Lagrangian Solution Research Articles

An Improved Scheme for the Finite Difference Approximation of the Advective Term in the Heat or Solute Transport Equations

AbstractTransport equations are widely used to describe the evolution of scalar quantities subject to advection, dispersion and, possibly, reactions. Numerical methods are required to solve these equations in applications, adopting either the advective or conservative formulations. Conservative formulations are usually preferred in practice because they conserve mass. Advective formulations do not, but have received more mathematical attention and are required for Lagrangian solution methods. To obtain an advective formulation that conserves mass, we subtract the discretized fluid flow equation, multiplied by concentration, from the conservative form of the transport equation. The resulting scheme not only conserves mass, but is also elegant in that it can be interpreted as averaging the advective term at cell interfaces, instead of approximating it at cell centers as in traditional centered schemes. The two schemes are identical when fluid velocity is constant, and both have second-order convergence, but the truncation errors are slightly different. We argue that the error terms appearing in the proposed scheme actually imply an improved representation of subgrid spreading/contraction and acceleration/deceleration caused by variable velocity. We compare the proposed and traditional schemes on several problems with variable velocity caused by recharge, discharge or evaporation, including two newly developed analytical solutions. The proposed method yields results that are slightly, but consistently, better than the traditional scheme, while always conserving mass (i.e., mass at the end equals mass at the beginning plus inputs minus outputs), which the traditional centered finite differences scheme does not. We conclude that this scheme should be preferred in finite difference solutions of transport.

A meta-heuristic extension of the Lagrangian heuristic framework

Lagrangian heuristics for discrete optimization work by modifying Lagrangian relaxed solutions into feasible solutions to an original problem. They are designed to identify feasible, and hopefully also near-optimal, solutions and have proven to be highly successful in many applications. Based on a primal-dual global optimality condition for non-convex optimization problems, we develop a meta-heuristic extension of the Lagrangian heuristic framework. The optimality condition characterizes (near-)optimal solutions in terms of near-optimality and near-complementarity measures for Lagrangian relaxed solutions. The meta-heuristic extension amounts to constructing a weighted combination of these measures, thus creating a parametric auxiliary objective function, which is a close relative to a Lagrangian function, and embedding a Lagrangian heuristic in a search procedure in the space of the weight parameters. We illustrate and make a first assessment of this meta-heuristic extension by applying it to the generalized assignment and set covering problems. Our computational experience show that the meta-heuristic extension of a standard Lagrangian heuristic can significantly improve upon solution quality.

Lattice Boltzmann simulations of unsteady Bingham fluid flows

Transient flows of viscoplastic fluids have very peculiar characteristics. The startup and cessation flows of viscoplastic materials have been subject to many theoretical and numerical investigations. The most challenging aspect of numerical solutions of viscoplastic fluids is the viscosity singularity during the transition from yielded to unyielded material. Hence, the proper representation of yield surfaces is the most critical aspect of numerical methods in viscoplastic fluid flow. In the present work, we use a lattice Boltzmann scheme to solve an ideal Bingham fluid’s startup and cessation flows. This numerical scheme advantage is that can represent infinite viscosity without noticeable numerical instabilities, producing yield surfaces with more accuracy and quality. Theoretical solutions for the startup flow are available in the literature. However, it is unclear which is more accurate and what their validity ranges are. Nonetheless, these solutions served as a reference for the present simulations. The overall aspect of the numerical solutions agreed with the theoretical models. The cessation flow of the Bingham fluid was also simulated. Unlike a Newtonian fluid, this type of flow is known to have a finite period until cessation. The simulations correctly reproduced this behavior. The transient yield surfaces matched very well with augmented Lagrangian solutions.

Overall constitutive properties of stratified lattices with alternating chirality.

The present research focuses on a continuum description of stratified metamaterials achieved through the superposition of layers with alternating chirality. Each layer is constructed as a periodic assembly of centre-symmetric periodic cells, formed by a recurring arrangement of rigid circular discs connected by elastic ligaments. The layers are interconnected through elastic pins passing through the centres of aligned discs, allowing for either restrained or free relative rotation. A micropolar continuum model is used to describe each individual layer. The overall response of the metamaterials to in-plane forces is derived using a multi-field non-local model, expressed in terms of the average and difference of displacement and rotational fields. The overall micropolar and standard (Cauchy) constitutive tensors have been determined in closed form. The validity of the equivalent generalized micropolar model has been confirmed through comparison with discrete Lagrangian solutions of representative examples. In addition, a detailed analysis of a pseudo-indentation test has been carried out. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'.

Existence and stability of weak solutions of the Vlasov–Poisson system in localised Yudovich spaces

We consider the Vlasov–Poisson system both in the repulsive (electrostatic potential) and in the attractive (gravitational potential) cases. Our first main theorem yields the analog for the Vlasov–Poisson system of Yudovich’s celebrated well-posedness theorem for the Euler equations: we prove the uniqueness and the quantitative stability of Lagrangian solutions f=f(t,x,v) whose associated spatial density ρf=ρf(t,x) is potentially unbounded but belongs to suitable uniformly-localised Yudovich spaces. This requirement imposes a condition of slow growth on the function p↦‖ρf(t,⋅)‖Lp uniformly in time. Previous works by Loeper, Miot and Holding–Miot have addressed the cases of bounded spatial density, i.e. ‖ρf(t,⋅)‖Lp≲1 , and spatial density such that ‖ρf(t,⋅)‖Lp∼p1/α for α∈[1,+∞) . Our approach is Lagrangian and relies on an explicit estimate of the modulus of continuity of the electric field and on a second-order Osgood lemma. It also allows for iterated-logarithmic perturbations of the linear growth condition. In our second main theorem, we complement the aforementioned result by constructing solutions whose spatial density sharply satisfies such iterated-logarithmic growth. Our approach relies on real-variable techniques and extends the strategy developed for the Euler equations by the first and fourth-named authors. It also allows for the treatment of more general equations that share the same structure as the Vlasov–Poisson system. Notably, the uniqueness result and the stability estimates hold for both the classical and the relativistic Vlasov–Poisson systems.

Dynamic Mesh Simulations in OpenFOAM: A Hybrid Eulerian–Lagrangian Approach

The past few decades have witnessed a growing popularity in Eulerian–Lagrangian solvers due to their significant potential for simulating aerodynamic flows, particularly in cases involving strong body–vortex interactions. In this hybrid approach, the two component solvers are mutually coupled in a two-way fashion. Initially, the Lagrangian solver can supply boundary conditions to the Eulerian solver, while the Eulerian solver functions as a corrector for the Lagrangian solution in regions where the latter cannot achieve high accuracy. To utilize such tools effectively, it is vital for them to be capable of handling dynamic mesh movements. This study builds upon the previous research conducted by our team and extends the capabilities of the hybrid solver to handle dynamic meshes. While OpenFOAM, the Eulerian component of this hybrid code, incorporates built-in dynamic mesh properties, certain modifications are necessary to ensure its compatibility with the Lagrangian solver. More specifically, the evolution algorithm of the pimpleFOAM solver needs to be divided into two discrete steps: first, updating the mesh, and later, evolving the solution. This division enables a proper coupling between pimpleFOAM and the Lagrangian solver as an intermediate step. Therefore, the primary objective of this specific paper is to adapt the OpenFOAM solver to meet the demands of the hybrid solver and subsequently validate that the hybrid solver can effectively address dynamic mesh challenges using this approach. This approach introduces a pioneering method for conducting dynamic mesh simulations within the OpenFOAM framework, showcasing its potential for broader applications. To validate the approach, various test cases involving dynamic mesh movements are employed. Specifically, all these cases employ the Lamb–Oseen diffusing vortex, but each case incorporates different types of mesh movements, including translational, rotational, oscillational, and combinations thereof. The results from these cases demonstrate the effectiveness of the proposed OpenFOAM algorithm, with the maximum relative errors —when compared to the analytical solution across all presented cases—capped at 2.0% for the worst-case scenario. This affirms the algorithm’s capability to successfully handle dynamic mesh simulations with the proposed solver.

Strong convergence of the vorticity and conservation of the energy for the α-Euler equations

In this paper, we study the convergence of solutions of the α-Euler equations to solutions of the Euler equations on the two-dimensional torus. In particular, given an initial vorticity ω 0 in Lxp for p∈(1,∞) , we prove strong convergence in Lt∞Lxp of the vorticities q α , solutions of the α-Euler equations, towards a Lagrangian and energy-conserving solution of the Euler equations. Furthermore, if we consider solutions with bounded initial vorticity, we prove a quantitative rate of convergence of q α to ω in Lp , for p∈(1,∞) .

Novel Exact Solution of Elastic Catenary and Applications on Floating Wind Turbine Mooring Systems

Abstract This research addresses the mathematical solution of the elastic catenary, a fundamental problem in offshore mooring engineering. A novel exact solution in a non-Lagrangian form is developed through rigorous mathematical derivation, distinguishing it from classical Lagrangian solutions. The procedure is reported in detail, and the resulting expressions are applied to analyze the mooring system of a reference floating turbine. A general approach is introduced to solve the transcendental equations associated with catenary mooring problems. The newly derived formulae exhibit greater applicability to geometry-to-force problems compared to existing Lagrangian expressions, making them particularly useful for conceptual designs and front-end engineering. In summary, this work provides valuable new insights into the exact solution of the elastic catenary, enhancing understanding and enabling practical applications in the field of floating wind turbines.

On Resonance Values of Parameters in the Problem of Stability of Lagrange Solutions in a Restricted Three-Body Problem Close to a Circle

On Resonance Values of Parameters in the Problem of Stability of Lagrange Solutions in a Restricted Three-Body Problem Close to a Circle

A Method for Finding Exact Solutions to the 2D and 3D Euler–Boussinesq Equations in Lagrangian Coordinates

We study the Boussinesq approximation for the incompressible Euler equations using Lagrangian description. The conditions for the Lagrangian fluid map are derived in this setting, and a general method is presented to find exact fluid flows in both the two-dimensional and the three-dimensional case. There is a vast amount of solutions obtainable with this method and we can only showcase a handful of interesting examples here, including a Gerstner type solution to the two-dimensional Euler–Boussinesq equations. In two earlier papers we used the same method to find exact Lagrangian solutions to the homogeneous Euler equations, and this paper serves as an example of how these same ideas can be extended to provide solutions also to related, more involved models.

ANALISIS LAGRANGIAN NULL NONSTANDAR DAN FUNGSI GAUGE UNTUK HUKUM INERSIA NEWTON : SEBUAH REVIEW

This paper describes a review of a journal entitled 'Nonstandard Null Lagrangians and Gauge Functions for Newtonian Law of Inertia' which discusses Nonstandard Lagrangian Null solutions for Newton's Law of Inertia. The purpose of this study is to present in detail the Lagrangian Formalism method for generating Nonstandard Lagrangian Null and its Gauge function for Newton's Law of Inertia, as well as the role of action invariant in generating Lagrangian Null and Exact Gauge functions, by deriving a one-dimensional oscillator arm using the basic Lagrangian equations. The Nonstandard Null Lagrangian is derived from the Nonstandard Lagrangian, then the two Lagrangian Null are entered into the action invariant to make it Exact, after the Nonstandard Lagrangian Null is declared Exact, it is substituted into which represents Newton's Law of Inertia. The results of this research show that an Exact Nonstandard Lagrangian Null can be generated by making the Gauge Function Invariant, so that the first Nonstandard Lagrangian Null and Gauge Function for Newton's Law of Inertia are obtained.

Coupling SPH with a mesh-based Eulerian approach for simulation of incompressible free-surface flows

This paper presents a coupling algorithm for the simulation of incompressible free-surface flows over the near and far fields by solving the Navier-Stokes equations. The coupling combines the Smoothed Particle Hydrodynamics (SPH) with the Finite Difference (FD) method implemented on structured Eulerian grids. Based on the domain decomposition, the SPH is applied to the near-field core region for modeling with high-fidelity the severe free-surface deformation and even fragmentation of flow, while the FD method is used to resolve the remaining far-field region with relatively high efficiency. The primary purpose of the hybrid approach is to mitigate the computational load of SPH applications through combination with a mesh-based Eulerian method while maintaining high accuracy for both the near-field and far-field simulations. Two-way coupling between the FD and SPH models is achieved by the overlapping region between the computational subdomains, in which the Eulerian solutions and the Lagrangian solutions are coupled with each other through interpolation in the overlapping region for the fluid transfer and free-surface movement between the near-field and far-field regions. The proposed algorithm is validated by several test cases, showing favorable accuracy, convergence, and applicability in the simulation of wave interactions with complex bathymetry and structures.

Regional Constellation Reconfiguration Problem: Integer Linear Programming Formulation and Lagrangian Heuristic Method

A group of satellites, with either homogeneous or heterogeneous orbital characteristics and/or hardware specifications, can undertake a reconfiguration process due to variations in operations pertaining to Earth observation missions. This paper investigates the problem of optimizing a satellite constellation reconfiguration process against two competing mission objectives: 1) the maximization of the total coverage reward, and 2) the minimization of the total cost of the transfer. The decision variables for the reconfiguration process include the design of the new configuration and the assignment of satellites from one configuration to another. We present a novel biobjective integer linear programming formulation that combines constellation design and transfer problems. The formulation lends itself to the use of generic mixed-integer linear programming (MILP) methods such as the branch-and-bound algorithm for the computation of provably optimal solutions; however, these approaches become computationally prohibitive even for moderately sized instances. In response to this challenge, this paper proposes a Lagrangian relaxation-based heuristic method that leverages the assignment problem structure embedded in the problem. The results from the computational experiments attest to the near-optimality of the Lagrangian heuristic solutions and a significant improvement in the computational runtime as compared to a commercial MILP solver.

On Resonant Values of Parameters in the Problem on the Stability of Lagrangian Solutions in the Near-Circular Restricted Three-Body Problem

The restricted problem of three bodies (material points) moving under the action of gravitational attraction according to Newton’s law is considered. The orbits of the main attracting bodies are considered to be ellipses with a small eccentricity, and a passively gravitating body can perform arbitrary spatial motion near the triangular libration point. For the Hamiltonian function corresponding to such a motion, the structure of the normal form is pointed in the cases of third-order resonances. In the planar restricted three-body problem, the equations up to the second degree of eccentricity for resonance curves for all resonances up to the sixth order inclusive are obtained.

A new Lagrangian solution scheme for non-decomposable multidisciplinary design optimization problems

A new Lagrangian solution scheme for non-decomposable multidisciplinary design optimization problems

Lagrangian solutions to the transport–Stokes system

In this paper we consider the transport–Stokes system, which describes the sedimentation of a particles in a viscous fluid in inertialess regime. We show existence of Lagrangian solutions to the Cauchy problem with L1 initial data. We prove uniqueness of solutions as a corollary of a stability estimate with respect to the 1-Wasserstein distance for solutions with initial data in a Yudovich-type refinement of L3, with finite first moment. Moreover, we describe the evolution starting from axisymmetric initial data. Our approach is purely Lagrangian.

Quasi-conservative Integration Method for Restricted Three-body Problem

The simplest restricted three-body problem, in which two massive points and a massless point particle attract one another according to Newton’s law of inverse squares, has pulsating Hill’s regions where the massless particle moves inside the closed regions surrounding only one of the massive points. Until now, no numerical integrator is known to maintain these regions, making it challenging to reproduce the phenomenon of gravitational capture of massless particles. In this article, we propose a second-order integrator that preserves Hill’s regions to accurately simulate this phenomenon. Our integrator is based on a logarithmic Hamiltonian leapfrog method developed by Mikkola and Tanikawa and features a parameter that is adjusted to preserve a second-order approximation of an invariant integration relation of this restricted three-body problem. We analytically and numerically clarify that this integrator has the following properties: (i) it retains the collinear and triangular Lagrangian solutions regardless of the eccentricity of the relative orbit of the two massive points, (ii) it has the same Hill stability criterion for satellite-type motion of the massless point particle as the original problem, and (iii) it conserves the Jacobi integral for zero eccentricity.

Power Optimization of UOWC-MIMO-OFDM System Based on PSO-WF Algorithm

Under the constraint that the total transmission power is constant, when multiple input multiple output orthogonal frequency division multiplexing (MIMO-OFDM) technology is applied to the signal transmission design in the underwater optical wireless communication system (UOWC), the traditional water-filling algorithm (WF) uses mathematical derivation to solve the optimization problem, which requires multiple Lagrangian optimization solutions. In order to reduce the computational complexity and accelerate the convergence speed, a particle swarm optimization (PSO-WF) algorithm for water injection is proposed. It randomly selects particles in the target area, calculates the fitness, and gradually approximates the optimal solution to avoid negative power distribution. The simulation results show that, compared with the average power allocation scheme, the algorithm improves the system capacity and operation efficiency by about 80%; Compared with WF algorithm, it greatly reduces the complexity.

A semi-Lagrangian relaxation heuristic algorithm for the simple plant location problem with order

The Simple Plant Location Problem with Order (SPLPO) is a variant of the Simple Plant Location Problem (SPLP) where the customers have preferences over the facilities that will serve them. In particular, customers define their preferences by ranking each of the potential facilities from the most preferred to the least preferred. Even though the SPLP has been widely studied in the literature, the SPLPO, which is a harder model to deal with, has been studied much less and the size of instances that can be solved is very limited. In this article, we study a Lagrangian relaxation of the SPLPO model and we show that some properties previously studied and exploited to develop efficient SPLP solution procedures can be extended for their use in solving the SPLPO. We propose a heuristic method that takes a Lagrangian relaxation solution as the starting point of a semi-Lagrangian relaxation algorithm. Finally, we include some computational studies to illustrate the good performance of this method, which quite often ends with the optimal solution.

Newly Developed Multiparameter Bulk Cloud Schemes. Part I: A New Triple-Moment Condensation Scheme and Tests

Abstract Double-moment schemes cannot accurately describe the evolution of the cloud droplet spectrum during condensation. Hence, a new triple-moment condensation scheme is developed to describe the evolution of cloud droplet spectra. In this scheme, a three-parameter gamma distribution function of the cloud droplet mass is adopted, and the prognostic equations of the spectral shape parameter and slope parameter are derived by means of the number concentration, cloud water content, and reflectivity factor of cloud droplets. The new parameterization scheme is compared with high-resolution Lagrangian and Eulerian bin schemes, double-moment schemes, and existing triple-moment schemes by performing simulations under different supersaturation values. The new scheme can reduce the cloud spectral error in the cloud water content and reflectivity factor caused by the fixed shape parameter in some bulk schemes. The spectra simulated with the new scheme match the Lagrangian analytical solutions well, with errors within approximately 1% in the cloud water content and reflectivity factor. The effects of curvature and solution on condensation growth are also tested using the new scheme, and a method of using multiple gamma distribution functions to characterize the multimodal spectrum of cloud droplets is proposed in the new condensation scheme. Ultimately, the formation of rain embryos from giant aerosols can be simulated via the new scheme.