Two-scale Model Research Articles

Modeling the Polarization Ratio in the Upper Microwave Band for Sea Clutter Analysis

The two-scale model (TSM) based on Bragg tilting is widely employed to model the normalized radar cross section (NRCS) from the sea surface in the microwave bands. It also leads to the compound distribution model when used to represent the distribution of intensity in high-resolution sea clutter. While the TSM satisfactorily describes the dynamical behavior of the polarized NRCS at moderate incidence angles for various sea states and radar frequencies, it fails to consistently describe the polarized sea clutter distribution in the upper microwave band. The hidden reason is an overestimation of the Bragg polarization ratio (PR) at larger local incidence angles. Following the concept of universal analytical scattering models and based on the experimental evidence of two independent X-band airborne data sets, we propose a simple correction to the facet NRCS which brings the PR in closer agreement to the observations and makes it analytically very close to the popular Thompson empirical model. This amended model preserves the simple structure of Bragg scattering; it admits one single extra parameter which makes a dynamical transition between the asymptotic Bragg and Kirchhoff regime depending on the sea state and the radar frequency. This allows to correctly describe the angular variations of the PR in the upper microwave regime (C-, X-, and Ku-bands) with minimal <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> information regarding the sea surface. Once this correction is incorporated in the TSM, a consistent modeling of the polarized sea clutter statistics can be obtained with the compound distribution model.

Improvement of remeshed Lagrangian methods for the simulation of dissolution processes at pore-scale

This article shows how to consistently and accurately manage the Lagrangian formulation of chemical reaction equations coupled with the superficial velocity formalism introduced in the late 80s by Quintard and Whitaker. Lagrangian methods prove very helpful in problems in which transport effects are strong or dominant, but they need to be periodically put back in a regular lattice, a process called remeshing. In the context of digital rock physics, we need to ensure positive concentrations and regularity to accurately handle stagnation point neighborhoods. These two conditions lead to the use of kernels resulting in extra-diffusion, which can be prohibitively high when the diffusion coefficient is small. This is the case especially for reactive porous media, and the phenomenon is reinforced in porous rock matrices due to Archie’s law. This article shows how to overcome this difficulty in the context of a two-scale porosity model applied in the Darcy-Brinkman-Stokes equations, and how to obtain simultaneous sign preservation, regularity and accurate diffusion, and apply it to dissolution processes at the pore scale of actual rocks.

Size-dependent two-scale topological design for maximizing structural fundamental eigenfrequency

A two-scale concurrent topology optimization method based on the couple stress theory is proposed for maximizing structural fundamental eigenfrequency. Because of the fact that the classical mechanics theory cannot reveal the size effect because of neglecting the influence of microstructure, the theory of couple stress including the microscopic properties of materials can be used to describe the size effect in deformations. On the foundation of the couple stress theory, the two-scale optimization model for finding optimal configurations of macrostructures and their periodic composite material microstructures is built. And the fundamental eigenfrequency of the macrostructure is maximized. The effective macroscopic couple stress constitutive constants of macrostructures are calculated by the representative volume element method. And a modified solid isotropic material with a penalization model is used to effectively avoid the localized mode. The optimization algorithm based on the bidirectional evolutionary structural optimization method is proposed. The optimal results of numerical examples show that the optimal topologies and natural frequencies obtained by the couple stress theory may differ significantly from those obtained by the typical Cauchy theory. It is obvious that couple stress theory can effectively describe the size effect in topology optimization.

Upscaled Model for Multicomponent Gas Transport in Porous Media Incorporating Slip Effect

A two-scale model for multicomponent gas transport in porous media is developed. At the pore-scale, Stefan–Maxwell formulation is used to describe the multi-gas transport together with the mass and momentum conservation equations, whereas on the solid/fluid interface, a slip velocity according to the Kramers–Kistemaker condition is taken into account. The pore-scale equations are then upscaled using a formal homogenization procedure. The macroscopic model shows that the total average velocity is modified by a slip velocity which depends mostly on the diffusive flux of the light gas in the mixture. Application to hydrogen transport in electrochemical devices such as fuel cell, electrochemical hydrogen purifier/compressor, in which considerable contrast of molar mass between the gases occurs, is carried out. As a result, the gas slip effect can modify considerably the gas transport behavior within the porous medium of the devices. This is an important result because the gas transport mechanisms play a crucial role in their efficiency.

Constitutive model of multiphase ceramics based on microcrack damage evolution

abstract The damage evolution of microcracks that dispersed in eutectic composite ceramics is an important factor leading to fracture failure. We propose a meso damage model (CMMC), which uses our proposed two-scale cellular model with microcracks, and fully considers the characteristics of the microstructure on the eutectic composite ceramics, thus providing a more objective structural modeling. In addition, there are two significant features existing in CMMC: One is to use the interaction direct derivative (IDD) estimate method to calculate the additional flexibility increment caused by a single open elliptical microcrack, which considers the spatial orientation of the microcrack. The other is that the additional flexibility tensor from the microcrack system is obtained by introducing probability density function and selecting appropriate microcrack growth criteria. Experiments on the related datasets show that CMMC achieves the consistent results compared with the experimental data. Furthermore, the effects of microcrack aspect ratio and microstructure parameters on the effective properties of multiphase ceramics were analyzed by using numerical method. • Damage evolution of microcracks in composite ceramics leading to fracture failure. • A meso damage model based on microcrack damage evolution is proposed. • Microcrack aspect ratio affects effective properties of multiphase ceramics. • Microstructure parameters influences effective properties of multiphase ceramics. • The additional flexibility tensor from the microcrack system is obtained.

A multiscale micromorphic model with strain rate relationship between MD simulations and macroscale experimental tests and dynamic heterogeneity for glassy polymers

This paper proposes a micromorphic simulation model with novel strain rate relationships between MD simulations and macroscale experiments at the ductile-brittle (DB) transition of glassy polymer and dynamic heterogeneity of deformation. Novelties of this paper are on the strain rate relationship between two different scale regions and investigations on the effects of molecular weight on the relationship. From molecular dynamics model having realistically entangled polymer networks and experiments, the strain rate relationships are derived and then realized through a two-scale multiscale continuum model based on the micromorphic theory. The investigation on the effects of molecular weights on the relationship yields that the relationship is maintained when the molecular weight is higher than the entanglement length. The relationship confirmed that the dynamic heterogeneity alleviates as the polymer enters into the plastic flow above the glass transition temperature and disappears above the melting temperature. These findings will be a basis for developing the constitutive relationship of glassy polymers through a multiscale approach that can incorporate the heterogeneous dynamics of molecules.

Bistatic Scattering From Anisotropic Rough Surfaces via a Closed-Form Two-Scale Model

Bistatic radars have been a topic of increasing interest in recent years, thanks to the introduction of new bistatic (and multistatic) configurations, including those based on the opportunistic exploitation of global navigation satellite systems (GNSSs). The research on bistatic electromagnetic scattering models plays an important role in the analysis of these systems, in their simulation, and in the prediction of their performance. The two-scale model (TSM) is a widely used approach for the computation of scattering from rough surfaces, since it is able to account for depolarization effects due to surface tilting. However, in its original formulation, it requires a computationally intensive numerical integration, in order to perform appropriate average over surface random slopes. To overcome this limitation, a closed-form polarimetric TSM (PTSM) was developed, which has been also recently extended to the case of anisotropic rough surfaces (A-PTSM), with a focus on the sea surface. The A-PTSM can be efficiently used to compute the backscattering from anisotropic rough surfaces and can support the development and analysis of monostatic radar missions. In order to extend its scope to the general case of bistatic and multistatic configurations, in this article, we extend the A-PTSM to the case of bistatic electromagnetic scattering, presenting the evaluation of all the elements of the bistatic polarimetric covariance matrix. Due to the relevance of circularly polarized signals in opportunistic GNSS reflectometry applications, both the linear and the circular polarization bases are considered. The behavior of the obtained elements is discussed, and simplified expressions of the elements of the covariance matrix are provided for the case of scattering within the incidence plane. Relevant numerical examples are provided and compared to those obtained by the more refined, but more computationally intensive, second-order small-slope approximation (SSA2) method. In the examples, we consider both a wind-driven sea surface and a tilled soil, and both L-band and X-band frequencies. However, the presented method can be used at all frequencies of interest for microwave remote sensing and for all observation geometries, except for near grazing incidence and/or scattering.

Modification of two-scale continuum model and numerical studies for carbonate matrix acidizing

Matrix acidizing is one of the most effective stimulation techniques to achieve industrial development of carbonate reservoirs. In this work, a two-scale continuum model is modified to consider mass conversion between liquid and solid phases. The results show that the mass conversion between liquid and solid phases is very important and nonnegligible. Compared to modified model, original model overestimates amount of acid required for core breakthrough. A series of numerical cases based on modified model are presented for sensitivity studies on acid injection conditions, core geometries, acid, and rock properties. The modified model can well capture optimum acid injection velocity and typical dissolution patterns observed in experimental studies. Pore volume to breakthrough (PVBT) is very sensitive to acid injection concentration and acid surface reaction rate. Increasing injected acid concentration substantially reduces the amount of solute required but acid injection mass does not change significantly. Strong acid can effectively reduce least pore volume to breakthrough (PVBT) but meanwhile optimum acid injection velocity obviously increases. The simulation results are compared with available experimental and numerical data and found to attain a consistent match.

Single-Stage Stimulation of Anhydrite-Rich Carbonate Rocks Using Chelating Agent: An Experimental and Modeling Investigation

SummaryCalcium sulfate (anhydrite) exists in most of the carbonate reservoirs, and its content can reach more than 20%. The high content of anhydrite affects the efficiency of the acidizing process because of the low solubility of calcium sulfate in different acids. Carbonate stimulation, matrix acidizing, or acid fracturing is carried out mostly using hydrochloric acid (HCl)-based fluids. The solubility of calcium sulfate in HCl or HCl-based fluids and organic acids is very low.In this study, a new formulation was developed to stimulate carbonate rocks with high anhydrite content. A formulation was developed to dissolve both anhydrite and carbonate at the same time. Anhydrite dissolution was achieved by converting anhydrite to calcite using the newly developed formulation. This treatment can be conducted in a single stage, and the formulation consists of a high pH chelating agent such as ethylenediaminetetraacetic acid (EDTA) in addition to potassium carbonate as a converter. Coreflooding experiments were conducted for carbonate rocks with varying anhydrite content. The effluent samples of the coreflooding experiments were analyzed for cations and anions concentration using inductively coupled (IC) and inductively coupled plasma (ICP) measurements. Computed tomography scans were conducted to show the wormholes generated in the samples. Rock mineralogical analysis using X-ray diffraction, thin section, and the QEMSCAN® automated mineralogy solution (FEI Company, Hillsboro, Oregon, USA) was conducted as well. The reaction kinetics were investigated using a rotating disk apparatus (RDA) using a disk that consists of almost 24% anhydrite and 71% dolomite. Also, a two-scale continuum model was built to simulate the reaction between the new formulation and carbonate rocks. The model could capture the experimental outcomes such as the number of pore volumes (PV) to break through and the shape of the wormholes. This provided confidence in the reaction parameters obtained because the model could reproduce the experimental outcomes.Coreflooding experiments showed that the new formulations were very effective in stimulating carbonate rocks with high anhydrite content in a single stage. The effluent analysis showed high sulfate concentration that indicates the dissolution of anhydrite compared to conventional acid treatments. Reaction kinetics results showed that the new formulation increased the reaction rate with anhydrite through a mass transfer reaction regime. The model reproduced the wormholing behavior obtained from the experiments, indicating that the branched wormhole was due to the anhydrite conversion.

A Stochastic Two-Scale Model for Rarefied Gas Flow in Highly Heterogeneous Porous Media

This paper presents the development of a model enabling the analysis of rarefied gas flow through highly heterogeneous porous media. To capture the characteristics associated with the global- and the local-scale topology of the permeable phase in a typical porous medium, the heterogeneous multi-scale method, which is a flexible framework for constructing two-scale models, was employed. The rapid spatial variations associated with the local-scale topology are accounted for stochastically, by treating the permeability of different local-scale domains as a random variable. The results obtained with the present model show that an increase in the spatial variability in the heterogeneous topology of the porous medium significantly reduces the relevance of rarefaction effects. This clearly shows the necessity of considering a realistic description of the pore topology and questions the applicability of the results obtained for topologies exhibiting regular pore patterns. Although the present model is developed to study low Knudsen number flows, i.e. the slip-flow regime, the same development procedure could be readily adapted for other regimes as well.

Software Tool for Soil Surface Parameters Retrieval from Fully Polarimetric Remotely Sensed SAR Data.

The retrieval of soil surface parameters, in particular soil moisture and roughness, based on Synthetic Aperture Radar (SAR) data, has been the subject of a large number of studies, of which results are available in the scientific literature. However, although refined methods based on theoretical/analytical scattering models have been proposed and successfully applied in experimental studies, at the operative level very simple, empirical models with a number of adjustable parameters are usually employed. One of the reasons for this situation is that retrieval methods based on analytical scattering models are not easy to implement and to be employed by non-expert users. Related to this, commercially and freely available software tools for the processing of SAR data, although including routines for basic manipulation of polarimetric SAR data (e.g., coherency and covariance matrix calculation, Pauli decomposition, etc.), do not implement easy-to-use methods for surface parameter retrieval. In order to try to fill this gap, in this paper we present a user-friendly computer program for the retrieval of soil surface parameters from Polarimetric Synthetic Aperture Radar (PolSAR) imageries. The program evaluates soil permittivity, soil moisture and soil roughness based on the theoretical predictions of the electromagnetic scattering provided by the Polarimetric Two-Scale Model (PTSM) and the Polarimetric Two-Scale Two-Component Model (PTSTCM). In particular, nine different retrieval methodologies, whose applicability depends on both the used polarimetric data (dual- or full-pol) and the characteristics of the observed scene (e.g., on its topography and on its vegetation cover), as well as their implementation in the Interactive Data Language (IDL) platform, are discussed. One specific example from Germany’s Demmin test-site is presented in detail, in order to provide a first guide to the use of the tool. Obtained retrieval results are in agreement with what was expected according to the available literature.

Hierarchy of beam models for lattice core sandwich structures

A discrete-to-continuum transformation to model 2-D discrete lattices as energetically equivalent 1-D continuum beams is developed. The study is initiated in a classical setting but results in a non-classical two-scale micropolar beam model via a novel link within a unit cell between the second-order macrorotation-gradient and the micropolar antisymmetric shear deformation. The shear deformable micropolar beam is reduced to a couple-stress and two classical lattice beam models by successive approximations. The stiffness parameters for all models are given by the micropolar constitutive matrix. The four models are compared by studying stretching- and bending-dominated lattice core sandwich beams under various loads and boundary conditions. A classical 4th-order Timoshenko beam is an apt first choice for stretching-dominated beams, whereas the 6th-order micropolar model works for bending-dominated beams as well. The 6th-order couple-stress beam is often too stiff near point loads and boundaries. It is shown that the 1-D micropolar model leads to the exact 2-D lattice response in the absence of boundary effects even when the length of the 1-D beam (macrostructure) equals that of the 2-D unit cell (microstructure), that is, when L=l.

Development of a two-scale damage model for incorporating the fatigue crack nucleation from surface inclusions

In high cycle fatigue, damage and plasticity occur at the microscale, a scale smaller than the representative volume element (mesoscale). Therefore, the two-scale damage model is used for fatigue life assessment. In this model, a spherical inclusion is assumed to be fully embedded in the matrix. However, experimental observations show that the microplasticity and microcracks nucleate more easily in grains on the free surface rather than in the bulk. Hence, the aim of this work is to extend the two-scale damage model to take into account the nucleation of fatigue cracks on free surfaces. In order to do this, a surface scale transition law is derived to link the micro and mesoscales, considering a hemispherical damaged inclusion at the surface of an elasto-plastic matrix. Furthermore, a numerical scheme is proposed by developing an in-house Python code in conjunction with Abaqus software to compute the damage. The material parameters required in model are identified using monotonic and fatigue test results of smooth specimens. As a validation, for the butt-welded specimens subjected to fatigue loading, the obtained results of the proposed surface law are compared with the results of the pre-existing non-surface laws and the experimental fatigue data. It is confirmed that the use of surface localization law leads to a noticeable reduction in the number of cycles to crack initiation of about 25%.

Two-scale thermomechanical damage model for dynamic shear failure in brittle solids

A coupled thermomechanical damage approach for dynamic shear failure in brittle solids is proposed in the present contribution. The model is constructed by asymptotic homogenization from microstructures with dynamically evolving microcracks, in mode II, with unilateral contact and friction conditions on their lips. Crack-tip and frictional heating effects assumed at the small scale give rise to distributed heat sources in the macroscopic temperature equation and specific dissipation terms in the upscaled damage law. The analysis of the effective thermomechanical response of the model reveals strain rate and size effects and the influence of friction and growth of microcracks on the macroscopic thermal evolutions.

Multiscale modeling of tensile fracture in fiber reinforced composites

Intralaminar fiber-matrix fracture, also known as splitting, is encountered frequently during the tensile failure of fiber reinforced polymer matrix composites (FRPMCs). The prediction of this failure mode and its influence on a macroscopically homogenized model brings associated challenges in numerical modeling. An efficient two-scale model is presented, based on the micromechanics 2-concentric cylinder (2CYL) [1] analytical method for pre-peak nonlinearity, interfaced with the 3D orthotropic smeared crack approach (SCA) [2,3] for post-peak softening behavior. A homogenized ply is modeled using standard 3D finite elements, such that a matrix crack can be captured explicitly with respect to the orientation of the ply angle. The capability of the proposed method is demonstrated by predicting the grip-to-grip longitudinal split in a 00 ply, mixed-mode matrix cracking in 450 ply, the transverse matrix cracking in 900 ply and the failure of laminate containing these plies. The predictions of the proposed method are found to be in very good agreement with corresponding experimental data [4, 5].

Prediction of residual stresses of second kind in deep drawing using an incremental two-scale material model

ABSTRACT For modern engineering applications, the prediction of residual stresses is a starting-point for the subsequent optimisation of the design with regard to the process induced stresses on the macro and the grain scale. In this paper, a new approach for the numerically efficient prediction of phase-specific residual stresses (residual stresses of second kind) in two-phase materials is presented. The proposed model allows for a two-scale simulation of complex forming processes, solely based on the phase-specific constitutive equations. This enables the calculation of the average stresses and strains in each phase during the entire process and the prediction of the macroscopic material response. The numerical results are compared to experimental diffraction-based data of a bending bar and a deep drawn cup and are aligned better compared to existing models.

Dynamic Damage Law with Fragmentation Length: Strain-Rate Sensitivity of the Tensile Response

A damage law for dynamic failure in brittle solids is proposed in the present note. The new evolution equation is obtained by incorporating the Grady–Glenn–Chudnovsky average fragment size expression as the microstructural length of a two-scale damage model. In this construction, the internal length of the model explicitly depends on the local strain rate and the material parameters. The capacity of the new approach to account for the strain-rate sensitivity of the tensile strength is illustrated by comparison of the predicted material and structural responses with experimental data for concrete and ceramic materials.

Research progresses in formation mechanism of complex fracture network for unconventional reservoir

The practice of oilfield development promotes the development theory, and the breakthrough of development theory brings about the revolution of development technology. Using horizontal wells and volume fracturing technology, volume fracturing has been applied in unconventional reservoir, and the formation mechanism of complex fracture network will bring innovation to development technology. At present, the research on the formation mechanism mainly focuses on theoretical research, experimental research, and numerical simulation. Theoretically, the mechanical model of the intersection of HF and NF is established, and some judgment criteria are put forward. Activation of NF by HF is a necessary condition for the formation of complex fracture networks; the activation conditions of NF are verified by true triaxial experiment in terms of horizontal stress difference, approximation angle, and fracturing fluid viscosity; finite element method, finite difference method, boundary element method, and discontinuous displacement method have their own advantages and disadvantages on simulation. Using digital speckle correlation technique, the authors explained the mechanism of shear action on the formation of fracture network, established a macro-meso two-scale non-uniform damage constitutive model, and formed a computing method for dynamic expansion of fractured network in tight reservoir. A field well example is calculated using this method.

Comprehensive Vector Radiative Transfer Model for Estimating Sea Surface Salinity From L-Band Microwave Radiometry

Sea surface salinity (SSS) retrieval from satellite-based microwave radiometer is hampered by uncertainties due to atmospheric and surface scattering contributions. This study presents a comprehensive vector radiative transfer model (VRTM) for estimation of brightness temperature (TB) (for SSS retrieval) from L-band radiometry based on a matrix-operator method. It includes an efficient two-scale model (TSM) that combines a geometrical optics (GO) model and a small-perturbation model (SPM) for computing both the small- and large-scale scattering components of the sea surface. Moreover, it considers the influence of rain effects on TB by including the radiation extinction term (scattering and attenuation). The simulation results using the VRTM were validated with those obtained from the RT4 model for flat sea surface conditions and the RTTOV model for rough sea surface conditions. The relative difference in the estimated TB among the models was small (<; 1%) for low wind speeds (<; 1 m/s) and increased up to 3% for high wind speeds and observation angles. Simulations on the influence of wind speed on TB with various parameterizations were further examined. Compared with SMOS-MIRAS and Aquarius measurements, the VRTM simulations agreed well with satellite measurements for both vertically and horizontally polarized TBs with biases of less than 2.2 K for observation angles from 20° to 65°. The binned TBs showed even better results, with a standard deviation of less than 1.45 K and an absolute mean error of less than 1.2 K.

Effects of action at a distance in water

Dipole–dipole interaction between molecules of hydrogen-bonding polar liquids (HBPLs), which has a collective and long-range nature, determines the basic large-scale properties of such liquids. We present a two-scale phenomenological vector model of polar liquids (VMPLs), wherein the liquid is described by a polarization vector. The simplest version of this model satisfactorily reproduces the well-known properties of HBPLs and interaction between macroscopic objects in a liquid. The possible existence of a ferroelectric phase transition (FPT) in supercooled liquid water is discussed. Near the FPT, fluctuations of the polarization vector increase, which may be the cause of the so-called ‘anomalous’ properties of water. We propose a quantitative classification of body surfaces based on the properties of their wettability by polar liquids. The ordering of dipoles of molecules located in the near-surface layers of HBPLs and phase transitions in these layers are discussed. The proposed model enables a significant reduction in computer time in numerical simulations of systems that contain a large number of water molecules.