Lagrange Multiplier Research Articles

Hybrid TBETI domain decomposition for huge 2D scalar variational inequalities

AbstractThe unpreconditioned H‐TFETI‐DP (hybrid total finite element tearing and interconnecting dual‐primal) domain decomposition method introduced by Klawonn and Rheinbach turned out to be an effective solver for variational inequalities discretized by huge structured grids. The basic idea is to decompose the domain into non‐overlapping subdomains, interconnect some adjacent subdomains into clusters on a primal level, and enforce the continuity of the solution across both the subdomain and cluster interfaces by Lagrange multipliers. After eliminating the primal variables, we get a reasonably conditioned quadratic programming (QP) problem with bound and equality constraints. Here, we first reduce the continuous problem to the subdomains' boundaries, then discretize it using the boundary element method, and finally interconnect the subdomains by the averages of adjacent edges. The resulting QP problem in multipliers with a small coarse grid is solved by specialized QP algorithms with optimal complexity. The method can be considered as a three‐level multigrid with the coarse grids split between primal and dual variables. Numerical experiments illustrate the efficiency of the presented H‐TBETI‐DP (hybrid total boundary element tearing and interconnecting dual‐primal) method and nice spectral properties of the discretized Steklov–Poincaré operators as compared with their finite element counterparts.

Investigation of cavitation with air injection using Eulerian-Lagrangian method

Abstract Air injection can be used to reduce the cavitation erosion in fluid machinery. However the inverse result that the injected air indeed increases the size of cavitation may be obtained. To find out the interaction between air injection and cavitation, the present work proposes a Eulerian-Lagrangian method to investigate the effect of air injection on the characteristics of cavitating flow. The Eulerian method is used to describe the liquid and vapor distribution, and the morphology of large cavities are tracked by the Volume of Fluid (VOF) method in the Eulerian frame. On the other side, the Lagrangian method is used to track both air and vapor bubbles in small scale. With the Eulerian-Lagrangian coupling, discrete micro bubbles in sub-grid scale as well as the large cavities can be well captured and simulated. This work takes two cavitation situations in different cavitation numbers of σ = 0.81 and 0.70 into account. The cavitation features in σ = 0.81 are mainly characterized by the intermittent shedding of small-scale cavities under re-entrant flow mechanism. For σ =0.70, the injected air flow plays an important part which enhances the scale of cavity. For both cases, simulated results of the cavity shapes and distribution of discrete bubbles anastomoses well with experimental results, which verifies the accuracy of the algorithm. The interaction between injected air and vapor is also analysed.

Research on Internal Flooding Analysis of Small Modular Reactors Based on Particle Method

Abstract The analysis of internal water flooding in nuclear power plants is a necessary part of safety review in the approval process of nuclear power plant construction projects. According to relevant regulations, it is necessary to conduct water flooding spread analysis on pipelines, water tanks, or pools filled with liquid in nuclear power plant buildings, analyze the spread path and water flooding height. This article analyzes and calculates the flow characteristics of water in a nuclear power plant building using a fluid calculation method based on the Lagrangian method (particle method). Taking the small modular reactor JR building as a case study, it analyzes the spread and height of four water flooding sources inside the building. According to the simulation time of 1800S, the results show that the flooding caused by the No. 2 water source is the most severe, with a flooding height of 1.367m. Drainage or preventive measures need to be taken to prevent the pipeline from breaking accidents; The flooding sources 1, 2, and 4 have a relatively mild impact, resulting in a lower flooding height and will not have a significant impact on important safety equipment, with little impact on the safety of nuclear power plants. The CFD analysis method was used to conduct water flooding analysis for nuclear power plants, which has the characteristics of strong adaptability and good visualization. It can dynamically simulate the entire process of water flooding analysis and better guide the design of water flooding protection for nuclear power plants.

An improved implicit direct-forcing immersed boundary method (DF-IBM) around arbitrarily moving rigid structures

An improved implicit direct-forcing immersed boundary method (DF-IBM) is presented for simulating incompressible flows around complex rigid structures undergoing arbitrary motion. The current approach harnesses the pressure implicit with splitting of operators algorithm to handle the fluid–solid system's dual constraints in a segregated manner. As a result, the divergence-free condition is preserved throughout the Eulerian domain, and the no-slip velocity boundary condition is exactly enforced on the immersed boundary. A new pressure Poisson equation (PPE) is derived, incorporating the boundary force where the no-slip condition is already met, enabling the use of fast iterative PPE solvers without modifications. The improvement involves integrating Lagrangian weight methods having better reciprocity over the IBM-related linear operators with the implicit formulation. An additional force initialization scheme is introduced to further boost the algorithm's performance. The method's accuracy, efficiency, and capability are verified through various stationary and moving immersed boundary cases. The results are validated against experimental and numerical data from the literature. The proposed improvements seamlessly integrate into existing incompressible fluid solvers with minimal adjustments to the original system equations, highlighting their ease of implementation.

The universal multiplicity function: counting haloes and voids

We present a novel combination of the excursion-set approach with the peak theory formalism in Lagrangian space and provide accurate predictions for halo and void statistics over a wide range of scales. The set-up is based on an effective moving barrier. Besides deriving the corresponding numerical multiplicity function, we introduce a new analytical formula reaching the percent level agreement with the exact numerical solution obtained via Monte Carlo realisations down to small scales, ∼ 1012 h -1M⊙. In the void case, we derive the dependence of the effective moving barrier on the void formation threshold, δ v, by comparison against the Lagrangian void size function measured in the DEMNUni simulations. We discuss the mapping from Lagrangian to Eulerian space for both haloes and voids; adopting the spherical symmetry approximation, we obtain a strong agreement at intermediate and large scales. Finally, using the effective moving barrier, we derive Lagrangian void density profiles accurately matching measurements from cosmological simulations, a major achievement towards using void profiles for precision cosmology with the next generation of galaxy surveys.

VSHARP: Variable Splitting Half-quadratic ADMM algorithm for reconstruction of inverse-problems

Medical Imaging (MI) tasks, such as accelerated parallel Magnetic Resonance Imaging (MRI), often involve reconstructing an image from noisy or incomplete measurements. This amounts to solving ill-posed inverse problems, where a satisfactory closed-form analytical solution is not available. Traditional methods such as Compressed Sensing (CS) in MRI reconstruction can be time-consuming or prone to obtaining low-fidelity images. Recently, a plethora of Deep Learning (DL) approaches have demonstrated superior performance in inverse-problem solving, surpassing conventional methods. In this study, we propose vSHARP (variable Splitting Half-quadratic ADMM algorithm for Reconstruction of inverse Problems), a novel DL-based method for solving ill-posed inverse problems arising in MI. vSHARP utilizes the Half-Quadratic Variable Splitting method and employs the Alternating Direction Method of Multipliers (ADMM) to unroll the optimization process. For data consistency, vSHARP unrolls a differentiable gradient descent process in the image domain, while a DL-based denoiser, such as a U-Net architecture, is applied to enhance image quality. vSHARP also employs a dilated-convolution DL-based model to predict the Lagrange multipliers for the ADMM initialization. We evaluate vSHARP on tasks of accelerated parallel MRI Reconstruction using two distinct datasets and on accelerated parallel dynamic MRI Reconstruction using another dataset. Our comparative analysis with state-of-the-art methods demonstrates the superior performance of vSHARP in these applications.

An MP-Newton method for computing nonlinear eigenpairs and its application for solving a semilinear Schrödinger equation

ln Yao and Zhou (2008), a minimax method for computing nonlinear eigenpairs by calculating critical points of the Lagrange multiplier function is presented. But, the method is slow and can find limited amount of eigenpairs. In this paper, a new general characterization, orthogonal-max characterization, for critical points of the Lagrange multiplier function is suggested. An MP-Newton method for finding orthogonal-max type critical points is designed through analyzing how the minimax method works. The new method becomes fast and able to calculate more nonlinear eigenpairs. Numerical experiment confirms these two progresses. Also, the MP-Newton method inherits the advantages of the minimax method. A convergence result for the method is established. Finally, an application for solving a semilinear Schrödinger equation is discussed.

Adaptive Neurodynamic Approach to Multiple Constrained Distributed Resource Allocation.

In this article, an adaptive neurodynamic approach over multiagent systems is designed to solve nonsmooth distributed resource allocation problems (DRAPs) with affine-coupled equality constraints, coupled inequality constraints, and private set constraints. It is to say, agents focus on tracking the optimal allocation to minimize the team cost under more general constraints. Among the considered constraints, multiple coupled constraints are dealt with by introducing auxiliary variables to make Lagrange multipliers reach consensus. Furthermore, aiming to address private set constraints, an adaptive controller is proposed with the aid of the penalty method, thus avoiding the disclosure of global information. Through using the Lyapunov stability theory, the convergence of this neurodynamic approach is analyzed. In addition, to reduce the communication burden of systems, the proposed neurodynamic approach is improved by introducing an event-triggered mechanism. In this case, the convergence property is also explored, and the Zeno phenomenon is excluded. Finally, a numerical example and a simplified problem on a virtual 5G system are implemented to demonstrate the effectiveness of the proposed neurodynamic approaches.

Normalized solutions for [formula omitted]-supercritical Schrödinger equations

This paper is concerned with the following L2-supercritical Schrödinger equation −Δu−λu=f(u),∫RNu2dx=c,where N≥3, f∈C(R,R), c>0 is a given mass and λ∈R will arise as a Lagrange multiplier depending on the solution u∈H1(RN). By introducing new weak L2-supercritical conditions on f, we develop robust arguments to establish the existence of normalized solutions to the above equation. Our result complements the ones of Chen and Tang (2024).

Dynamic allocation of display advertising impressions in dual sales channels

We study a multi-period ad allocation problem faced by an online publisher who sells ad impressions on websites through two sales channels. In the guaranteed sales channel, advertisers submit heterogeneous offers for contracts under which the publisher guarantees delivery of a certain number of ad impressions over a certain period; in the real-time bidding (RTB) sales channel, the publisher runs an RTB auction to sell ad impressions. In each period, the publisher decides whether to accept or reject contract proposals; how to allocate ad impressions across existing contracts; and how many impressions to sell via RTB. The publisher faces uncertain demand from advertisers and an uncertain supply of impressions, which are generated by viewers visiting the publisher’s websites. We formulate the problem as a finite-horizon stochastic dynamic program, which poses significant methodological challenges. We first present structural properties of optimal policies under certain cases. To avoid the curse of dimensionality in dynamic programming, we develop an approach involving Lagrangian relaxations. We decompose the problem into a series of solvable subproblems and derive optimal policies. We further develop Lagrangian policies with performance guarantees. We show that when Lagrange multipliers depend on more signal history, the linear term’s weight of the number of contract types in the performance upper bound decreases. Furthermore, if the Lagrange multipliers depend on the full signal history, the corresponding Lagrangian policies will be asymptotically optimal to the number of contract types. We also explore a more suitable case for large-scale real-time ad allocation and create Lagrangian policies that yield comparable performance guarantees. Finally, we extend our main results to four new scenarios.

De-mixing variational mode decomposition and its application on operational modal analysis in the presence of closely spaced modes

Variational mode decomposition (VMD), and its derivative methods have been demonstrated to be promising signal decomposition tools and widely utilized in modal analysis. However, they inevitably suffer from mode mixing sometimes, greatly impairing the quality of decomposition. This paper proposes a pair of novel VMD-based methods for univariate signal and multivariate signal respectively, referred to as de-mixing VMD (D-VMD) and de-mixing multivariate VMD (D-MVMD), which can greatly alleviate mode mixing in decomposition. The ensemble correlation coefficient is defined to evaluate the correlatedness between one mode and others and then involved in the variational formulation as an additional Lagrangian multiplier item to enhance the constraint in terms of mode mixing. Additionally, a new scheme is proposed to robustly identify modal parameters from the decomposed modal responses. Numerical simulations and a pedestrian bridge were investigated to validate the effectiveness of the proposed methods in identifying closely spaced modes.

A neurodynamic optimization approach to distributed nonconvex optimization based on an HP augmented Lagrangian function

This paper develops a neurodynamic model for distributed nonconvex-constrained optimization. In the distributed constrained optimization model, the objective function and inequality constraints do not need to be convex, and equality constraints do not need to be affine. A Hestenes–Powell augmented Lagrangian function for handling the nonconvexity is established, and a neurodynamic system is developed based on this. It is proved that it is stable at a local optimal solution of the optimization model. Two illustrative examples are provided to evaluate the enhanced stability and optimality of the developed neurodynamic systems.

An efficient numerical method for the distributed‐order time‐fractional diffusion equation with the error analysis and stability properties

In this manuscript, we study and examine the time‐fractional modified anomalous sub‐diffusion model of distributed‐order. Two numerical approaches are used to study the approximate solutions of the presented model. For the first approach, we use a second‐order difference method based on the L1 formula for the temporal variable. In this case, stability and convergence analysis for time discretization using the L1 formula is displayed. For the second approach, we display and demonstrate the numerical method for the full‐discrete based on the Galerkin weak form based on various kernels and shape functions of reproducing kernel particle method which do not have the ‐Kronecker property. Moreover, in this manuscript, in order to be able to use the necessary boundary conditions, the two straight strategies are used: one is the Lagrange multiplier method, and the other one is the penalty method. By using penalty method, we can the main boundary value model is changed to a new BVP with Robin boundary condition. At the end of the article, some numerical examples are presented and analyzed for the efficiency of the proposed numerical method. Also, the presented numerical method is compared with other numerical approaches, and the results of this comparison are reported in the form of a table.

OPTIMIZATION PROBLEM FOR PROPYLENE GLYCOL PRODUCTION

Currently, in the rapid development era of new technologies and in order to increase the economic efficiency of production, more and more attention is paid to the current automation, control, and optimization problems of technological processes in the oil refining, petrochemical, and chemical industries [1-11]. It is known that the production of commercial propylene glycol is one of the most important industries in modern chemical technology. Propylene glycol can act as one of the components in the obtaining drug's process. In addition, it is used for lubrication and preservation of food packaging machines, is used as a plasticizer in the production of cellophane and polyvinyl chloride films, as a solvent for natural and synthetic substances in the preparation of ointments, pastes, creams, shampoos, etc., in the pharmaceutical and cosmetic industries, in the food industry as a solvent for food additives and a tobacco humectant, etc. Based on the relevance of the above, this article, based on a comprehensive study of the features of the chemical-technological process for obtaining propylene glycol, in particular, the chemical reaction of obtaining commercial propylene glycol in one of the main apparatuses for the studied block - the hydrator, a physically substantiated mathematical statement of the optimal control problem of this apparatus is formulated. The optimal control problem of the studied technological apparatus is reduced to obtaining the maximum amount of the target product, considering the restrictions imposed on the control parameters. It is established that by its nature and specificity, it is a nonlinear programming problem, for the solution of which in the presented article an algorithm for optimizing the modes of the technological process is proposed, based on the use of the Lagrange multiplier method. An analysis of the results for solving the optimal control problem of the obtaining propylene glycol process showed that when maintaining the optimal values of the control parameters obtained by us, the amount of propylene glycol obtained increases by 3.5% compared to the actual amount. Keywords: technological process, commercial propylene glycol, optimal control problem, control parameters, Lagrange multiplier method, technological apparatus.

SIMULATION HEATING PROCESS FOR WEAKLY REACTIVE COAL IN THE HORIZONTAL REACTOR

In this study results of the active medium parameters influence formed by hot air on the thermal heating process of weakly reactive coal from the formation «Maikubensk 3B» for the extraction of additional heat are represented. In the course of the research the regularities of the hygroscopic moisture and volatile combustible substances release for the obtaining additional heat have been determined. Optimal temperature ranges of the stage heating taking into account thermal destruction of coal in the temperature range from 20 C to 600 C, the main steps of the insulated heating process, the moisture content gradient propagation lines and dependence of the heat transfer and mass transfer coefficients on the temperature and the humidity during the thermal heating process in the horizontal coal heating reactor. The results show that it is possible to release volatile combustible gases with a gradual increase of temperatures in the obtained mode. The heating mode comprises a pre-heating stage with hydroscopic moisture release, in the temperature range from 20 C to 105 C, the stage of isothermal heating in the temperature range from 105 C to 400 C and the phase of flame-free combustion in the temperature range from 400 C to 600 C with the temperature rise above the set value. Process simulation was carried out in the Comsol Multiphisics software environment for the weakly reactive coal real-world heating conditions. Keywords: coal heating, weakly reactive coal, isothermal heating, step-by-step heating, Lagrange method, non-liner equation, non-stationary process.

Equation-Oriented Meanline Method for Axial Turbine Performance Prediction Under Choking Conditions

Abstract Meanline models play a crucial role in turbine design and system-level analyses, facilitating rapid evaluation of design concepts and prediction of off-design performance. Most of the existing meanline methods are inadequate in predicting turbine performance under choking conditions. These models either neglect the impact of losses on choking, or increases the computational complexity significantly. This limitation is addressed in this work, presenting a novel meanline model. The choking state at each cascade is determined by maximizing the mass flow rate, while taking into account the effect of losses. Leveraging the method of Lagrange multipliers, the optimization problems are transformed into a set of equations that seamlessly integrate with the rest of the meanline model. The resulting system of equations is then solved simultaneously using efficient root-finding algorithms, resulting in fast and reliable convergence. Validation against experimental data from three different turbines demonstrates the model's ability to accurately predict mass flow rate, torque, and exit flow angles across single- and multi-stage turbines, with errors typically within ±2.5 % and ±5.0 % for mass flow rate and torque respectively, and within ±5 degrees for flow angles. The proposed approach represents a significant advancement in meanline modeling, offering improved accuracy and computational efficiency.

Bend-twist adaptive control for flexible wind turbine blades: Principles and experimental validation

This study proposes a new methodology for optimizing the power curve of a wind turbine at low wind speeds. The principles of bend-twist coupling and the mechanism of energy exchange between the structure and inflow are analyzed. For the blade’s geometric nonlinearity, the virtual displacements and strain fields are described using the Green-Lagrange strain theorem, retaining third-order terms in the energy expressions. The equations of motion are derived using Hamilton’s principle. The bend-twist coupling effects and large deformations of the blade are analyzed using the Updated Lagrange method. Notably, the angle of attack for a single blade section is influenced by bend-twist deformation, causing variations in the rotor’s maximum power coefficient from its optimal value. Additionally, the projection length of the blade, influenced by centrifugal forces, also affects the bend-twist deformation. Based on these findings, an aero-elastic coupling control strategy, termed “Bend-twist Adaptive Control”, is proposed and validated through experiments. The results demonstrate that the proposed control strategy could increase annual power production by 2.3 %. These conclusions offer a promising outlook for future wind turbine blade design and power optimization.

Analytical, semi-analytical and numerical solutions for global stability analysis in buildings: Double-Beam Systems Timoshenko

In this article, two analytical solutions, one semi-analytical solution, and one numerical solution are proposed to calculate the global critical buckling load of buildings employing a continuum model parallel coupling two Timoshenko beams. The parallel coupling of two Timoshenko beams represents the four types of building behavior: global bending, global shear, local bending, and local shear, the latter of which has been widely ignored in the literature. The model is associated with one translational kinematic field and two rotational fields. The equilibrium equations, constitutive laws, and boundary conditions are derived using a variational approach by applying Hamilton’s principle to the relevant Lagrangian function. Specifically, to solve the classical problem of buildings with uniform properties along their height, the first analytical solution proposes exact closed-form equations for a vertical load applied at the top and a closed-form analytical solution based on the decomposition of two subsystems for a polynomial vertical load uniformly applied along the height of the beam. Meanwhile, the semi-analytical solution exactly solves the differential equation associated with a polynomial vertical load profile and is based on coefficient iteration, converging with only two or three iterations. To address the challenge of buildings with variable properties along their height, the combined application of the continuous method and the transfer matrix method allows us to propose an efficient numerical method leading to a 6 × 6 transfer matrix, constant regardless of the number of discretizations and derived analytically, which results in a computational advantage by drastically reducing computational costs. The numerical results are encouraging and show good agreement with the results of the finite element method, suggesting that the proposed method can be used for engineering purposes in both the preliminary and final phases of the structural analysis of tall buildings.

Robust joint multiple resources allocation algorithm for cooperative underwater acoustic communication networks

AbstractIn this article, a joint relay selection, power control and time allocation problem is studied to maximize the energy efficiency for the cooperative underwater acoustic communication networks. The joint optimization problem is full of challenges due to the strong coupling of multiple resources and the uncertain characteristics of the underwater acoustic communication scenario. To address this issue, the worst‐case method is employed to transform an original uncertain problem into a deterministic problem. Furthermore, we propose the block coordinate descent‐based method to decouple the strongly coupling multi‐resource allocation problem into three relatively independent sub‐problems. The coupling of multiple resources is completely decoupled, thereby greatly reducing the solving difficulty. In addition, given that the sub‐problems with the fractional objective function are still non‐convex and hard to solve, the Dinkelbach‐based method is proposed to transform the fractional objective function into a subtractive form. At last, the relay selection sub‐problem is transformed into a integer programming problem, and the time allocation sub‐problem is transformed into a linear programming problem, whose optimal solutions can be obtained by some well‐established solution methods. The power allocation problem is transformed into a convex optimization problem, which can be solved by the Lagrangian dual method. Finally, in the proposed iteration structure, the three sub‐problems are alternatingly solved until convergence. Simulation results are presented to demonstrate the efficiency and robustness of the proposed algorithm.

Mass minimization using the reaction-diffusion level set method by considering local stress constraints in an integral form

Research in topology optimization with stress constraints has shown that defining the stress constraint locally yields better performance compared to global methods. However, an examination of the formulas for local stress constraints reveals a limitation - the constraint is only satisfied at points where it is explicitly defined, failing to guarantee satisfaction across the entire design domain. To address this shortcoming, this paper proposes utilizing an integral form of the stress constraint. The integral formulation theoretically ensures that the stress constraint is satisfied over whole design domain. The objective is to minimize the mass of plane stress structures using reaction-diffusion level set method while incorporating local stress constraints in this integral form. The methodology utilizes finite element approximations for geometry and displacements, defining local stress constraints through an integral formulation. A Lagrangian function combines objective and constraint functions, with sensitivity analysis performed during optimization based on level set function changes. Structural boundaries are updated using the Hamilton-Jacobi equation. The paper presents numerical examples with varying loads and support conditions to demonstrate the effectiveness of this integral stress constraint approach. Results illustrate the capability of the proposed method to generate optimal topologies while satisfying stress constraints across the entire design space.