System Of Differential-algebraic Equations Research Articles

Mathematical Approach for Directly Solving Air–Water Interfaces in Water Emptying Processes

Emptying processes are operations frequently required in hydraulic installations by water utilities. These processes can result in drops to sub-atmospheric pressure pulses, which may lead to pipeline collapse depending on soil characteristics and the stiffness of a pipe class. One-dimensional mathematical models and 3D computational fluid dynamics (CFD) simulations have been employed to analyse the behaviour of the air–water interface during these events. The numerical resolution of these models is challenging, as 1D models necessitate solving a system of algebraic differential equations. At the same time, 3D CFD simulations can take months to complete depending on the characteristics of the pipeline. This presents a mathematical approach for directly solving air–water interactions in emptying processes involving entrapped air, providing a predictive tool for water utilities. The proposed mathematical approach enables water utilities to predict emptying operations in water pipelines without needing 2D/3D CFD simulations or the resolution of a differential algebraic equations system (1D model). A practical application is demonstrated in a case study of a 350 m long pipe with an internal diameter of 350 mm, investigating the influence of air pocket size, friction factor, polytropic coefficient, pipe diameter, resistance coefficient, and pipe slope. The mathematical approach is validated using an experimental facility that is 7.36 m long, comparing it with 1D mathematical models and 3D CFD simulations. The results confirm that the derived mathematical expression effectively predicts emptying operations in single water installations.

Optimal control of a model for dynamic district heating networks

In the present paper an optimal control problem for a concrete system of differential-algebraic equations (DAEs) is considered. This problem arises in the dynamic optimization of unsteady district heating networks. Based on the Carathéodory theory existence of a unique solution to the specific DAE system is proved using specific properties of the considered district heating network model. Moreover, it is shown that the optimal control problem possesses optimal solutions. For the numerical experiments different networks are considered including also data from a real district heating network.

Model-based workflow for sustainable production of high-quality spirits in packed column stills

This study addresses water scarcity in Chilean distilleries by developing a model-based engineering workflow. Prolonged droughts, likely driven by global warming, have intensified this problem. The primary challenge is maintaining high-quality spirits while reducing cooling water and energy use during batch distillation in packed columns. This process was modeled using mass and energy balances, resulting in a system of partial differential algebraic equations (PDAE). Our workflow includes mechanistic modeling, disturbance modeling, multiobjective optimization, model predictive control, and Monte Carlo simulations. Our findings show that cooling water consumption can be reduced by up to 35 % and energy consumption by up to 14.4 % while maintaining product quality. The proposed system is robust against operational disturbances and model mismatch, ensuring consistent distillate quality. This research demonstrates the integration of model-based optimization and control strategies in batch distillation processes, which can be replicated in other fruit wine distillation processes for improved sustainability.

A numerical framework for modeling evaporation and combustion of isolated, spherically-symmetric, multi-component fuel droplets

This paper presents a comprehensive numerical framework for simulating the evaporation and combustion of isolated, spherically-symmetric, multi-component fuel droplets. The framework incorporates a detailed description of chemical reactions in the gaseous phase and is capable of modeling pure evaporation, autoignition, and hot-wire ignition scenarios.The transport equations for mass, species, and energy are solved in both the liquid (droplet) and gaseous (surrounding atmosphere) phases. Diffusion in the liquid phase is described using the Stefan–Maxwell theory, while in the gaseous phase, both molecular and thermal diffusion are considered. The model also includes the thermophoretic effect for carbonaceous particles and accounts for gas radiation through various models, ranging from optically-thin approximations to more complex methods like the P1 and discrete ordinate methods. Non-gray radiative effects are handled using the Weighted-Sum-of-Gray-Gases Model (WSGGM). Liquid/gas interface conditions are evaluated by imposing flux continuity of mass and energy, along with thermodynamic equilibrium for species. Deviations from ideal thermodynamic behavior in the liquid droplet are managed by incorporating a suitable activity coefficient or by using a proper cubic equation of state. Additionally, the presence of supporting fibers is modeled using a simplified one-dimensional approach. The transport equations are solved using the method of lines, with spatial discretization performed via the finite difference method on a body-fitted grid. The resulting system of Differential–Algebraic Equations (DAEs) is then solved using a fully-coupled approach.Thanks to its generality in terms of kinetic and thermodynamic descriptions and the reduced computational time, the proposed framework offers significant potential for advancing our understanding of the complex combustion processes of multi-component liquid fuels and for enhancing the planning and execution of experiments involving isolated fuel droplets.

Physics-informed neural networks for dynamic process operations with limited physical knowledge and data

In chemical engineering, process data are expensive to acquire, and complex phenomena are difficult to fully model. We explore the use of physics-informed neural networks (PINNs) for modeling dynamic processes with incomplete mechanistic semi-explicit differential–algebraic equation systems and scarce process data. In particular, we focus on estimating states for which neither direct observational data nor constitutive equations are available. We propose an easy-to-apply heuristic to assess whether estimation of such states may be possible. As numerical examples, we consider a continuously stirred tank reactor and a liquid-liquid separator. We find that PINNs can infer immeasurable states with reasonable accuracy, even if respective constitutive equations are unknown. We thus show that PINNs are capable of modeling processes when relatively few experimental data and only partially known mechanistic descriptions are available, and conclude that they constitute a promising avenue that warrants further investigation.

On functions of finite analytical complexity

We construct examples of polynomials and analytic functions of any predetermined finite analytical complexity n n . We obtain an estimate of the order of derivative of the differential-algebraic criteria for membership in the class C l n Cl_{n} of functions of analytical complexity not higher than n n . We find uniform estimates for finite values d n d_{n} of the analytic spectrum { d n } \{d_{n}\} for systems of differential-algebraic equations of fixed order of derivative δ \delta .

H∞ stabilization of singular networked cascade control systems subject to periodic DoS attack

In this paper, a [Formula: see text] controller with an event-triggered mechanism is designed for a singular networked cascade control system (SNCCS) under periodic denial-of-service (DoS) attacks. The SNCCS is treated as a system of differential-algebraic equations (DAEs) with an index of two and is modeled as a switched system. The paper first establishes sufficient stability conditions, which are formulated as linear matrix inequalities based on multiple Lyapunov functions. These conditions ensure that the system remains regular, free from impulse modes, and stable. Next, state feedback controllers are designed for both the inner loop and the outer loop. However, the outer loop is subjected to periodic DoS cyber-attacks. Finally, the proposed method’s effectiveness is demonstrated through simulation results on a practical thermal cascade control system.

A grey-box model with neural ordinary differential equations for the slow voltage dynamics of lithium-ion batteries: Application to single-cell experiments

Lithium-ion batteries exhibit a complex, nonlinear and dynamic voltage behaviour. Modelling their slow dynamics is a challenge due to the multiple potential causes involved. We present here a neural equivalent circuit model for lithium-ion batteries including slow voltage dynamics. The model uses an equivalent circuit with voltage source, series resistor, and diffusion element. The series resistance is parameterized using neural networks. The diffusion element is based on a discretized form of Fickian diffusion, parameterized using a neural network and learnable parameters. It is flexible to represent not only Warburg behaviour, but also resistor-capacitor-type dynamics. Mathematically, the resulting model is given by a differential–algebraic equation system combining ordinary and neural differential equations. Therefore, the model combines concepts of both physical theory (white-box model) and artificial intelligence (black-box model) to a combined framework (grey-box model). We apply this approach to a lithium iron phosphate based lithium-ion battery cell. The experimental voltage behaviour during constant-current cycles as well as the dynamics during pulse tests are well reproduced by the model. Only at very high and very low states of charge the simulation significantly deviates from experiments, which might result from insufficient training data in these regions.

The importance of incorporating ventricular–ventricular interaction (VVI) in the study of pulmonary hypertension

Ventricular ventricular interaction (VVI) affects blood volume and pressure in the right and left ventricles of the heart due to the location and balance of forces on the septal wall separating the ventricles. In healthy patients, the pressure of the left ventricle is considerably higher than the right, resulting in a septal wall that bows into the right ventricle. However, in patients with pulmonary hypertension, the pressure in the right ventricle increases significantly to a point where the pressure is similar to or surpasses that of the left ventricle during portions of the cardiac cycle. For these patients, the septal wall deviates towards the left ventricle, impacting its function. It is possible to study this effect using mathematical modeling, but existing models are nonlinear, leading to a system of algebraic differential equations that can be challenging to solve in patient-specific optimizations of clinical data. This study demonstrates that a simplified linearized model is sufficient to account for the effect of VVI and that, as expected, the impact is significantly more pronounced in patients with pulmonary hypertension.

Nonlinear state estimation of a power plant superheater by using the extended Kalman filter for differential algebraic equation systems

A nonlinear estimator was developed for process monitoring of a power plant superheater system to estimate the transient temperature profiles of the critical measured and unmeasured variables under load-following operations. The model used in the estimator was a detailed, distributed, first-principles, differential–algebraic equation model. The superheater geometry is characterized based on an operating natural gas combined cycle power plant. The model included mass and energy balances for the operating fluids, i.e., steam and flue gas, with due consideration of tube wall temperature dynamics. The dynamic model was used for state and parameter estimation by using a modified extended Kalman filter. When results are compared with the data from a natural gas power plant, the estimator yielded root mean squared error of 0.2 °C for the main steam temperature compared to 1 °C obtained from the first-principles model. While the first-principles model yielded as high as 2.5 °C absolute instantaneous error for the tube temperatures, the estimator reduced the absolute instantaneous error below 0.3 °C for most variables. Performance of the estimator for estimating critical variables like the main steam temperature and flue gas temperature was evaluated in the presence of high model mismatch and noisy measurement data by simulating load-following operation of the plant. The estimator yielded maximum absolute instantaneous error of 3.4 °C for the flue gas and 1 °C for the main steam temperature.

Calculation of Plates of Different Shapes Using Generalized Equations of the Finite Difference Method

Statement of the problem. The article reviewed nonaxisymmetric tasks for calculating slabs in shape of a circle sector, wedge and a slab formed by a connection of rectangular and circular areas. The solution is based on generalized equations of the finite differences method. The algorithm allows us to take into account finite discontinuities of the desired function, the right side of the original differential equations, as well as discontinuities of derivatives of these functions without accumulation of the grid and using points outside the contour. It resolves oneself into a system of differential algebraic equations generated for field and contour points. In this paper the initial differential equations and their numerical approximation are provided. Results. Calculations of a semicircle and a wedge under different boundary conditions are presented, as well as an uncut slab in the form of a semicircle with an annular support and a composite slab are consi-dered. The solution of the above problems has been carried out on the minimum possible computational grid, comparison with the available solutions has been provided, and static checks of the calculation have been carried out. Conclusions. Along with the most common numerical method for calculating buildings and structures, it is possible and quite effective to use these equations. It is proved by means of some examples that such an approximation has a number of advantages: the results allow us to talk about the sufficient accuracy of the solution with a small number of partitions; fundamental simplicity and universality of the method.

A modified cell-centered nodal integral scheme for the convection-diffusion equation

The nodal integral methods (NIMs) are very efficient and accurate coarse-mesh methods for solving partial differential equations. The cell-centered NIM (CCNIM) is a simplified variant of the NIMs that has recently shown its efficiency in solving fluid flow problems but has been hampered by issues such as inapplicability to one-dimensional problems, complexities in handling Neumann boundary conditions and the formulation of a system of differential-algebraic equations (DAEs) for discrete unknowns. Here, we present a modified version of the CCNIM designed to overcome the challenges associated with its previous version. Our novel development retains the essence of CCNIM while resolving these issues. The proposed scheme, grounded in the nodal framework, achieves second-order accuracy in both spatial and temporal dimensions. Unlike its precursor, the proposed method formulates algebraic equations for discrete variables per node, eliminating the cumbersome DAE system. Neumann boundary conditions are seamlessly incorporated through a straightforward flux definition, and applicability to one-dimensional problems is now feasible. We successfully apply our approach to one and two-dimensional convection-diffusion problems with known analytical solutions to validate our approach. The simplicity and robustness of the approach lay the foundation for its seamless extension to more complex fluid flow problems.

Index‐aware learning of circuits

AbstractElectrical circuits are present in a variety of technologies, making their design an important part of computer aided engineering. The growing number of parameters that affect the final design leads to a need for new approaches to quantify their impact. Machine learning may play a key role in this regard; however, current approaches often make suboptimal use of existing knowledge about the system at hand. In terms of circuits, their description via modified nodal analysis is well‐understood. This particular formulation leads to systems of differential‐algebraic equations (DAEs), which bring with them a number of peculiarities, for example, hidden constraints that the solution needs to fulfill. We use the recently introduced dissection index that can decouple a given system of DAEs into ordinary differential equations, only depending on differential variables, and purely algebraic equations, that describe the relations between differential and algebraic variables. The idea is to then only learn the differential variables and reconstruct the algebraic ones using the relations from the decoupling. This approach guarantees that the algebraic constraints are fulfilled up to the accuracy of the nonlinear system solver, and it may also reduce the learning effort as only the differential variables need to be learned.

Physics-Informed Neural Networks for Solving High-Index Differential-Algebraic Equation Systems Based on Radau Methods

As is well known, differential algebraic equations (DAEs), which are able to describe dynamic changes and underlying constraints, have been widely applied in engineering fields such as fluid dynamics, multi-body dynamics, mechanical systems, and control theory. In practical physical modeling within these domains, the systems often generate high-index DAEs. Classical implicit numerical methods typically result in varying order reduction of numerical accuracy when solving high-index systems. Recently, the physics-informed neural networks (PINNs) have gained attention for solving DAE systems. However, it faces challenges like the inability to directly solve high-index systems, lower predictive accuracy, and weaker generalization capabilities. In this paper, we propose a PINN computational framework, combined Radau IIA numerical method with an improved fully connected neural network structure, to directly solve high-index DAEs. Furthermore, we employ a domain decomposition strategy to enhance solution accuracy. We conduct numerical experiments with two classical high-index systems as illustrative examples, investigating how different orders and time-step sizes of the Radau IIA method affect the accuracy of neural network solutions. For different time-step sizes, the experimental results indicate that utilizing a 5th-order Radau IIA method in the PINN achieves a high level of system accuracy and stability. Specifically, the absolute errors for all differential variables remain as low as 10−6, and the absolute errors for algebraic variables are maintained at 10−5. Therefore, our method exhibits excellent computational accuracy and strong generalization capabilities, providing a feasible approach for the high-precision solution of larger-scale DAEs with higher indices or challenging high-dimensional partial differential algebraic equation systems.

On the update of algebraic states during state estimation of differential–algebraic equation (DAE) systems

This manuscript presents a discussion on the algebraic state update step performed during recursive filtering of the differential–algebraic equation (DAE) systems. Existing DAE state estimation approaches follow a two-step state update procedure at each sampling instant. In particular, they first estimate the differential states using the Kalman update, and then update algebraic states by explicitly solving the algebraic equations. Specifically, for the case of DAE systems involving linear algebraic equations though the differential equations are nonlinear, we show that when appropriately initialized, this two-step state update procedure is not needed. It can instead be replaced with a one-step state update procedure that computes the differential and algebraic state estimates simultaneously through the Kalman update. The satisfaction of algebraic equations is guaranteed by this one-step update without it being explicitly enforced. Towards this end, we show that the error covariance matrix of augmented states, when properly initialized, satisfies a null-space property after prediction and update step at each sampling instant. This property ensures that the state estimates obtained using the proposed one-step update approach, satisfy the algebraic equations. This holds for both analytical linearization based extended Kalman filtering and statistical linearization based sigma-point filtering approaches. We also propose a heuristic-based update procedure for state estimation of DAE systems that involve nonlinear algebraic equations. This procedure draws out inferences from the case of DAE systems involving linear algebraic equations and is based on the analysis of algebraic equations residuals obtained from the updated differential and algebraic state estimates with a one-step state update. The efficacy of the proposed state update procedures is demonstrated by performing simulation studies on a benchmark drum boiler system case study. Results demonstrate that the proposed update procedures satisfactorily estimate the differential and algebraic states of a DAE system when compared to the traditional two-step update procedure.

Plasticity in nonlinear elastic metamaterials: Low-order dynamic modeling

Nonlinear elastic metamaterials have been shown to admit a variety of rich, dynamical features that can be leveraged to tailor the propagation of mechanical waves. Since these materials derive their properties from intricate, subwavelength geometries, direct numerical simulations are often prohibitively expensive at scales of interest. To overcome this limitation, reduced-order models, typically in the form of effective continua or discrete lattices that capture the essential features of the material at sufficiently long wavelengths, have been developed. While many prior studies have implemented these models successfully, the vast majority have considered only recoverable elastic deformations with linear damping and neglected history-dependent effects, such as plasticity and friction. In this presentation, we introduce an effective lattice modeling framework for nonlinear elastic metamaterials undergoing plastic deformation. Due to the history-dependent nature of plasticity, this framework generally yields a system of differential-algebraic equations whose computational cost is significantly greater than a purely elastic system of similar size. We apply the method to several examples of interest and explore means to obtain phenomenological elastic-plastic models for general material architectures.

Solving differential‐algebraic equations in power system dynamic analysis with quantum computing

AbstractPower system dynamics are generally modeled by high dimensional nonlinear differential‐algebraic equations (DAEs) given a large number of components forming the network. These DAEs' complexity can grow exponentially due to the increasing penetration of distributed energy resources, whereas their computation time becomes sensitive due to the increasing interconnection of the power grid with other energy systems. This paper demonstrates the use of quantum computing algorithms to solve DAEs for power system dynamic analysis. We leverage a symbolic programming framework to equivalently convert the power system's DAEs into ordinary differential equations (ODEs) using index reduction methods and then encode their data into qubits using amplitude encoding. The system nonlinearity is captured by Hamiltonian simulation with truncated Taylor expansion so that state variables can be updated by a quantum linear equation solver. Our results show that quantum computing can solve the power system's DAEs accurately with a computational complexity polynomial in the logarithm of the system dimension. We also illustrate the use of recent advanced tools in scientific machine learning for implementing complex computing concepts, that is, Taylor expansion, DAEs/ODEs transformation, and quantum computing solver with abstract representation for power engineering applications.

On solving system of differential-algebraic equations using adomian decomposition method.

In this paper, we focus on an efficient and easy method for solving the given system of differential-algebraic equations (DAEs) of second order. The approximate solutions are computed rapidly and efficiently with the help of a semi-analytical method known as Adomian decomposition method (ADM). The logic of this method is simple and straightforward to understand. To demonstrate the proposed method, we presented several examples and the computations are compared with the exact solutions to show the efficient. One can employ this logic to different mathematical software tools such as Maple, SCILab, Mathematica, NCAlgebra, Matlab etc. for the problems in real life applications. In this paper, we offered a method for solving the given system of secondorder nonlinear DAEs with aid of the ADM. We shown that the proposed method is simple and efficient, also one can obtain the approximate solutions quickly using this method. A couple of examples are discussed for illustrating this method and graphical and mathematical assessments are discussed with the analytical solutions of the given problems.

Modeling carbon nutrition and photosynthesis taking into account plant stomatal resistance

A differential-algebraic system of equations of carbon dioxide transport and photosynthesis in vegetation is formulated for mathematical analysis of the influence on productivity and water regime of plant populations. A simple semi-empirical algebraic approximation of the true photosynthesis function is constructed for the quasi-stationary solution of the system, including the influence of stomatal regulation, leaf index, absorbed photosynthetically active radiation, and temperature.

CSPlib: A performance portable parallel software toolkit for analyzing complex kinetic mechanisms

Computational singular perturbation (CSP) is a method to analyze dynamical systems. It targets the decoupling of fast and slow dynamics using an alternate linear expansion of the right-hand side of the governing equations based on eigenanalysis of the associated Jacobian matrix. This representation facilitates diagnostic analysis, detection and control of stiffness, and the development of simplified models. We have implemented CSP in a C++ open-source library CSPlib1 using the Kokkos2 parallel programming model to address portability across diverse heterogeneous computing platforms, i.e., multi/many-core CPUs and GPUs. We describe the CSPlib implementation and present its computational performance across different computing platforms using several test problems. Specifically, we test the CSPlib performance for a constant pressure ignition reactor model on different architectures, including IBM Power 9, Intel Xeon Skylake, and NVIDIA V100 GPU. The size of the chemical kinetic mechanism is varied in these tests. As expected, the Jacobian matrix evaluation, the eigensolution of the Jacobian matrix, and matrix inversion are the most expensive computational tasks. When considering the higher throughput characteristic of GPUs, GPUs performs better for small matrices with higher occupancy rate. CPUs gain more advantages from the higher performance of well-tuned and optimized linear algebra libraries such as OpenBLAS. Program summaryProgram Title: CSPlibCPC Library link to program files:https://doi.org/10.17632/p9gb7z54sp.1Developer's repository link:https://github.com/sandialabs/csplibLicensing provisions: BSD 2-clauseProgramming language: C++Nature of problem: Dynamical systems can involve coupled processes with a wide range of time scales. The computational singular perturbation (CSP) method offers a reformulation of these systems which enables the use of dynamically-based diagnostic tools to better comprehend the dynamics by decoupling fast and slow processes. CSPlib is an open-source software library for analyzing general ordinary differential equation (ODE) and differential algebraic equation (DAE) systems, with specialized implementations for detailed chemical kinetic ODE/DAE systems. It relies on CSP for the analysis of these systems. CSPlib has been used in gas kinetic and heterogeneous catalytic kinetic models.Solution method: CSP analysis seeks a set of basis vectors to linearly decompose the right-hand side (RHS) of a dynamical system in a manner that decouples fast and slow processes. The CSP basis vectors are often well approximated with the right eigenvectors of the RHS Jacobian. And the left basis vectors are found by the inversion of the matrix, whose columns are the CSP basis vectors. Accordingly, the right and left CSP basis vectors are orthonormal. CSP defines mode amplitudes as the projections of the left basis vectors on the RHS; the time scales as the reciprocals of the RHS Jacobian eigenvalue magnitudes; and the CSP pointers, which are the element-wise multiplication of the transpose of the right CSP basis vectors with the left CSP basis vectors. For kinetic models that can be cast as the product of a generalized stoichiometric matrix and a rate of progress vector, CSP defines the participation index, which represents the contribution of a chemical reaction to each mode. Further, it defines the slow and fast importance indices, which describe the contribution of a chemical reaction to the slow and fast dynamics of a state variable, respectively. These indices are useful in diagnostic studies of dynamical systems and the construction of simplified models.Additional comments including restrictions and unusual features: CSPlib is a portable library that carries out many CSP analyses in parallel and can be used in modern high-performance platforms.