Discrete-continuum Model Research Articles

On differentiability of the membrane-mediated mechanical interaction energy of discrete–continuum membrane–particle models

We consider a discrete–continuum model of a biomembrane with embedded particles. While the membrane is represented by a continuous surface, embedded particles are described by rigid discrete objects which are free to move and rotate in lateral direction. For the membrane we consider a linearized Canham–Helfrich energy functional and height and slope boundary conditions imposed on the particle boundaries resulting in a coupled minimization problem for the membrane shape and particle positions. When considering the energetically optimal membrane shape for each particle position we obtain a reduced energy functional that models the implicitly given interaction potential for the membrane-mediated mechanical particle–particle interactions. We show that this interaction potential is differentiable with respect to the particle positions and orientations. Furthermore, we derive a fully practical representation of the derivative only in terms of well defined derivatives of the membrane. This opens the door for the application of minimization algorithms for the computation of minimizers of the coupled system and for further investigation of the interaction potential of membrane-mediated mechanical particle–particle interaction. The results are illustrated with numerical examples comparing the explicit derivative formula with difference quotient approximations. We furthermore demonstrate the application of the derived formula to implement a gradient flow for the approximation of optimal particle configurations.

Dynamic analysis and non-standard continualization of a Timoshenko beam lattice

In this paper, a Timoshenko beam lattice, made up of a chain of masses and straight segments, is proposed, considering bending and shear deformation by means of linear rotational and transverse springs, respectively. Different standard and non-standard continualization methods are applied to it, highlighting here for the first time the suitability of taking the coupled discrete governing equations as a starting point for deriving new continuum models. Several novel low order non-classical continuum models are obtained, with the aim of reliably capturing size-effects and reflecting the dispersive behaviour of the discrete system. Low order governing equations prevents the need for extra boundary conditions when finite (bounded) solids are treated. An extensive analysis of the transition frequency, which initiates the shear propagation spectrum, has been carried out, examining its influence for the discrete and non-standard continuum models. The natural frequencies of a finite solid with two different boundary conditions are obtained through an edge treatment applied here for the first time to this kind of lattices, thus making it possible to solve the clamped-free edges configuration. The reliability of these approaches is evaluated by comparing their dynamic behaviours with that of the discrete system (taken as a reference), through both dispersion and vibration analyses, some of the new proposed continuum models successfully capturing the behaviour of the discrete one, even for high wavenumbers. Moreover, the appearance of physical inconsistencies is examined.

Investigation on the behavior of hybrid pile foundation and its surrounding soil during cyclic lateral loading

With increasing demand and requirements for offshore wind turbines (OWTs) construction, research on the hybrid pile foundation is gaining increasing popularity. In this paper, a series of 1-g model tests are conducted to investigate the lateral behavior of monopile and hybrid pile foundations under one-way cyclic loading. The influence of pile stiffness, pile diameter for the hybrid foundation and hybrid foundation type on the lateral behavior of the structure is analyzed. To visualize particle movements during the loading process, several colored sand bands are arranged at ground surface around the structure. Continuous monitoring of sand surface during the loading process and excavation of sand sample reveal different particle migration patterns around different structures. Experimental results verify the superiority of hybrid foundation in resisting lateral loads and reducing accumulated lateral displacement. Based on the coupled discrete-continuum modelling of cyclic load tests, the deformation pattern of monopile and hybrid structure and the particle displacement contour at different sections are provided. The numerical results further point out the potential influence of the soil plug on the lateral capacity of the hybrid pile foundation, which could be the focus of future research.

Modelling transient discharge into deep-buried tunnels in karst area based on a coupled discrete-continuum model

Karst aquifers are characteristic of complex hydrogeological conditions and highly developed void structures, often causing serious water and mud inrush, karst collapse and other geological disasters during construction of deep-buried tunnels. Accurate prediction of groundwater discharge into tunnels has become a key issue for tunnel engineering in karst formations. Motivated by safety assessment of tunnel construction in the Jiayan Water Diversion Project located in Guizhou Province, China, we perform groundwater flow simulations to predict transient discharge into the long, deep-buried tunnels. Two numerical models, an equivalent porous medium model and a coupled discrete-continuum model, are used. In the coupled discrete-continuum model, both the tunnels and karst conduits are treated as discrete channels, which are coupled with the surrounding matrix through water exchange. Based on hydrogeological data and site characterization, the model parameters are calibrated by using the PEST inversion algorithm. It is shown that numerical predictions of tunnel discharge during construction agree well with the measurements, indicating reliability of the numerical models. Comparison between the two models shows that the coupled discrete-continuum model can more accurately reflect the changes in the groundwater level and can better estimate the sources of groundwater discharge into the tunnels during the construction. This work suggests that the coupled discrete-continuum model is advantageous over the equivalent porous medium model as the former is a better representation of the karst system. It is demonstrated that the simulation methodology provides a reliable method for assessment of water discharge into tunnels and the optimization of construction scheme. The results are of practical significance for safe construction of tunnels in karst areas.

Comprehensive Evaluation of Hydrogeological Impact of Tunnel Construction in Karst Aquifers by 3D Numerical Simulations and Water Balance Models

Underground engineering activities such as tunnelling are commonly involved in large-scale hydraulic and hydropower projects. In karst regions, tunnel engineering often encounters geological disasters such as water inrush and ground collapse. Water discharge during tunnel construction can also strongly disturb the karst groundwater systems. The above issues require a comprehensive evaluation of the hydrogeological impact of tunnel engineering. In this study, we adopted both a 3D numerical approach based on coupled discrete-continuum models and a modified lumped-parameter water balance model to comprehensively evaluate the hydrogeological impact of tunnel engineering during implementation of the Jiayan Water Diversion Project in Guizhou Province. In the numerical approach, a high-resolution 3D model was built to quantitatively evaluate the spatial and temporal characteristics of the tunnel construction impact on a karst groundwater system. It was shown that during construction, the groundwater drainage through the tunnels accounted for about 11% of the total groundwater discharge in the studied area. The total discharge through karst conduits decreases by about 30% in response to the tunnel construction. Meanwhile, the lumped-parameter water balance model provided an avenue for fast evaluation the impact of tunnel construction on the discharge from the aquifer system. The discharge through the spring and tunnels evaluated by water balance model are in good agreement with the results of the numerical model. On a whole, our work demonstrates the use of different modelling approaches to study karst groundwater flow affected by tunnel engineering that can be of values for different stages of an engineering project.

Multi-Ion Bridged Pathway of N-Oxides to 1,3-Dipole Dilithium Oxide Complexes.

Roussi's landmark work on the generation of 1,3-dipoles from tertiary amine N-oxides has not reached its full potential since its underlying mechanism is neither well explored nor understood. Two competing mechanisms were previously proposed to explain the transformation involving either an iminium ion or a diradical intermediate. Our investigation has revealed an alternative mechanistic pathway that explains experimental results and provides significant insights to guide the creation of new N-oxide reagents beyond tertiary alkylamines for direct synthetic transformations. Truhlar's M06-2x functional and Møller-Plesset second-order perturbation theory with Dunning's [jul,aug]-cc-pv[D,T]z basis sets and discrete-continuum solvation models were employed to determine activation enthalpies and structures. During these mechanistic explorations, we discovered a unique multi-ion bridged pathway resulting from the rate-determining step, which was energetically more favorable than other alternate mechanisms. This newly proposed mechanism contains no electrophilic intermediates, strengthening the reaction potential by broadening the reagent scope and limiting the possible side reactions. This thoroughly defined general mechanism supports a more direct route for improving the use of N-oxides in generating azomethine ylide-dilithium oxide complexes with expanded functional group tolerance and breadth of chemistry.

Experimental and numerical investigations on transient multiscale bubble behaviors in CuSO4 aqueous solution electrolysis cell

The present investigation demonstrates the transient multiscale bubble behaviors including the bubble homogeneous nucleation, continuous growth, disordered collision and coalescence, and disturbed detachment under the anode in the CuSO4 aqueous solution electrolysis process by experimental and numerical methods. The combined approach of the volume of fluid (VOF) method and discrete phase model (DPM) is applied to solve the macro bubbles’ surfaces and track the micro bubbles’ trajectories, respectively. The link between the macro continuous and micro dispersed bubbles is achieved by the discrete-continuum transition model (DCTM). The dumbbell and the gourd morphologies are revealed in both experiment and numerical simulation during the binary bubble collision and coalescence. The predicted maximum bubble thickness and maximum bubble coverage present reasonable matches to the experimental data. The results show that the large coalescing bubbles can extend the bubble collision period and increase the fluctuation amplitude of the bubble interface. Increasing the current will increase the bubble collision velocity and shorten the contact time before bubble coalescence, moreover, the recovery time of the oscillating bubble from coalescence to the stable state is also prolonged. Increasing inclined angles of the anode can increase the bubble velocity and decrease the bubble diameter.

Breakthrough curves of dye tracing tests in karst aquifers: Review of effective parameters based on synthetic modeling and field data

Breakthrough curves (BTCs) are known as adequate evidence to characterize karst aquifers which mainly specified by dual flow systems (matrix and conduit) as well as some degree of connection between dual media. Conduit Flow Process version 2 (CFPv2) is a discrete conduit continuum model applying which it is possible to simulate such characteristics. In this research, first CFPv2 was employed to evaluate the effect of several factors including hydraulic conductivity, exchange flow, tracing distance, hydraulic gradient, and conduit diameter and roughness on BTCs in a synthetic karst aquifer. Then, 253 BTCs from 24 dye tracing tests, conducted in karstic aquifers of Iran, were analyzed to evaluate the effect of hydraulic gradient and tracing distance on the BTCs with taking into account the effect of medium type (matrix or conduit) and karst development. These tests were conducted under either natural or elevated man-made hydraulic gradient conditions. The results revealed that peak concentration decreased with enlarging conduit diameter because of considerable flow rate. Likewise, it decreases over longer distances specifically in the elevated man-made hydraulic conditions. However, with increasing hydraulic gradient, peak concentration and its timing have upward and downward trends, respectively. When BTCs start to demonstrate tailing and become flattened, it could be an indication of low matrix hydraulic conductivity or lower hydraulic gradient in conduit as a result of less exchange flow from the matrix into conduits. Based on this study a new intermixture of flow and solute transport parameters was engineered to classify karst development. High peak concentration, short peak time, high velocity particularly over long distances and under the natural hydraulic gradient conditions and feasible conduit flow regime whether laminar or turbulent suggested as indication of well-developed karstic system.

How acid can become a dihydrogen complex in water? A DFT study

To accurate know the acidity of a dihydrogen molecule coordinated to a transition metal ion in water medium is an issue of interest in many areas, from electrochemistry to enzymes and catalysis. However, experimental determination of this magnitude is challenging, and very few values have been reported. In this article we describe a computational protocol, based on DFT calculations and employing a discrete-continuum solvent representation, to estimate pKawater of transition metal dihydrogen complexes. In this approach the number of solvent molecules explicitly included in the calculations is determined by the convergence with the solvation Gibbs energy of the proton in the solvent. The approach has been initially validated with experimental data in tetrahydrofuran (THF) solvent. Using (THF)3 clusters a mean absolute deviation from experiments of only 1.4 pKa unit is achieved. In water the convergence is reached with (H2O)10 clusters. Using them in a discrete-continuum model, the pKawater of twelve dihydrogen complexes experimentally characterized in water have been computed. pKawater values span a wide range, from 23 to -4, illustrating how coordination to a transition metal modifies the dihydrogen acidity. Decomposition of the ∆G of the acid-base equilibrium in two contributions, one intrinsic to the complex and another one accounting for solvent effects enables a deeper analysis of the dihydrogen acidities.

Indentation resistance of brittle auxetic structures: Combining discrete representation and continuum model

This work investigates the indentation resistance of the auxetic/non-auxetic honeycomb ceramics layer by combining a discrete numerical representation and a continuum analytical model. The indentation depth of honeycombs with varying cell-wall angle and cell-wall thickness are measured. The indentation depth of auxetic honeycombs with negative Poisson’s ratio (NPR) is smaller than that of conventional ones in the premise of the same cell-wall thickness. But on the condition that the honeycombs possess the same relative density, there exists an inverse tendency. The corresponding mechanisms responsible for those trends are revealed. The stress intensity factor (SIF) is defined to characterize the stress singularity at the corner of the punch. The influence of NPR on the magnitudes of SIF is evaluated. A punch toughness concept is defined by a novel multi-scale method as the failure criterion of brittle honeycomb ceramics in indentation tests. The effects of NPR on the punch toughness of honeycombs are discussed. And the empirical formulas of punch toughness varying with the cell-wall thickness and the relative density are summarized. Finally, the NPR effects on the collinear punches problem are detected; the results show that the auxetic performance can significantly weaken the interaction between the punches.

Hybrid discrete-continuum modeling of shear localization in granular media

Shear localization is a frequent feature of granular materials. While the discrete element method can properly simulate such a phenomenon as long as the grain representation is accurate, it is computationally intractable when there are a large number of grains. The continuum-based finite element method is computationally tractable, yet struggles to capture many grain-scale effects, e.g., shear band thickness, because of mesh dependence, unless the constitutive model has a length scale. We propose a hybrid discrete-continuum technique that combines the speed of the continuum method with the grain-scale accuracy of the discrete method. In the case of shear localization problems, we start the simulation using the continuum-based material point method. As the simulation evolves, we monitor an adaptation oracle to identify the onset of shear bands and faithfully enrich the macroscopic continuum shear bands into the microscopic-scale grains using the discrete element method. Our algorithm then simulates the shear band region with the discrete method while continuing to simulate the rest of the domain with the continuum method so that the computational cost remains significantly cheaper than a purely discrete solution. We validate our technique in planar shear, triaxial compression, and plate indentation tests for both dry and cohesive granular media. Our method is as accurate as a purely discrete simulation but over 100 times faster than a discrete simulation that would require tens of millions of grains.

Temperature-dependent oxidation of BSCAPE molecule in methanol medium

Temperature-dependent solvation free energy and oxidation by free energy of ionization of 2-Phenylethyl (2E)-3-(1-benzenesulfonyl-4,5-dihydroxyphenyl) acrylate (BSCAPE) in methanol medium are the concerns of the present work. This molecule is a relevant phenolic acid enclosing multiple bioactivities. The explicit, implicit and discrete-continuum models of solvation were used. The methanol molecules were coordinated to this acid to form cluster complexes. The dual method M06-2X/6–31++G(d,p)//B3LYP/6-31G(d) was employed along with basis set superposition error correction. The results show that, the free energy of coordination and solvation are distant. Both quantities increase with temperature. From discrete-continuum treatment, there is non-spontaneity of solvation process, while coordination yielded spontaneity and non-spontaneity at cold and hot room temperatures, respectively. The ionization potential in gas phase, decreases with temperature. All the solvation models yielded lower ionization potential than that of gas phase. Thus, it follows that, the increase of temperature and methanol medium favours the oxidation of BSCAPE. Consequently, this favours its metabolism processes.

A predictive discrete-continuum multiscale model of plasticity with quantified uncertainty

Multiscale models of materials, consisting of upscaling discrete simulations to continuum models, are unique in their capability to simulate complex materials behavior. The fundamental limitation in multiscale models is the presence of uncertainty in the computational predictions delivered by them. In this work, a sequential multiscale model has been developed, incorporating discrete dislocation dynamics (DDD) simulations and a strain gradient plasticity (SGP) model to predict the size effect in plastic deformations of metallic micro-pillars. The DDD simulations include uniaxial compression of micro-pillars with different sizes and over a wide range of initial dislocation densities and spatial distributions of dislocations. An SGP model is employed at the continuum level that accounts for the size-dependency of flow stress and hardening rate. Sequences of uncertainty analyses have been performed to assess the predictive capability of the multiscale model. The variance-based global sensitivity analysis determines the effect of parameter uncertainty on the SGP model prediction. The multiscale model is then constructed by calibrating the continuum model using the data furnished by the DDD simulations. A Bayesian calibration method is implemented to quantify the uncertainty due to microstructural randomness in discrete dislocation simulations (density and spatial distribution of dislocations) on the macroscopic continuum model prediction (size effect in plastic deformation). The outcomes of this study indicate that the discrete-continuum multiscale model can accurately simulate the plastic deformation of micro-pillars, despite the significant uncertainty in the DDD results. Additionally, depending on the macroscopic features represented by the DDD simulations, the SGP model can reliably predict the size effect in plasticity responses of the micropillars with below 10% of error.

Reproducing laboratory earthquakes with a discrete-continuum model

We present a novel numerical model allowing to take the best part of continuum-based and discrete modelling in a single framework. This model is applied to the reproduction of laboratory earthquakes in a high-pressure triaxial cell. It allows to represent most of the relevant phenomena at stake, including elastic stress build-up during loading, fast and slow sliding events, seismic waves emission in the surrounding elastic medium and evolution of fault gouge on the sliding interface. We review here some illustrative results obtained with this model and propose further research avenues.

ПРОЕКТУВАННЯ ОПТИМАЛЬНИХ ТРАЄКТОРІЙ СВЕРДЛОВИН

The problem of optimizing the trajectories of deep curved oil and gas wells, in which the total curvature of the well and its length is minimized, is discussed. For the first time, a discrete-continuum model of the well geometry was proposed, based on the method of projection of a gradient on the hyperplane of linearized constraints, a method was developed for minimizing the corresponding target functionals, which would reduce the risks of emergency drilling situations. An algorithm for reducing the problem of nonlinear optimal control to the problem of nonlinear programming is shown. Such a transition is achieved by approximating the well trajectories with a system of cubic splines, analytically integrating differential equations in separate sections of the trajectory, and further applying the methods of nonlinear programming theory. The considered approach is more algorithmic and allows solving problems of well trajectory optimization under more complex constraints. KEYWORDS: WELL TRACKING, OPTIMAL CONTROL PROBLEM, OBJECTIVE FUNCTIONS, NONLINEAR PROGRAMMING.

Numerical model development for investigating hydrologic pathways in shallow fluviokarst

Karst pathways and fluvial pathways control hydrology in shallow fluviokarst basins, and numerical modelling of fluviokarst is seldom reported. In this study, we developed a combined discrete-continuum fluviokarst numerical model by simulating surface river routing, in-stream swallet sources and sinks, epikarst storage and dynamic transfer, matrix bedrock interactions, and closed conduit phreatic flow. We applied the model to the Cane Run-Royal Spring basin in Kentucky, USA. Model evaluation indicated that spring discharge alone inadequately constrained model pathways and uncertainty. Instead multi-objective calibration, integrating riverine discharge and well-head data from multiple locations, assisted in identifying sensitive parameters (p < 0.05). Multi-objectives improved representation of stream-cave connectivity and limited prior knowledge biases of the system but was computationally expensive with 168 h required on a high-performance cluster. Results provided evidence for a mature fluviokarst basin with well-defined fracture-conduit network and phreatic aquifer. Residence times of karst pathways vary by five orders of magnitude, ranging from less than one hour in vertical swallets, 12.7 h in longitudinal conduits, 12.7 days in the vadose zone and epikarst, and 142.7 days in the bedrock matrix. Results suggest arrival of source waters to the subsurface systems is disconnected in time from the springflow response. Model simulations show a dimensionless vertical to longitudinal conveyance ratio helps predict swallets linking the fluvial and karst systems. Transferability of the developed model, and other karst models, is discussed relative to availability of information for karst basins.

Evaluation of hydrogeological impact of tunnel engineering in a karst aquifer by coupled discrete-continuum numerical simulations

Anthropogenic disturbance of karst groundwater systems by large-scale underground engineering activities is an important topic but is difficult to address due to the compound influence of the heterogeneity and construction progression. In this study, we adopt the coupled discrete-continuum modeling approach where both the tunnels and karst conduits are treated as discrete channels. Based on extensive site characterization, groundwater observations, and tunnel discharge data, we establish a three-dimensional coupled model for a karst aquifer where deeply-buried, long tunnels are being constructed. We perform high-resolution simulations to quantitatively evaluate the spatial and temporal characteristics of the tunnel construction impact on a karst groundwater system. The simulation results satisfactorily reproduce the tunnel discharge time-series. We show that the excavation and lining schedule is an important factor to consider when evaluating the hydrogeological impact. During tunnel construction, the groundwater drainage through the tunnels (5.3 × 107 m3) accounts for a significant part of the total groundwater discharge (~11%) in the studied area. The continuous water influx to the tunnels results in widespread drawdown with an area of impact (with drawdown >5 m) covering 64.0% of the entire study area after four years of construction. The total discharge through karst conduits decreases by about 30% in response to the tunnel construction. Our simulation results predict that after completion of lining the groundwater system will slowly approach a groundwater level 5–10 m below the undisturbed level. This work demonstrates the use of coupled discrete-continuum models to study regional groundwater flow in karst aquifers affected by tunnel engineering activities and also provides a guidance for the impact assessment of other underground engineering projects in mining and transportation industries. The simulation methodology may be exploited for optimization of the construction plans to minimize hydrogeological impact and to ensure construction safety.

Raman Spectra of Glycine and Their Modeling in Terms of the Discrete–Continuum Model of Their Water Solvation Shell

Raman spectra of glycine in a crystalline form and in a water environment have been obtained. B3LYP/6-311++G(3df,2p) quantum-chemical calculations have shown that an adequate description of the Raman spectra of glycine is achieved using the discrete–continuum model of its water solvation shell. A model is proposed for the stabilization of the zwitterionic state of glycine involving the inclusion of a single water molecule that is hydrogen-bonded with the –СOO– and –NH $$_{3}^{ + }$$ groups. Calculations of the Raman spectra of glycine complexes Gly + nH2O (where n = 1–3), as well as of a glycine dimer, yield good agreement with experiment.

Optimal Design of Bus Stop Locations Integrating Continuum Approximation and Discrete Models

Although transit stop location problem has been extensively studied, the two main categories of modeling methodologies, i.e., discrete models and continuum approximation (CA) ones, seem have little intersection. Both have strengths and weaknesses, respectively. This study intends to integrate them by taking the advantage of CA models’ parsimonious property and discrete models’ fine consideration of practical conditions. In doing so, we first employ the state-of-the-art CA models to yield the optimal design, which serves as the input to the next discrete model. Then, the stop location problem is formulated into a multivariable nonlinear minimization problem with a given number of stop location variables and location constraint. The interior-point algorithm is presented to find the optimal design that is ready for implementation. In numerical studies, the proposed model is applied to a variety of scenarios with respect to demand levels, spatial heterogeneity, and route length. The results demonstrate the consistent advantage of the proposed model in all scenarios as against its counterparts, i.e., two existing recipes that convert CA model-based solution into real design of stop locations. Lastly, a case study is presented using real data and practical constraints for the adjustment of a bus route in Chengdu (China). System cost saving of 15.79% is observed by before-and-after comparison.

Karst aquifer characterization by inverse application of MODFLOW-2005 CFPv2 discrete-continuum flow and transport model

This research presents the first application of a distributed discrete-continuum numerical model for simultaneous inversion of long-term spring flow in conjunction with solute- and heat-transport signatures, in a large-scale karst system. The MODFLOW-2005 CFPv2 code was adopted for this reason, utilizing PEST for automated inversion. Taking the principal of model parsimony into account, simulations were performed at two stages, testing likely conceptual models based on the hydro-geological understandings. At the first stage, simple yet realistic conceptual models were setup considering different probable variants of conduit geometries. Results of this stage highlights the importance of sinkhole-spring dynamics, in the recorded spring signatures; such that, successful simultaneous calibration of all spring signatures could not be obtained unless high linkage of conduits to the sinkholes all over the catchment was adopted. Subsequently, some additional rational complexity was added in order to achieve a more demonstrable model of the karst system, as the second stage. Model selection criteria, fitting statistics and reasonableness of calibrated parameters were assessed, at this stage. Although one of the model variants could statistically represent all observations with reasonable parameter values and high agreement with catchment hydrogeological perspective, it still cannot be approved as a predictive tool, unless field measurements on conduit system characteristics and matrix hydraulic parameters, as well as long-term observation of aquifer physicochemical signatures be collected for further validation/improvement of conceptual and numerical model. Achieved simultaneous inversion results can serve as a proof of concept for applicability of physically based discrete-continuum modeling approach for flow and transport modeling in karst.