Lagrangian Approach Research Articles

A Lagrangian Approach for Trajectory Tracking of Underactuated Marine Vessels

This paper attempts to explore the trajectory tracking problem for underactuated marine vessels from a Lagrangian perspective. First, we derive the vessel’s dynamics from Euler-Lagrangian equations and its kinematics from the mechanical constraint, both in the polar frame. The benefit of using polar frame is that we can transform the two-input-three-output underactauted system into a two-input-two-output form such that it becomes easy for us to solve the control problem using general backstepping method. In this paper, the main focus is taken on that how to improve the control performance by fully utilizing the features of the vessel’s nonlinear dynamics from the Lagrangian perspective. Some numerical studies are also carried out to investigate and analyse the effectiveness of the proposed scheme.

A comparison of Eulerian and semi-Lagrangian approaches for modeling stream water quality

ABSTRACT This paper describes and compares some of the advantages and limitations of Eulerian and Lagrangian approaches to water quality modeling and introduces a mixed Eulerian-Lagrangian (or semi-Lagrangian) methodology that captures the strengths of both approaches. The semi-Lagrangian modeling approach is applied to advection-dominated rivers, and flexibly ensures unconditional stability for all time step durations and grid segmentations. The semi-Lagrangian modeling approach is demonstrated by applying it to estimate the dissolved oxygen concentrations in the Sava River in Slovenia, focusing on aspects of the methodology and findings that would be of broad interest to managers of water quality in fluvial water bodies. Results of comparisons of the semi-Lagrangian model with the Eulerian-based QUAL2K model in steady and non-steady scenarios demonstrate that while both models are fully capable of producing satisfactory results when optimally configured, the semi-Lagrangian approach offers accuracy and stability without sensitivity to the interaction of time step size and computational grid segmentation scheme.

Wing tip vortices dynamics with crosswind effect using the large eddy simulation (LES) theory

In the operational control of airports it is imperative to reduce the interval between landing and takeoff operations. To this end, it is necessary to address the risk of aircraft vortex wake encounter during these operations. This paper describes a refined numerical method developed to study the phenomenon of aircraft vortex wake interaction with the airport runway, employing the Lagrangian approach with accelerated processing by parallel computing.The numerical methodology relies on a vortex method incorporating the Large Eddy Simulation (LES) theory for the two-dimensional domain.The Image method is used to guarantee the condition of impenetrability on the airport runway. In this context, open multiprocessing (OpenMP) and an algorithm that subdivides the fluid domain into box structures to expedite the calculation of eddy viscosity locally are used in a Fortran code. Quantitative and qualitative results are presented at various Reynolds numbers, in dimensionless crosswind conditions. In addition, preliminary results are presented for a Boeing 757-200, representative of practical aviation problems. The results are in agreement with experimental results and other numerical simulations. It is possible to satisfactorily capture the centroid trajectories of the vortical structures shed from the wingtips, the temporal deformation of these structures, as well as the dynamics of the primary and secondary structures in interaction with the airport runway.

Realistic fuel spray modeling for gasoline direct injection engine applications

Fuel spray modeling plays a critical role during modern gasoline direct injection (GDI) engine development due to fuel injection’s dominant impact on engine performance and emissions as well as the complex physical processes involved. In engineering three-dimensional (3D) computational fluid dynamics (CFD) simulations, the liquid-phase fuel atomization, evaporation, and mixing are usually modeled with the discrete droplet model (DDM) adopting a Lagrangian approach for multiphase CFD simulations. To this end, general practices heavily depend on the reduced order characterization of the injector nozzle flow. However, such simplified injector modeling may lead to insufficient representations of the complex spray dynamics. To tackle this problem, this study proposes a novel workflow to numerically evaluate GDI sub-cooled and flash-boiling sprays under engine-relevant conditions using a side-mounted GDI injector together with real gasoline fuel properties. The workflow introduces a one-way coupling (OWC) method leveraging high-fidelity nozzle flow simulations to provide realistic boundary conditions to the Lagrangian injector model. The proposed workflow was first verified in a constant volume chamber (CVC) environment and then implemented in a practical GDI engine setup to study spray morphology, fuel-air mixing, and wall-wetting propensity. In addition, detailed comparison was performed between the OWC method and the conventional rate of injection (ROI) routine. Quantitative analysis of spray characteristics was conducted to highlight possible source of discrepancies of the conventional ROI method.

Numerical simulation of particle flow around the impulsive monopile

In order to explore the particle flow field characteristics around the impulsive monopile, the particle flow field and particle distribution around impulsive monopile were simulated based on the classical impulsive flow around a monopile at medium and low Reynolds number (Re = 550). The Large Eddy Simulation (LES) technique was harnessed to capture the dynamic flow field around the monopile and to facilitate in-depth analysis of vortex evolution. Concurrently, the Lagrangian approach was employed to track and characterize particle trajectories, shedding light on the patterns of particle diffusion and congregation in the vicinity of the monopile under varying Reynolds numbers. Additionally, this investigation unveils the distinctive behavior of particle flow during the impulsive starting and stopping phases adjacent to the monopile. Leveraging these insights, the study elucidates the intricate influence of the unsteady flow field dynamics around the monopile on the progression of particle flow.

Optimum-pursuing method for constrained optimization and reliability-based design optimization problems using Kriging model

This paper proposes a new active learning method named as optimum-pursuing method (OPM) from the viewpoint of optimization theory, which aims to provide an effective tool for solving constrained optimization and reliability-based design optimization (RBDO) problems with low computation cost. It uses the cheap Kriging metamodel to replace the expensive physical response. The novelty of the proposed OPM primarily lies in two aspects. First, the OPM utilizes the advantage of the optimization theory rather than sampling technology. By using the augmented Lagrangian approach, it comprehensively considers the objective, constraints, and their relations, thereby automatic identification of important region in the vicinity of the optimum. Second, the accordingly optimum-pursuing function consists of three parts: Kriging mean, Kriging standard deviation, and merit function. Also, the target reliability surface is further considered to enhance the local accuracy of the reliability analysis. The performance of OPM is tested for both deterministic optimization and problems, in which two mathematical and three real-world engineering examples are selected to showcase the feasibility and validity. The results demonstrate that OPM is promising for solving both deterministic optimization and RBDO problems by comparing with the well-known active learning methods.

A Review of CO2 Plume Dispersion Modeling for Application to Offshore Carbon Capture and Storage

The exponential increase in greenhouse gas emissions necessitates urgent measures to mitigate climate change impacts. Carbon capture and storage (CCS) has emerged as a promising solution, capturing CO2 from industrial processes and storing it underground. However, CCS implementation poses risks that demand sophisticated modeling. This review focuses on the numerical modeling of CO2 plume dispersion from reservoir leaks during offshore CCS projects, including near- and far-field modeling and the comparison between Lagrangian and Eulerian modeling in particular. Near-field modeling examines CO2 behavior in jet plume, considering depth-related changes. Far-field modeling, employing Eulerian and Lagrangian methods, evaluates dispersion in marine environments. Case studies illustrate the complexity and uniqueness of CO2 dispersion events. The Lagrangian approach emphasizes gas bubble tracking, while the Eulerian approach employs fixed grid systems for detailed hydrodynamic modeling. Both approaches contribute valuable insights, with Eulerian models excelling in site-specific complexities and Lagrangian models offering computational efficiency. A hybrid approach may offer a comprehensive understanding of CO2 dispersion.

Operationalizing modular autonomous customised buses based on different demand prediction scenarios

ABSTRACT This paper presents a novel framework for customised modular bus systems that leverages travel demand prediction and modular autonomous vehicles to optimise services proactively. The proposed framework addresses two prediction scenarios with different forward-looking operations: optimistic operation and pessimistic operation. A mixed integer programming model in a space-time-state network is developed for the optimistic operation to determine module routes, schedules, formations and passenger-to-module assignments. For the pessimistic case, a two-stage optimisation procedure is introduced. The first stage involves two formulations (i.e., deterministic and robust) to generate cost-saving plans, and the second stage adapts plans with control strategies periodically. A Lagrangian heuristic approach is proposed to solve formulations efficiently. The performance of the proposed framework is evaluated using smartcard data from Beijing and two state-of-the-art machine learning algorithms. Results indicate that the proposed framework outperforms the real-time approach in operating costs and highlights the role of module capacity and time dependency.

Fully finite volume method on a curvilinear grid-based arbitrary Lagrangian Eulerian approach for computational fluid flexible-structure interaction

For Computational Fluid flexible-Structure Interaction (CFfSI), this paper in on proposition of a unified fluid-dynamic and elastic stresses-based fully Finite Volume Method (FVM)—for both fluid and structural dynamics—on a curvilinear grid for both fluid and structure subdomains. Further, a hybrid method is proposed for the dynamic curvilinear-grid generation in the fluid subdomain. Arbitrary Lagrangian Eulerian (ALE) framework-based governing equations are presented for fluid-flow while those for the structural deformations are presented in the Lagrangian framework; along with the fluid-solid interfacial conditions. Solution methodology is presented, for the ALE method on the curvilinear grid, using a fully implicit pressure projection method for the flow solver and a pseudo-transient time stepping method for the structural solver; along with a fully implicit coupling between the two solvers. For the ALE method-based in-house CFfSI solver, an excellent agreement with the published numerical results is demonstrated (for steady and periodic flow-profiles/structural-deformations) on four different CFfSI problems. Further, a II-order accuracy in space as well as time is demonstrated for the present ALE method. After the validation and order of accuracy studies, a performance study is presented for our CFfSI solver on a dynamic-grid—generated by the proposed hybrid method as compared to a traditional method. The hybrid method resulted in up to 30% reduction in computational time for the four CFfSI problems; used in the validation study. An ALE method, based on a novel fully-FVM and hybrid method for dynamic curvilinear-grid generation, is presented here—for an easy-to-implement, stable, and computationally-efficient CFfSI development.

A Lagrangian Perspective on the Atlantic and Pacific Precipitation‐Evaporation Asymmetry

AbstractTotal precipitation minus evaporation (P − E) for the Atlantic is negative while it is approximately neutral for the Pacific. This has frequently been attributed to westward Atlantic‐to‐Pacific moisture flux across Central America. However, this Eulerian perspective has limited scope as it does not consider the origins of the water crossing ocean drainage basin boundaries and the possibility that it has remote sources. By using an airmass trajectory model, we take a Lagrangian approach to investigate the origin of the moisture contributing to fluxes, Qn, across these boundaries. Qn is partitioned into contributions from each basin, the stratosphere and trajectories not assigned an origin. The total Qn across each basin boundary are mainly composed of contributions from the two adjacent basins while remote or stratospheric origins make small contributions. Partitioning Qn shows that the atmosphere exports ∼1 Sv water vapor from the Atlantic, Indian and Pacific basins with a similar quantity imported to the Pacific. However, Atlantic and Indian atmospheric imports are ∼0.5–0.6 Sv. Normalizing by drainage basin perimeters reveals that the import to these basins is half as efficient as Pacific import. Partitioning P − E into contributions from other basins shows that Pacific moisture import is dominated by trajectories with Indian basin origin (∼38%, or 0.43 Sv, of total Pacific import). The import is greatest in boreal summer due to the Asian monsoon flow and stronger westward flux across Central America. These anomalies dominate the difference in annual imports between the Pacific, Atlantic and Indian basins.

Application of the Euler–Lagrange Approach and Immersed Boundary Method to Investigate the Behavior of Rigid Particles in a Confined Flow

The presence of particles with a small but finite size, suspended in viscous fluids with low volumetric concentrations, is observed in many applications. The present study focuses on the tridimensional and incompressible lid-driven flow of Newtonian fluids through the application of the immersed boundary method and the Euler–Lagrange approach. These methods are used to numerically predict three-dimensional particle motion by considering nearly neutrally buoyant conditions as well as all relevant elementary processes (drag and lift forces, particle rotation, particle–wall interactions, and coupling between phases). Considering the current stage of the numerical platform, two coupling approaches between phases are considered: one-way and two-way coupling. A single particle is inserted in the cavity after steady-state conditions are achieved. Its three-dimensional motion is obtained from numerical simulations and compared with research data, considering the same conditions, evidently showing that the particle trajectory follows the experimental data until the first collision with a solid surface. After this first contact, there is a deviation between the results, with the two-way coupling results better representing the experimental data than the one-way coupling results. The dimensionless forces’ peaks acting on the particles are associated with the relative velocity of the particle near the wall–particle collision position. In terms of magnitude, in general, the drag force has shown greater influence on the particle’s motion, followed by the rotation-induced and shear-induced lift forces. Finally, a special application is presented, in which 4225 particles are released into the domain and their dynamic is evaluated throughout dimensionless time, showing similar behavior for both couplings between phases, with variations in local concentrations observed in certain regions. The mean square displacement used to quantify the dispersion evolution of the particles showed that the particulate flow reaches an approximately homogeneous distribution from the moment of dimensionless time tU/S = 130.

Aspiration efficiency and respiratory tract deposition of indoor suspended micro-particles during steady and transient breathings

The assessment of potential health risks from exposure to airborne contaminants and the inhalability of microparticles through human nasal breathing is crucial for the design of ventilation systems. In this study, using a standing computer-simulated person (CSP) directly linked to a numerical model of the respiratory tract, we investigated the aspiration efficiency (AE) of microparticles ranging between 1 and 80 μm under steady and transient breathing conditions. Two ventilation scenarios with displacement and mixed ventilation systems (DV and MV) were assumed for AE calculations. To determine the appropriate particle injection location, we investigated the position of highly inhaled particles corresponding to the breathing zone using a reverse particle-tracking simulation with reversed time progression. Additionally, the total and regional deposition fractions (TDF and RDF) of the inhaled microparticles on the respiratory wall surfaces were evaluated using the Lagrangian approach. The results indicate that the transient breathing cycle showed relatively higher AE and TDF rates compared to the assumed steady inhalation conditions, particularly for the size range 1–10 μm. An insignificant variation was observed in the AE and RDF results between different ventilation systems. However, microparticles in the MV room showed slightly higher AE than the DV system. Moreover, AE and penetration ratio were highly sensitive to the initially assumed particle density. For a comprehensive and accurate assessment of inhalation exposure to microparticles, we should predict the heterogeneous flow field formed around the human body and then analyze the AE and TDF in the respiratory tract during realistic transient breathing.

A mechanistic analysis of particle flow in a pulsatile circulating dual fluidized bed for chemical looping technology based on CDF-DEM simulation

This paper presents a three-dimensional computational fluid dynamic–discrete element method (CFD-DEM) model using Geldart class D particles for studying the fluid flow pattern in a circulating dual fluidized bed (CDFB) for chemical looping dry reforming of methane (CLDRM). The system design comprised a high-velocity air reactor (AR) also known as riser fluidized bed, a riser tube, a cyclone chamber, a fluidizing bed fuel reactor (FR), and a connecting loop, which closes the loop between AR and FR. The particles distribution, volume fraction, flow pattern, velocity and their circulation between the AR (riser) and FR (second fluidized bed) in CDFB were deeply investigated. The results showed that the CLDRM system experiences a pressure imbalance between the reduction and oxidation zones, causing solid particles to undergo a highly intricate and turbulent pulse flow. As a result, the operation of the system is alternately governed by two flow regimes: bubbling fluidization and pulsatile circulating fluidization. The burst of a high pressure pulses provide the pressure head required for circulation of the particles. The pulses also led to the intermittent transportation of particles ― in the form of particles clusters ― from high-pressure area to the region with a lower pressure.

Dynamics analysis of a rigid-flexible-liquid coupled satellite antenna system via absolute nodal coordinate formulation curvature continuity constraints

Typically, Euler and Lagrangian approaches are primarily used in fluid dynamics and multibody system (MBS) dynamics models, respectively. Absolute nodal coordinate formulation (ANCF), as a Lagrangian nonlinear finite element method (FEM), can effectively handle large deformation problems and incorporate the arbitrary Lagrange-Eulerian (ALE) method for fluids turbulence and breaking. Therefore, we have established a rigid-flexible-liquid coupled satellite antenna system dynamics model via ANCF. Navier-Stokes equations and virtual work principle are applied to formulate the fluid-structure interaction (FSI) dynamics equations. A penalty function is used to describe liquid incompressibility. Bi-directional FSI effects are solved simultaneously and brick fluid element elastic force is validated with reference experiments. Moreover, we introduce curvature continuity constraints to make the liquid free surface continuous. Results indicate that curvature continuity constraints can smoother liquid free surface and eliminate the protrusions and depressions caused by assembling brick fluid elements. Furthermore, we achieve simulation results consistent with reference experiments by using fewer elements with curvature continuity constraints. In addition, due to bi-directional FSI effects, liquid sloshing will magnify satellite antenna vibrations and simultaneously oscillate the advancing tank.

Search for the anomalous quartic gauge couplings through Z γ production at e − e + colliders

Spontaneous breaking of the electroweak symmetry of the standard model (SM) sets the constraints on triple and quartic gauge couplings. Therefore, the measurement of multiboson production in e − e + collisions allows us to directly test the predictions of the SM and to indirectly probe new physics beyond the SM. In this paper, we focus on the process e − e + → e − Z γ e + with Z boson decaying into neutrinos to study the anomalous quartic gauge couplings using the effective Lagrangian approach at the compact linear collider (CLIC). We obtain the sensitivities on the anomalous f T,i /Λ4 (i = 0, 2, 5, 6, 7, 8, 9) couplings taking into account the systematic uncertainties of 3, 5% at 95% Confidence Level for the CLIC with TeV. Our results show that the sensitivities on some anomalous couplings without systematic errors are up to two orders of magnitude better than the current experimental limits. Considering a realistic systematic uncertainty such as 5% from possible experimental sources, the sensitivity of all anomalous quartic couplings becomes worse by about 10% compared to those without systematic uncertainty for the CLIC.

Numerical simulation of rime ice accretion on a three-dimensional wind turbine blade using a Lagrangian approach

Numerical simulation of rime ice accretion on a three-dimensional wind turbine blade using a Lagrangian approach

Numerical Simulation of Particulate Matter Transport in the Atmospheric Urban Boundary Layer Using the Lagrangian Approach: Physical Problems and Parallel Implementation

Numerical Simulation of Particulate Matter Transport in the Atmospheric Urban Boundary Layer Using the Lagrangian Approach: Physical Problems and Parallel Implementation

Detailed simulations of one-dimensional detonation propagation in dilute n-heptane spray and air mixtures

One-dimensional numerical simulations based on the hybrid Eulerian–Lagrangian approach are performed to investigate detonation dynamics in two-phase gas-droplet n-heptane/air mixtures with and without liquid fuel pre-vaporization. The reactive Navier–Stokes equations considering the two-way coupling for interphase exchanges of mass, momentum, energy, and species are solved with a skeletal mechanism consisting of 44 species and 112 reactions. The effects of n-heptane droplet diameter and equivalence ratio (ER) on average detonation speed and mode are studied. For pre-vaporization cases, the average detonation speed first decreases and then increases with droplet diameter ranging from 2.5 to 40 μm, which is minimum at 7.5–10 μm due to the competition between fuel vapor addition and droplet evaporative heat absorption. However, the average speed increases monotonically as the droplet ER increases from 0.2 to 1.2. A further increase in the droplet ER (e.g., 2.4) would lead to detonation suppression in the presence of large droplets (e.g., above 30 μm). The detonation is fully quenched when the droplet ER is 3.2. Similar observations are also made for the pure sprayed cases without n-heptane pre-vaporization, where the average speed increases rapidly for droplet ER of 0.2–0.8 and slowly for ER of 0.8–1.6. Various detonation modes are observed with respect to droplet diameter and equivalence ratio, either with or without fuel pre-vaporization. Generally, the pure sprayed cases show more irregular behaviors in detonation propagation. The laden droplets provide a new approach to control the intrinsically unstable or highly irregular behaviors of pure gas or pure sprayed detonations. The finite, small disturbances from the spatially non-uniform droplets, and enrichment from the droplet evaporative mass addition, are two essential mechanisms for the mitigation of the pulsating detonation.

An augmented Lagrangian approach for cardinality constrained minimization applied to variable selection problems

To solve convex constrained minimization problems, that also include a cardinality constraint, we propose an augmented Lagrangian scheme combined with alternating projection ideas. Optimization problems that involve a cardinality constraint are NP-hard mathematical programs and typically very hard to solve approximately. Our approach takes advantage of a recently developed and analyzed continuous formulation that relaxes the cardinality constraint. Based on that formulation, we solve a sequence of smooth convex constrained minimization problems, for which we use projected gradient-type methods. In our setting, the convex constraint region can be written as the intersection of a finite collection of convex sets that are easy and inexpensive to project. We apply our approach to a variety of over and under determined constrained linear least-squares problems, with both synthetic and real data that arise in variable selection, and demonstrate its effectiveness.

On Distributed Detection in EH-WSNs With Finite-State Markov Channel and Limited Feedback

We consider a network of N sensors, tasked with solving binary distributed detection, a fusion center (FC), and a feedback channel from the FC to sensors. Each sensor is capable of harvesting energy and is equipped with a finite-size battery to store randomly arrived energy. Sensors process their observations and transmit their symbols to the FC over orthogonal and Markovian time correlated fading channels. The FC fuses the received symbols and makes a global binary decision. We aim at developing adaptive channel-dependent transmit power control policies such that J-divergence based detection metric is maximized at the FC, subject to total transmit power constraint. Modeling quantized fading channel, energy arrival, and battery dynamics as time-homogeneous finite-state Markov chains, and the network lifetime as a geometric random variable, we formulate our power control optimization problem as a discounted infinite-horizon constrained Markov decision process (MDP) problem, where sensors’ transmit powers are functions of the battery states, quantized channel gains, and the arrived energies. We utilize stochastic dynamic programming and Lagrangian approach to find the optimal and sub-optimal power control policies. We demonstrate that our sub-optimal policy provides a close-to-optimal performance with a reduced computational complexity and without imposing signaling overhead on sensors.