arXiv subjects

Joscha Podlesny

Publications and source records attributed to Joscha Podlesny.

3 recordsLinked to original sources

arXiv2022

Numerical simulation of multiscale fault systems with rate- and state-dependent friction

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.

math.NA↗

arXiv2020

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.

math.NA↗

arXiv2019

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.

math.AP↗