Strong Discontinuities Research Articles

Geometrically exact 3D beam theory with embedded strong discontinuities for modeling of localized failure in bending

In this work, we propose the elastoplastic model of a three-dimensional (3D) geometrically exact beam, which can capture localized bending deformation leading to a plastic hinge. This is a follow-up work to Tojaga et al. (2023), where we only studied the fracture of fiber-like beam structures that break in mode I or mode II, here extended to handle beam-like behavior with bending failure and the main difficulty of the non-vectorial character of large 3D rotations. The Reissner model is chosen for representing the large elastic and large plastic deformations of such a beam model, which leads to the multiplicative decomposition of the rotation tensor. We shown how to replace this with the corresponding additive decomposition of the rotation vector derivatives, to be performed in the material description in the tangent space of SO(3) manifold in the initial configuration. The plasticity model for which we give a more detailed implementation employs a Rankin-like multi-surface plasticity criteria, where each moment vector component (i.e. torsion or bending) is considered separately. The non-local variational formulation is proposed to deal with the softening phenomena characteristic of localized bending failure, where the fracture energy is introduced as the main parameter for softening. The discrete approximation is built in terms of the embedded-discontinuity finite element method (ED-FEM), which introduces a jump in rotation vector that can be handled at the level of a particular beam element. This kind of approach builds upon the best approximation property of the FEM discrete approximation in the energy norm and provides the best-approximation property for the dissipated energy computations, and thus optimal computational accuracy if the quantity of interest is inelastic dissipation. The computations are carried out by the operator split method, which separates the computations of global state variables (displacements and moments) from local (plastic curvature) variables. Several illustrative examples are provided to confirm an excellent performance of the proposed methodology.

Gauging Iron-Sulfur Cubane Reactivity from Covalency: Trends with Oxidation State.

We investigated room-temperature metal and ligand K-edge X-ray absorption (XAS) spectra of a complete redox series of cubane-type iron-sulfur clusters. The Fe K-edge position provides a qualitative but convenient alternative to the traditional spectroscopic descriptors used to identify oxidation states in these systems, which we demonstrate by providing a calibration curve based on two analytic methods. Furthermore, high energy resolution fluorescence detected XAS (HERFD-XAS) at the S K-edge was used to measure Fe-S bond covalencies and record their variation with the average valence of the Fe atoms. While the Fe-S(thiolate) covalency evolves linearly, gaining 11 ± 0.4% per bond and hole, the Fe-S(μ3) covalency evolves asystematically, reflecting changes in the magnetic exchange mechanism. A strong discontinuity manifested for superoxidation to the all-ferric state, distinguishing its electronic structure and its potential (bio)chemical role from those of its redox congeners. We highlight the functional implications of these trends for the reactivity of iron-sulfur cubanes.

Numerical Analysis of Unsteady Vibrations of a Plate Resting on an Elastic Isotropic Half-Space

The paper is devoted to the numerical solution of the problem of vibrations of an infinite elastic plate resting on an elastic isotropic half-space using the analytical method based on the ray method with its numerical realization via the Maplesoft package. Unsteady oscillations are caused by the action of instantaneous loads on the plate, resulting in the appearance of two plane wave surfaces of strong discontinuity in the elastic half-space, behind the fronts of which, up to the contact boundary, the solution is constructed using ray series. The unknown functions entering the coefficients of the ray series and the equation of plate motion are determined from the boundary conditions of the contact interaction between the plate and the half-space. Previously, the approximate solution of this problem was obtained analytically without using mathematical packages, and the dynamic deflection of the plate involving only the first three terms of the ray series was written down. In this work, a two-layer medium with different properties was investigated using an algorithm developed to solve contact dynamic problems related to the occurrence and propagation of strong and weak discontinuity surfaces.

A closed-form Timoshenko beam element with multiple localized singularities for nonlinear material behavior using the strong discontinuity approach

The development of a novel closed-form beam element for nonlinear material behavior in structures is described, which is formulated according to Timoshenko beam theory and the embedded strong discontinuity approach. Multiple strong transverse and rotation discontinuities are considered to model several dissipative mechanisms due to localized singularities in the shear strain and curvature fields. Issues of dependence on numerical techniques, as occurs in displacement- and forced-based beam elements with the finite element method, are overcome since the formulated element is derived from a pure mathematical treatment. Theoretical principles are stemming from a Lagrangian variational model, from which singularities are naturally derived. The proposed formulation is solved by the conventional methods of calculus of variations, resulting in the boundary value problem. Then, closed-form expressions for any boundary conditions are obtained. By particularizing closed-form expressions for a basic system, a symmetric closed-form flexibility matrix is obtained. This matrix naturally leads to a symmetric closed-form stiffness matrix for any type of loading and unloading process, in which the nodeless degrees of freedom of the multiple embedded strong discontinuities are naturally condensed. Since Gauss integration methods are not used, the multiple localized singularities are not influenced by predefined positions of the integration points. However, the accuracy depends on the chosen number of singularities and their positions. Then, the proposed element is mesh independent when sufficient discontinuities are considered, which does not imply an increase in the degrees of freedom. Softening discontinuities are modeled at arbitrary positions in a single element, which overcomes the drawbacks of other embedded beam theories where more than one element is necessary to identify failure since strong discontinuities are generally located at the midpoint of the beam element. The obtained closed-form stiffness matrix is not always positive definite for strain-softening problems. Nevertheless, there are no major convergence issues as it is a symmetric matrix that does not require a constraint equation, so even the complex snap-back behavior in structures is adequately modeled. The capability of the formulated beam element for nonlinear material behavior in structures is validated with representative examples.

Physics-constrained and flow-field-message-informed graph neural network for solving unsteady compressible flows

With the increasing use of deep neural networks as surrogate models for accelerating computational simulations in mechanics, the application of artificial intelligence in computational fluid dynamics has seen renewed interest in recent years. However, the application of deep neural networks for flow simulations has mainly concentrated on relatively simple cases of incompressible flows. The strongly discontinuous structures that appear in compressible flows dominated by convection, such as shock waves, introduce significant challenges when approximating the nonlinear solutions or governing equations. In this work, we propose a novel physics-constrained, flow-field-message-informed (FFMI) graph neural network for spatiotemporal flow simulations of compressible flows involving strong discontinuities. To enhance the nonlinear approximation capability of strong discontinuities, a shock detector method is leveraged to extract the local flow-field messages. These messages are embedded into the graph representation to resolve the discontinuous solutions accurately. A new FFMI sample-and-aggregate-based message-passing layer, which aggregates the edge-weighted attributes with node features on different hop layers, is then developed to diffuse and process the flow-field messages. Furthermore, an end-to-end paradigm is established within the encoder–decoder framework to transform the extracted information from the flow field into latent knowledge about the underlying fluid mechanics. Finally, a variety of one- and two-dimensional cases involving strong shock waves are considered to demonstrate the effectiveness and generalizability of the proposed FFMI graph neural network.

High-order finite volume multi-resolution WENO schemes with adaptive linear weights on triangular meshes

This paper presents high-order finite volume multi-resolution weighted essentially non-oscillatory schemes with adaptive linear weights to solve hyperbolic conservation laws on triangular meshes. They are abbreviated as the ALW-MR-WENO schemes. The novel third-order, fourth-order, and fifth-order ALW-MR-WENO schemes are designed by applying two unequal-sized hierarchical central stencils in comparison to the classical WENO schemes which utilize many equal-sized upwind biased/central stencils. With the application of one simple condition, only two linear weights are automatically adjusted to be positive values on condition that their summation is one. The novel finite volume ALW-MR-WENO schemes could maintain the designed order of accuracy in smooth areas and reduce to the first-order accuracy so as to keep essentially non-oscillatory properties around strong discontinuities. So it is the first time that any high-order WENO schemes with the application of only two unequal-sized stencils are obtained on triangular meshes. And the major benefits are their efficiency, compactness, and simplicity in large scale engineering applications on unstructured meshes. Finally, several tests are used to indicate the effectiveness of these new finite volume WENO schemes.

Peridynamic analysis of thermal behaviour of PCM composites for heat storage

One possibility to utilize excess energy from electricity generation or other industrial processes is to use thermal energy storage systems based on phase change materials (PCM). These systems can accumulate and release significant amounts of heat energy during the phase transitions. The volume and properties of PCM undergo rapid changes during the transitions, creating strong physical non-linearities and geometric discontinuities. Describing this complexity with local formulations of the physical processes, i.e., formulations with differential equations, makes the numerical solutions of the corresponding problems challenging. This paper presents a numerical approach for the analysis of such physically complicated systems based on Peridynamics. The approach is illustrated by modelling and analysing the thermal behaviour of a PCM. The computational challenges associated with simulating composite structures with large differences in the heat conductivity of their components are addressed by investigating the influence of different types of peridynamic kernels and their numerical implementation. The results demonstrate that the solution is significantly influenced by the definition of the average thermal conductivity of the peridynamic bonds. The proposed modelling framework provides the base for developing a precise thermo-hydro-mechanical description of PCM composites that can fully represent the complex physical behaviour of thermal energy storage systems.

A new random walk method for simulating heat flow and temperature distribution in three-dimensional discontinuous systems

Numerical methods for analyzing diffusion phenomena involving strong discontinuities and complicated interfaces are of great scientific and technical importance. In this paper, we extend the existing step-balanced random walk, which overcomes the problem of detailed balance of a random walker in three-dimensional (3D), diffusion-tensor discontinuous systems, to particle density (ρ) discontinuous systems, and introduce volumetric heat capacity C and temperature T by considering ρ as heat density, i.e., ρ=CT, to apply it to thermal problems. Two types of thermophysical simulations are demonstrated: one is steady-state heat flow due to a temperature difference at the end surfaces on a two-phase slab model unit cell with periodic boundary conditions; the other is the time variation of temperature distribution due to heat diffusion from a point heat source in a repeated two-phase slab model with no periodic boundary conditions. We see the correct behavior of heat and temperature expected in 3D discontinuous systems composed of two phases with anisotropic thermal conductivity. The applicability to problems other than pure diffusion and the limitations of the method are also discussed.

Novel hydrodynamic schemes capturing shocks and contact discontinuities and comparison study with existing methods

We present a new hydrodynamic scheme named Godunov Density-Independent Smoothed Particle Hydrodynamics (GDISPH), that can accurately handle shock waves and contact discontinuities without any manually tuned parameters. This is in contrast to the standard formulation of smoothed particle hydrodynamics (SSPH), which requires the parameters for an artificial viscosity term to handle the shocks and struggles to accurately handle the contact discontinuities due to unphysical repulsive forces, resulting in surface tension that disrupts pressure equilibrium and suppresses fluid instabilities. While Godunov SPH (GSPH) can handle the shocks without the parameters by using solutions from a Riemann solver, it still cannot fully handle the contact discontinuities. Density-Independent Smoothed Particle Hydrodynamics (DISPH), one of several schemes proposed to handle contact discontinuities more effectively than SSPH, demonstrates superior performance in our tests involving strong shocks and contact discontinuities. However, DISPH still requires the artificial viscosity term. We integrate the Riemann solver into DISPH in several ways, yielding some patterns of GDISPH. The results of standard tests such as the one-dimensional Riemann problem, pressure equilibrium, Sedov–Taylor, and Kelvin–Helmholtz tests are favourable to GDISPH Case 1 and GDISPH Case 2, as well as DISPH. We conclude that GDISPH Case 1 has an advantage over GDISPH Case 2effectively handling shocks and contact discontinuities without the need for specific parameters or introducing any additional numerical diffusion.

Discrete Lattice Element Model for Fracture Propagation with Improved Elastic Response

This research presents a novel approach to modeling fracture propagation using a discrete lattice element model with embedded strong discontinuities. The focus is on enhancing the linear elastic response within the model followed by propagation of fractures until total failure. To achieve this, a generalized beam lattice element with an embedded strong discontinuity based on the kinematics of a rigid-body spring model is formulated. The linear elastic regime is refined by correcting the stress tensor at nodes within the domain based on the internal forces present in lattice elements, which is achieved by introducing fictitious forces into the standard internal force vectors to predict the right elastic response of the model related to Poisson’s effect. Upon initiation of the first fractures, the procedure for the computation of the fictitious stress tensor is terminated, and the embedded strong discontinuities are activated in the lattice elements for obtaining an objective fracture and failure response. This transition ensures a shift from the elastic phase to the fracture propagation phase, enhancing the predictive capabilities in capturing the full fracture processes.

Demonstration and STEM Analysis of Ferroelectric Switching in MOCVD‐Grown Single Crystalline Al0.85Sc0.15N

AbstractWurtzite‐type Al1−xScxN solid solutions grown by metal organic chemical vapor deposition are for the first time confirmed to be ferroelectric. The film with 230 nm thickness and x = 0.15 exhibits a coercive field of 5.5 MV cm−1 at a measurement frequency of 1.5 kHz. The single crystal quality and homogeneous chemical composition of the film are confirmed by X‐ray diffraction and spectroscopic methods such as time of flight secondary ion mass spectrometry. Annular bright field scanning transmission electron microscopy serves to prove the ferroelectric polarization inversion at the unit cell level. The single crystal quality further allows to image the large‐scale domain pattern of a wurtzite‐type ferroelectric for the first time, revealing a predominantly cone‐like domain shape along the c‐axis of the material. As in previous work, this again implies the presence of strong polarization discontinuities along this crystallographic axis, which can be suitable for current transport. The domains are separated by narrow domain walls, for which an upper thickness limit of 3 nm is deduced but which can potentially be atomically sharp. The authors are confident that these results will advance the commencement of the integration of wurtzite‐type ferroelectrics to GaN as well as generally III‐N‐based heterostructures and devices.

Разрывное конически симметричное течение идеальной несжимаемой жидкости

The Euler equations are considered in spherical coordinates to describe the steady flow of an ideal incompressible fluid. An exact conically symmetric solution based on the source/sink and strong conic discontinuity with an apex in zero of the coordinate system is obtained. The northern region of the flow is situated in the finite vicinity of the symmetry axis, i.e., within a discontinuity cone, where the solution is regular and vortex-free. On the other side of the discontinuity, the flow adjoins the permeable equator plane. In this region, the flow is vortex-like, and its properties are determined by a density jump. The fluid flowing through the discontinuity is governed by the increase of entropy principle. The thermal field corresponding to the flow is presented. It shows that the spatial heterogeneity of temperature results from the interaction between the azimuth vorticity component and the meridional velocity component. A strong discontinuity cone angle is revealed to be a significant parameter of the problem. A comparative analysis of the source and sink properties is performed. The considered flows qualitatively differ in terms of pressure and velocity behavior.

A PWDG method for the Maxwell system in anisotropic media with piecewise constant coefficient matrix

In this paper we are concerned with plane wave discontinuous Galerkin (PWDG) methods for time-harmonic Maxwell equations in three-dimensional anisotropic media, for which the coefficients of the equations are piecewise constant symmetric matrices, where each constant symmetric matrix is defined on a medium (subdomain). By using suitable scaling transformations and coordinate (complex) transformations on every subdomain, the original Maxwell equation in anisotropic media is transformed into a Maxwell equation in isotropic media occupying a union domain of specific subdomains of complex Euclidean space. Based on these transformations, we define anisotropic plane wave basis functions and discretize the considered Maxwell equations by PWDG method with the proposed plane wave basis functions. We derive error estimates of the resulting approximate solutions, and further introduce a practically feasible local hp-refinement algorithm, which substantially improves accuracies of the approximate solutions. Numerical results indicate that the approximate solutions generated by the proposed PWDG methods possess high accuracy for the case of strong discontinuity media.

A unified hp-HDG framework for Friedrichs' PDE systems

This work proposes a unified hp-adaptivity framework for hybridized discontinuous Galerkin (HDG) method for a large class of partial differential equations (PDEs) of Friedrichs' type. In particular, we present unified hp-HDG formulations for abstract one-field and two-field structures and prove their well-posedness. In order to handle non-conforming interfaces we simply take advantage of HDG built-in mortar structures. With split-type mortars and the approximation space of trace, a numerical flux can be derived via Godunov approach and be naturally employed without any additional treatment. As a consequence, the proposed formulations are parameter-free. We perform several numerical experiments for time-independent and linear PDEs including elliptic, hyperbolic, and mixed-type to verify the proposed unified hp-formulations and demonstrate the effectiveness of hp-adaptation. Two adaptivity criteria are considered: one is based on a simple and fast error indicator, while the other is rigorous but more expensive using an adjoint-based error estimate. The numerical results show that these two approaches are comparable in terms of convergence rate even for problems with strong gradients, discontinuities, and singularities.

High speed flows with particles on demand: Boundary conditions

The particles on demand (PonD) method is a new kinetic theory model that allows for simulation of high speed compressible flows. While the standard lattice-Boltzmann method is limited by a fixed reference frame, significantly reducing the range of Mach numbers, PonD takes advantage of adaptive reference frames to get rid of the restrictions of standard LB and is able to simulate flows at high speeds and with large temperature variation. Previously, PonD has been shown to be a viable alternative for simulation of flows with strong discontinuities and for detonation modelling. However, treatment of flows with complex boundaries has been lacking. Here, we present PonD augmented with a non-equilibrium extrapolation based boundary condition. We present several compressible test cases such as shock-vortex interaction in the Schardin’s problem and supersonic flow over a two-dimensional cylinder at Mach numbers up to 5. We observe that the results agree well with literature, paving the way for a kinetic theory based approach for simulating compressible flows in realistic scenarios.

New Third-Order Finite Volume Unequal-Sized WENO Lagrangian Schemes for Solving Euler Equations

In this paper, new third-order finite volume unequal-sized weighted essentially non-oscillatory (US-WENO) Lagrangian schemes are designed to solve Euler equations in two and three dimensions. The spatial reconstruction procedures are implemented by using a convex combination of a quadratic polynomial with several linear polynomials specified on unequal-sized stencils, so the new US-WENO Lagrangian schemes can achieve the designed third-order accuracy and maintain an essentially non-oscillatory property near strong discontinuities in multi-dimensions. Unlike the traditional WENO reconstruction procedures specified on unstructured meshes, the linear weights of these new two-dimensional and three-dimensional US-WENO spatial reconstructions can be selected as any positive numbers as long as their summation equals one and they are not related to the local mesh topology or the location of quadrature points. Moreover, the linear weights do not have to be recalculated even if the grid moves with the fluid, avoiding the appearance of negative linear weights, thus improving computation efficiency and robustness in multi-dimensional Lagrangian numerical simulations. Finally, extensive benchmark numerical cases are employed to display the excellent capability of the presented US-WENO Lagrangian schemes.

Evaluation of a high-order central-difference solver for highly compressible flows out of thermochemical equilibrium

A high-order shock-capturing central finite-difference scheme is evaluated for numerical simulations of hypersonic high-enthalpy flows out of thermochemical equilibrium. The scheme is an extension to thermochemical out-of-equilibrium flows of the technique presented in Sciacovelli et al. (2021) for high-speed flows in chemical nonequilibrium. It relies on a standard tenth-order accurate central-difference approximation of the inviscid fluxes, supplemented with a high-order accurate nonlinear artificial dissipation term of ninth-order accuracy in smooth flow regions. Close to flow discontinuities, a shock-capturing low-order term is activated based on a highly selective shock sensor. To enable robust and non oscillatory solutions in regions of strong discontinuities of the thermodynamic variables, including the vibrational temperature, a shock detector consisting of a combination of a pressure-based term and Ducros’ vorticity/dilatation sensor is applied to all conservation equations except that of vibrational energy. For the latter, a shock sensor based on the vibrational temperature itself is adopted instead, to account for the loose coupling between vibrational energy and pressure. The accuracy and robustness of the proposed approach is demonstrated for a variety of thermochemical non-equilibrium configurations, ranging from one-dimensional benchmarks to three-dimensional turbulent flows, for which the ILES capabilities of the selective high-order numerical dissipation are also showcased.

An iterative plane stress integrated Transformation Field Analysis for Equivalent Homogeneous Medium characterization and localization — Application to structural holes

Full-scale structure models of high-energy situations exhibited insufficient localization when connector elements were employed to model the assemblies. An alternative approach using a super-element featuring a hole was proposed and successful in mechanical fields localization. However this approach was restrained to linear problems. An extension to materially non-linear problems was thus looked for, and in particular perforated plate materially non-linear fields.An iterative plane stress integrated TFA method is thus developed to relocate the micro-scale elastic–plastic mechanical fields of a plate containing an extremely soft medium/a strong discontinuity: a hole. To this end, a RVE is defined and discretized, the accuracy of the localization relation is assessed, and a radial return algorithm is set-up to integrate the constitutive equation of the micro-scale problem. The results of the proposed TFA approach are compared to an analytical solution and to non-linear FEA. The results show that the localization of elastic–plastic perforated plate mechanical fields is accurate, provided that the method is iterative and that the hole is discretized in the RVE. It is foreseen that both the sequential and integrated TFA approaches will be combined to evaluate criteria in full-scale structure computations.

Wyprowadzenie fenomenologicznych równań hydromechaniki przepływów wielofazowych

In the article, a multi-phase (non-homogeneous, heterogeneous) medium is considered as a macrosystem (continuum) composed of several (at least two) phases, such as a carrier phase (liquid, vapor or gas) and a carried phase (solid particles, bubbles or drops).The masses and mixtures of these phases undergo continuous changes over time due to the addition or separation of new masses to or from both phases. The model takes into account interphase transitions, discontinuities inside the mixture, and the possibility of phases being either continuous or discrete, depending on their location. A method for preliminary smoothing of discontinuities has been developed, leveraging the fact that the location in space, as well as the shape and size of the discrete phase are random. A function, denoted as φi (x, y, z, t), has been introduced, which indicates the probability of the presence of the i-th phase in the vicinity of a given point in space at time t, or that the given point of space x, y, z at time t belongs to the set of points of the i-th phase. On the other hand, this probability can be interpreted as the volumetric concentration of the i-th phase at a given point in space (i.e., the ratio of the measure of the set of points belonging to the i-th phase in the vicinity of the point under consideration at time t to the measure of the entire set of points in the surrounding area). This hypothetical medium, being equivalent to the original one, serves as a model for a multi-phase (inhomogeneous, heterogeneous, two-phase) medium. The uniqueness of the model arises from its construction. In addition, this paper considers several main areas of theoretical and experimental research concerning the hydrodynamics of a multi-phase (two-phase suspension-carrying) flow of a continuous medium. It also discusses the most important results achieved in existing works. A critical analysis of known theories for mathematically describing the motion of multi-phase (two-phase) systems and methods for averaging the hydrodynamic characteristics of a turbulent flow are given. The procedure for closing the equations systems of hydromechanics of multi-phase flows proposed in existing works is carried out within the framework of semi-empirical theories of turbulence. In nature, the vast majority of multi-phase (two-phase, inhomogeneous) mixtures exhibit turbulent behavior, making its study a crucial practical task. The mathematical description of the motion of a turbulent multi-phase flow relies on stylized laws of mechanics. The methods of operational analysis proposed at various times by different researchers for the mathematical description of the motion of a multi-phase (two-phase) flow have varying degrees of approximation and certain limited areas of application. One of the main challenges in formulating differential equations for the motion of a turbulent multi-phase (two-phase, suspension-carrying) flow is the fact that in a turbulent flow of a mixture, where the characteristics of the flow change chaotically and randomly over time and at each point in space, both in magnitude and in direction, there are surfaces with weak and strong discontinuities. Consequently, the actual values of velocity and pressure of a multi-phase flow, strictly speaking, cannot be considered continuous functions of the coordinates of space and time throughout the entire region occupied by the mixture.

Geochemical studies of the gold mineralization of Kodiarani-(Doko permit). Prefecture of Siguiri, Administrative region of Kankan, Republic of Guinea

West Africa is now a major player in world gold production with numerous gold deposits hosted in Birimien green belts (2.2-2.0 Ga). However, these belts and their geological formations remain unexplored, as does their evolution during the Eburnean orogeny. This is why, after acquiring its exploration Licence, TOROGOLD focused its work on Air Core drilling activities in the Doko-Kodiarani Licence in north-eastern Guinea. During these activities, geochemical studies will be carried out on rock samples taken from the six (6) boreholes pre-selected for this purpose. The aim of this research work is to carry out a geological study in order to determine the chemical associations between the mineralization, the nature of the host rock and others for future work. To achieve this objective, we used a number of methods, including a statistical approach, field reconnaissance, and macroscopic and lithogeochemical analyses. Geochemical studies reveal that the rocks studied are meta-sediments. They belong to the schist groups. Their protholites are moderately altered (60 < CIA < 80) and the source is thought to be andesitic and basaltic, emplaced in a geotectonic oceanic island arc. As for the various AC drilling sections, our observations reveal that the gold mineralization is found in two types of deposit: disseminated, present in all lithology’s, and vein-type, resulting from the filling of fracture zones, associated with the highest concentrations. We noted a strong discontinuity of mineralization in these veins, due to a tectonic phase that may have played an important role in the remobilization of gold in these structures.