### Project Summary

Earthquake statistics reveal scale invariance over 10 orders of magnitude of the earthquake strength as expressed, e.g., by the famous Gutenberg–Richter power law. Wellestablished explanations for earthquake activity in the lithosphere are based on strain accumulation and stress release along fault networks chie.y in plate boundary zones as described by rate and state dependent (RSD) friction models. However, because of the incompleteness of the real-world record of earthquakes and deformation accumulation beyond the instrumental and historical time scales (decades to centuries), there is a fundamental lack of insight into the multiscale nature of these processes.

We propose to explore the scaling properties of the process of deformation accumulation in fault networks by means of an original simulation strategy involving complementary laboratory scale analogue as well as mathematical modelling and numerical simulations. More precisely, the goal of the project is to derive, analyze, numerically approximate, and experimentally validate a multiscale model for fault networks consisting of a hierarchy of effective rate and state dependent friction models on increasing spatial and temporal scales. We will explore whether deformation patterns emerge exhibiting transience, episodicity or random distribution, clustering or supercycles and how internal damping or external forcing in a mechanically layered lithosphere in.uence the observed pattern at various time scales. Development and validation will involve numerical computations, analytical considerations and laboratory experiments. Our goal is to reproduce the seismic behavior of interacting individual faults as well as earthquake statistics and long-term deformation patterns in large fault network zones over long time scales as well as to determine scale ranges featuring self-similar and nonself-similar behavior. Based on these simulation results and real world data we plan to identify the responsible controlling parameters and feedback mechanisms. In this way, we will help to bridge the gap between mathematically consistent descriptions derived from a fundamental root model, i.e. fault networks with RSD friction on the laboratory scale, and large-scale phenomenological relationships such as the Gutenberg–Richter law.

The project combines analytical considerations, numerical computations, and laboratory experiments. In the .rst period of funding, we will concentrate on existence, uniqueness, and numerical analysis of single faults with RSD friction in thermomechanically consistent formulations (e.g., as gradient systems or in the GENERIC framework), numerical computations for coplanar fault networks with simpli.ed (Tresca) friction, and the derivation of .rst two-scale models in 2D. All numerical computations will be accompanied by laboratory experiments providing the effective parameters as well as acting as a testbed for numerically predicted patterns. In the second period of funding, we derive multiscale models for fault networks with RSD friction in 3D in a realistic setting. while the third period of funding will be devoted to practical applications and further extensions including, e.g., complex fault geometries, crack formation and healing, fault opening, nonlinear materials and plasticity.

### Project publications

Kornhuber, R. and Youett, E.
(2018)
*Adaptive Multilevel Monte Carlo Methods for Stochastic Variational Inequalities.*
SIAM Journal on Numerical Analysis, 56
(4).
pp. 1987-2007.
ISSN 0036-1429

Kornhuber, R. and Peterseim, D. and Yserentant, H.
(2018)
*An analysis of a class of variational multiscale methods based on subspace decomposition.*
Mathematics of Computation, 87
(314).
pp. 2765-2774.
ISSN 1088-6842 (online)

Heida, M. and Kornhuber, R. and Podlesny, J.
(2017)
*Fractal homogenization of multiscale interface problems.*
SFB 1114 Preprint in arXiv
.
pp. 1-17.
(Submitted)

Ritter, M. and Rosenau, M. and Oncken, O.
(2017)
*Growing Faults in the Lab: Insights into the Scale Dependence of the Fault Zone Evolution Process.*
Tectonics, 36
.
pp. 1-32.

Rosenau, M. and Horenko, I. and Corbi, F. and Rudolf, M. and Kornhuber, R. and Oncken, O.
(2017)
*Synchronization of great subduction megathrust earthquakes: Insights from scale model analysis.*
SFB 1114 Preprint in EarthArXiv
.
pp. 1-35.
(Unpublished)

Rudolf, M. and Rosenau, M. and Oncken, O.
(2017)
*Interseismic deformation transients and precursory phenomena: Insights from stick-slip experiments with a granular fault zone.*
SFB 1114 Preprint in EarthArXiv:10.17605/OSF.IO/6MWRX
.
pp. 1-26.
(Unpublished)

Ritter, M. and Santimano, T.N. and Rosenau, M. and Leever, K. and Oncken, O.
(2017)
*Sandbox rheometry: Coevolution of stress and strain in riedel- and critical wedge-experiments.*
Tectonophysics, 722
.
pp. 400-409.

Corbi, F. and Funiciello, F. and Brizzi, S. and Lallemand, S. and Rosenau, M.
(2017)
*Control of asperities size and spacing on seismic behavior of subduction megathrusts.*
Geophysical Research Letters, 44
(16).
pp. 8227-8235.

Kornhuber, R. and Podlesny, J. and Yserentant, H.
(2017)
*Direct and Iterative Methods for Numerical Homogenization.*
In:
Domain Decomposition Methods in Science and Engineering.
Lecture Notes in Computational Science and Engineering, XXIII
(116).
SpringerLink, pp. 217-225.
ISBN 978-3-319-52389-7

Pipping, E.
(2017)
*Existence of long-time solutions to dynamic problems of viscoelasticity with rate-and-state friction.*
SFB 1114 Preprint
.
(Submitted)

Mielke, A. and Roubícek, T.
(2016)
*Rate-Independent elastoplasticity at finite strain and its numerical approximation.*
Mathematical Models and Methods in Applied Sciences, 26
(12).
pp. 2203-2236.
ISSN 1793-6314

Rudolf, M. and Boutelier, D. and Rosenau, M. and Schreurs, G. and Oncken, O.
(2016)
*Rheological benchmark of silicone oils used for analog modeling of short- and long-term lithospheric deformation.*
Tectonophysics, 684
.
pp. 12-22.

Pipping, E. and Kornhuber, R. and Rosenau, M. and Oncken, O.
(2016)
*On the efficient and reliable numerical solution of rate-and-state friction problems.*
Geophysical Journal International, 204
(3).
pp. 1858-1866.
ISSN Online: 1365-246X Print: 0956-540X

Horenko, I. and Gerber, S.
(2015)
*Improving clustering by imposing network information.*
Science Advances, 1
(7).
ISSN 2375-2548