Full Tensor Research Articles

A new high order finite volume element solution on arbitrary triangular and quadrilateral meshes

In this work, for the two dimensional anisotropic diffusion problem, the classical kth order finite element solution is postprocessed to obtain a new finite volume element solution, such that the new solution satisfies the local conservation property on a certain dual mesh, and converges to the analytic solution with optimal rates. The postprocessing algorithm has a local nature and can be conducted element by element. The novelty of this paper is the introduction of a new bubble function, which enables us to prove the existence and uniqueness of the postprocessed solution on arbitrary triangular or convex quadrilateral meshes with full anisotropic diffusion tensor. The theoretical findings are also verified by some numerical results.

SQUIDs for magnetic and electromagnetic methods in mineral exploration

Research on quantum sensors for the detection of magnetic fields (quantum magnetometers) is one of the fast-moving areas of Quantum Technologies. Since there exist expectations about their use in geophysics, this work will provide a brief overview on the various developing quantum technologies and their individual state of the art for implementing quantum magnetometers. As one example, the developments on superconducting quantum interference devices, so-called SQUIDs as a specific implementation of a quantum magnetometer, are presented. In the course of this, SQUID instrument implementations and associated demonstrations and case studies will be presented. An airborne vector magnetometer with ultra-low noise (<10 mathrm{fT}/sqrt{mathrm{Hz}}) and high dynamic range of >32 mathrm{bit} will be introduced which has the prospect to be applied for the magnetic method in parallel with electromagnetic methods such as passive audio-frequency magnetics or semi-airborne methods using active transmitters such as elongated grounded dipole sources. The according signals are separated in the frequency domain. A second implementation is an airborne full tensor gradiometer instrument will be discussed which has shown already a number of successful case studies and which turned into commercial operation in the past years. Besides the airborne instrument, a very successful implementation of quantum magnetometers is the SQUID-based receiver for the ground-based transient electromagnetic method. Today it is a mature technology which has been in commercial use for more than a decade and has led to a number of discoveries of conductive ore bodies. One case study will be presented which demonstrates the performance of this instrument. Finally, future prospects of using quantum magnetometers, including SQUIDs and new optically pumped magnetometers, in geophysical exploration will be discussed. Particular applications for both sensor types will be introduced.

An Adaptive Dynamical Low Rank Method for the Nonlinear Boltzmann Equation

Efficient and accurate numerical approximation of the full Boltzmann equation has been a longstanding challenging problem in kinetic theory. This is mainly due to the high dimensionality of the problem and the complicated collision operator. In this work, we propose a highly efficient adaptive low rank method for the Boltzmann equation, concerning in particular the steady state computation. This method employs the fast Fourier spectral method (for the collision operator) and the dynamical low rank method to obtain computational efficiency. An adaptive strategy is introduced to incorporate the boundary information and control the computational rank in an appropriate way. Using a series of benchmark tests in 1D and 2D, we demonstrate the efficiency and accuracy of the proposed method in comparison to the full tensor grid approach.

Numerical convergence of the family of flux-continuous schemes with variable quadrature (qu1,qu2) for single phase flow in porous media

Finite-volume schemes, which honor pressure and flux-continuity conditions, is developed using double quadrature (qu1,qu2), referred as double family scheme. The scheme is applicable to solve the elliptic pressure equation used in reservoir simulation. Schemes are applicable on both regular cartesian and unstructured triangular meshes. The scheme is defined over a control-volume distributed formulation. The scheme can be applied to both diagonal and full permeability tensor elliptic pressure equation with discontinuous coefficients. The scheme removes the first order errors, which are introduced by standard reservoir simulation schemes when applied to full tensor flow. The scheme is quantified with help of a quadrature rule. When the scheme is applied to highly heterogeneous and anisotropic porous media it does not honor maximum principle resulting in unstable solution with oscillatory behavior. The numerical solution is termed non-monotonicity for high anisotropy ratios with results showing oscillations in the numerical pressure solution. In this paper a double (qu1,qu2) quadrature flux continuous schemes is presented, which with specific choice of quadrature (qu1,qu2) helps in improved stability of the numerical solutions. Numerical convergence of the scheme is also demonstrated with help of a number of numerical test cases and schemes impact on monotonicity behavior is also demonstrated with numerical examples.

Prospective Evaluation of Geothermal Resources in the Shangqiu Uplift of the Southern North China Basin with Magnetotelluric Detection

The magnetotelluric sounding (MT) method is used to detect and study the deep stratigraphic structure and hidden faults in the Shangqiu Uplift. A total of 4 MT profiles are arranged, and 97 stations are collected. The nonlinear conjugate gradient (NLCG) two-dimensional inversion method is used to jointly invert data from both the TE and TM modes after a dimensionality analysis and impedance tensor decomposition; reliable two-dimensional resistivity models are produced since the data quality is excellent. Three-dimensional inversion is carried out to full impedance tensor as well to produce a three-dimensional resistivity model of the study area, which shows good consistency with the two-dimensional models. The results show that the electrical structure of the Shangqiu Uplift has typical layered characteristics, which can be divided into three layers from top to bottom, namely low-, medium-to-high-, and high-resistivity layers. According to the resistivity models and in combination with the gravity, aeromagnetic, seismic, and regional geological data of the study area, a geological map of the basement rock of the Shangqiu Uplift is produced. Two prospective geothermal anomaly areas are proposed according to the distribution of the high-resistivity anomaly formed by the basement uplift, which has a good corresponding relationship with the high-value area of the regional geothermal field. A geothermal exploration well (SR-1) is constructed in one of the inferred prospective geothermal anomaly areas. The well is 1702 m deep, with a water output of 1500 m3/day and a wellhead water temperature of 51.5 °C. This is the geothermal well with the largest water yield in the Shangqiu area at present, which provides a new basis for future geothermal exploration, development, and utilization.

Lattice dynamics and elastic properties of black phosphorus

We experimentally determine the lattice dynamics of black phosphorus layered crystals through a combination of x-ray diffuse scattering and inelastic x-ray scattering, and we rationalize our experimental findings using $\textit{ab initio}$ calculations. From the phonon dispersions at terahertz frequencies, we derive the full single-crystal elastic tensor and relate it to the macroscopic elastic response of black phosphorus, described by the elastic moduli. The elastic stiffness coefficients obtained here provide an important benchmark for models of black phosphorus and related materials, such as phosphorene and black phosphorus nanotubes, recently emerged quantum materials, disclosing a huge potential for nanophotonics, optoelectronics, and quantum sensing.

Triclinic tricalcium silicate: Structure and thermoelastic properties from molecular simulations

Triclinic tricalcium silicate is the stable form of pure C3S at ambient conditions. Atomic-level understanding of the structure and properties of this polymorph may provide some insights for innovative solutions to control its reactivity or to optimize its dopage. Here, we perform classical molecular dynamics simulations to obtain the full stiffness, thermal expansion, and thermal conductivity tensors as well as the heat capacities of triclinic tricalcium silicate (T1). In agreement with previous work, semi-classical computations are required to get experimentally consistent heat capacity. Phonon scattering and acoustics are also analyzed. This work provides fundamental component property data in cement systems, which is crucial to well understand the contribution of polymorphs in alite.

Estimating Explosion Yields Using Moment Tensor Solutions and Seismic Moment

Abstract Seismic moment, a measurable and well-understood quantity of seismic sources, is used to estimate the yield of explosions. Application of such a method in the past, as in the manner of mb-derived yields, has been complicated by the effect of variations in the explosion working point, depth, and secondary source effects (such as spalling and tectonic release) on the observed moment. We start using the full (six-element) moment tensor solution, which can capture the relevant source physics and, at least in theory, better isolate the primary explosion source. The moment-to-yield ratio is then estimated using an explosion source model which, provided with emplacement conditions, can relate the two parameters. We discuss the major sources of uncertainty associated with the method, and calibrate it with chemical and nuclear explosions at the Nevada National Security Site. We then apply the method to published moment tensor solutions for the six declared North Korean nuclear explosions that occurred between 2006 and 2017. The results are mostly consistent with other yield estimates made using a variety of high-frequency methods. This technique is a new approach to estimating explosive yield and simple to implement, as much of the complexity is captured by the source models.

Moment tensor catalogue of earthquakes in West Bohemia from 2008 to 2018

Abstract. We present a unique catalogue of full moment tensors (MTs) of earthquakes with ML between 0.5 and 4.4 that occurred in West Bohemia, Czech Republic, in the period from 2008 to 2018 (Vavryčuk et al., 2022a, b). The MTs were calculated from vertical components of P-wave amplitudes. The MT inversion was based on principal component analysis applied to optimally filtered velocity records of local seismic stations deployed in the West Bohemia area. The minimum number of inverted stations is 15, and the rms between theoretical and observed amplitudes is lower than 0.5. The catalogue is exceptional in several aspects: (1) it represents an extraordinary, extensive dataset of more than 5100 MTs; (2) it covers a long period of seismicity in the studied area, during which several prominent earthquake swarms took place; (3) the locations and retrieved MTs of earthquakes are of a high accuracy. Additionally, we provide three-component records at the West Bohemia (WEBNET) seismic stations, the velocity model in the region, and the technical specification of the stations. The dataset is ideal for being utilized by a large community of researchers for various seismological purposes, e.g. for studies of (1) the migration of foci and the spatiotemporal evolution of seismicity, (2) redistribution of stress during periods of intense seismicity, (3) the interaction of faults, (4) the Coulomb stress along the faults and local stress anomalies connected to fault irregularities, (5) diffusivity of fluids along the activated faults, or (6) the time-dependent seismic risk due to the migration of seismicity in the region. In addition, the dataset is optimum for developing and testing new inversions for MTs and for tectonic stress. Since most of the earthquakes are non-shear, the dataset can contribute to studies of non-double-couple components of MTs and their relation to shear–tensile fracturing and/or seismic anisotropy in the focal zone.

Deep learning model as an inversion tool for resonant ultrasound spectroscopy of piezoelectric materials

Device fabrication based on piezoelectric materials requires prior characterization of full matrix constants. For this, the Institute of Electrical and Electronics Engineers standard on piezoelectricity suggests the use of ultrasonic pulse-echo and electric resonance methods. However, these techniques tend to provide inconsistent characterization, because they require multiple samples with drastically different sizes. Resonant ultrasound spectroscopy (RUS) is a promising alternative, because it uses only a single sample for characterization, thus ensuring self-consistent results. The inverse problem of finding material constants from resonant frequencies is often solved using the nonlinear least squares method despite its being a time-consuming algorithm. Herein, deep learning (DL) is introduced in the inversion procedure of RUS. After the DL network is trained, the material constants are determined with high efficiency. The practicability and reliability of the combination of DL and RUS are demonstrated by characterizing the full tensor constants of LiNbO3 single crystals.

Structural interpretation of Southern Main Ethiopian Rift basin using constrained full tensor gravity inversion of the basement morphology

The Southern Main Ethiopian Rift is part of the great East African Rift system situated within the limit of 37o–38.5o east and 5.5o–7o north. Unlike the Central and Northern Main Ethiopian Rift where numerous geophysical studies have been conducted, the subsurface geology of Southern Main Ethiopian Rift is poorly constrained. Geological field work on Amaro Horst shows the basement outcrops are dominated by High grade gneisses and Intrusive with an average density of 6 gm/cm3 overlain by Mesozoic sedimentary rocks possibly extend into the studied sub-basins. In addition, field structural measurements and extracted lineaments from digital elevation model show NE–SW, NW–SE and N–S directions in agreement with fault block geometries identified by Invariance Tensor analysis. The Precambrian basement morphology of the area is delineated using constrained Tensor Gravity Inversion by applying Parker–Oldenburg algorisms. The basement morphology depicts the sub-basins; Northern Abaya, Southern Abaya, Chamo and Gelana basins where these areas are characterized by low Bouguer anomaly below -175mGal and lower Invariance Tensor anomaly may show the sediment infill of the sub-basins. The deepest depression exceeds 2500m below sea level in northern Abaya sub-basin where the maximum sediment thickness is more than 3700m. The result agrees with Spectral and Euler source depths solutions that show the limits of the crystalline basement vary between 2500m and 3500m. Amaro Horst, Chencha and Agere Selam are the basement structural high with average sediment thickness of 500m. The basement morphology revealed the N–S orientation of Gelana basin and other sub-basins aligns with NE–SW Tertiary Rifting. This result agrees with the structural analysis as well as previous regional studies on directions of prominent geological structures. These structurally controlled sub-basins having thick sedimentary sections are favorable zones for follow-up hydrocarbon exploration.

Measurement of Full Diffusion Tensor Distribution Using High-Gradient Diffusion MRI and Applications in Diffuse Gliomas

Diffusion MRI is widely used for the clinical examination of a variety of diseases of the nervous system. However, clinical MRI scanners are mostly capable of magnetic field gradients in the range of 20–80 mT/m and are thus limited in the detection of small tissue structures such as determining axon diameters. The availability of high gradient systems such as the Connectome MRI scanner with gradient strengths up to 300 mT/m enables quantification of the reduction of the apparent diffusion coefficient and thus resolution of a wider range of diffusion coefficients. In addition, biological tissues are heterogenous on many scales and the complexity of tissue microstructure may not be accurately captured by models based on pre-existing assumptions. Thus, it is important to analyze the diffusion distribution without prior assumptions of the underlying diffusion components and their symmetries. In this paper, we outline a framework for analyzing diffusion MRI data with b-values up to 17,800 s/mm2 to obtain a Full Diffusion Tensor Distribution (FDTD) with a wide variety of diffusion tensor structures and without prior assumption of the form of the distribution, and test it on a healthy subject. We then apply this method and use a machine learning method based on K-means classification to identify features in FDTD to visualize and characterize tissue heterogeneity in two subjects with diffuse gliomas.

Anisotropic temperature-dependent elastic constants and thermal conductivities of TRISO particle coatings

Tristructural isotropic (TRISO) particles show great promise as a candidate fuel for use in several next-generation high-temperature nuclear reactor designs due to their structural integrity and fuel performance at high temperatures and burnups. Computational codes exist that can simulate TRISO fuel performance characteristics and failure probabilities under extreme conditions which require knowledge of the TRISO coatings’ thermophysical properties. The thermophysical descriptions of the TRISO particle's layers (i.e., buffer, pyrolytic carbon, and silicon carbide) currently used in fuel performance codes, however, assume that many of these properties are constant with respect to temperature or texture. In this paper, we obtain the full elastic stiffness tensors of the carbon and silicon carbide layers, which have transversely isotropic symmetry. The calculated elastic properties of each of the coating layers are in remarkable agreement with the current models at room temperature and correct orientations. Additionally, the textured 3C-SiC layer was found to exhibit novel auxetic behavior above 1500 °C. The anisotropic temperature-dependent thermal conductivities of the carbon layers were calculated using acoustical Grüneisen-Debye theory which are in excellent agreement with current models at room temperature and correct orientations. These texture- and temperature-dependent relationships can be incorporated into the thermophysical description of TRISO particles in order to more accurately model fuel performance and failure probabilities under extreme conditions in forthcoming high-fidelity computational simulations.

Tidewater glacier response to individual calving events

AbstractTidewater glaciers have been observed to experience instantaneous, stepwise increases in velocity during iceberg-calving events due to a loss of resistive stresses. These changes in stress can potentially impact tidewater glacier stability by promoting additional calving and affecting the viscous delivery of ice to the terminus. Using flow models and perturbation theory, we demonstrate that calving events and subsequent terminus readvance produce quasi-periodic, sawtooth oscillations in stress that originate at the terminus and propagate upstream. The stress perturbations travel at speeds much greater than the glacier velocities and, for laterally resisted glaciers, rapidly decay within a few ice thickness of the terminus. Consequently, because terminus fluctuations due to individual calving events tend to be much higher frequency than climate variations, individual calving events have little direct impact on the viscous delivery of ice to the terminus. This suggests that the primary mechanism by which calving events can trigger instability is by causing fluctuations in stress that weaken the ice and lead to additional calving and sustained terminus retreat. Our results further demonstrate a stronger response to calving events in simulations that include the full stress tensor, highlighting the importance of accounting for higher order stresses when developing calving parameterizations.

Simple positivity-preserving nonlinear finite volume scheme for subdiffusion equations on general non-conforming distorted meshes

We propose a positivity-preserving finite volume scheme on non-conforming quadrilateral distorted meshes with hanging nodes for subdiffusion equations, where the differential equations have a sum of time-fractional derivatives of different orders, and the typical solutions of the problem have a weak singularity at the initial time \(t=0\) for given smooth data. In this paper, a positivity-preserving nonlinear method with centered unknowns is obtained by the two-point flux technique, where a new method to handling vertex unknown including hanging nodes is the highlight of our paper. For each time derivative, we apply the L1 scheme on a temporal graded mesh. Especially, the existence of a solution is strictly proved for the nonlinear system by applying the Brouwer’s fixed point theorem. Numerical results show that the proposed positivity-preserving method is effective for strongly anisotropic and heterogeneous full tensor subdiffusion coefficient problems.

Evaluation of Destress Blasting Effectiveness Using the Seismic Moment Tensor Inversion and Seismic Effect Methods

The effectiveness evaluation of long-hole destress blasting in an underground hard coal mine was made via the seismic effect method and the seismic moment tensor inversion. The seismic effect method is based on the difference between seismic and explosive energies, indicating that additional processes appeared to change the stress state in the rock mass, however, the seismic effect does not consider the focal mechanism. Therefore, the seismic moment tensor inversion was applied. Tremors that were an effect of long-hole destress blasting during the extraction of coal seam No. 507 in one of the hard coal mines localized in the Polish part of the Upper Silesian Coal Basin were analyzed. The results of the preceding applied methods were compared. High values of seismic effect indicate stress relaxation and thus the occurrence of additional processes, leading to a new stress equilibrium state in the rock mass. It has been shown that in analyzed cases, achieving such a state corresponded to the presence of a nonexplosive mechanism, with a large share or domination of the double couple component in the full seismic moment tensor solution. The reverse slip mechanism was present in the foci of provoked tremors, which can be associated with the displacement of rock blocks under conditions of high horizontal stress. Convergent results of the methods used confirm that the appropriate design of blasting parameters in relation to specific geological and mining conditions make it possible to provoke additional geomechanical processes in the rock mass.

The crustal structure of the Cantabrian Mountains revealed by new magnetotelluric soundings

The upper crustal structure of the central Cantabrian Mountains consists of a block of Variscan basement uplifted over the Duero basin as a consequence of a southward displacement along a major thrust during the convergence between Europe and Iberia in the Eocene-Oligocene (Alpine orogeny). We present the results of the 3-D modeling of thirteen new magnetotelluric sites, five of them being completed by long period data, over a 100 km-long, N-S oriented profile across the central Cantabrian Mountains and the Duero basin. Dimensionality analyses indicated a dominant E-W direction but with influence of 3-D structures at long periods and locally in the Cantabrian Mountains. Accordingly, we performed a 3-D joint inversion of the full impedance tensor and the geomagnetic transfer function following a sequential inversion workflow. The inverse model presents similarities with existing lithospheric models. In the southern part of the area, the conductive sediments of the Duero basin over a high resistive and homogeneous Iberian lithosphere are well delineated. Towards the north, beneath the Cantabrian Mountains, the model reveals a heterogeneous and conductive lithosphere, in which various elongated and dipping conductors in the upper and middle crust are associated with major Alpine thrusts, one being the frontal thrust of the Cantabrian Mountains over the Duero basin. At deeper depths (between 20 and 35 km depth), the Iberian crust appears as subducting to the north beneath a conductive zone interpreted as the hydrated mantle wedge of the north-Iberian continental margin.

Anomalous elasticity of talc at high pressures: Implications for subduction systems

Talc is a layered hydrous silicate mineral that plays a vital role in transporting water into Earth’s interior and is crucial for explaining geophysical observations in subduction zone settings. In this study, we explored the structure, equation of state, and elasticity of both triclinic and monoclinic talc under high pressures up to 18 GPa using first principles simulations based on density functional theory corrected for dispersive forces. Our results indicate that principal components of the full elastic constant tensor C11 and C22, shear components C66, and several off-diagonal components show anomalous pressure dependence. This non-monotonic pressure dependence of elastic constant components is likely related to the structural changes and is often manifested in a polytypic transition from a low-pressure polytype talc-I to a high-pressure polytype talc-II. The polytypic transition of talc occurs at pressures within its thermodynamic stability. However, the bulk and shear elastic moduli show no anomalous softening. Our study also shows that talc has low velocity, extremely high anisotropy, and anomalously high VP/VS ratio, thus making it a potential candidate mineral phase that could readily explain unusually high VP/VS ratio and large shear wave splitting delays as observed from seismological studies in many subduction systems.

Cooperative P-Wave Velocity Measurement with Full Waveform Moment Tensor Inversion in Transversely Anisotropic Media.

Precise stochastic approaches to quantitatively calculate the source uncertainties offers the opportunity to eliminate the influence of anisotropy on moment tensor inversion. The effects of ignoring anisotropy were tested by using homogeneous Green’s functions. Results indicate the influence of anisotropy and noise on fault plane rotation is very small for a pure shear source whether it is restricted to double couple solution or full moment tensor solution. Green’s functions with different prior rough anisotropy information were tested, indicating that the complex source is more sensitive to velocity models than the pure shear source and the fault plane rotation caused by full moment tensor solution is larger than the pure double couple solution. Collaborative P-wave velocity inversion with active measurements and passive acoustic emission data using the fast-marching method were conducted, and new Green’s functions established based on the tomography results. The resolved fault plane solution rotated only 3.5° when using the new Green’s functions, but the presence of spurious isotropic and compensated linear vector dipole components was not completely eliminated. It is concluded that the cooperative inversion is capable of greatly improving the accuracy of the fault plane solutions and reducing the spurious components in the full moment tensor solution.

Design of Triaxial Tests with Polymer Matrix Composites.

Multiaxial testing in composites may generate failure modes which are more representative of what occurs in a real structure submitted to complex loading conditions. However, some of its main handicaps include the need for special facilities, the correct design of the experiments, and the challenging interpretation of the results. The framework of this research is based on a triaxial testing machine with six actuators which is able to apply simultaneous and synchronized axial loads in the three space directions. Then, the aim was to design from a numerical point of view a triaxial experiment adapted to this equipment. The methodology proposed could allow for an adequate characterization of the triaxial response of a polymer-based composite with apparent isotropic behaviour in the testing directions. The finite element method (FEM) is applied in order to define the geometry of the triaxial specimen. The design pursues to achieve homogeneous stress and strain states in the triaxially loaded region, which should be accessible for direct measurement of the strains. Moreover, a fixing system is proposed for experimentally reproducing the desired boundary conditions imposed on the numerical simulations. The procedure to determine the full strain tensor in the triaxially loaded region is described analytically and with the help of FEM virtual testing. The hydrostatic component and the deviatoric part of the strain tensor are proposed for estimating the susceptibility of the polymer-based composite to fail due to the triaxial strain state imposed. Then, the loading scenarios that cause higher values of the deviatoric components in the triaxially loaded region are considered to be more prone to damage the region of interest. Nevertheless, the experimental failure is expected to be produced in the arms of the specimen which are uniaxially loaded, since in all of the loading cases the simulations show higher levels of stress concentration out of the triaxially loaded region. Thus, although the triaxial strength could not be accurately determined by the proposed tests, they can be utilized for observing the triaxial response before failure.