System Of Differential-algebraic Equations Research Articles

Mathematical modeling of an unmanned object motion control system in SimInTech environment

In today's world robotic devices are more and more often used to help people. They solve both domestic and industrial problems. When designing any object inevitably have to deal with testing under different conditions. To do this can build a test model, but if the object is quite complex and several models need to be built at once, to save labor and material resources can help mathematical modeling. This article presents mathematical modeling of processes based on typical functional blocks in the form of systems of differential-algebraic equations. Mathematical modeling and control algorithm as a set of interrelated structures are considered. The robot motion is simulated for the forward direction of rotation of wheels, the reverse direction of rotation, and the opposite directions of rotation. A model of the control device, which forms the control actions on the wheel motor according to the value of the deviation of the current orientation of the wheeled robot from the preset one, is constructed. These influences allow you to bring the orientation of the robot to the desired orientation. Quality metrics are obtained for various values of the rotational speed of the work. Although this model neglects the action of many forces that arise during the motion, it allows us to identify the influence on the motion and trajectory of the robot of such factors as the radius of the wheels, the distance between them, the magnitude of the voltage applied to the motors during the turn.

Bulk–surface Lie splitting for parabolic problems with dynamic boundary conditions

AbstractThis paper studies bulk–surface splitting methods of first order for (semilinear) parabolic partial differential equations with dynamic boundary conditions. The proposed Lie splitting scheme is based on a reformulation of the problem as a coupled partial differential–algebraic equation system, i.e., the boundary conditions are considered as a second dynamic equation that is coupled to the bulk problem. The splitting approach is combined with bulk–surface finite elements and an implicit Euler discretization of the two subsystems. We prove first-order convergence of the resulting fully discrete scheme in the presence of a weak CFL condition of the form $\tau \leqslant c h$ for some constant $c>0$. The convergence is also illustrated numerically using dynamic boundary conditions of Allen–Cahn type.

A Reduced and Linearized High Fidelity Waveboard Multibody Model for Stability Analysis

AbstractIn this paper, the robustness of a recently validated linearization approach is demonstrated with the linear stability analysis of a waveboard, a human-propelled two-wheeled vehicle consisting in two rotatable platforms, joined by a torsion bar and supported on two caster wheels. A multibody model with holonomic and nonholonomic constraints is used to describe the system. The nonlinear equations of motion, which constitute a differential-algebraic system of equations (DAE system), are linearized along the steady forward motion. With this approach, the minimal set of linearized equations of motion of the waveboard multibody model with toroidal wheels is derived. The procedure enables the generation of the Jacobian matrix in terms of the geometric and dynamic parameters of the multibody system, and the eigenvalues of the system are parameterized in terms of the design parameters. The resulting minimum set of linear equations leads to the elimination of null eigenvalues, while retaining all the stability information in spite of the reduction of the Jacobian matrix. The linear stability results of the waveboard obtained in previous work are validated with this approach. The procedure shows an excellent computational efficiency with the waveboard, its utilization being highly advisable to linearize the equations of motion of complex constrained multibody systems.

New approach in solving derivative causality problems in the bond-graph method

From the bond-graph model of a dynamic system, the state equations can be obtained. When the system contains only energy storing elements in integral causality, the system of state equations is immediately and directly determined. When the model contains at least one energy storing element in derivative causality, the resulted system of differential-algebraic equations arises some difficulties in finding the final form of the system of state equations. The work presents a new method of deducing the state equations in case of bond-graph models with one, or several energy storing elements in derivative causality. This method is based on the kinetic energy of the system and offers the possibility to avoid a difficult mathematical calculus for the transition from a system of differential algebraic equations to the system of state equations.

Dynamic modeling of a topside process plant with modified black-oil approach

Modeling and simulation of oil and gas facilities has been done by several authors in closed-source commercial applications, with low flexibility for model adaptation. A few authors used free modeling environments, which results in a differential-algebraic equations (DAE) system and lets the model be adaptable for plantwide simulation. Still, it has limitations in obtaining the model's initial conditions and to perform dynamic simulations when the compositional approach is used. The Modified Black-Oil (MBO) approach, which models the thermodynamic equilibria in a simpler way, has been used by several authors for reservoir modeling but not for topside facilities. This work proposes the use of the MBO approach for topside dynamic modeling. This was done in EMSO©, obtaining a large index-1 DAE system and successful obtention of consistent initial conditions. Disturbances on specific variables were applied and shows the potential use of the MBO approach for new dynamic studies.

Nonlinear dynamic analysis and numerical continuation of periodic orbits in high-index differential–algebraic equation systems

Nonlinear dynamic analysis and numerical continuation of periodic orbits in high-index differential–algebraic equation systems

An efficient extended block Arnoldi algorithm for feedback stabilization of incompressible Navier-Stokes flow problems

Navier-Stokes equations are well known in modeling of an incompressible Newtonian fluid, such as air or water. This system of equations is very complex due to the non-linearity term that characterizes it. After the linearization and the discretization parts, we get a descriptor system of index-2 described by a set of differential algebraic equations (DAEs). The two main parts we develop through this paper are focused firstly on constructing an efficient algorithm based on a projection technique onto an extended block Krylov subspace, that appropriately allows us to construct a reduced system of the original DAE system. Secondly, we solve a Linear Quadratic Regulator (LQR) problem based on a Riccati feedback approach. This approach uses numerical solutions of large-scale algebraic Riccati equations. To this end, we use the extended Krylov subspace method that allows us to project the initial large matrix problem onto a low order one that is solved by some direct methods. These numerical solutions are used to obtain a feedback matrix that will be used to stabilize the original system. We conclude by providing some numerical results to confirm the performance of our proposed method compared to other known methods.

FORMATION OF A DESIGN MODEL AND ANALYSIS OF THE STRESS-STRAIN STATE OF STRUCTURAL ELEMENTS UNDER ALTERNATING LOADING

The article gives geometric and physical relationships for shell structure elements. To describe the elastoplastic behavior of the shell element, the theory of variable plasticity is used, which allows one to take into account hardening-softening, cyclic anisotropy, and material damage. Applying the linearization method, after a series of transformations, a canonical system of differential-algebraic equations is obtained that describes the elastoplastic behavior of a composite shell structure. Boundary-value problems are solved using the orthogonal sweep method.

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.

Impact of the diffusion coefficient calculation on predicting Fe2B boride layer thickness

Abstract In this study, a single-phase boride layer thickness Fe2B is predicted on two different substrates (Armco iron and XC38 steel) by following the integral method. This method is a mathematical model based on a system of differential algebraic equations that help to deduce the diffusion coefficient, which is the key factor on predicting the layer thickness. Literatures cover different diffusion coefficients for each substrate, albeit researchers usually extract from experimental data, variations of growth rate constants within only one time treatment and deduce the diffusion coefficient from them. This deduction is done via an estimation of a frequency factor and an activation energy from the growth rate constants. Therefore, our main aim is to illustrate the impact of the deduction of the diffusion coefficient on predicting the boride layer thickness. Lastly, the impact with and without incubation time on the boriding kinetics of both substrates was also examined.

System Analysis of Target-Mediated Drug Disposition (TMDD) Models

In this paper we have performed system analysis of target-mediated drug disposition (TDMM) models and study their stability, steady state (long term) behavior, and complexity. The original one- and two- compartment models are represented by systems of nonlinear differential equations. It has been found that these models are unstable at three different sets of their steady state values, which limits their applicability only to short time intervals. As an extension of the two-compartment TMDD model, the quasi-equilibrium model (QE-TMDD) appear to be very complex since it is represented by the system of differential-algebraic equations. Similar conclusion holds for its variant known as the quasi-steady state QSS-TMDD model. Moreover, it was shown analytically that when the initial value of the drug-receptor complex is zero, both QE-TMDD and QSS-TMDD models produce the zero values for the drug-receptor complex at all times, which appear to be a serious drawback for the applicability of these models.

Computer simulation of boronizing kinetics for a TB2 alloy

Abstract The boron diffusion at the surface of a TB2 alloy was simulated via two mathematical models relying on the numerical resolutions of the system of differential algebraic equations (DAE) for the integral method and ordinary differential equations for the mean diffusion coefficient (MDC) method. Both approaches allowed us to compute the boron diffusion coefficients in TiB2 and TiB for a maximum boron content of 31.10 wt.-% in TiB2 at 1223, 1273, 1323 and 1373 K. The boron activation energies in TiB2 and TiB were evaluated and compared with the data published in the literature. Finally, an experimental validation of both models was made through a comparison of the thicknesses of the experimental layers with the predicted values. Consequently, the simulated thicknesses were in line with the experimental values.

Direct single shooting for dynamic optimization of differential-algebraic equation systems with optimization criteria embedded

Differential-algebraic equation systems with embedded optimization criteria (DAEOs) arise in a number of applications, most prominently dynamic models of separation processes based on phase equilibrium and microbial transformation processes modeled via the dynamic flux balance analysis approach.We consider the optimization of DAEOs. To this end, we first reformulate the DAEO using first-order optimality conditions yielding a nonsmooth DAE system. This solution approach tracks a stationary point of the embedded optimization problem that may not be globally or locally optimal on the total time horizon for non-convex problems.We use adjoint sensitivity analysis for the resulting nonsmooth DAEs for gradient-based optimization in a direct single-shooting approach. We apply the solution method to the optimization of a single flash unit, calculation of an optimal start-up scenario of a rectification column, and the optimization of biomass growth in a microbial transformation process.

The boundary value problems for the linear singularly perturbed systems of differential-algebraic equations

The article considers the main stages of the development of the theory of asymptotic integration of boundary-value problems for linear singularly perturbed differential-algebraic systems. The need of developing constructive methods of approximate integration of boundary-value problems for differential-algebraic systems is due to the importance of their practical application in the theory of nonlinear oscillations, stability of motion, control theory, radio engineering, and biology.
 In the present paper the authors offer a review of literary sources, which consider the methods of constructing asymptotic solutions of singularly perturbed systems of differential equations with a degenerate matrix with derivatives under the condition of stability of the spectrum of the limit pencil of matrices. It is noted that the problem of constructing asymptotic solutions of boundary-value problems for systems of this type is poorly studied, and therefore relevant. In particular, the question of the conditions for the existence and uniqueness of the solutions of these problems and the development of methods for constructing their asymptotics in various cases related to the behavior of the spectrum of the limit pencil of matrices has been poorly researched.

Evaluating the performance of an Inexact Newton method with a preconditioner for dynamic building system simulation

Efficiently, robustly, and accurately solving systems of nonlinear differential algebraic equations for dynamic building system simulation is becoming more important due to the increasing need to simulate large-scale problems. The focus of this paper is an investigation of methods for solving the nonlinear algebraic equations encountered in dynamic building system simulations. Dynamic building system simulations employ a myriad of solution techniques, many of which use derivative information. As the problem size grows, so do the memory requirements, leading to challenges in solving large-scale problems. Newton-Krylov methods are a promising candidate for large-scale simulation. The performance of these methods is often improved, sometimes drastically, when employed with a preconditioning technique. Here, a cost-effective preconditioning technique is applied to a Newton-Krylov method and tested on a series of problems. To illustrate the benefits of the approach, comparison is made to the frequently employed Powell’s Hybrid Method.

Magnetic field simulations using explicit time integration with higher order schemes

PurposeA transient magneto-quasistatic vector potential formulation involving nonlinear material is spatially discretized using the finite element method of first and second polynomial order. By applying a generalized Schur complement the resulting system of differential algebraic equations is reformulated into a system of ordinary differential equations (ODE). The ODE system is integrated in time by using explicit time integration schemes. The purpose of this paper is to investigate explicit time integration for eddy current problems with respect to the performance of the first-order explicit Euler scheme and the Runge-Kutta-Chebyshev (RKC) method of higher order.Design/methodology/approachThe ODE system is integrated in time using the explicit Euler scheme, which is conditionally stable by a maximum time step size. To overcome this limit, an explicit multistage RKC time integration method of higher order is used to enlarge the maximum stable time step size. Both time integration methods are compared regarding the overall computational effort.FindingsThe numerical simulations show that a finer spatial discretization forces smaller time step sizes. In comparison to the explicit Euler time integration scheme, the multistage RKC method provides larger stable time step sizes to diminish the overall computation time.Originality/valueThe explicit time integration of the Schur complement vector potential formulation of eddy current problems is accelerated by a multistage RKC method.

Using computational singular perturbation as a diagnostic tool in ODE and DAE systems: a case study in heterogeneous catalysis

We have extended the computational singular perturbation (CSP) method to differential algebraic equation (DAE) systems and demonstrated its application in a heterogeneous-catalysis problem. The extended method obtains the CSP basis vectors for DAEs from a reduced Jacobian matrix that takes the algebraic constraints into account. We use a canonical problem in heterogeneous catalysis, the transient continuous stirred tank reactor (T-CSTR), for illustration. The T-CSTR problem is modelled fundamentally as an ordinary differential equation (ODE) system, but it can be transformed to a DAE system if one approximates typically fast surface processes using algebraic constraints for the surface species. We demonstrate the application of CSP analysis for both ODE and DAE constructions of a T-CSTR problem, illustrating the dynamical response of the system in each case. We also highlight the utility of the analysis in commenting on the quality of any particular DAE approximation built using the quasi-steady state approximation (QSSA), relative to the ODE reference case.

Differential-algebraic boundary-value problems with the variable rank of leading-coefficient matrix

Conditions for the solvability of the linear boundary-value problem for systems of differential-algebraic equations with the variable rank of the leading-coefficient matrix and the corresponding solution construction procedure have been found.

Differential-algebraic boundary-value problems with the variable rank of leading-coefficient matrix

Conditions for the solvability of the linear boundary-value problem for systems of differential-algebraic equations with the variable rank of the leading-coefficient matrix and the corresponding solution construction procedure have been found.

A Robust and Sleek Electrochemical Battery Model Implementation: A MATLAB® Framework

A Doyle-Fuller-Newman electrochemical battery model implementation in a robust and sleek MATLAB® framework for lithium-ion batteries as an open-access MATLAB code is presented. The Doyle-Fuller-Newman (DFN) model, in the form of partial differential equations, is first numerically discretized then converted to a differential algebraic equation (DAE). The most efficient way to implement a DAE system is through the adoption of standard DAE solvers provided by available commercial software. MATLAB® is a widely used software in the control community, and to the best of our knowledge, its standard solvers have failed to successfully simulate the DFN model when the battery undergoes high C-rates of operations. One critical issue with DFN model simulation is related to the inconsistency of initial conditions. In 2015, a robust single-step iteration-free initialization approach, enabling solving DAE systems using a standard solver using Maple® symbolic environment, was proposed by Lawder et al. A symbolic environment enables direct and efficient derivation of implicit ordinary differential equations from algebraic equations during initialization. We perform the single-step iteration-free initialization approach in MATLAB® environment by adopting the MATLAB® symbolic toolbox and simulate the DFN model with the ode15s solver. This framework allows users to robustly simulate the DFN model and identify model parameters directly after numerical discretization, utilizing a standard solver.