the application of well-known approaches to numerical homogenization, such as localized orthogonal decomposition (LOD) for the construction of multiscale discretizations with optimal a priori error estimates or subspace correction methods, that lead to algebraic solvers with mesh- and scale-independent convergence rates.
Finally, numerical experiments with a single fault and the layered multiscale fault system illustrate
the properties of the mathematical model as well as the efficiency, reliability and scale-independence of the suggested algebraic solver
Fractal homogenization of multiscale interface problems
Inspired by continuum mechanical contact problems with geological fault
networks, we consider elliptic second order differential equations with jump
conditions on a sequence of multiscale networks of interfaces with a finite
number of non-separating scales. Our aim is to derive and analyze a description
of the asymptotic limit of infinitely many scales in order to quantify the
effect of resolving the network only up to some finite number of interfaces and
to consider all further effects as homogeneous. As classical homogenization
techniques are not suited for this kind of geometrical setting, we suggest a
new concept, called fractal homogenization, to derive and analyze an asymptotic
limit problem from a corresponding sequence of finite-scale interface problems.
We provide an intuitive characterization of the corresponding fractal solution
space in terms of generalized jumps and gradients together with continuous
embeddings into L2 and Hs, s<1/2. We show existence and uniqueness of the
solution of the asymptotic limit problem and exponential convergence of the
approximating finite-scale solutions. Computational experiments involving a
related numerical homogenization technique illustrate our theoretical findings
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Fractal homogenization of a multiscale interface problem
Inspired from geological problems, we introduce a new geometrical
setting for homogenization of a well known and well studied problem of an
elliptic second order differential operator with jump condition on a
multiscale network of interfaces. The geometrical setting is fractal and
hence neither periodic nor stochastic methods can be applied to the study of
such kind of multiscale interface problem. Instead, we use the fractal nature
of the geometric structure to introduce smoothed problems and apply methods
from a posteriori theory to derive an estimate for the order of convergence.
Computational experiments utilizing an iterative homogenization approach
illustrate that the theoretically derived order of convergenceof the
approximate problems is close to optimal
Numerical Homogenization of Fractal Interface Problems
We consider the numerical homogenization of a class of fractal elliptic
interface problems inspired by related mechanical contact problems from the
geosciences. A particular feature is that the solution space depends on the
actual fractal geometry. Our main results concern the construction of
projection operators with suitable stability and approximation properties. The
existence of such projections then allows for the application of existing
concepts from localized orthogonal decomposition (LOD) and successive subspace
correction to construct first multiscale discretizations and iterative
algebraic solvers with scale-independent convergence behavior for this class of
problems
Fractal homogenization of a multiscale interface problem
Inspired from geological problems, we introduce a new geometrical setting for homogenization of a well known and well studied problem of an elliptic second order differential operator with jump condition on a multiscale network of interfaces. The geometrical setting is fractal and hence neither periodic nor stochastic methods can be applied to the study of such kind of multiscale interface problem. Instead, we use the fractal nature of the geometric structure to introduce smoothed problems and apply methods from a posteriori theory to derive an estimate for the order of convergence. Computational experiments utilizing an iterative homogenization approach illustrate that the theoretically derived order of convergence of the approximate problems is close to optimal
NUMERICAL HOMOGENIZATION OF FRACTAL INTERFACE PROBLEMS
We consider the numerical homogenization of a class of fractal elliptic interface
problems inspired by related mechanical contact problems from the geosciences. A particular
feature is that the solution space depends on the actual fractal geometry. Our main
results concern the construction of projection operators with suitable stability and approximation
properties. The existence of such projections then allows for the application of
existing concepts from localized orthogonal decomposition (LOD) and successive subspace
correction to construct first multiscale discretizations and iterative algebraic solvers with
scale-independent convergence behavior for this class of problems
NUMERICAL SIMULATION OF MULTISCALE FAULT SYSTEMS WITH RATE- AND STATE-DEPENDENT FRICTION
Abstract. We consider the deformation of a geological structure with non-intersecting
faults that can be represented by a layered system of viscoelastic bodies satisfying rate- and
state-depending friction conditions along the common interfaces. We derive a mathematical
model that contains classical Dieterich- and Ruina-type friction as special cases and accounts
for possibly large tangential displacements. Semi-discretization in time by a Newmark scheme
leads to a coupled system of non-smooth, convex minimization problems for rate and state
to be solved in each time step. Additional spatial discretization by a mortar method and
piecewise constant finite elements allows for the decoupling of rate and state by a fixed
point iteration and efficient algebraic solution of the rate problem by truncated non-smooth
Newton methods. Numerical experiments with a spring slider and a layered multiscale system
illustrate the behavior of our model as well as the efficiency and reliability of the numerical
solver
Numerical homogenization of fractal interface problems
We consider the numerical homogenization of a class of fractal elliptic interface problems inspired by related mechanical contact problems from the geosciences. A particular feature is that the solution space depends on the actual fractal geometry. Our main results concern the construction of projection operators with suitable stability and approximation properties. The existence of such projections then allows for the application of existing concepts from localized orthogonal decomposition (LOD) and successive subspace correction to construct first multiscale discretizations and iterative algebraic solvers with scale-independent convergence behavior for this class of problems
Numerical homogenization of fractal interface problems
We consider the numerical homogenization of a class of fractal elliptic interface problemsÂ
inspired by related mechanical contact problems from the geosciences.
A particular feature is that the solution space depends on the actual fractal geometry.
Our main results concern the construction of projection operators with suitable stability and approximation properties.Â
The existence of such projections then allows for the application of existing concepts from localized orthogonal decomposition (LOD)
and successive subspace correction to construct first multiscale discretizations and iterative algebraic solvers with scale-independent convergence behavior for this class of problems