A monolithic approach toward coupled electrodynamic–thermomechanical problems with regard to weak formulations

Weak formulations of boundary value problems are the basis of various numerical discretization schemes. They are classically derived applying the method of weighted residuals or a variational principle. For electrodynamical and caloric problems, variational approaches are not straightforwardly obtained from physical principles like in mechanics. Weak formulations of Maxwell’s equations and of energy or charge balances thus are frequently derived from the method of weighted residuals or tailored variational approaches. Related formulations of multiphysical problems, combining mechanical balance equations and the axioms of electrodynamics with those of heat conduction, however, raise the additional issue of lacking consistency of physical units, since fluxes of charge and heat intrinsically involve time rates and temperature is only included in the heat balance. In this paper, an energy-based approach toward combined electrodynamic–thermomechanical problems is presented within a classical framework, merging Hamilton’s and Jourdain’s variational principles, originally established in analytical mechanics, to obtain an appropriate basis for a multiphysical formulation. Complementing the Lagrange function by additional potentials of heat flux and electric current and appropriately defining generalized virtual powers of external fields including dissipative processes, a consistent formulation is obtained for the four-field problem and compared to a weighted residuals approach.

The Post-Covid-19 Era Requires Maximum Urban Resilience

This paper reviews the low-resilience problem in many cities, poor designs of cities to cope with disasters, and the need for tolerance of urban constructions. It explores answers concerning the question of how shall we build cities resiliently? The method of this applied research is a multiphase process that considers all physical and socioeconomic elements of a city. It introduces six indicator groups of urban management (M), economy (E), built environments (U), Infrastructures (I), natural environments (N), and health protection (H). The groups include 55 indicators as variables in the mathematical calculations in this paper. This paper builds a mathematical model to maximize the profitability of resilient buildings by optimizing investments in the required projects. The projects will upgrade the firmness and tolerance of cities against nature-based and human-made dangers and risks. There is a linear programming in 55 variables to select optimal solutions from fifty-five factorial alternatives. Then, the programming will develop into non-linear programming. The unique innovation of this paper is its linear programming interpretation by non-linear to give optimal solutions for the problem. Applying the Lagrange function in the Kuhn-Tucker conditions proves the accuracy of the hypothesis that post-COVID urbanization requires maximum resilience. Only in this way, the urban economies will be free of risks. Outcomes in this paper will assist in the pre-planning, design, and building of built environments everywhere resilient and sustainable.

Evaluation Method of Synergy Degree for Comprehensive Benefits System of Hydropower Projects

The hydropower project had comprehensively benefited people from aspects of the economy, society, ecology, etc. The comprehensive benefit is a key indicator for evaluating a project’s performance. However, the existing studies only evaluated the comprehensive benefit and ignored the relationship among different benefits, which is of great significance for the sustainable development of a project. Therefore, in the framework of the complex system composed of economic, social, and ecological benefit subsystems, a synergy degree evaluation method is constructed based on the evaluation index system of the comprehensive benefits, and the compound weight is determined by using the non-linear model and the Lagrange function. Thus, the changing rules of the order degree and the synergy degree for the subsystems in different years can be obtained. The proposed method is applied to a gate dam (named SG) to appraise the relationship among benefits. The results show that the economic, social, and ecological benefits of the SG dam from 2011 to 2018 are gradually to be a better state, but the synergy degrees of the complex system belong to “bottom synergy” and “moderate synergy” level, which indicates that there is no close cooperation among the three benefit subsystems.

A Two-sided Matching Decision-making Approach based on PROMETHEE Under the Probabilistic Linguistic Environment

Matching problems in daily life can be effectively solved by two-sided matching decision-making (TSMDM) approaches. The involved matching intermediary is to match two sides of subjects. This paper proposes a TSMDM approach based on preference ranking organization method (PROMETHEE) under the probabilistic linguistic environment. The probabilistic linguistic evaluations are firstly normalized and transformed to the benefit types. Then, the preference degrees of a subject over other subjects from the same side are obtained by using six types of preference function. Afterwards, groups of preference degrees of a subject are aggregated to the preference indexes by considering the weights of criteria. Hereafter, the preference degrees of a subject over other subjects from the same side are aggregated to the outgoing flow, while the preference degrees of other subjects from the same side over this subject are aggregated to the incoming flow. Furthermore, the net-flows, which is recognized as the satisfaction degrees are calculated by using outgoing flows to minus incoming flows. On the basis of this, the multi-objectives TSMDM model is built by considering the matching aspirations. A model with respect to the matching aspirations is built and solved by using the Lagrange function. The multi-objectives TSMDM model is further transformed to the single-objective model, the solution of which is the matching scheme. A matching problem related to the intelligent technology intermediary is solved to verify the effectiveness and the feasibility of the proposed approach.

Hamiltonian structures for integrable hierarchies of Lagrangian PDEs

Many integrable hierarchies of differential equations allow a variational description, called a Lagrangian multiform or a pluri-Lagrangian structure. The fundamental object in this theory is not a Lagrange function but a differential $d$-form that is integrated over arbitrary $d$-dimensional submanifolds. All such action integrals must be stationary for a field to be a solution to the pluri-Lagrangian problem. In this paper we present a procedure to obtain Hamiltonian structures from the pluri-Lagrangian formulation of an integrable hierarchy of PDEs. As a prelude, we review a similar procedure for integrable ODEs. We show that exterior derivative of the Lagrangian $d$-form is closely related to the Poisson brackets between the corresponding Hamilton functions. In the ODE (Lagrangian 1-form) case we discuss as examples the Toda hierarchy and the Kepler problem. As examples for the PDE (Lagrangian 2-form) case we present the potential and Schwarzian Korteweg-de Vries hierarchies, as well as the Boussinesq hierarchy.

Parametric study of flapwise and edgewise vibration of horizontal axis wind turbine blades

In this paper, flapwise and edgewise vibrations of a horizontal axis wind turbine (HAWT) blade are studied. Rayleigh-Ritz method is used in which; orthogonal mode functions of the Euler-Bernoulli beam having fixed-free boundary are introduced into the Lagrange function and then the dynamic equations are derived. Effect of gravity, pitch angle, centrifugal stiffening, and rotary inertia are considered. Nondimensional equations are obtained by defining nondimensional parameters like; natural frequency, blade rotation, slenderness ratio, and simple pendulum frequency. The stiffness term of the natural frequency has two speed dependent elements and it is shown that, for small pitch angles, flapwise natural frequencies of the blade are increased by the increasing blade speed while the edgewise natural frequencies of the blade are decreased with the increasing blade speed. Pitch angle values ranging from 0° to 15° has negligible effect on the nondimensional natural frequencies of the blade up to the nondimensional blade speed of 4. Since the natural frequencies are the function of the blade speed, rotor critical speeds should be calculated with Campbell diagrams. Vibrational response of the blade tip to the gravity is dominant and much greater than that of the wind speed in the edgewise and flapwise vibration.

The sufficient condition to find an exact dual bound in a separable quadratic optimization problem

The paper considers nonconvex separable quadratic optimization problems subject to inequality constraints. A sufficient condition is given for finding the value and the point of the global extremum of a problem of this type by calculating the Lagrange dual bound. The peculiarity of this condition is that it is easily verified and requires from the Hessian matrix of the Lagrange function only that its region of positive definiteness is not empty. The result obtained for the dual bound also holds for the bound obtained using SDP relaxation.

The Generalized Hamilton Principle and Non-Hermitian Quantum Theory

The Hamilton principle is a variation principle describing the isolated and conservative systems, its Lagrange function is the difference between kinetic energy and potential energy. By Feynman path integration, we can obtain the Hermitian quantum theory, i.e., the standard Schrodinger equation. In this paper, we have generalized the Hamilton principle to the generalized Hamilton principle, which can describe the open system (mass or energy exchange systems) and nonconservative force systems or dissipative systems, and given the generalized Lagrange function, it has to do with the kinetic energy, potential energy and the work of nonconservative forces to do. With the Feynman path integration, we have given the non-Hermitian quantum theory of the nonconservative force systems. Otherwise, with the generalized Hamilton principle, we have given the generalized Hamiltonian for the particle exchanging heat with the outside world, which is the sum of kinetic energy, potential energy and thermal energy, and further given the equation of quantum thermodynamics. PACS: 03.65.-w, 05.70.Ce, 05.30.Rt

The Second-Order Differential Equation System with the Controlled Process for Variational Inequality with Constraints

In this paper, the variational inequality with constraints can be viewed as an optimization problem. Using Lagrange function and projection operator, the equivalent operator equations for the variational inequality with constraints under the certain conditions are obtained. Then, the second-order differential equation system with the controlled process is established for solving the variational inequality with constraints. We prove that any accumulation point of the trajectory of the second-order differential equation system is a solution to the variational inequality with constraints. In the end, one example with three kinds of different cases by using this differential equation system is solved. The numerical results are reported to verify the effectiveness of the second-order differential equation system with the controlled process for solving the variational inequality with constraints.

Energy Management Scheme in Hierarchical Integrated Energy Systems with Coalition

Often, coalitions are formed by the hierarchical integrated energy systems (HIESs) and their evolutionary process which is driven by the benefits of stakeholders and consolidate energy consumers and producers. Several literature have failed to analyze the operation of HIES under the impact of multiple coalitions. At the lower level, multiple users, in the middle level, the multiple distributed energy stations (DESs) and at the upper level, one natural gas and one electricity utility company structure is used for analyzing the HIES operation with a trading scheme. The Lagrange function is used for deriving the optimal operation strategy based analytical function for each probable coalition and each market participant comprising of users and the DESs. It is evident from the results that in a single coalition, the profits linked to other DESs will decrease while increasing the profit of one DES with technological enhancements, users show an aversion towards DESs with high generation coefficient while they are attracted to the ones that enable reduction of heat and electricity price. Maintaining their isolation is preferred by high heat and electricity consuming DESs at the same energy price. Other coalitions and their operations are not affected by the change in parameters of one coalition.