Three-dimensional dynamic rupture simulation with a high-order discontinuous Galerkin method on unstructured tetrahedral meshes

Accurate and efficient numerical methods to simulate dynamic earthquake rupture and wave propagation in complex media and complex fault geometries are needed to address fundamental questions in earthquake dynamics, to integrate seismic and geodetic data into emerging approaches for dynamic source inversion, and to generate realistic physics-based earthquake scenarios for hazard assessment. Modeling of spontaneous earthquake rupture and seismic wave propagation by a high-order discontinuous Galerkin (DG) method combined with an arbitrarily high-order derivatives (ADER) time integration method was introduced in two dimensions by de la Puente et al. (2009). The ADER-DG method enables high accuracy in space and time and discretization by unstructured meshes. Here we extend this method to three-dimensional dynamic rupture problems. The high geometrical flexibility provided by the usage of tetrahedral elements and the lack of spurious mesh reflections in the ADER-DG method allows the refinement of the mesh close to the fault to model the rupture dynamics adequately while concentrating computational resources only where needed. Moreover, ADER-DG does not generate spurious high-frequency perturbations on the fault and hence does not require artificial Kelvin-Voigt damping. We verify our three-dimensional implementation by comparing results of the SCEC TPV3 test problem with two well-established numerical methods, finite differences, and spectral boundary integral. Furthermore, a convergence study is presented to demonstrate the systematic consistency of the method. To illustrate the capabilities of the high-order accurate ADER-DG scheme on unstructured meshes, we simulate an earthquake scenario, inspired by the 1992 Landers earthquake, that includes curved faults, fault branches, and surface topography

Synthesis, Characterization, and Reactivity Studies of Low-valent 3d Metal Complexes with N-Heterocyclic Carbene and α-Diimine Ligands

Chapter 1. The Chemistry of Mononuclear Phosphane and N-Heterocyclic Carbene Nickel(I) Complexes: Synthesis, Structural Motifs, and Reactivity The introductory chapter reviews the chemistry of the dynamically emerging field of mononuclear NHC nickel(I) complexes and related phosphane compounds. The aim of this chapter is to highlight the analogies and differences between and within the two classes. Besides the synthesis and structural motifs of the monomers, this chapter covers the reactivity of the nickel(I) species. After a brief introduction about synthetic access strategies, this work is subdivided in phosphane nickel(I) halides and related cationic nickel(I) complexes with weakly coordinating counteranions, followed by phosphane complexes with ancillary N-donors, pincer complexes, phosphane complexes with additional S-donors, and a brief section on piano-stool hydrocarbyl complexes. Finally the chemistry of NHC nickel(I) complexes will be described. Chapter 2. Selective P4 Activation by an Organometallic Nickel(I) Radical: Formation of a Dinuclear Nickel(II) Tetraphosphide and Related Di- and Trichalcogenides[1] The main goal of this thesis was to synthesize and characterize new monomeric cyclopentadienyl (Cp) nickel(I) complexes supported by NHCs and explore their reactivity. The results are summarized in Schemes 1−5. Complexes 1−3 were synthesized according to Scheme 1 by reacting nickel(II) chlorides of type [(η5-C5R5)Ni(NHC)] with potassium graphite in THF. Compounds 1−3 exhibit typical metalloradical character as indicated by EPR spectroscopy and cyclic voltammetry. Interestingly, these d9 species show broad but characteristic 1H NMR signals. Furthermore, the X-ray diffraction analysis revealed a bent structure in the solid-state as indicated by the Ccarbene–Ni–(C5H5)centroid angle, which is probably caused by the asymmetric spin density distribution at the nickel center. Scheme 1. Synthesis of new nickel(I) metalloradicals 1−3. In an initial study, we investigated the reactivity of the new nickel(I) radical 1 toward small molecules and typical radical traps (Scheme 2). The reaction of 1 with the mild oxidizing agent ferrocenium hexafluorophosphat gave the nickel(II) cation 4 with a two-legged piano stool coordination environment for nickel. Upon converting 1 with the persistent radical TEMPO the square-planar nickel(II) complex 5 was formed in a very selective fashion. Scheme 2. Radical reactivity of complex 1. It is well known that radicals can be trapped by element-element bonds. Therefore, we tested the reactivity of 1 toward elemental chalcogenides sulfur, grey selenium, and grey tellurium, as well as white phosphorus (P4). While the reactions with sulfur and selenium resulted in mixtures of di- and trichalcogenide-bridged complexes 6-E and 7-E, the only product observed was 6-Te in case of tellurium. Remarkably, the reaction of 1 with P4 resulted in the high-yield synthesis of new complex [{(C5H5)Ni(IDipp)}2(-η1:η1P4)] (8) with a bridging “butterfly” P42− ligand, which represents the first example of a nickel complex bearing such ligand. All complexes were thoroughly characterized by X-ray crystallography, multinuclear NMR and UV-Vis spectroscopy. Chapter 3. Half-Sandwich Nickel(I) Complexes of Ring-Expanded N-Heterocyclic Carbenes: A Structural and Quantum Chemical Study[2] In a follow-up study, we were interested in designing new nickel(I) radicals by utilizing ring-expanded N-heterocyclic carbenes (RE-NHCs) and compare the influence of these unconventional carbenes on the structural and electronic properties of [CpNi(NHC)] complexes, as well as their reactivity toward P4. The three novel complexes 9−11 were prepared by reducing the nickel(II) bromides 9Br−11Br with KC8 in a similar manner as described for compounds 1−3 (Scheme 3). The nickel(II) precursors were obtained in the reaction of [CpNiBr(PPh3)] with the respective RE-NHC. Scheme 3. Synthesis of cyclopentadienyl nickel(I) complexes with ring-expanded carbenes. The X-ray diffraction analysis revealed that complexes 9−11 possess similar, though less bent structures as 1−3 (increased Ccarbene−Ni−Cpcentroid angles) in comparison with 1 and 2 and decreased Ccarbene−Ni−Cpcentroid angle as compared with 3. According to DFT calculations complexes 9−11 also showed comparable electronic structures as 1−3. Nevertheless, diamidocarbene complex 11 (Scheme 3) exhibits a distinct UV-Vis and 1H NMR spectrum as compared with 1−3, 9, and 10. Compound 9 and 10 readily react with white phosphorus at room temperature. The reaction of 9 affords tetraphosphide 12. A mixture of the analogous complex 13 and unidentified products was obtained using 10 according to 31P{1H} NMR spectroscopy. In marked contrast, compound 11 did not react with P4, further demonstrating its distinct electronic properties. Scheme 3. Reactivity of complexes 9−11 toward white phosphorus. Chapter 4. Formation of Heteronickelacycles through the Reductive Coupling of Phenyl Iso(thio)cyanate[3] In extension to our reactivity studies toward element−element bonds in chapters 2 and 3, the reactivity of complex 1 toward heteroallenes was investigated. The reaction of 1 with one equivalent phenyl isothiocyanate afforded compound 14, which features the new {SC(NPh)N(Ph)CS}2− dianion as a result of the dimerization of the substrate in the coordination sphere of the metal atom (Scheme 4). The nickel(II) complex [(η5 Cp)(η1 Cp)Ni(IDipp)] (15) is a stoichiometric by-product in this reaction. We presume that the reaction pathway probably involves the transfer of a Cp radical to a second equivalent of complex 1, which then leads to a reactive intermediate that essentially acts as nickel(0) surrogate. In fact, complex 14 was cleanly produced in the reaction of the nickel(0) compound [(IDipp)Ni(styrene)2] (A) with two equivalents PhNCS. Interestingly, the reaction of A with phenyl isocyanate yields the new γ-lactam-nickellacycle [(IDipp)Ni{N(Ph)C(O)CH2CHPh}] (rac-16) as a result of the reductive coupling of PhNCO and a styrene ligand. The solid-state molecular structure of the unsaturated complex rac-16 displays a rare T-shaped structure for nickel. In contrast, the reaction of PhNCO with the nickel(I) complex [(η5-Cp)Ni(IDipp)] gave an inseparable mixture of diamagnetic products, showing that this complex has a distinct reactivity in this case with PhNCO and PhNCS. Scheme 4. Comparison of the reactivity of 1 and A toward heteroallenes. Chapter 5. Insertion of Phenyl Isothiocyanate into a P−P Bond of a Nickel-substituted Bicyclo[1.1.0]tetraphosphabutane[4] Encouraged by the good accessibility of phosphide complex 8, we were interested in functionalizing the P4 moiety in this complex. Therefore, we investigated the reactivity of 8 and derivative 17, bearing the slightly less bulky IMes carbene instead of IDipp, toward phenyl isothiocyanate. Complex 17 was obtained in an analogous fashion as described for 8 (chapter 2) and features a similar molecular structure and spectroscopic properties (Scheme 5). Scheme 5. Functionalization of the P4 fragment in 17. Reactions of 8 and 17 with CS2 gave products which feature ADMX spin systems in the 31P{1H} NMR spectra, but these products could not be isolated. Interestingly, the reaction of 17 with an excess of PhNCS yielded a mixture of two main products (ADMX spin systems) and one minor species (A2MX spin system). Both main products, 18a and 18b, were obtained as pure compounds and fully characterized by X-ray diffraction as well as UV-Vis and multinuclear NMR spectroscopy. In addition, DFT calculations were performed in order to get further insight into the reaction mechanism. The novel complexes 18a and 18b feature unprecedented bicyclo[3.1.0]heterohexane fragments. Moreover, this reaction represent the first example of an insertion of a heteroallene into a bicyclo[1.1.0]tetraphosphabutane ligand. The investigations of chapter 2-5 demonstrate that the presented NHC nickel(I) complexes exhibit typical metalloradical properties. Their high reactivity and radical nature enables mild reaction procedures for element-element bond activations, in particular P−P bonds of P4. Moreover, the use of NHCs and cyclopentadienyl ligands enables the modification of the electronic and steric properties of these compounds. In case of complexes 9−11 our initial investigations indicate the presence of a distinct electronic structure of compound 11 in comparison with 1−3, 9, and 10. Further investigations by EPR and ENDOR spectroscopy are ongoing to confirm this presumption and get deeper insight into the electronic structure. In future work, it will also be of interest to investigate whether reactions of metal-substituted cyclopolyphosphanes with polar multiple bonds offer a general route toward “functionalized” polyphosphanes as exemplified by the reaction of complex 17 with PhNCS. Chapter 6. Synthesis and Structural Characterization of Iron(II), Cobalt(II), and Nickel(II) Complexes of a Cyclic (Alkyl)(amino)carbene[5] In the context of utilizing unconventional carbenes in 3d metal chemistry, we report on new cyclic (alkyl)(amino)carbene (CAAC) complexes of iron(II), cobalt(II) and nickel(II), which are potentially useful starting materials for further applications in catalysis and small molecule activation. Complexes [FeCl(μ-Cl)(CAAC1)]2 (19), [CoBr(μ-Br)(CAAC1)]2 (20), and [NiBr(μ-Br)(CAAC1)]2 (21) were synthesized by reacting Bertrand’s CAAC1 with divalent iron, cobalt and nickel halides (Scheme 6). They represent the first CAAC complexes with these metals and are paramagnetic and highly air-sensitive, but appear to be thermally stable under an inert atmosphere. X-ray diffraction analyses revealed dimeric halide-bridged structures with distorted tetrahedral metal environments. Scheme 6. Synthesis of CAAC complexes 1921. Chapter 7. Preparation of a Trigonal Planar Manganese(II) Amido Complex Supported by an N-Heterocyclic Carbene[6] In chapter 7 we describe a convenient synthesis of the NHC-stabilized manganese(II) chloride 22 from commercially available starting materials and investigated its ability as precursor for low-coordinate manganese(II) NHC complexes. The X-ray diffraction analysis of 1-THF indicate the dissociation of the dimer 22 in tetrahydrofuran solutions (Scheme 7). Scheme 7. Synthesis of the dimeric manganese(II) chloride 22 and reactivity toward MeMgI and LiN(SiMe3)2. Furthermore, the lability of the manganese(II)-carbon bond in 22 was demonstrated in the reaction of 22 with four equivalents of MeMgI, which yielded the dinuclear compound [(IDipp)MgMeCl]2 (23). As shown by single-crystal X-ray analysis of 23, the IDipp and the chloride ligands were transferred from manganese in 22 to the magnesium atom. The reaction of 22 with four equivalents lithium hexamethyldisilazide gave the monomeric amide 24 (Scheme 7), which features a trigonal planar coordination environment. Compound 24 represents a rare example of a three-coordinate NHC-stabilized manganese(II) complex. Solution magnetic moments showed the presence of d5 high spin manganese(II) centers in 22 and 24. This initial study paves the way for further investigations regarding the use of complex 22 as precursor for further coordinatively and electronically unsaturated complexes. Future work should also focus on the exploration of the reactivity of 24 toward nucleophilic reagents. Chapter 8. Synthesis and Reactivity Studies of a Heteroleptic α-Diimine Cobalt Anion[7] Another aim of this thesis was to synthesize and characterize novel highly reduced cobalt complexes with redox-active α-diimine ligands, and investigate their reactivity toward small molecules. The novel BIAN (BIAN = bis(2,6-diisopropylphenylimino)acenaphthene) cobalt complex 25 was synthesized from [Co(cod)2]− via straightforward substitution of one cod ligand (Scheme 8). Scheme 8. Preparation of the heteroleptic BIAN complex 25 and its reactivity toward small molecules. The reaction of 25 with carbon disulfide yielded the dinuclear complex 26 with a bridging ethylene tetrathiolate by the reductive dimerization of the substrate. Furthermore, complex 27 was obtained by converting 25 with two equivalents of tert-butylphosphaalkyne, containing an η4-1,3-disphosphacyclobutadiene ligand. The dianionic complex 28, which features an unusual rectangular P44- framework, was formed by the selective activation of P4. Its oxidation afforded monoanion 29 with a rare rhombic cyclo-P44− ligand. Our investigations revealed that the use of the redox-active BIAN ligand appears crucial for achieving a high degree of P4 reduction. Phosphide anions 28 and 29 are potentially useful starting complexes for the functionalization of the P4 moiety in the coordination sphere of the cobalt centers with electrophilic inorganic and organic molecules. In addition, the synthesis of complexes related to 25 with modified electronic and steric properties is promising in order to tune the highly reduced cobalt complexes for further reactivity studies toward other small molecules as well as potential catalytic applications

Regional wave propagation using the discontinuous Galerkin method

We present an application of the discontinuous Galerkin (DG) method to regional wave propagation. The method makes use of unstructured tetrahedral meshes, combined with a time integration scheme solving the arbitrary high-order derivative (ADER) Riemann problem. This ADER-DG method is high-order accurate in space and time, beneficial for reliable simulations of high-frequency wavefields over long propagation distances. Due to the ease with which tetrahedral grids can be adapted to complex geometries, undulating topography of the Earth's surface and interior interfaces can be readily implemented in the computational domain. The ADER-DG method is benchmarked for the accurate radiation of elastic waves excited by an explosive and a shear dislocation source. We compare real data measurements with synthetics of the 2009 L'Aquila event (central Italy). We take advantage of the geometrical flexibility of the approach to generate a European model composed of the 3-D <i>EPcrust</i> model, combined with the depth-dependent <i>ak135</i> velocity model in the upper mantle. The results confirm the applicability of the ADER-DG method for regional scale earthquake simulations, which provides an alternative to existing methodologies

Verification of an ADER-DG method for complex dynamic rupture problems

We present results of thorough benchmarking of an arbitrary high-order derivative discontinuous Galerkin (ADER-DG) method on unstructured meshes for advanced earthquake dynamic rupture problems. We verify the method by comparison to well-established numerical methods in a series of verification exercises, including dipping and branching fault geometries, heterogeneous initial conditions, bimaterial interfaces and several rate-and-state friction laws. We show that the combination of meshing flexibility and high-order accuracy of the ADER-DG method makes it a competitive tool to study earthquake dynamics in geometrically complicated setups

Tilt effects on moment tensor inversion in the near field of active volcanoes

Dynamic tilts (rotational motion around horizontal axes) change the projection of local gravity onto the horizontal components of seismometers. This causes sensitivity of these components to tilt, especially at low frequencies. We analyse the consequences of this effect onto moment tensor inversion for very long period (vlp) events in the near field of active volcanoes on the basis of synthetic examples using the station distribution of a real deployed seismic network and the topography of Mt. Merapi volcano (Java, Indonesia). The examples show that for periods in the vlp range of 10-30 s tilt can have a strong effect on the moment tensor inversion, although its effect on the horizontal seismograms is significant only for few stations. We show that tilts can be accurately computed using the spectral element method and include them in the Green's functions. The (simulated) tilts might be largely influenced by strain-tilt coupling (stc). However, due to the frequency dependence of the tilt contribution to the horizontal seismograms, only the largest tilt signals affect the source inversion in the vlp frequency range. As these are less sensitive to stc than the weaker signals, the effect of stc can likely be neglected in this application. In the converse argument, this is not necessarily true for longer periods, where the horizontal seismograms are dominated by the tilt signal and rotational sensors would be necessary to account for it. As these are not yet commercially available, this study underlines the necessity for the development of such instrument

On the initiation of sustained slip-weakening ruptures by localized stresses

Numerical simulations of dynamic earthquake rupture require an artificial initiation procedure, if they are not integrated in long-term earthquake cycle simulations. A widely applied procedure involves an ‘overstressed asperity’, a localized region stressed beyond the static frictional strength. The physical properties of the asperity (size, shape and overstress) may significantly impact rupture propagation. In particular, to induce a sustained rupture the asperity size needs to exceed a critical value. Although criteria for estimating the critical nucleation size under linear slip-weakening friction have been proposed for 2-D and 3-D problems based on simplifying assumptions, they do not provide general rules for designing 3-D numerical simulations. We conduct a parametric study to estimate parameters of the asperity that minimize numerical artefacts (e.g. changes of rupture shape and speed, artificial supershear transition, higher slip-rate amplitudes). We examine the critical size of square, circular and elliptical asperities as a function of asperity overstress and background (off-asperity) stress. For a given overstress, we find that asperity area controls rupture initiation while asperity shape is of lesser importance. The critical area obtained from our numerical results contrasts with published theoretical estimates when background stress is low. Therefore, we derive two new theoretical estimates of the critical size under low background stress while also accounting for overstress. Our numerical results suggest that setting the asperity overstress and area close to their critical values eliminates strong numerical artefacts even when the overstress is large. We also find that properly chosen asperity size or overstress may significantly shorten the duration of the initiation. Overall, our results provide guidelines for determining the size of the asperity and overstress to minimize the effects of the forced initiation on the subsequent spontaneous rupture propagation