Two-scale Model Research Articles

Study on the mechanism of reactive acid transport in fractured two-mineral carbonate rocks

Centering on the mechanism of reactive acid transport in fractured two-mineral carbonate rocks, this paper established a two-mineral model for fractured media with the extended finite element method based on the two-scale continuum model. A series of examples are presented with the numerical model, the numerical results show that the variation of volume fraction of dolomite, the surface reaction constant of dolomite as well as the diffusion coefficient can alter the dissolution morphology and dissolution performance. As the volume fraction of dolomite increases or surface reaction constant of dolomite decreases, the dissolution patterns would transform form wormhole to uniform dissolution. And the bedding distribution and porphyritic distribution of dolomite will cause a fragmentation phenomenon for the dissolution morphology. Besides, the bedded dolomite can bring about an increased acid consumption. It can be inferred that the reaction rate difference between limestone and dolomite is the key factor that affects the dissolution morphology, and the retarded acid with a low reaction rate in dolomite rocks does not perform well.

Two-scale random field model for quasi-brittle materials

A new idea of two-scale random field is presented by taking the mechanical behaviors of quasi-brittle materials as an example. In this way, the random fluctuation of material damage at the micro-meso-scale and the spatial correlation of mechanical properties at the macro-scale are both well described by the random distribution of fracture strain. Moreover, it is proved that by introducing a transform matrix, the mesoscopic random fields can be described by a two-scale random field. Based on this, a two-step numerical implementation of the two-scale random field is proposed, and the mechanical behaviors of quasi-brittle materials can be analyzed by incorporating the probability density evolution theory. Finally, an experimental example is presented to demonstrate the capability of the proposed model. Ideal agreements are achieved against the experimental results. Particularly, the randomness of material properties at both scales can be well reproduced.

Insight into Liquid Polymorphism from the Complex Phase Behavior of a Simple Model.

We systematically explored the phase behavior of the hard-core two-scale ramp model suggested by Jagla [Phys. Rev. E 63, 061501 (2001)PRESCM1539-375510.1103/PhysRevE.63.061501] using a combination of the nested sampling and free energy methods. The sampling revealed that the phase diagram of the Jagla potential is significantly richer than previously anticipated, and we identified a family of new crystalline structures, which is stable over vast regions in the phase diagram. We showed that the new melting line is located at considerably higher temperature than the boundary between the low- and high-density liquid phases, which was previously suggested to lie in a thermodynamically stable region. The newly identified crystalline phases show unexpectedly complex structural features, some of which are shared with the high-pressure ice VI phase.

LOW FREQUENCY PROPERTY OF A FRACTAL VIBRATION MODEL FOR A CONCRETE BEAM

A concrete beam is a basic element for many applications in architectural engineering, and its vibration property plays an important role in safety and life-span. Previous vibration models were based on continuum beam theories, which could not take into account the porous property of the concrete beam. This paper adopts a two-scale fractal vibration model, and its low frequency property is revealed. This new finding can explain the vibration attenuation and vibration absorption of the porous concrete, and the present theory sheds a new light on the concrete design.

Study on Non-Bragg Microwave Backscattering from Sea Surface Covered with and without Oil Film at Moderate Incidence Angles

In the past decades, Bragg scattering has been considered to be an important scattering mechanism of microwave backscattering from sea surfaces. However, as reported in many recent literatures, non-Bragg scattering (which is often attributed to wave breaking) also makes a significant impact on radar scattering, especially for Horizontal–Horizontal (HH) polarized radar signals. To date, we know far less about non-Bragg scattering than Bragg scattering. Herein, this paper carries out an investigation on non-Bragg scattering and its effect on radar echoes at moderate incidence angles, both for oil-free and oil-covered sea surfaces. This paper firstly presents a systematic comparison of several sea spectra commonly used for the simulation of microwave scattering from sea surfaces. It is found that none of them perform well for the description of Bragg waves. Then, the “pure” Bragg wave spectra are inverted in the framework of the two-scale model (TSM) and geophysical model functions (GMFs). The normalized radar cross sections (NRCS) related to total scattering, non-Bragg scattering, and “pure” Bragg scattering in C, X, and Ku-bands are simulated under various conditions (i.e., incidence angles, wind speeds, and wind directions). Quantitative assessments of the relative contributions of non-Bragg scattering to total scattering are conducted. We also perform a survey on the non-Bragg scattering from the oil-covered sea surface. This article provides some new insights for a better understanding of the non-Bragg microwave scattering from rough sea surfaces at moderate incidence angles.

Dual percolations of electrical conductivity and electromagnetic interference shielding in progressively agglomerated CNT/polymer nanocomposites

Recent experiments have revealed two distinct percolation phenomena in carbon nanotube (CNT)/polymer nanocomposites: one is associated with the electrical conductivity and the other is with the electromagnetic interference (EMI) shielding. At present, however, no theories seem to exist that can simultaneously predict their percolation thresholds and the associated conductivity and EMI curves. In this work, we present an effective-medium theory with electrical and magnetic interface effects to calculate the overall conductivity of a generally agglomerated nanocomposite and invoke a solution to Maxwell’s equations to calculate the EMI shielding effectiveness. In this process, two complex quantities, the complex electrical conductivity and complex magnetic permeability, are adopted as the homogenization parameters, and a two-scale model with CNT-rich and CNT-poor regions is utilized to depict the progressive formation of CNT agglomeration. We demonstrated that there is indeed a clear existence of two separate percolative behaviors and showed that, consistent with the experimental data of poly-L-lactic acid (PLLA)/multi-walled carbon nanotube (MWCNT) nanocomposites, the electrical percolation threshold is lower than the EMI shielding percolation threshold. The predicted conductivity and EMI shielding curves are also in close agreement with experimental data. We further disclosed that the percolative behavior of EMI shielding in the overall CNT/polymer nanocomposite can be illustrated by the establishment of connective filler networks in the CNT-poor region. It is believed that the present research can provide directions for the design of CNT/polymer nanocomposites in the EMI shielding components.

Temperature impact on linear and radial wormhole propagation in limestone, dolomite, and mixed mineralogy

The temperature has a significant impact on acid stimulation efficiency in carbonate reservoirs. To fully understand its effect, this study utilizes a two-scale continuum model coupled with heat transfer. The study investigated the importance of the heat of reaction, injected acid temperature, and reservoir temperature on wormhole propagation and treatment efficiency. This research's novelty stems from its application to limestone, dolomite, and combined mineralogy in radial and linear flow patterns. It also investigated the role of vugs on acidizing efficiency in dolomite formations, showing that vugs improve wormhole penetration distance. The model reproduced the outcomes from the literature, showing that the heat of reaction has a negligible effect on wormhole propagation. On the contrary, increasing the injected acid temperature reduced the wormhole radius in limestone but improved its efficiency in dolomite. The acid could not create efficient wormholes at low acid temperatures in dolomite formation. It diverted itself towards the limestone in sequence mineralogy cases. Nevertheless, it equally stimulated the limestone and dolomite sections at elevated acid temperatures. The reservoir temperature was less critical than the acid temperature in determining the wormhole size and treatment efficiency. The model can be used to select the optimum temperature at which the most efficient wormhole is created.

Modelling the influence of process parameters on precipitate formation in powder-bed fusion additive manufacturing of IN718

In this work, we have developed a two-scale (finite element + phase field) model for simulating the precipitate evolution in IN718 in powder-bed fusion additive manufacturing. The thermal simulations at the scale of the component are resolved at the level of a single stripe of the scan path. This resolution allows to model various process parameters (or scan strategies) in complex geometry domains, here demonstrated by an impeller, and predicts the complete thermal history within each finite element layer. The predicted thermal history is in-full passed to multiple phase-field simulation representative volume elements (RVEs) that explicitly model the evolution of the γ′ and γ″ precipitates in various locations of the impeller. We simulate the precipitate microstructures in different regions of the impeller and demonstrate the influence of process parameters on the volume fraction, size, and shape of the precipitates. Although no precipitates are found to form during the standard printing conditions (for instance EOS M290), we show that changing the process parameters such as stripe width or chamber temperature may induce significant precipitation in the as-built component.

Analytical Method for Evaluating Calcium Diffusion Coefficient of Partially Leached Cement Paste

This paper aims at presenting an analytical method for evaluating the calcium diffusion coefficient of partially leached cement paste. The analytical method is developed based on a two-scale model of partially leached cement paste. On the first scale, the solid phase in partially leached cement paste is modeled as a two-phase composite material, composed of a matrix of calcium silicate hydrate (C─S─H) gel and inclusions of unhydrated cement and undissolved calcium hydroxide (CH) crystal. The relative calcium diffusion coefficient is derived analytically by the differential effective medium scheme. On the second scale, partially leached cement paste is treated as a system consisting of the solid phase and spheroidal capillary pores; the percolation behavior of capillary pores near the critical volume fraction is considered. In combination with percolation theory, the effective medium approach is modified to evaluate the calcium diffusion coefficient of partially leached cement paste. The depolarization factor is expressed in terms of the critical volume fraction of capillary pores, and the percolation exponents are calibrated from computer simulation data. Finally, the validity of the analytical method is verified with computer simulation data and experimental results collected from the literature. It is concluded that the proposed analytical method can predict the calcium diffusion coefficient of partially leached cement paste with reasonable accuracy.

A refined two-scale model for Newtonian and non-Newtonian fluids in fractured poroelastic media

A refined model is presented which numerically resolves the fluid flow within a fracture inside a saturated poroelastic material. At the discontinuity, the mass balance of the fluid is solved using the velocity profile inside the fracture, with the velocity profile being determined numerically from the momentum balance in the integration points at the discontinuity. The resolution of the mass balance and the velocity profile in the integration points at the discontinuity is coupled to surrounding poroelastic material model in a two-scale approach. The resulting monolithic scheme allows for complex fluid behaviour to be included, which is demonstrated via the inclusion of the fluid inertia and the use of a Carreau fluid. The governing equations are discretised using T-splines, while the fracture is represented by spline-based interface elements, and an implicit time discretisation scheme is used. Mesh refinement studies are carried out for a typical case, which contains a pressurised and propagating fracture. A coarse macro-scale mesh is sufficient to obtain the correct propagation velocities, but a finer mesh is needed to prevent pressure oscillations. Time step refinement studies show the capabilities of the model to capture pressure waves within the fracture. Finally, it is shown that if inertial effects are limited and a Newtonian fluid is used, a fracture scale discretisation using a single element is sufficient. However, for a Carreau fluid or when the problem is inertia-dominated, smaller elements are needed to correctly represent the velocity profile.

Two-scale mathematical model for tsunami wave

Scale or dimension is critical for all physical laws, and different scales might result in distinct and sometimes contradicting laws for the same phenomenon. In this study, the idea of the two-scale fractal is utilized to model the time-fractional tsunami wave traveling on an unsmooth surface. Mass and momentum conservation equations are established in a fractal space. To covert the considered fractional model into a differential equation, the fractional complex transform is used, then He’s variational iteration method is adopted to solve the resultant equation. The fractal model helps in studying the dynamic behavior of tsunami wave and their prevention through boundary control. The present study sheds new light on two-scale fluid mechanics.

A probabilistic approach for high cycle fatigue of Wire and Arc Additive Manufactured parts taking into account process-induced pores

Wire and Arc Additive Manufacturing (WAAM) is a direct-energy deposition technique (unlike SLM or EBM) that builds up a part in a layer-by-layer fashion, each layer being constituted of interlaced weld beads. It is the best suited Additive Manufacturing (AM) technique for large structures thanks to its high deposition rate (5 kg/h). The resulting material shows a rough surface, strong residual stress induced by its complex thermal history, a heterogeneous microstructure marked by the different weld passes as well as defects formed by gas pockets. Despite their rarity, pores are found to have a first-order influence on the fatigue life of machined specimens. The discrepancy in their size (> 100 μm) and position is responsible for a considerable scatter that makes classical fatigue tests ineffective. The aim of this study is to propose a novel approach to take into account the effect of rare WAAM-induced defects in high cycle fatigue. To achieve this, numerical porous structures are generated from the knowledge of the real pore population determined by tomography. Their fatigue performances are predicted via a two-scale probabilistic model identified on experimental self-heating results, on which pores have no influence. In that sense, the probabilistic model describes the behavior of a virtually healthy material. Then, by computing a database of representative pore cases, the whole bundle of Wöhler curves for each numerical porous structure is determined. Finally, the numerical fatigue scatter is in close agreement with experimental data, and it is shown that the ranking in pore criticality according to the model matches the fractography observations.

A multiscale DEM\u2013FEM coupled approach for the investigation of granules as crash-absorber in ship building

This paper covers a numerical analysis of a novel approach to increasing the crashworthiness of double hull ships. As proposed in Schöttelndreyer (Füllstoffe in der Konstruktion: ein Konzept zur Verstärkung vonSchiffsseitenhüllen, Technische Uni-versitt Hamburg, Hamburg, 2015), it involves the usage of granular materials in the cavity of the double hull ship. For the modeling of this problem, the discrete element method (DEM) is used for the granules while the finite element method is used for the ship’s structure. In order to account for the structural damage caused by collision, a gradient-enhanced ductile damage model is implemented. In addition to avoid locking, an enhanced strain-based formulation is used. For large-scale problems such as the one in the current study, modeling of all granules with realistic size can be computationally expensive. A two-scale model based on the work of Wellmann and Wriggers (Comput Methods Appl Mech Eng 205:46–58, 2012) is applied—and the region of significant localization is modeled with the DEM, while a continuum model is used for the other regions. The coupling of both discretization schemes is based on the Arlequin method. Numerical homogenization is used to estimate the material parameters of the continuum region with the granules. This involves the usage of meshless interpolation functions for the projection of particle displacement and stress onto a background mesh. Later, the volume-averaged stress and strain within the representative volume element is used to estimate the material parameters. At the end, the results from the combined numerical model are compared with the results from the experiments given in Woitzik and Düster (Ships Offshore Struct 1–12, 2020). This validates both the accuracy of the numerical model and the proposed idea of increasing the crashworthiness of double hull vessels with the granular materials.

A two-scale time dependent damage model for preferential gas flow in clayey rock materials

Understanding of the gas migration within the host rock is critical for the safety assessment of a deep geological repository for radioactive wastes. However, localized gas flow in clayey host rock materials is a complex behavior associated with dynamic and unstable network of dilatant pathways, which are accompanied by micro-cracking that indicates macroscopic tensile fractures. The subcritical crack propagation at the microscale may represent the mechanism of time dependent damage observed at the macroscale. A two-scale damage model is developed in this study to explicitly simulate the preferential gas flow in clayey rock materials. A homogenization method based on asymptotic developments is employed to deduce the macroscopic damage behavior coupled with the poroelastic system, which initiates from the periodically distributed microstructures with micro-cracks. A time dependent damage evolution law is constructed based on the microscopic phenomena and the corresponding intrinsic permeability model is proposed which implicitly accounts for the fracture opening induced permeability change. The local macroscopic response of the model is analyzed and validated against the experimentally measured direct tensile strength, which illustrates the homogenized elastic and permeability coefficients, and highlights the influence of several parameters, i.e., the initial damage, the microstructural size and the strain rate. Numerical examples are presented in order to illustrate the global macroscopic response, in which a pure mechanical test, i.e., uniaxial tension test is simulated and verified against the laboratory results, then the simulation of preferential gas flow is illustrated together with the comparison of experimental explanation. The numerical results showed that the proposed two-scale model can explicitly simulate the gas induced fracturing, in which the damage propagation and the dilatant gas pathways are well captured.

Two level simulation of Schmidt number effect on passive scalar transport in wall-bounded turbulent flows

In this study, the two-level simulation (TLS) modeling strategy developed earlier for momentum mixing is assessed in turbulent flows with passive scalar mixing and transport. Compared to the conventional algebraic and transport equation based subgrid-scale closures for large-eddy simulation (LES), where the effects of the unresolved small scales (SS) of motion on the large scales (LS) are modeled, in the TLS model, both large and small scales are explicitly evolved in a coupled two-scale decomposition method called TLS. Earlier studies demonstrated the ability of the SS model in the TLS to simulate high Reynolds number wall-bounded flows with a reasonable LS grid resolution. In the present study, this two-scale strategy is extended to model passive scalar mixing in turbulent channel flows. A comprehensive a priori assessment of the scalar SS model is performed using direct numerical simulation datasets for passive scalar transport in a fully developed turbulent channel flow at frictional Reynolds number Reτ=395 and at Schmidt numbers Sc= 1 and 4. This assessment is carried out to study the evolution of the large and the small scales of the scalar field in physical and spectral space and also to assess the SS prediction of quantities relevant for modeling of scalars such as scalar dissipation rate, counter-gradient transport, and backscatter. A TLS model is, then, used in a posteriori near-wall study using a hybrid TLS–LES approach to assess the two-scale scalar model.

A STUDY ON GAS DRAINAGE CONSIDERING COUPLING PROCESS OF FRACTURE-PORE MICROSTRUCTURE AND COAL DEFORMATION

Coal seam contains a large number of fractures, whose shape and structure of fissures are complicated, and they are the main channels for gas migration. Therefore, the quantitative analysis of the internal relationship between the microstructure and the macroscopic permeability is the key issue to increase the drainage volume. Some investigators discussed the permeability of coal seams based on the fractal theory, but the mechanisms of gas migration under the influence of fissure microstructures and coal deformation are still unclear. Moreover, most of the multi-process coupling analysis in the stage of Coal Bed Methane (CBM) mining was not taken into account. In this paper, the effects of fissure structures on the coal gas drainage rate in multi-field coupling are studied. Aiming at the mechanisms of gas drainage under the joint action of fracture structure, in-situ stress and adsorption deformation, we propose a two-scale multi-field coupling model, which takes into account the influences of micro fracture and pore structure. On this basis, we obtained the pressure evolution rule of coal seam pore system and fissure system and the distribution rule of coal seam displacement with drilling positions. Meanwhile, the effects of coal seam maximum fracture length, maximum pore diameter, the fractal dimension for fractures and fractal dimension for pores on coal seam extraction are discussed.

Computational homogenisation of phase-field fracture

In this manuscript, the computational homogenisation of phase-field fractures is addressed. To this end, a variationally consistent two-scale phase-field fracture framework is developed, which formulates the coupled momentum balance and phase-field evolution equations at the macro-scale as well as at the Representative Volume Element (RVE11The term ‘RVE’ is used interchangeably with sub-scale domain or microstructure, in this manuscript.) scale. The phase-field variable represent fractures at the RVE scale, however, at the macro-scale, it is treated as an auxiliary variable. The latter interpretation follows from the homogenisation of the phase-field through volume or a surface-average. For either homogenisation choices, the set of macro-scale and sub-scale equations, and the pertinent macro-homogeneity satisfying boundary conditions are established. As a special case, the concept of selective homogenisation is introduced, where the phase-field is chosen to live only in the RVE domain, thereby eliminating the macro-scale phase-field evolution equation. Numerical experiments demonstrate the local macro-scale material behaviour of the selective homogenisation based two-scale phase-field fracture model, while its non-selective counterpart yields a non-local macro-scale material behaviour.

Microstructure dependent strain localization during primary hot working of TA15 titanium alloy: Behavior and mechanism

Strain localization, a cross-scale issue degrading homogeneity and formability during hot working of TA15 titanium alloy, is complicated by lamellar microstructure for hierarchical feature, spatial morphology, and orientation relationship. In this work, by taking advantage of hat-shaped specimens to maintain deformation mode, the effect of heterogeneous lamellar microstructure on strain localization is accessible and visualized by cross-section comparison enlightened by cake cutting. Moreover, the evolution behavior and mechanism are also investigated by the two-scale crystal plasticity model. The mechanism of strain localization during hot working of lamellar microstructure is re-recognized as the structural softening governed by an abrupt phase interface weakening and continuous geometrical softening. The driving force of strain localization is the ‘collapse’ of the stress gradient between constituent phases intensified by the progression of compelled uniformed deformation. The key to microstructure-dependent strain localization is the influence of lamellar orientation on germination and colony interface on intensification. Colonies with lamellar orientation deviated 45 o from the direction of maximum shear stress are endowed with the intrinsic potential of strain localization. Large-size and strongly-orientated colony interfaces intensify strain localization. These findings provide new insights into microstructure-dependent strain localization supporting a more efficient design of processing routing to control localization powerfully.

Combining data assimilation and machine learning to infer unresolved scale parametrization.

In recent years, machine learning (ML) has been proposed to devise data-driven parametrizations of unresolved processes in dynamical numerical models. In most cases, the ML training leverages high-resolution simulations to provide a dense, noiseless target state. Our goal is to go beyond the use of high-resolution simulations and train ML-based parametrization using direct data, in the realistic scenario of noisy and sparse observations. The algorithm proposed in this work is a two-step process. First, data assimilation (DA) techniques are applied to estimate the full state of the system from a truncated model. The unresolved part of the truncated model is viewed as a model error in the DA system. In a second step, ML is used to emulate the unresolved part, a predictor of model error given the state of the system. Finally, the ML-based parametrization model is added to the physical core truncated model to produce a hybrid model. The DA component of the proposed method relies on an ensemble Kalman filter while the ML parametrization is represented by a neural network. The approach is applied to the two-scale Lorenz model and to MAOOAM, a reduced-order coupled ocean-atmosphere model. We show that in both cases, the hybrid model yields forecasts with better skill than the truncated model. Moreover, the attractor of the system is significantly better represented by the hybrid model than by the truncated model. This article is part of the theme issue 'Machine learning for weather and climate modelling'.

A two-scale finite element model for the fatigue design of large welded structures

Weld toes and weld roots of continuously welded structures subjected to cyclic loading are critical zones in terms of the fatigue resistance. The finite element method coupled with a fatigue criterion is commonly used to ensure the correct sizing and fatigue design of welded structures. However, weld geometries are often simplified or idealized to limit computational cost. In this work, a numerical two-scale approach is proposed in order to calculate a non-local multiaxial equivalent stress at the weld toe and the weld root from a global finite element shell model.The influence of the parameters of the proposed model on the stiffness behaviour is investigated for three welded structures and for different loading cases. A comparison in terms of stiffness with other models from the literature is also proposed. The results show that the stiffness behaviour is not affected by the parameters of the proposed approach and that it is the most robust model for the different geometries and loading cases studied. The variation in the non-local multiaxial equivalent stress as a function of the parameters of the proposed approach was also studied. The comparison with full solid finite element models makes it possible to define minimum values for the different parameters studied and validates the potential of the proposed approach for the fatigue design of welded structures.