Lagrangian Coordinates Research Articles

A Comprehensive Model and Numerical Study of Shear Flow in Compressible Viscous Micropolar Real Gases

Understanding shear flow behavior in compressible, viscous, micropolar real gases is essential for both theoretical advancements and practical engineering applications. This study develops a comprehensive model that integrates micropolar fluid theory with compressible flow dynamics to accurately describe the behavior of real gases under shear stress. We formulate the governing equations by incorporating viscosity and micropolar effects and transform the obtained system into the mass Lagrangian coordinates. Two numerical methods, Faedo–Galerkin approximation and finite-difference methods, are used to solve it. These methods are compared using several benchmark examples to assess their accuracy and computational efficiency. Both methods demonstrate good performance, achieving equally precise results in capturing essential flow characteristics. However, the finite-difference method offers advantages in speed, stability, and lower computational demands. This research bridges gaps in existing models and establishes a foundation for further investigations into complex flow phenomena in micropolar real gases.

Global well-posedness of the three-dimensional free boundary problem for viscoelastic fluids without surface tension

In this paper, we consider the three-dimensional free boundary problem of incompressible and compressible neo-Hookean viscoelastic fluid equations in an infinite strip without surface tension, provided that the initial data is sufficiently close to the equilibrium state. By reformulating the problems in Lagrangian coordinates, we can get the stabilizing effect of elasticity. In both cases, we utilize the elliptic estimates to improve the estimates. Moreover, for the compressible case, we find there is an extra ODE structure that can improve the regularity of the free boundary, thus we can have the global well-posedness. To prove the global well-posedness for the incompressible case, we employ two-tier energy method introduced in [11][12][13] to compensate for the inferior structure.

Conservation laws of the one-dimensional relativistic magnetogasdynamics equations

Abstract The present paper offers a comprehensive analysis of the one-dimensional relativistic magnetogasdynamics equations in Lagrangian coordinates. These equations, describing the behavior of a relativistic magnetized gas, are reformulated in variational form to facilitate further analysis. A complete group analysis of the resulting Euler–Lagrange equation is performed, allowing us to systematically identify the symmetries inherent in the system. Using Noether’s theorem, we derive conservation laws in Lagrangian coordinates, offering critical insights into the invariants and conserved quantities of the system. This work expands the theoretical understanding of relativistic magnetogasdynamics.

Local smooth solutions to the Euler-Poisson equations for semiconductor in vacuum

In this paper, we study the initial-boundary value problem to the Euler-Poisson equations for semiconductors, which involves the vacuum for the electronic density, a challenging case because of its degeneracy and singularity. The main issue is to investigate the local well-posedness of smooth solutions to the isentropic system with an adiabatic exponent γ>1, a degenerate hyperbolic-elliptic system on the free boundary. By setting the system to the Lagrangian coordinates, we reduce it to the quasi-linear wave equation coupling the Poisson equations, where the initial degeneracy can be explicitly expressed by the function ρ0γ−1 of the initial density ρ0, which equals to the distance function near boundaries. By applying the Hardy inequality and weighted Sobolev spaces depending on the distance function, we can overcome the degeneracy and singularity of the system caused by the vacuum, and we technically establish some crucial priori estimates and then prove the existence and uniqueness of the local smooth solution. This is the first result on the smooth solution to the Euler-Poisson equations for semiconductors in vacuum.

Nonlinear dynamics of a flexible rod partially sliding in a rigid sleeve under the action of gravity and configurational force

We investigate various methods of analyzing systems with moving boundaries, using as an example a flexible rod sliding in an ideal frictionless sleeve in the field of gravity. Special attention is paid to the configurational force acting on the rod at the sleeve opening and thus determining the rod’s dynamics. The non-material kinematic description used in simulations is based on the re-parametrization of the Lagrangian arc length coordinate. The variational formulation uses the energy expressions written for the entire rod, comprising the free segment and the one inside the sleeve. A novel finite element scheme is efficient for highly flexible rods, which may undergo complete ejection. A simplified two degrees of freedom model, which accelerates simulations, shows a good agreement as the bending stiffness increases. An analytical study using Hamiltonian mechanics exploits the separation of variables into fast oscillations and slow axial motion. The adiabatic invariant approach leads to approximate closed-form solutions for the slow dynamics and yields the maximum injection depth of the rod into the sleeve.

Unsteady Lifting-Line Theory for Camber Morphing Wings State-Space Aeroelastic Modeling

A novel frequency-domain analytical-numerical model for the aerodynamic solution of camber morphing wings is developed. The model combines an unsteady lifting-line formulation with Küssner–Schwarz aerodynamic theory to provide the pressure distribution and thus the transcendental aerodynamic matrix that relates the generalized aerodynamic forces to the Lagrangian coordinates of a given wing structural dynamics model. The state-space form of the aerodynamic loads is obtained from the rational matrix approximation of the aerodynamic matrix. This, combined with the wing structural dynamics operator, can readily provide the state-space form of the aeroelastic problem. To assess the accuracy of the proposed aerodynamic solution method, numerical investigations are performed. These consist of its application to several conventional wing configurations and wings with camber morphing, followed by the comparisons of the corresponding solutions with the predictions given by a well-validated panel-method solver for potential flows. These results are shown to be in very good agreement, thus demonstrating that the proposed approach can capture the effects due to wing tapering, sweep angle, and camber morphing, while requiring a remarkably lower computational effort. The excellent accuracy of a few-pole finite-state approximation of the aerodynamic matrix is finally proven for a swept wing with camber morphing.

Relation Between the Single Orths of a Point on the Axis of the Deformed Thread in the Case of Euler and Lagrange Coordinates

The values of the position vectors and unit tangents have been determined by research, defining the positions of points on the axis of the deformed thread using Eulerian and Lagrangian coordinates. The position vector defines the point's position on the line relative to the moving coordinate system. It determines the point's position on the line relative to the stationary coordinate system. Consequently, when the thread is displaced, the point moves to a new position. The new position vector defines the point's position on the line. The displacement vector of the point on the thread axis characterizes the shift of points due to deformation at the thread intersection. This has made it possible to use these theoretical relation at the initial stage of designing the technological process to derive equations for determining thread tension in the fabric and knit forming zone, reduce thread breakage rates, and enhance the quality of the finished products

An energy‐based finite‐strain model for 3D heterostructured materials and its validation by curvature analysis

AbstractThis paper presents a comprehensive study of the intrinsic strain response of 3D heterostructures arising from lattice mismatch. Combining materials with different lattice constants induces strain, leading to the bending of these heterostructures. We propose a model for nonlinear elastic heterostructures such as bimetallic beams or nanowires that takes into account local prestrain within each distinct material region. The resulting system of partial differential equations (PDEs) in Lagrangian coordinates incorporates a nonlinear strain and a linear stress‐strain relationship governed by Hooke's law. To validate our model, we apply it to bimetallic beams and hexagonal hetero‐nanowires and perform numerical simulations using finite element methods (FEM). Our simulations examine how these structures undergo bending under varying material compositions and cross‐sectional geometries. In order to assess the fidelity of the model and the accuracy of simulations, we compare the calculated curvature with analytically derived formulations. We derive these analytical expressions through an energy‐based approach as well as a kinetic framework, adeptly accounting for the lattice constant mismatch present at each compound material of the heterostructures. The outcomes of our study yield valuable insights into the behavior of strained bent heterostructures. This is particularly significant as the strain has the potential to influence the electronic band structure, piezoelectricity, and the dynamics of charge carriers.

The Amazon River plume—a Lagrangian view

AbstractHydrographic data, nutrient data and bulk rates of nitrate uptake and primary production were determined in the Amazon River plume (ARP) in the Western Tropical North Atlantic (WTNA) during three cruises in May 2018, June/July 2019, with RV Endeavor and April/May 2021 with RV Meteor. Using daily quasi‐geostrophic surface velocity data from satellite observations, the geographical positions of the stations of observations were transformed onto Lagrangian coordinates to obtain a dynamically coherent and consistent spatial distribution. After the transformation, the observed surface salinity and temperature fields were consistent with the flow fields, the ARP formed a coherent structure and the retroflection of the North Brazil Current became visible. By transforming other surface variables such as nitrate concentration, photosynthetically available radiation, turbidity, bulk rates of nitrate uptake, and primary production onto Lagrangian coordinates, patterns became consistent with the physical variables at the surface. The use of “synchronous” fields as done here by transformation onto Lagrangian coordinates is essential for spatially structured analyses of data collected over tens of days in a highly dynamic region characterized by complex flow fields with low persistence such as the WTNA. Therefore, the use of the Lagrangian method provides a powerful tool for exploring spatial distributions of biologically relevant factors in regions with complex and dynamic flow patterns. These spatial distributions are qualitatively in agreement with satellite images of daily sea surface temperature and composites of monthly mean Chlorophyll a distributions.

On the well-posedness of the Cauchy problem for the two-component peakon system in Ck∩Wk,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^k\\cap W^{k,1}$$\\end{document}

This study focuses on the Cauchy problem associated with the two-component peakon system featuring a cubic nonlinearity, constrained to the class (m,n)∈Ck(R)∩Wk,1(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(m,n)\\in C^{k}(\\mathbb {R}) \\cap W^{k,1}(\\mathbb {R})$$\\end{document} with k∈N∪{0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\in \\mathbb {N}\\cup \\{0\\}$$\\end{document}. This system extends the celebrated Fokas–Olver–Rosenau–Qiao equation and the following nonlocal (two-place) counterpart proposed by Lou and Qiao: ∂tm(t,x)=∂x[m(t,x)(u(t,x)-∂xu(t,x))(u(-t,-x)+∂x(u(-t,-x)))],\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\partial _t m(t,x)= \\partial _x[m(t,x)(u(t,x)-\\partial _xu(t,x)) (u(-t,-x)+\\partial _x(u(-t,-x)))], \\end{aligned}$$\\end{document}where m(t,x)=1-∂x2u(t,x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m(t,x)=\\left( 1-\\partial _{x}^2\\right) u(t,x)$$\\end{document}. Employing an approach based on Lagrangian coordinates, we establish the local existence, uniqueness, and Lipschitz continuity of the data-to-solution map in the class Ck∩Wk,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^k\\cap W^{k,1}$$\\end{document}. Moreover, we derive criteria for blow-up of the local solution in this class.

Local existence of compressible magnetohydrodynamic equations without initial compatibility conditions

In this paper, we study the initial‐boundary value problem of three‐dimensional viscous, compressible, and heat conductive magnetohydrodynamic equations. Local existence and uniqueness of strong solutions are established with any such initial data that the initial compatibility conditions do not be required. The analysis is based on some suitable prior estimates for the strong coupling term and strong nonlinear term . Our proof of the existence and uniqueness of solutions is in the Lagrangian coordinates first and then transformed back to the Euler coordinates.

On the physical vacuum free boundary problem of the 1D shallow water equations coupled with the Poisson equation

This paper is concerned with the vacuum free boundary problem of the 1D shallow water equations coupled with the Poisson equation. We establish the local-in-time well-posedness of classical solutions to this system, and the solutions possess higher-order regularity all the way to the vacuum free boundary, though the density degenerates near the vacuum boundary. To deal with the force term generated by the Poisson equation, we make use of the structure of the momentum equation formulated in a fixed domain by the Lagrangian coordinates. The proof is built on some higher-order weighted energy functionals and weighted embeddings corresponding to the degeneracy near the initial vacuum boundary.

Symmetry Analysis of the Two-Dimensional Stationary Gas Dynamics Equations in Lagrangian Coordinates

This article analyzes the symmetry of two-dimensional stationary gas dynamics equations in Lagrangian coordinates, including the search for equivalence transformations, the group classification of equations, the derivation of group foliations, and the construction of conservation laws. The consideration of equations in Lagrangian coordinates significantly simplifies the procedure for obtaining conservation laws, which are derived using the Noether theorem. The final part of the work is devoted to group foliations of the gas dynamics equations, including for the nonstationary isentropic case. The group foliations approach is usually employed for equations that admit infinite-dimensional groups of transformations (which is exactly the case for the gas dynamics equations in Lagrangian coordinates) and may make it possible to simplify their further analysis. The results obtained in this regard generalize previously known results for the two-dimensional shallow water equations in Lagrangian coordinates.

A nonlocal Lagrangian traffic flow model and the zero-filter limit

In this study, we start from a Follow-the-Leaders model for traffic flow that is based on a weighted harmonic mean (in Lagrangian coordinates) of the downstream car density. This results in a nonlocal Lagrangian partial differential equation (PDE) model for traffic flow. We demonstrate the well-posedness of the Lagrangian model in the L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} sense. Additionally, we rigorously show that our model coincides with the Lagrangian formulation of the local LWR model in the “zero-filter” (nonlocal-to-local) limit. We present numerical simulations of the new model. One significant advantage of the proposed model is that it allows for simple proofs of (i) estimates that do not depend on the “filter size” and (ii) the dissipation of an arbitrary convex entropy.

Invariant conservative finite-difference schemes for the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates

Invariant finite-difference schemes for the one-dimensional shallow water equations in the presence of a magnetic field for various bottom topographies are constructed. Based on the results of the group classification recently carried out by the authors, finite-difference analogues of the conservation laws of the original differential model are obtained. Some typical problems are considered numerically, for which a comparison is made between the cases of a magnetic field presence and when it is absent (the standard shallow water model). The invariance of difference schemes in Lagrangian coordinates and the energy preservation on the obtained numerical solutions are also discussed.

Fluid Structure Interaction Using Modal Superposition and Lagrangian CFD

This study investigates the impact of fluid loads on the elastic deformation and dynamic response of linear structures. A weakly coupled modal solver is presented, which involves the solution of a dynamic equation of motion with external loads. The mode superposition method is used to find the dynamic response, utilizing predetermined mode shapes and natural frequencies associated with the structure. These essential parameters are pre-calculated and provided as input for the simulation. Integration of the weakly coupled modal solver is accomplished with the Lagrangian Differencing Dynamics (LDD) method. This method can directly use surface mesh as boundary conditions, so it is much more convenient than other meshless CFD methods. It employs Lagrangian finite differences, utilizing a strong formulation of the Navier–Stokes equations to model an incompressible free-surface flow. The elastic deformation of the structure, induced by fluid forces obtained from the flow solver, is computed within the modal coupling algorithm through direct numerical integration. Subsequently, this deformation is introduced into the flow solver to account for changes in geometry, resulting in updated flow pressure and velocity fields. The flow particles and vertices of the structure are advected in Lagrangian coordinates, resulting in Lagrangian–Lagrangian coupling in spaces with weak or explicit coupling in time. The two-way coupling between fluid and structure is successfully validated through various FSI benchmark cases. The efficiency of the LDD method is highlighted as it operates directly on surface meshes, streamlining the simulation setup. Direct coupling of structural deformation eliminates the conventional step of mapping fluid results onto the structural mesh and vice versa.

On the stability of elastic structures subjected to follower forces

Within the framework of a second-order theory, this study revisits classical stability problems characterized by critical loads associated with dynamic instability, previously explored by Feriani and Carini (in: Zingoni (ed) Insights and innovations in structural engineering, mechanics and computation, 2016) . The primary focus remains on systems exclusively comprising a single lumped mass. This simplification, together with the assumption of a negligible axial strain and the adoption of a second-order theory, reduces the analysed systems to problems featuring a single Lagrangian coordinate. Consequently, static methods become applicable, facilitating the derivation of analytical expressions for stiffness coefficients and enabling an investigation into dynamic stability. The study begins by examining a well-established case: a cantilever beam with a lumped mass positioned at its free end, subjected to a follower load, as presented in Panovko and Gubanova (Stability and oscillations of elastic systems: models, paradoxes and errors, Nunka Press, Moscow, 1967). In this current work, a novel lumped mass system is explored. The system consists of a straight-axis beam characterized by a constant cross-section area and stiffness. The distributed mass of the beam is neglected and the beam is hinged at one end, simply supported at an intermediate point, and left free at the other end, where a lumped mass is introduced. Various loading scenarios are scrutinized, including: (a) The application of a follower force to the free end; (b) The imposition of two forces—one conservative and one follower—both applied to the free end; and (c) The application of a uniformly distributed follower force along the length of the beam. As seen in the examples introduced in Feriani and Carini (2016), the new examples considered in the present paper reveal that the first asymptote of the stiffness coefficient corresponds to the critical load. This critical load corresponds to the phenomenon known as divergence at infinity, as described by Felippa (Nonlinear finite element methods, 2014). It is also confirmed that, in all cases, the first dynamical critical load equals the minimum value between the static buckling load of the original structure and the static buckling load of an auxiliary structure. This last differs from the original one because the concentrated mass is replaced by a constraint fixing the corresponding Lagrangian coordinate.

An efficient forward semi-Lagrangian model

An efficient forward trajectory model is proposed, in which the property and position of the fluids advected from the Euler coordinates to the Lagrangian coordinates can be accurately evaluated. After sorting and aligning those fluid elements on the irregular Lagrangian curves, we apply the cubic or other high-degree polynomials to interpolate the properties of the elements from the irregular curves to the regular grids. There is no need to solve the cubic equations and the associated coefficients as proposed previously. The model is quite simple, accurate, and much more efficient than the previous models. It also allows higher-order polynomials to be employed in the interpolations. It is suitable for simulating the multi-dimensional fast-moving flows with large Courant Numbers, the transport of pollutants in the atmosphere and ocean, and movement of raindrops in atmospheric models.

Local well-posedness of 1D degenerate drift diffusion equation

<abstract><p>This paper proves the well-posedness of locally smooth solutions to the free boundary value problem for the 1D degenerate drift diffusion equation. At the free boundary, the drift diffusion equation becomes a degenerate hyperbolic-Poisson coupled equation. We apply the Hardy's inequality and weighted Sobolev spaces to construct the appropriate a priori estimates, overcome the degeneracy of the system and successfully establish the existence of solutions in the Lagrangian coordinates.</p></abstract>

POLARIS-LC: A Multi-Class traffic Flow Model in Lagrangian Coordinates for Large-Scale Simulation

Agent-based modeling has become increasingly prevalent in transportation systems simulation as the scenarios around new technologies and policies become increasingly complex. This increased complexity, in terms of traveler behavior, traffic flow, modal operations, system management, and so on, requires increasingly sensitive and detailed representation of the core components of the simulation, especially as it relates to traffic flow. Many future mobility solutions rely on connectivity, communication, advanced sensing, and detailed information flows, while the simulation also needs to remain computationally tractable. In this paper, we propose a new Lagrangian Coordinate within the POLARIS Agent-Based-Modeling Framework (LC-POLARIS) of traffic flow, which combines computational efficiency while adding microscopic features. This model allows multi-class traffic flow by setting different speed-spacing relationships. In this modeling, automated vehicles have lower reaction time, while light-duty trucks have lower free-flow speed and higher jam spacing. The model captures the reduction in capacity and higher queue spillback because of higher jam spacing of trucks. Also, the model yields higher capacity for both passenger and light duty automated vehicles on expressways. The LC-ABM traffic flow model is validated in Bloomington, IL, USA, in a scenario with different rates of four vehicle types.