Continuum Systems Research Articles

From variational to bracket formulations in nonequilibrium thermodynamics of simple systems

A variational formulation for nonequilibrium thermodynamics was recently proposed in Gay-Balmaz and Yoshimura (2017a, 2017b) for both discrete and continuum systems. This formulation extends the Hamilton principle of classical mechanics to include irreversible processes. In this paper, we show that this variational formulation yields a constructive and systematic way to derive, from a unified perspective, several bracket formulations for nonequilibrium thermodynamics proposed earlier in the literature, such as the single generator bracket and the double generator bracket. In the case of a linear relation between the thermodynamic fluxes and the thermodynamic forces, the metriplectic or GENERIC bracket is recovered. We also show how the processes of reduction by symmetry can be applied to these brackets. In the reduced setting, we also consider the case in which the coadjoint orbits are preserved and explain the link with double bracket dissipation. A similar development has been presented for continuum systems in Eldred and Gay-Balmaz (2020) and applied to multicomponent fluids.

Assessment of Nitrogen Uptake and Biological Nitrogen Fixation Responses of Soybean to Nitrogen Fertiliser with SPACSYS

Chemical fertiliser nitrogen addition will inhibit biological nitrogen fixation (BNF) for soybean (Glycine max [L.] Merr) growth. The optimal balance of these two nitrogen input sources has been a key issue for sustainable development in Northeast China. We used the data collected from a four-year experiment with varied irrigation and fertiliser treatments from 2007 to 2010 to evaluate the SPACSYS (Soil-Plant-Atmosphere Continuum SYStem) model. The validated model was run to investigate the responses to different management practices in seed yield, BNF, protein yield and soil nitrogen budgets. Scenario testing showed average yield increase of 2.4–5.2% with additional 50–100 kg N/ha application. Irrigation at the reproductive stage improved seed yield in drier years with an increase of 12–33% compared with the rain-fed treatment. BNF was suppressed by fertiliser nitrogen application and drought stress with a decrease of 6–33% and 8–34%, respectively. The average nitrogen budget without fertilization indicated a deficit of 39 kg N/ha. To attain higher seed yield, applying fertiliser at 25–30 and 15–20 kg N/ha before sowing is advised in drier and wetter years, respectively. To achieve a higher seed nitrogen content, an application rate of 55–60 and 45–50 kg N/ha is recommended for drier and wetter years, respectively.

Finding flows of a Navier\u2013Stokes fluid through quantum computing

There is great interest in using quantum computers to efficiently simulate a quantum system’s dynamics as existing classical computers cannot do this. Little attention, however, has been given to quantum simulation of a classical nonlinear continuum system such as a viscous fluid even though this too is hard for classical computers. Such fluids obey the Navier–Stokes nonlinear partial differential equations, whose solution is essential to the aerospace industry, weather forecasting, plasma magneto-hydrodynamics, and astrophysics. Here we present a quantum algorithm for solving the Navier–Stokes equations. We test the algorithm by using it to find the steady-state inviscid, compressible flow through a convergent-divergent nozzle when a shockwave is (is not) present. We find excellent agreement between numerical simulation results and the exact solution, including shockwave capture when present. Finally, we compare the algorithm’s computational cost to deterministic and random classical algorithms and show that a significant speed-up is possible. Our work points to a large new application area for quantum computing with substantial economic impact, including the trillion-dollar aerospace industry, weather-forecasting, and engineered-plasma technologies.

The Development of Ultrafast Electron Microscopy

Time-resolved electron microscopy is based on the excitation of a sample by pulsed laser radiation and its probing by synchronized photoelectron bunches in the electron microscope column. With femtosecond lasers, if probing pulses with a small number of electrons—in the limit, single-electron wave packets—are used, the stroboscopic regime enables ultrahigh spatiotemporal resolution to be obtained, which is not restricted by the Coulomb repulsion of electrons. This review article presents the current state of the ultrafast electron microscopy (UEM) method for detecting the structural dynamics of matter in the time range from picoseconds to attoseconds. Moreover, in the imaging mode, the spatial resolution lies, at best, in the subnanometer range, which limits the range of observation of structural changes in the sample. The ultrafast electron diffraction (UED), which created the methodological basis for the development of UEM, has opened the possibility of creating molecular movies that show the behavior of the investigated quantum system in the space-time continuum with details of sub-Å spatial resolution. Therefore, this review on the development of UEM begins with a description of the main achievements of UED, which formed the basis for the creation and further development of the UEM method. A number of recent experiments are presented to illustrate the potential of the UEM method.

Assessing healthcare service quality using routinely collected data: Linking information systems in emergency care

Emergency departments need to continuously calculate quality indicators in order to perform structural improvements, improvements in the daily routine, and ad-hoc improvements in everyday life. However, many different actors across multiple disciplines collaborate to provide emergency care. Hence, patient-related data is stored in several information systems, which in turn makes the calculation of quality indicators more difficult. To address this issue, we aim to link and use routinely collected data of the different actors within the emergency care continuum. In order to assess the feasibility of linking and using routinely collected data for quality indicators and whether this approach adds value to the assessment of emergency care quality, we conducted a single case study in a German academic teaching hospital. We analyzed the available data of the existing information systems in the emergency continuum and linked and pre-processed the data. Based on this, we then calculated four quality indicators (Left Without Been Seen, Unplanned Reattendance, Diagnostic Efficiency, and Overload Closure). Lessons learned from the calculation and results of the discussions with staff members that had multiple years of work experience in the emergency department provide a better understanding of the quality of the emergency department, the related challenges during the calculation, and the added value of linking routinely collected data.

An approach to calculating large strain accumulation for discrete element simulations of granular media

SummaryFor research on granular materials, establishing a method to calculate continuum strain from particle displacements is necessary for understanding the material behaviour at macro‐level and developing continuum constitutive models. Existing methods are generally based on constructing a mesh or background grid to calculate strain from particle motions. These methods offer rigorous ways to measure strain for granular materials; however, they suffer from several problems such as mesh distortion and lacking grid‐to‐particle strain mapping procedure, which hinders their capability of calculating strain accumulation during large deformation processes of granular media. To address this issue, this study proposes a new strain calculation method for discrete element simulations of granular materials. This method describes a particle assembly as an equivalent continuum system of material points, each of which corresponds to a particle centre and represents a continuous region with its initial volume/area presumably equal to the volume/area of Voronoi cells generated in accordance with the particle assembly configuration. Smooth Particle Hydrodynamics (SPH) interpolation functions are then employed to calculate strain for these material points. This SPH‐based method does not require any mesh or background grid for computation, leading to advantages in calculating strain accumulation under large deformation. Simulations of granular materials in both uniform and heterogeneous gradations were carried out, and strain results obtained by the proposed method indicate good agreements with analytical and numerical solutions. This demonstrates its potential for strain calculations in discrete element simulations of granular materials involving large deformations and/or large displacements.

Topological Elasticity of Flexible Structures

Flexible mechanical metamaterials possess repeating structural motifs that imbue them with novel, exciting properties including programmability, anomalous elastic moduli and nonlinear and robust response. We address such structures via micromorphic continuum elasticity, which allows highly nonuniform deformations (missed in conventional elasticity) within unit cells that nevertheless vary smoothly between cells. We show that the bulk microstructure gives rise to boundary elastic terms. Discrete lattice theories have shown that critically coordinated structures possess a topological invariant which determines the placement of low-energy modes on edges of such a system. We show that in continuum systems a new topological invariant emerges which relates the difference in the number of such modes between two opposing edges. Guided by the continuum limit of the lattice structures, we identify macroscopic experimental observables for these topological properties that may be observed independently on a new length scale above that of the microstructure.

A stress analysis method for molecular dynamics systems

As the development of micro/nanoelectromechanical devices continues at a fast pace, there is a growing need to bridge the gap between material behavior at the atomic and molecular levels and material behavior at length scales relevant to most engineering and industrial applications. Since stress is one of the most fundamental quantities in continuum mechanics (CM), it is desirable to introduce stress methods that are applicable to both continuum and discrete systems, such as those modeled by molecular dynamics (MD). Thus, the objective of this study was to demonstrate how a traction vector-based stress method that is compatible with CM can be used to examine MD systems of crystalline solids undergoing small lattice distortion. In the bulk of face-centered-cubic (FCC) and body-centered-cubic solids, the traction vector-based atomic stress definition used in this study is shown to be equivalent to the classical energy-based virial stress that is commonly used for small deformations and low temperatures, i.e., negligible thermal vibration. However, contrary to the virial stress, the components of the present atomic stress diminish in the region close to a free surface, consistent with the traction-free boundary condition. The validity of the stress method developed herein is demonstrated by MD results of the bulk modulus of FCC copper, the surface tension of an FCC solid, and the subsurface stress field of an FCC half-space indented by a rigid flat punch.

The stable isotopic composition variation characteristics of desert plants and water sources in an artificial revegetation ecosystem in Northwest China

Characterizing the dynamics of plant water availability and the mechanisms through which plants access available water sources are key challenges in ecohydrology, especially in arid and semi-arid regions. The establishment of artificial revegetation of different ages has formed vegetation landscapes with different chronological sequences in the southeast margin of the Tengger Desert. As an important regulatory factor of the Soil Plant Atmosphere Continuum (SPAC) system, changes in vegetation characteristics will affect the water uptake of the entire ecosystem. In this study, the δ2H and δ18O isotope ratios in precipitation, plant water, soil moisture, and underground water, and the δ13C isotope ratio of plant leaves were measured using the stable isotope technique for artificial revegetation used to fix the sand dunes in the Tengger Desert. The purpose of this study was to establish the isotope content and characteristics of potential water sources for plants and leaf carbon isotopes, to clarify whether the soil moisture condition and duration of sand-fixing influenced the isotopic signal, and to identify the main water source for desert plants. The results indicated that air temperature and amounts of precipitation were two important influencing factors in precipitation isotope ratio change. Soil water isotopic ratios were different under different water conditions, and under the same water conditions, the vertical variation in soil water isotopes in a single area and spatial variation at any depth for different revegetation areas were all significant (p < 0.01). Shallow soil waters were enriched in isotope ratios and the value increased with an increase of the sand-fixing duration (as found in older revegetation areas). The δ2H and δ18O values in Caragana korshinskii leaves were higher than they were in Artemisia ordosica leaves because the former has a higher transpiration rate. The higher δ13C value of Artemisia ordosica plant leaves indicated high water use efficiency, which is a result of better adapting to the arid environment compared to Caragana korshinskii. Overall, water uptake of these two shrubs was mainly (>70%) concentrated at a soil depth of 20–150 cm. For Caragana korshinskii plants, the 60–150 cm soil layer was the main water supply layer, while the water contribution layer of Artemisia ordosica plants focused on the 20–100 cm soil layer and the contribution rate of shallow soil water increased with the increase of sand fixation time in older revegetation areas, regardless of the species used for revegetation.

TMTDyn: A Matlab package for modeling and control of hybrid rigid–continuum robots based on discretized lumped systems and reduced-order models

A reliable, accurate, and yet simple dynamic model is important to analyzing, designing, and controlling hybrid rigid–continuum robots. Such models should be fast, as simple as possible, and user-friendly to be widely accepted by the ever-growing robotics research community. In this study, we introduce two new modeling methods for continuum manipulators: a general reduced-order model (ROM) and a discretized model with absolute states and Euler–Bernoulli beam segments (EBA). In addition, a new formulation is presented for a recently introduced discretized model based on Euler–Bernoulli beam segments and relative states (EBR). We implement these models in a Matlab software package, named TMTDyn, to develop a modeling tool for hybrid rigid–continuum systems. The package features a new high-level language (HLL) text-based interface, a CAD-file import module, automatic formation of the system equation of motion (EOM) for different modeling and control tasks, implementing Matlab C-mex functionality for improved performance, and modules for static and linear modal analysis of a hybrid system. The underlying theory and software package are validated for modeling experimental results for (i) dynamics of a continuum appendage, and (ii) general deformation of a fabric sleeve worn by a rigid link pendulum. A comparison shows higher simulation accuracy (8–14% normalized error) and numerical robustness of the ROM model for a system with a small number of states, and computational efficiency of the EBA model with near real-time performances that makes it suitable for large systems. The challenges and necessary modules to further automate the design and analysis of hybrid systems with a large number of states are briefly discussed.

Characteristic Analysis of Droughts and Waterlogging Events for Maize Based on a New Comprehensive Index through Coupling of Multisource Data in Midwestern Jilin Province, China

Frequent droughts and waterlogging events are a threat to food security. An understanding of the spatial and temporal variations that occur during such events is essential when conducting a regional and/or global drought and waterlogging assessment. The goal of this study is to define a comprehensive index that considers the continuum system of atmosphere, crops, and soil moisture impacts on droughts and waterlogging events, and to analyze the temporal and spatial distribution of such events in the development of maize. The results show that the proposed comprehensive drought and waterlogging index (CDWI) can bring together the advantages of a single drought and waterlogging index and reasonably describe its range. During the study period, the annual trends of the CDWI decreased at different growth stages from 1982 to 2015, whereas the CDWI did not show significant spatial heterogeneity during any particular stage. Increasing trends of CDWI over 0.019/year were found in the northern part of Midwestern Jilin Province from the emergence to tasseling stages. In addition, decreasing trends were observed in the study area from the tasseling to maturation stages. Slight drought and waterlogging events occurred more frequently than moderate and serious drought and waterlogging events.

Ordinary and Slow Earthquakes Reproduced in a Simple Continuum System With Stochastic Temporal Stress Fluctuations

AbstractEarthquakes are complex and diverse, ranging from energetic and destructive megathrust earthquakes to slow earthquakes. Complex earthquake rupture processes are generally modeled by prescribing a deterministic system and solving deterministic differential equations. However, such approaches cannot adequately capture the temporal evolution of a complex fault system because of the inherent unpredictability that results from various unprescribed processes. Such unpredictability may be better represented by time‐dependent stochastic fluctuations in addition to the deterministic estimates. Here, we demonstrate that the consideration of time‐dependent stochastic stress fluctuations in ordinary crack simulations can reproduce a variety of ruptures, including both crack‐like and pulse‐like ordinary (fast) earthquakes and slow earthquakes, by simply changing the strength drop and the initial stress level. The results indicate that stochasticity is effective for reproducing and better understanding the diversity of earthquakes, including slow earthquakes.

Modified precise direct time integration method for the transient response analysis of viscoelastic systems using an internal variable model

Engineering systems for different levels of energy dissipation use internal variable models, which may lead to tremendous problems in accurate analysis. This article aims to provide an alternative direct integration method for the analysis of systems involving an anelastic displacement field model. A new state-space formulation built on an augmented set of anelastic variables for asymmetric systems is developed. Then, a precise time integration method based on state-space matrix formulation is proposed by introducing a Legendre–Gauss quadrature. The new integration method in terms of numerical stability and its implementation is discussed. The effect of sensitivity of the selection of the time-step and computational time on the performance of the new method is investigated by using a multi-degree-of-freedom system. The performance of the new method is also evaluated in terms of both computational accuracy and efficiency at higher degrees of freedom by using a continuum system. It is demonstrated that the computational accuracy and efficiency of the new method on large-scale problems are higher than that of the direct integration linear displacement–velocity method.

Hierarchical multiscale finite element method for multi-continuum media

Simulation in media with multiple continua where each continuum interacts with every other is often challenging due to multiple scales and high contrast. One needs some type of model reduction. One of the approaches is a multi-continuum technique, where every process in each continuum is modeled separately and an interaction term is added. Direct numerical simulation in multiscale media is usually not practicable. For this reason, one constructs the corresponding homogenized equations which approximate the solutions to the multiscale equations when the microscopic scales tend to 0. Computing the effective coefficients of the homogenized equations can be expensive because one needs to solve local cell problems for a large number of macroscopic points. The paper considers a two scale two continuum system where the interaction terms between the continua are scaled as O(1∕ϵ2) where ϵ is the microscopic scale. We prove that in the homogenization limit, we obtain the same limit for both continua. We derive the homogenized equation for the limit function; and prove the homogenization convergence rigorously. The homogenized coefficients are established from solutions of the cell problems which are systems of equations of a similar form as the two continuum system. We develop a hierarchical approach for solving these cell problems at a dense network of macroscopic points with an essentially optimal computation cost. The method employs the fact that neighboring representative volume elements (RVEs) share similar features; and effective properties of the neighboring RVEs are close to each other. The hierarchical approach reduces computation cost by using different levels of resolution for cell problems at different macroscopic points. Solutions of the cell problems which are solved with a higher level of resolution are employed to correct the solutions at neighboring macroscopic points that are computed by approximation spaces with a lower level of resolution. The method requires a hierarchy of macroscopic grid points and corresponding nested approximation spaces with different levels of resolution. Each level of macroscopic points is assigned to an approximation finite element (FE) space which is used to solve the cell problems at the macroscopic points from that level. We prove rigorously that this hierarchical method achieves the same level of accuracy as that of the full solve where cell problems at every macroscopic point are solved using FE spaces with the highest level of resolution, but at the essentially optimal computation cost. Numerical implementation that computes effective permeabilities of a two-scale multicontinuum system via the numerical solutions of the cell problems supports the analytical results.

Murray’s law for discrete and continuum models of biological networks

We demonstrate the validity of Murray’s law, which represents a scaling relation for branch conductivities in a transportation network, for discrete and continuum models of biological networks. We first consider discrete networks with general metabolic coefficient and multiple branching nodes and derive a generalization of the classical 3/4-law. Next we prove an analogue of the discrete Murray’s law for the continuum system obtained in the continuum limit of the discrete model on a rectangular mesh. Finally, we consider a continuum model derived from phenomenological considerations and show the validity of the Murray’s law for its linearly stable steady states.

Impact of polydispersity of particle shape and size on percolation threshold of 3D particulate media composed of penetrable superellipsoids

A versatile class of superellipsoids which smoothly interpolate from octahedron via ellipsoid to cuboid is introduced. By approximating the shapes of pore elements in the particulate networks by these superellipsoids with different shapes or sizes, we investigate the percolation performance of 3D disordered particulate media consisting of the binary penetrable superellipsoidal pores. To derive the critical percolation threshold ϕc of these continuum systems, a finite-size scaling method is applied here. By combining the finite-size scaling theory with different cubic models of bidisperse superellipsoids, the impact of polydispersity of particle shape and size on the value of ϕc for different particulate networks is independently studied. Based on the obtained dataset, the relationship between the percolation thresholds and the pore-structure characteristics is further analyzed and quantified by the generalized formulas. The comparisons reveal the proposed empirical formulas can provide an effectively approximate estimate of ϕc for a series of 3D binary superellipsoid dispersions.

Simulation of Phosphorus Chemistry, Uptake and Utilisation by Winter Wheat.

The phosphorus (P) supply from soils is crucial to crop production. Given the complexity involved in P-cycling, a model that can simulate the major P-cycling processes and link with other nutrients and environmental factors, e.g., soil temperature and moisture, would be a useful tool. The aim of this study was to describe a process-based P module added to the SPACSYS (Soil Plant and Atmosphere Continuum System) model and to evaluate its predictive capability on the dynamics of P content in crops and the impact of soil P status on crop growth. A P-cycling module was developed and linked to other modules included in the SPACSYS model. We used a winter wheat (Triticum aestivum, cv Xi-19) field experiment at Rothamsted Research in Harpenden to calibrate and validate the model. Model performance statistics show that the model simulated aboveground dry matter, P accumulation and soil moisture dynamics reasonably well. Simulated dynamics of soil nitrate and ammonium were close to the observed data when P fertiliser was applied. However, there are large discrepancies in fields without P fertiliser. This study demonstrated that the SPACSYS model was able to investigate the interactions between carbon, nitrogen, P and water in a single process-based model after the tested P module was implemented.

Continuum mechanics for thermo-chemo-mechanical coupling system based on decomposition of internal energy

In this paper, based on the decomposition of internal energy into free internal energy and dissipative energy, some conservation laws of a thermodynamically open continuum system as well as related constitutive relations and evolution equations are derived when considering heat conduction, mass transfusion, chemical reactant and mechanical deformation. Upon establishing a weak form of corresponding governing differential equations and boundary conditions under above theoretical framework, a simple example is then numerically implemented to demonstrate the solution procedure.

Many-body localization in continuum systems: Two-dimensional bosons

We demonstrate that many-body localization of two-dimensional weakly interacting bosons in disorder remains stable in the thermodynamic limit at sufficiently low temperatures. Highly energetic particles destroy the localized state only above a critical temperature, which increases with the strength of the disorder. If the particle distribution is truncated at high energies, as it does for cold atom systems, the localization can be stable at any temperature.

A novel model of water-heat coupling for water-saving irrigated rice fields based on water and energy balance: Model formulation and verification

It is necessary to quantify the hydrothermal processes by develop a water-heat coupling model in rice fields where the transfer between water and energy is dramatically violent, under the circumstance that water-saving irrigation (WSI) was widely used in rice paddy due to the increasing water scarcity. Based on the theory of energy and water balance, considering the interaction of impedance near the ground, a simplified holistic recursive model of water-heat coupling was proposed in the soil–plant–atmosphere continuum (SPAC) system at hourly and daily scales, and the parameters were fitted based on water and energy balance components measured in WSI rice fields. The model, takes various soil moisture conditions into consideration, integrates all the terms in the surface energy balance and the water balance with conventional meteorological data as inputs. Additionally, the model accounts for the specific drying-wetting cycles condition for WSI rice field in the calculation of the canopy resistance (rc) and the recursive process. Variations in crop growth and meteorology were also considered. The results showed that the model performed well in simulating the water and heat fluxes, and temperature profiles during the growth of the rice. This is encouraging for modelling the response of hydrothermal process to drying-wetting cycles or irrigation regimes, which is help for improving the water use efficiency in rice field. Meanwhile, this study indicates that the systemic model is efficient and useful for a better understanding the interaction between hydrological and energy processes in the WSI rice ecosystem.