An α-Model Parametrization Algorithm for Optimization with Differential-Algebraic Equations

An optimization task with nonlinear differential-algebraic equations (DAEs) was approached. In special cases in heat and mass transfer engineering, a classical direct shooting approach cannot provide a solution of the DAE system, even in a relatively small range. Moreover, available computational procedures for numerical optimization, as well as differential- algebraic systems solvers are characterized by their limitations, such as the problem scale, for which the algorithms can work efficiently, and requirements for appropriate initial conditions. Therefore, an αDAE model optimization algorithm based on an α-model parametrization approach was designed and implemented. The main steps of the proposed methodology are: (1) task discretization by a multiple-shooting approach, (2) the design of an α-parametrized system of the differential-algebraic model, and (3) the numerical optimization of the α-parametrized system. The computations can be performed by a chosen iterative optimization algorithm, which can cooperate with an outer numerical procedure for solving DAE systems. The implemented algorithm was applied to solve a counter-flow exchanger design task, which was modeled by the highly nonlinear differential-algebraic equations. Finally, the new approach enabled the numerical simulations for the higher values of parameters denoting the rate of changes in the state variables of the system. The new approach can carry out accurate simulation tests for systems operating in a wide range of configurations and created from new materials.

A new clique polynomial approach for fractional partial differential equations

This paper generates a novel approach called the clique polynomial method (CPM) using the clique polynomials raised in graph theory and used for solving the fractional order PDE. The fractional derivative is defined in terms of the Caputo fractional sense and the fractional partial differential equations (FPDE) are converted into nonlinear algebraic equations and collocated with suitable grid points in the current approach. The convergence analysis for the proposed scheme is constructed and the technique proved to be uniformly convegant. We applied the method for solving four problems to justify the proposed technique. Tables and graphs reveal that this new approach yield better results. Some theorems are discussed with proof.

Chebyshev Polynomials of Sixth Kind for Solving Nonlinear Fractional PDEs with Proportional Delay and Its Convergence Analysis

This work devotes to solving a class of delay fractional partial differential equations that arises in physical, biological, medical, and climate models. For this, a numerical scheme is implemented that applies operational matrices to convert the main problem into a system of algebraic equations; then, solving the resultant system leads to an approximate solution. The two-variable Chebyshev polynomials of the sixth kind, as basis functions in the proposed method, are constructed by the one-variable ones, and their operational matrices are derived. Error bounds of approximate solutions and their fractional and classical derivatives are computed. With the aid of these bounds, a bound for the residual function is estimated. Three illustrative examples demonstrate the simplicity and efficiency of the proposed method.

Optimizing Water Consumption in Richards’ Equation Framework with Step-Wise Root Water Uptake: A Simplified Model

This work arises from the need of exploring new features for modeling and optimizing water consumption in irrigation processes. In particular, we focus on water flow model in unsaturated soils, accounting also for a root water uptake term, which is assumed to be discontinuos in the state variable. We investigate the possibility of accomplishing such optimization by computing the steady solutions of a $$\theta$$ θ -based Richards equation revised as equilibrium points of the ODEs system resulting from a numerical semi-dicretization in the space; after such semi-discretization, these equilibrium points are computed exactly as the solutions of a linear system of algebraic equations: the case in which the equilibrium lies on the threshold for the uptake term is of particular interest, since the system considerably simplifies. In this framework, the problem of minimizing the water waste below the root level is investigated. Numerical simulations are provided for representing the obtained results. Article Highlights Root water uptake is modelled in a Richards’ equation framework with a discontinuous sink term. After a proper semidiscretization in space, equilibrium points of the resulting nonlinear ODE system are computed exactly. The proposed approach simplifies a control problem for optimizing water consumption.

An analytical approach for LQR design for improving damping performance of multi-machine power system

In a multi-machine environment, the inter-area low-frequency oscillations induced due to small perturbation(s) has a significant adverse effect on the maximum limit of power transfer capacity of power system. Conventionally, to address this issue, power systems were equipped with lead-lag power system stabilizers (CPSS) for damping oscillations of low-frequency. In recent years the research was directed towards optimal control theory to design an optimal linear-quadratic-regultor (LQR) for stabilizing power system against the small perturbation(s). The optimal control theory provides a systematic way to design an optimal LQR with sufficient stability margins. Hence, LQR provides an improved level of performance than CPSS over broad-range of operating conditions. The process of designing of optimal LQR involves optimization of associated state (Q) and control (R) weights. This paper presents an analytical approach (AA) to design an optimal LQR by deriving algebraic equations for evaluating optimal elements for weight matrix ‘Q’. The performance of the proposed LQR is studied on an IEEE test system comprising 4-generators and 10-busbars.

Features of driving of the steering wheel driving cars

Existing methods of diagnosing steering can be characterized by low efficiency. For various reasons, both declarative and actual (supported by the equipment) methods, as a rule, have low accuracy and inability to localize faults. The car's built-in diagnostics cannot affect the situation due to the small number of sensors in the steering system. The reasons for the low accuracy of the methods include design features, low availability of components (low maintainability). Difficulties in localization of malfunctions are caused by the structural scheme which is characterized by parallel - consecutive construction. The parameters of diagnostic methods are analyzed, the proposed method is based on the structure of the steering, in the implementation of which test effects are applied to the steered wheels. In total it is necessary to carry out three measurements of backlashes and as a result of mathematical processing of results it becomes possible to localize malfunction in three links of consecutive elements of the steering mechanism or a steering drive. In accordance with this approach, steering is considered as a set of three structures - parallel and two sequential. Rack and pinion steering was used as a model. Here, the parallel structure includes elements of the steering linkage: swing arm, left and right; steering rod, left and right; steering rack - left and right hinges. The sequential structure - left, includes a swing arm, left; steering rod, left; steering rack hinge, left; steering gear, steering shaft, steering wheel. Accordingly, the sequential structure of the right includes similar elements with the attribute "right". The structure of the steering play is considered in a similar way. As a result, it becomes possible to obtain a transformed system of three algebraic equations connecting clearances in three groups of mates and backlashes in parallel and two sequential steering structures. To measure the backlash, the turntables of the BOSCH FWA 4410 stand were used; in another version, the wheels were hung out. As a result of tests carried out on VW GOLF, VW PASSAT and RENAULT 25 vehicles with significant mileage, data was obtained indicating the need for technical interventions on localized groups of interfaces.

Haar wavelet method for solution of variable order linear fractional integro-differential equations

<abstract><p>In this paper, we developed a computational Haar collocation scheme for the solution of fractional linear integro-differential equations of variable order. Fractional derivatives of variable order is described in the Caputo sense. The given problem is transformed into a system of algebraic equations using the proposed Haar technique. The results are obtained by solving this system with the Gauss elimination algorithm. Some examples are given to demonstrate the convergence of Haar collocation technique. For different collocation points, maximum absolute and mean square root errors are computed. The results demonstrate that the Haar approach is efficient for solving these equations.</p></abstract>

ADG: automated generation and evaluation of many-body diagrams

The goal of the present paper is twofold. First, a novel expansion many-body method applicable to superfluid open-shell nuclei, the so-called Bogoliubov in-medium similarity renormalization group (BIMSRG) theory, is formulated. This generalization of standard single-reference IMSRG theory for closed-shell systems parallels the recent extensions of coupled cluster, self-consistent Green’s function or many-body perturbation theory. Within the realm of IMSRG theories, BIMSRG provides an interesting alternative to the already existing multi-reference IMSRG (MR-IMSRG) method applicable to open-shell nuclei. The algebraic equations for low-order approximations, i.e., BIMSRG(1) and BIMSRG(2), can be derived manually without much difficulty. However, such a methodology becomes already impractical and error prone for the derivation of the BIMSRG(3) equations, which are eventually needed to reach high accuracy. Based on a diagrammatic formulation of BIMSRG theory, the second objective of the present paper is thus to describe the third version (v3.0) of the code that automatically (1) generates all valid BIMSRG(n) diagrams and (2) evaluates their algebraic expressions in a matter of seconds. This is achieved in such a way that equations can easily be retrieved for both the flow equation and the Magnus expansion formulations of BIMSRG. Expanding on this work, the first future objective is to numerically implement BIMSRG(2) (eventually BIMSRG(3)) equations and perform ab initio calculations of mid-mass open-shell nuclei.

Two-level mathematical models for determining the stress state and life plate with a hole

Modern trends in the development of mechanical engineering and other industries related to the production of materials and structures with a given set of physical, mechanical, and technological properties are aimed at reducing material consumption, energy consumption, increasing accuracy, reliability, and competitiveness of the manufactured product. Therefore, the creation of mathematical methods for assessing the stress state of structural elements based on the analysis of the elastic characteristics of a material, taking into account the peculiarities of its internal microstructure, is an actual task. The considered algorithm includes the following stages: identification of strength parameters using data obtained from images of the material microstructure; study of the stress-strain state of the model based on the variational-difference finite element method; formation of a system of linear algebraic equations for solving the problem of analyzing the elastic properties of a material using the plane problem of the theory of elasticity; construction of the material yield surface for a series of tests based on the strength criteria of composite materials, taking into account the different resistance of the material under tensile and compressive loads. Based on the developed mathematical model, the SSS and the yield surface of the plate with a hole are estimated. Structural analysis is performed at the macro and micro levels. The occurrence of plastic deformations at the micro-level can lead to the development of cracks and structural damage at the macro level. As a result of the study, the probability of plastic deformation in the plate is determined, and the critical zones of the model are established. The practical significance of the results obtained is to create an approach to assessing the mechanical properties of a material, such as elastic modulus, shear modulus, Poisson's ratio, and their probabilistic characteristics following the internal material structure. The proposed approach contributes to the expansion of knowledge about the material and allows to increase the valuable information obtained by modeling. To assess the probability of plastic deformations, the generated method uses the entire set of probabilistic characteristics of the yield surface.