Lagrangian Coordinate System Research Articles

Convective Phenomenes in the Context of Meso-β-scale Convective Structures

This review article presents the results of radar studies of convective phenomena in Moldavia and the North Caucasus using Eulerian (ECS) and Lagrangian (LCS) coordinate systems. Application of the Lagrangian approach allowed us to exclude the influence of the tropospheric displacement and to obtain integral grid patterns of thunderstorm-hail processes. These structures are especially well manifested at small wind shears (up to 1 m/sec/km). At large shears, the mesh structures are transformed predominantly into linear structures. The methodology for obtaining integral pictures of radio echoes of thunderstorm processes is described. Intersections of linear elements, which we call faces, occur at nodes. The latter plays a particularly important role in the dynamics and kinematics of convective storms. The latter play a particularly important role in the dynamics and kinematics of convective storms. The development of storms occurs along the facets and at the nodes of meso-β-scale convective structures (MMCS), which explains the mechanisms of splitting and merging of storms: in the first case the facets diverge, in the second case they converge. The relations of motion vectors for different types of storms are obtained. It is shown that the direction of the radio echo canopy coincides with the storm motion trajectory; the evolution vector (propagation) for the most powerful storms deviates from the storm displacement direction by 80°–135°. The structure of the updated band within which the Flanking Line is formed for supercells and multicells is studied. Mnemonic rules have been derived that allow one to infer from instantaneous patterns of anvil orientation and the mutual location of storms whether they are converging or diverging, and to identify left- or right-moving storms. A hypothesis on the internal structure of the cold front of the 2nd kind is stated. The main conclusion of the work is that the evolution of storms is determined by the configuration of meso-β-scale convective structures. This explains various convective phenomena from unified positions. The results are applicable in works on modification of convective cloudiness, for ultra-short-term forecasts of dangerous phenomena, storm warnings of the population, rescue services, etc.

RFWBS Model: Multilevel Hybrid Mapping Solution Framework for Designing Neurorehabilitation Physiotherapy Devices

Healthcare services and rehabilitation equipment have entered a phase of rapid development driven by user requirements. However, the development of corresponding design models for rehabilitation equipment is lacking. A general framework and development process are urgently needed for neurorehabilitation physiotherapy equipment. To address problems such as inadequate knowledge representation in the design process and difficulties in modeling the functional structure of the product, we combined a decomposition topology model of neurorehabilitation physiotherapy equipment with the modular design method. We proposed a general model for the design of neurorehabilitation physiotherapy equipment comprising the following modules: requirements (R), function (F), principle workspace (W), behavior (B), and structure (S), i.e., the RFWBS model. Using the neurorehabilitation robotic glove as an example for design practice, in terms of kinematics, the mechanism is analyzed by establishing a Lagrangian coordinate system for resolution. The mechanism has three degrees of freedom and can achieve the natural flexion and extension angles of each finger joint. In terms of dynamics, during the entire finger extension, the angular acceleration is almost zero, and the average angular velocity is approximately 30~50°/s. This indicates that the mechanism is suitable for wearable use, validating the scientific and effective nature of the RFWBS expanded model.

Lagrangian steady-state discrete Boltzmann model for non-equilibrium flows at micro–nanoscale

In micro- and nanoscale channels, where the characteristic scale approaches or falls below the mean free path between gas molecules, flow characteristics exhibit pronounced discontinuity. In such scenarios, continuum-based models are inadequate, and adopting kinetic models based on statistical mechanics becomes imperative for accurately describing the gas transport phenomenon. This paper presents a novel steady-state discrete Boltzmann model (DBM) that is specifically tailored for non-equilibrium flows at the micro–nanoscale. Unlike the conventional DBM, the new model focuses on the evolution equation of the non-equilibrium component of the molecular velocity distribution function in the Lagrangian coordinate system, which follows the fluid micro-element. Through numerous numerical simulations, we demonstrate the effectiveness of our model in capturing gas flow characteristics across a wide spectrum of rarefaction parameters, ranging from slip flow to free molecular flow. Utilizing this new model, we initially examine the Onsager reciprocal relationship between heat flux generated by pressure gradients and mass flux resulting from temperature gradients in micro/nanochannels. Subsequently, we calculate higher-order non-equilibrium quantities up to the 10th order and make a comparison of their characteristics. Finally, we present and discuss the features of the non-equilibrium component of the molecular velocity distribution function.

Perturbation analysis of nonlinear partial reflected wave on a sloping bottom

In this paper, we use the perturbation method to develop a new mathematical derivation to describe nonlinear partial standing wave over uniformly sloping bottoms. In the Lagrangian coordinate system, the particle trajectories are obtained as a function of the nonlinear order parameter ɛ and the bottom slope α to the second order of the perturbation. The setups and mean return flow for an arbitrary bottom in the Lagrangian framework are also found as part of the solutions. The direct influences of reflection coefficient, wave steepness and bottom slope on the surface profiles of partially reflected waves are also derived. This nonlinear analytical solution is verified by reduction to the classical Stokes solution of progressive waves in both the deep water and constant depth limit. This solution allows to describe the successive deformation of partial reflected wave profiles and water particle trajectories prior to wave breaking to be described. The dynamic properties including mass transport, and Lagrangian mean level and radiation stress for nonlinear partially reflected waves on various sloping bottoms are investigated.

Numerical Simulation of Heat Transfer and Spread of Virus Particles in the Car Interior

The epidemic caused by the coronavirus infection SARS-CoV-2 at the beginning of 2022 affected approximately 500 million people in all countries. The source of infection is the particles of the virus, which, when breathing, talking, and coughing, are released with the respiratory droplets and aerosol dust of an infected person. Actions aimed at combating and minimizing the consequences of coronavirus infection led to taking measures in scientific areas to investigate the processes of the spread of viral particles in the air, in ventilation, and air conditioning systems of premises and transport, filtration through masks, the effect of partitions, face shields, etc. The article presents a mathematical model of the spread of viral particles in technological transport. Air intake diverters and the operator’s respiratory tract are the sources of the virus. The Euler–Lagrange approach was used to simulate liquid droplets in a flow. Here, the liquid phase is considered as a continuous medium using Navier–Stokes equations, the continuity equation, the energy equation, and the diffusion equation. Accounting for diffusion makes it possible to explicitly model air humidity and is necessary to consider the evaporation of droplets (changes in the mass and size of particles containing the virus). Liquid droplets are modeled using the discrete-phase model (DPM), in which each particle is tracked in a Lagrange coordinate system. The DPM method is effective, since the volume fraction of particles is small relative to the total volume of the medium, and the interaction of particles with each other can be neglected. In this case, the discrete and continuous phases are interconnected through the source terms in the equations. The averaged RANS equations are solved numerically using the k-ω turbulence model in the Ansys Fluent package. The task was solved in a static form and in the time domain. For a non-stationary problem, the stabilization time of the variables is found. The simulation results are obtained in the form of fields of pressures, velocities, temperatures and air densities, and the field of propagation of particles containing the virus. Various regimes were studied at various free flow rates and initial velocities of droplets with viral particles. The results of trajectories and velocities of particles, and particle concentrations depending on time, size, and on the evaporability of particles are obtained.

Large-deformation consolidation models of saturated–unsaturated materials under self-weight and evaporation

This paper develops the one-dimensional large-deformation consolidation theory for the porous medium under alternating saturated–unsaturated conditions, through a former generalization of the well-established Gibson’s nonlinear theory from saturated to unsaturated regime. With the differential relations between the coordinate systems, the equations governing the momentum\\mass balances in the unsaturated porous stratum are transported from the moving current to the initial Lagrangian coordinate system. Without loss of generality, the collective forms of various incremental constitutive laws are then incorporated for unsaturated materials, indicating that any particular constitutive law has been embraced. Two forms of final governing equations are derived depending on the primary independents: σv-uw (total vertical stress - pore fluid pressure) and w (gravimetric water content); which are solved by two distinct finite-element codes on the initial fixed configuration. Numerical examples demonstrate that the two resulting Total Lagrangian formulations are equivalent, yield virtually identical predictions and agrees well with an existing finite-differential code that solves the equations in the convective configuration. The dynamical consolidation process evolving from a weight-driven saturated stage to an evaporation-driven unsaturated stage is quantitatively captured very well by the predictions. The impacts of evaporation, material’s compressibility, and permeability in these two distinct stages are independently explored through several parametric analyses on the consolidation/dewatering rates and the penetration of drying front. Through one factor must affect, in concert with the others, such a hydro-mechanical coupling process, these results can provide a profound understanding of the entire consolidation process of a porous stratum under large deformations.

Formulation and Numerical Solution of the Stability Loss Problem of Elastic-Plastic Shells of Revolution with an Elastic Filler under Combined Axisymmetric and Torsional Loadings

A dynamic statement and a method for numerically solving the buckling problems of elastoplastic shells of revolution with filler in axisymmetric and non-axisymmetric shapes under quasi-static and dynamic loading are presented within the framework of two approaches. In the first approach, the problem of elastic-plastic deformation and buckling of shells of revolution with an elastic filler under combined axisymmetric loading with torsion is formulated in a two-dimensional (generalized axisymmetric) formulation based on the hypotheses of the shells theory of the Timoshenko type and the Winkler foundation. The constitutive relations are written in the cylindrical system of Euler coordinates. For each shell element, a local Lagrangian coordinate system is introduced. Kinematic relations are recorded in the current state metric. The distribution of the displacement velocity components over the shell thickness and strain rate tensors in the local basis is written as the sum of the momentless and moment components, which, in turn, are written as the sum of the symmetric and asymmetric parts in the local and in the general basis. The elastoplastic properties of the shell material are taken into account within the framework of the theory of flow with nonlinear isotropic hardening. To take into account non-axisymmetric forms of buckling, the desired functions (both displacements and forces, moments, contact pressure) are expanded into a Fourier series in the circumferential direction. The variational equations of shell motion are derived from the general equation of dynamics. The contact between the shell and the deformable filler is modeled based on the conditions of non-penetration along the normal and free slip along the tangent. The variational equations of shell dynamics for axisymmetric and nonaxisymmetric processes are interconnected through the physical relations of the theory of plasticity. They take into account large axisymmetric shape changes and the instantaneous stress-strain state of the shell. At the initial stage of the nonaxisymmetric buckling process, the deflections are small; therefore, the equations of nonaxisymmetric buckling are obtained as linearized with respect to nonaxisymmetric forms. To initiate nonaxisymmetric buckling modes, initial nonaxisymmetric deflections are introduced. To solve the defining system of equations, a finite-difference method and an explicit time integration scheme of the “cross” type are used. The second approach is based on continuum mechanics hypotheses and implemented in a three-dimensional setting. Both approaches make it possible to simulate the nonlinear subcritical deformation of shells of revolution with an elastic filler, to determine the ultimate (critical) loads in a wide range of loading rates, taking into account geometric shape imperfections, to study the processes of buckling in axisymmetric and non-axisymmetric shapes under dynamic and quasi-static complex loading by tension, compression, torsion, internal and external pressure. The results of numerical simulation are compared with experimental data on the torsion of steel cylindrical elastoplastic shells ( R / h = 1.45) with an elastic filler.

Analysis of the most efficient horizontal drainage position in the large strain consolidation considering the self-weight effect

To study the large-deformation settlement of soft soil foundations, an analysis of the large-strain consolidation of soil in a Lagrangian coordinate system based on continuum mechanics is performed, and the control equations with the displacement gradient are obtained in this paper. Based on the nonlinear permeability and compressibility of soft soil foundations, the equation of large-strain nonlinear consolidation is established. Then, the finite difference method is used to solve the governing equations. Under the condition of two-sided drainage while considering the soil self-weight stress, it is found that the distribution of excess pore pressure is not symmetric along the depth of the foundation. The position of the maximum excess pore pressure appears below the midpoint of the soil foundation. The deviation degree increases with increasing foundation thickness and semilogarithmic compression index Cc value, and it decreases with increasing semilogarithmic permeability index Ck value. The calculations show that setting the drainage layer position close to the position of maximum excess pore pressure can effectively improve the drainage ability of the upper and bottom soil layers, which is beneficial to consolidation, and that the deviation distance is related to the Cc and Ck values.

A novel approach for the reduction of powder deposition in spray drying chambers

This study aims to introduce a new dryer configuration to mitigate the particle deposition problem. Three new geometries (single-rib, double-ribs, and triple-ribs dryers) are compared with the typical cylinder-on-cone dryer. The equations governing the transient flow field are solved using the Eulerian approach, while particles are tracked through a Lagrangian coordinate system. The characteristic drying curve approach is used to model the drying of skim milk within the chamber. The sticky point curve of skim milk is chosen as a criterion for the detection of sticking or bouncing particles. The obtained results are validated using both experimental and numerical data, showing that particle deposition is significantly reduced in all-new geometries compared to the base design. More specifically, deposition is reduced by 56.8% in the double-ribs dryer, and the dryer’s yield has increased by 41.7% in the triple-ribs dryer.

A semi-discrete model of traffic flow in correspondence with a continuum model under Lagrange coordinate system

We propose a semi-discrete model by dividing traffic flow into moving particles, and establish its corresponding relation with a conserved high-order (CHO) continuum model under the Lagrange coordinate system. This suggests that they should share the same or similar properties and we prove the boundedness of solutions and analytically derive a traveling wave or wide moving jam in the CHO model. Indeed, the numerical simulation based on the semi-discrete model verifies these properties and especially indicates that the simulated wide moving jams converge to the analytical solution for sufficiently refined division of traffic flow.

An Alternative Algorithm for Simulating Flash Flood

Most of the approaches in numerical modeling techniques are based on the Eulerian coordinate system. This approach faces difficulty in simulating flash flood front propagation. This paper shows an alternative method that implements a numerical modeling technique based on the Lagrangian coordinate system to simulate the water of debris flow. As for the interaction with the riverbed, the simulation uses an Eulerian coordinate system. The method uses the conservative and momentum equations of water and sediment mixture in the Lagrangian form. Source terms represent deposition and erosion. The riverbed in the Eulerian coordinate system interacts with the flow of the mixture. At every step, the algorithm evaluates the relative position of moving nodes of the flow part to the fixed nodes of the riverbed. Computation of advancing velocity and depth uses the riverbed elevation, slope data, and the bed elevation change computation uses the erosion or deposition data of the flow on the moving nodes. Spatial discretization is implementing the Galerkin method. Furthermore, temporal discretization is implementing the forward difference scheme. Test runs show that the algorithm can simulate downward, upward, and reflected backward 1-D flow cases. Two-D model tests and comparisons with SIMLAR software show that the algorithm works in simulating debris flow.

The concept of vortex convective storms

It is proposed a method of analyzing the effects of the environmental vertical wind shear on the structure and dynamics of convective storms, the conditions for the occurrence of vortex storms based on the hydrodynamic laws of flow around of solid body in the Lagrangian coordinate system. There is described an example of the evolution of a supercell based on radar observations with the updated information of 3 minutes. It is shown the unity of the complex structure and dynamics, which allows proposing a typical model of vortex storms. The method of analysis affords to explain the sequential continuity of the types of storms, the prevalence of the right-propagation (in the Northern Hemisphere). It is proposed a classification of convective storms based on the interaction of the internally associated ascending flow and rotation with the environmental dynamic factor of wind shear.

A Novel Shape Finding Method for the Main Cable of Suspension Bridge Using Nonlinear Finite Element Approach

The determination of the final cable shape under the self-weight of the suspension bridge enables its safe construction and operation. Most existing studies solve the cable shape segment-by-segment in the Lagrangian coordinate system. This paper develops a novel shape finding method for the main cable of suspension bridge using nonlinear finite element approach with Eulerian description. The governing differential equations for a three-dimensional spatial main cable is developed before a one-dimensional linear shape function is introduced to solve the cable shape utilizing the Newton iteration method. The proposed method can be readily reduced to solve the two-dimensional parallel cable shape. Two iteration layers are required for the proposed method. The shape finding process has no need for the information of the cable material or cross section using the present technique. The commonly used segmental catenary method is compared with the present method using three cases study, i.e., a 1666-m-main-span earth-anchored suspension bridge with 2D parallel and 3D spatial main cables as well as a 300-m-main-span self-anchored suspension bridge with 3D spatial main cables. Numerical studies and iteration results show that the proposed shape finding technique is sufficiently accurate and operationally convenient to achieve the target configuration of the main cable.

Thermomechanical Behavior of Steels in Tension Studied with Synchronized Full-Field Deformation and Temperature Measurements

Despite tensile testing being commonly used for investigating the mechanical behavior of materials, the occurrence of heterogeneous strain and increasing temperature at high strain rates make the experiment much more complex. This work presents a method integrating synchronous full-field stereo Digital Image Correlation (DIC) and Infrared Thermography (IRT). This method enabled high resolution investigations of the development of local temperatures and strains of the specimen during tensile loading of four steels at strain rates ranging from 2.5·10−4 to 900 s−1. The tests were monitored by a stereo setup of optical cameras and an infrared camera. Data acquisition was synchronized, and a pinhole camera model was used to translate the images from all cameras to the same three-dimensional space. The displacement vector fields from DIC were subtracted from the IRT images to represent the temperature maps in a Lagrangian coordinate system. The overall thermomechanical response of the materials was shown as 3D waterfall plots, which represent localized strain and temperature as a function of position and engineering strain. The results show that temperature increased homogeneously during uniform deformation at higher strain rates (10−2-900 s−1) and increased markedly with the onset of necking on the region of localized strain. At these strain rates, the localized increase of strain and temperature during necking were observed at the same global engineering strain and position, evidencing the spatial and temporal synchronization. The described method was used to accurately investigate the evolution of localized strain and temperature in both low and high strain rate regime.

Modeling and Simulation Techniques Used in High Strain Rate Projectile Impact

A series of computational models and simulations were conducted for determining the dynamic responses of a solid metal projectile impacting a target under a prescribed high strain rate loading scenario in three-dimensional space. The focus of this study was placed on two different modeling techniques within finite element analysis available in the Abaqus software suite. The first analysis technique relied heavily on more traditional Lagrangian analysis methods utilizing a fixed mesh, while still taking advantage of the finite difference integration present under the explicit analysis approach. A symmetry reduced model using the Lagrangian coordinate system was also developed for comparison in physical and computational performance. The second analysis technique relied on a mixed model that still made use of some Lagrangian modeling, but included smoothed particle hydrodynamics techniques as well, which are mesh free. The inclusion of the smoothed particle hydrodynamics was intended to address some of the known issues in Lagrangian analysis under high displacement and deformation. A comparison of the models was first performed against experimental results as a validation of the models, then the models were compared against each other based on closeness to experimentation and computational performance.

Determination of acoustic radiation acting on a spherical rigid particle placed in an elastic tube filled with liquid

Acoustic radiation force effect upon a rigid spherical particle placed in the thin elastic tube is studied. The problem of determination of the acoustic radiation forces acting on an obstacle in an ideal liquid is formulated with respect to the Lagrange coordinate system. Thus, the radiation pressure is defined as time-averaged value of the acoustic pressure over the obstacle surface. This approach is adequate if, at determining of the acoustic pressure in a fluid, the deviation of the pressure from the harmonic law in time domain is taken into account in the obstacle vicinity. An action of the acoustic radiation force on the rigid spherical particle placed in the thin tube with elastic wall is studied here for the case of the incident plane sound wave propagating along the tube axis. Model is developed to describe the response of the system consisting of the compliant infinite thin circular cylindrical tube filled with the ideal compressible liquid and rigid spherical body which is immovable and located on the tube axis under the plane wave propagating along the tube axis. The problem of the hydrodynamic characteristics determination is reduced to the solution of the infinite system of algebraic equations that can be solved by the reduction method. The formula for the acoustic radiation force calculation is derived to characterize the force acting upon rigid spherical particle in the thin compliant elastic cylindrical tube.

The influence of a viaduct on PM dispersion in a typical street: Field experiment and numerical simulations

To investigate the effects of a viaduct on particle diffusion below the viaduct, field measurements for particulate matter (PM) concentration along roadsides with and without viaducts were conducted in this study. Air flow and particle (PM2.5) distribution around a viaduct were simulated by Computational Fluid Dynamic (CFD) equipped with the Discrete Phase Model in a Lagrange coordinate system. Experimental results of this case study indicate that the average mass concentrations of PM10 (156.7 μg m−3), PM2.5 (77.3 μg m−3) and PM1 (56.5 μg m−3) on an elevated highway exceeded those on the ground–level expressway by 15%, 10% and 12%, respectively. Simulation found that two main vortices as well as two or three secondary vortices are generated in the canyon when both a viaduct and a barrier are added, while there was only one main vortex in the street canyon without a viaduct. A viaduct causes the particulate matter to be distributed unevenly, and the concentration on the leeward side near the ground is the highest. The vortex center is elevated as the value of ΔT (ground–level air temperature) increases, but the concentrations are basically maintained. We concluded that a viaduct increases the concentration of particulate matter in the street canyons and greatly affects the airflow field. Both ΔT and a sound barrier on a viaduct reduce the PM concentrations, but the effects are limited.

Analysis of 1D large strain consolidation of structured marine soft clays

Structured soft clays are widely distributed in the coastal regions of China. The characteristics of the natural structure greatly influence the compressibility and permeability of marine soft clays and should be considered in the theory of their consolidation. When a large surcharge load is applied to a structured clay deposit, large strains can be induced in the clay layer due to the high compressibility, where the consolidation process follows the large strain assumption. However, there are few published theories of consolidation in which both the natural structure of marine soft clays and the large strain assumption can be considered simultaneously. In this study, a novel large strain consolidation model for structured marine soft clays was developed by considering the variation of structural yield stress with depth and using different calculation methods for initial effective stress of structured clay deposits in the Lagrangian coordinate system. The corresponding solution was derived by the finite difference method. Finally, the influences of the natural structure of soft clays and different geometric assumptions on consolidation behavior were investigated. The results show that the dissipation rate of excess pore water pressure of structured clays under the large strain assumption is expected to be faster than that under the small strain assumption, and the difference in consolidation behavior between the two assumptions increases with the strain level of natural structured clays. If the strain level in the clay layer is more than 15%, the difference in consolidation behavior between the large and small strain assumptions must be considered.

Damage evolution in the process of plane strain bending under tension at large strains

A semi-analytic solution for plane strain bending under tension of a sheet is proposed for elastic-plastic, isotropic, incompressible, strain hardening material with damage evolution at large strains using a Lagrangian coordinate system. Numerical treatment is only necessary to evaluate ordinary integrals and solve transcendental equations. No restriction is imposed on the hardening law. Quite a general uncoupled continuum damage evolution model independent of the third invariant of the stress tensor is used. It is shown that the solution for the model adopted is facilitated by choosing the equivalent plastic strain as one of the independent variables. An illustrative example is provided for Swift’s hardening law and two widely used damage evolution equations.

Численный анализ управляемой потери устойчивости пологих менисковых облицовок при высокоскоростном обжатии

Experimental and analytical research methods of losses of stability of meniscal form liners at their high-speed deformation, i.e., collapse, by products of detonation of explosive have limited opportunities. They are caused by the complicated nature of liner thickness change to control the operated loss of stability --- folding, high-speed deformations of liners, intensive drop of pressure acting on them and some other features. The paper introduces an approach to numerical three-dimensional modeling of collapse of meniscal form liners with variable thickness in the circumferential direction in the area of their periphery, the modeling being carried out by the finite element method in Lagrange coordinate system in LS-DYNA software package. The study also shows the main stages of implementing this approach and describes the key parameters of the materials models used, as well as the type of the final element and mechanism of adaptive updating of the computational grid. By the method of numerical simulation, we found the main regularities of liners collapse and folding of the afterbody of high speed elements formed during the collapse of the liners. The results of numerical calculations are confirmed by experimental data. The studies done are of interest to specialists involved in the analysis of the loss of stability of various structures under dynamic loads, as well as to specialists in the field of explosion and impact physics.