Differential-algebraic Equations Research Articles

Conditions of Solvability of the Problem Inverse to the Cauchy Problem for a Difference-Algebraic Equation

We establish conditions of solvability for the problem inverse to the Cauchy problem for a difference equation. In the case where the problem inverse to the Cauchy problem for the difference equation is not solvable, we propose conditions for the solvability of the problem inverse to the Cauchy problem for a difference-algebraic equation.

Discretization of inherent ODEs and the geometric integration of DAEs with symmetries

Discretization methods for differential-algebraic equations (DAEs) are considered that are based on the integration of an associated inherent ordinary differential equation (ODE). This allows to make use of any discretization scheme suitable for the numerical integration of ODEs. For DAEs with symmetries it is shown that the inherent ODE can be constructed in such a way that it inherits the symmetry properties of the given DAE and geometric properties of its flow. This in particular allows the use of geometric integration schemes with a numerical flow that has analogous geometric properties.

Modeling/simulation of transient linear heat conduction problems via integrating a wide variety of space/time methods and choices

We propose an implementation of a general purpose multi-spatial method, multi-time scheme subdomain Differential Algebraic Equations (DAE) framework allowing a mix of different space discretization methods while interfacing altogether different time integration algorithms on a single body analysis for the linear first order transient systems. With the concept of subdomains and DAE framework which allows constraint in time as well as space, we can divide a body into multiple subdomains and implement different spatial methods (finite elements, particles, etc.) in different regions of the body including altogether different time integrators and enable targeting an area with a specific method which can fully utilize its best features. The robust Generalized Single Step Single Solve (GS4) family of Linear Multistep (LMS) framework with second order time accuracy encompasses most of the developments over the last 50 years or so including new and optimal designs, and provides a wide variety of choices available to the analyst in a single analysis setting. We also advance, analyze and present the GS4 parameter studies regarding parameter selection. The GS4 readily allows flexibility in matching different time integration schemes, both existing traditional time schemes and other new and optimal competitive designs while maintaining second order time accuracy. Such implementation of coupling a wide variety of different spatial and time integration algorithms is not possible (especially implicit-implicit algorithm couplings) in the current state of the technology. Existing traditional practices of coupling multiple spatial and time algorithms show limitations in reduced order of accuracy, and consistency and the like in various field variables typical of, and including, temperature, temperature rate, and Lagrange multipliers. Various combinations of spatial methods and time algorithms between subdomains are tested with simple linear first order system problems for readers to readily mimic the results and demonstrate the overall robustness and efficacy for numerical computations.

Canonical Euler splitting method for parabolic partial functional differential algebraic equations

A novel canonical Euler splitting method is presented for semilinear composite stiff parabolic partial functional differential algebraic equations with initial and Dirichlet boundary conditions. The original partial differential problems are transformed into the semi-discrete problems by spatial discretization, and then the canonical Euler splitting method is employed to solve the resulting semi-discrete problems. Under appropriate assumptions, the stability and convergence theories of this method are established. A series of numerical experiments are given to illustrate the effectiveness of this method and the correctness of theoretical results.Numerical results also demonstrate that the constructed method can significantly improve the calculation speed.

A novel dynamic dry friction model for applications in mechanical dynamical systems

This work presents a novel continuously differentiable, dynamic dry friction model, with dependence on normal contact force, slip velocity, and static and dynamic coefficients of friction. The state-of-the-art, Brown and McPhee friction model depends on transitional velocity, which is an empirical parameter. A limitation with this model is that there is no generic approach to select the optimal value for the transitional velocity for a certain application. The simulation results presented in this work highlight the sensitivity of friction force with transitional velocity in Brown and McPhee’s model to obtain smooth solutions, which are supportive in control system applications. It is because control systems require jitter-free signals in order to save costs on low-pass filters. The proposed model overcomes such empirical dependence on transition velocity through a combination of an iterative methodology and an empirical parameterization. In this article, a comprehensive analysis of the proposed friction model and the contributing parameters is carried out. The algorithm to implement the proposed friction model is elaborated. The proposed friction model is then simulated and compared against Brown and McPhee’s model. using a stick-slip benchmark problem. Furthermore, the proposed force model is applied to two spatial multibody systems formulated using index 0 tangent space differential-algebraic equations. The results demonstrate how the proposed friction model can be employed in dynamical mechanical systems. The independence on transitional velocity in the proposed friction model is observed to be effective for obtaining smooth solutions that make it suitable for control system applications.

Numerical Simulation of Nonlinear and Non-Isothermal Liquid Chromatography for Studying Thermal Variations in Columns Packed with Core-Shell Particles

A high-resolution flux-limiting semi-discrete finite volume scheme (HR-FVS) is applied in this study to numerically approximate the nonlinear and non-isothermal flow of one-dimensional lumped kinetic model (1D-LKM), for a fixed-bed column loaded with core-shell particles. The developed model comprise a system of convection-dominated partial differential for mass and energy balances in the mobile phases coupled with differential equation and algebraic equation in the stationary phase. The solution of the model equations is obtained by utilizing a HR-FVS, the scheme has second-order accuracy even on the grid coarse and its explicit nature has the potential to resolve the arisen sharp discontinuities in the solution profiles. A second-order total variation diminishing (TVD) Runge-Kutta technique is used to solve the system of ODEs in time. Several forms of a single-solute mixture are produced to investigate the influences of the fractions of core radius on thermal waves and concentration fronts. Moreover, a particular criterion is introduced for analyzing the performance of the underlying process and to identify the optimal parameter values of the fraction of core radius.

Low-carbon economic dispatch of regional electro-thermal coupled system considering dynamic constraints of CHP units

The flexibility modification of Combined Heat And Power (CHP) units can enable the units to have fast electric output regulation capability, respond to grid auxiliary frequency regulation, and enhance the operational flexibility of regional electric and thermal coupling systems. Compared with the CHP unit constraint model established by the vertex convex combination method, the CHP dynamic constraint in the form of differential equations established by the mechanism modeling can reflect the unit variable states in real time and facilitate the mastering of the unit operation status. With the objective of optimal dispatch cost of the regional electro-thermal coupled system considering the carbon trading process, a low-carbon economic dispatch model of the regional electro-thermal coupled system considering the CHP dynamic constraint is established. Based on the sequential method framework, the finite element orthogonal configuration method is used to discrete the differential algebraic equations of the simulation layer, and the improved adaptive differential evolution algorithm is used to solve the nonlinear planning problem of the optimization layer. The algorithm verifies that the finite element orthogonal configuration method can obtain more accurate results with fewer discrete points, improve the solution speed, and verify that the CHP unit with fast electric output regulation can effectively improve the low-carbon operation of the system, which is also beneficial to the safe and stable operation of the unit.

Error estimates for Runge–Kutta schemes of optimal control problems with index 1 DAEs

In this paper we derive error estimates for Runge–Kutta schemes of optimal control problems subject to index one differential–algebraic equations (DAEs). Usually, Runge–Kutta methods applied to DAEs approximate the differential and algebraic state in an analogous manner. These schemes can be considered as discretizations of the index reduced system where the algebraic equation is solved for the algebraic variable to get an explicit ordinary differential equation. However, in optimal control this approach yields discrete necessary conditions that are not consistent with the continuous necessary conditions which are essential for deriving error estimates. Therefore, we suggest to treat the algebraic variable like a control, obtaining a new type of Runge–Kutta scheme. For this method we derive consistent necessary conditions and compare the discrete and continuous systems to get error estimates up to order three for the states and control as well as the multipliers.

Construction of Rosenbrock–Wanner method Rodas5P and numerical benchmarks within the Julia Differential Equations package

Rosenbrock–Wanner methods for systems of stiff ordinary differential equations are well known since the seventies. They have been continuously developed and are efficient for differential-algebraic equations of index-1, as well. Their disadvantage that the Jacobian matrix has to be updated in every time step becomes more and more obsolete when automatic differentiation is used. Especially the family of Rodas methods has proven to be a standard in the Julia package DifferentialEquations. However, the fifth-order Rodas5 method undergoes order reduction for certain problem classes. Therefore, the goal of this paper is to compute a new set of coefficients for Rodas5 such that this order reduction is reduced. The procedure is similar to the derivation of the methods Rodas4P and Rodas4P2. In addition, it is possible to provide new dense output formulas for Rodas5 and the new method Rodas5P. Numerical tests show that for higher accuracy requirements Rodas5P always belongs to the best methods within the Rodas family.

A novel Cell-Centered nodal integral method for the Convection-Diffusion equation

The nodal integral methods (NIMs) are coarse mesh methods for solving partial differential equations (PDEs) efficiently and accurately. Here, a new variant of the NIM is proposed for the solution of the two-dimensional convection–diffusion equation. The earlier versions of the NIM for fluid flow problems used surface-averaged variables. In contrast, the current scheme is developed in terms of cell-centered values using different treatments of cell interface conditions. The advantage of this scheme is that coupling with other physics is straightforward. Another distinctive feature of the current approach is that the temporal derivative is explicitly addressed, resulting in a system of differential–algebraic equations (DAEs) of index-1. The system of DAEs is then solved using Backward Difference Formula, which provides higher accuracy in time compared to traditional NIM. A few problems with the known analytical solution are solved to demonstrate the efficacy of the current simplified approach.

Parsimonious physics-informed random projection neural networks for initial value problems of ODEs and index-1 DAEs.

We present a numerical method based on random projections with Gaussian kernels and physics-informed neural networks for the numerical solution of initial value problems (IVPs) of nonlinear stiff ordinary differential equations (ODEs) and index-1 differential algebraic equations (DAEs), which may also arise from spatial discretization of partial differential equations (PDEs). The internal weights are fixed to ones while the unknown weights between the hidden and output layer are computed with Newton's iterations using the Moore-Penrose pseudo-inverse for low to medium scale and sparse QR decomposition with L 2 regularization for medium- to large-scale systems. Building on previous works on random projections, we also prove its approximation accuracy. To deal with stiffness and sharp gradients, we propose an adaptive step-size scheme and address a continuation method for providing good initial guesses for Newton iterations. The "optimal" bounds of the uniform distribution from which the values of the shape parameters of the Gaussian kernels are sampled and the number of basis functions are "parsimoniously" chosen based on bias-variance trade-off decomposition. To assess the performance of the scheme in terms of both numerical approximation accuracy and computational cost, we used eight benchmark problems (three index-1 DAEs problems, and five stiff ODEs problems including the Hindmarsh-Rose neuronal model of chaotic dynamics and the Allen-Cahn phase-field PDE). The efficiency of the scheme was compared against two stiff ODEs/DAEs solvers, namely, ode15s and ode23t solvers of the MATLAB ODE suite as well as against deep learning as implemented in the DeepXDE library for scientific machine learning and physics-informed learning for the solution of the Lotka-Volterra ODEs included in the demos of the library. A software/toolbox in Matlab (that we call RanDiffNet) with demos is also provided.

Liouvillian Solutions of Algebraic Ordinary Differential Equations of Order One of Genus Zero

Liouvillian Solutions of Algebraic Ordinary Differential Equations of Order One of Genus Zero

Switched hyperbolic balance laws and differential algebraic equations

Motivated by several applications, we investigate the well-posedness of a switched system composed by a system of linear hyperbolic balance laws and by a system of linear algebraic differential equations. This setting includes networks and looped systems of hyperbolic balance laws. The obtained results are globally in time, provided that the inputs have finite (but not necessarily small) total variation.

Accurate and efficient constrained molecular dynamics of polymers using Newton's method and special purpose code

In molecular dynamics simulations we can often increase the time step by imposing constraints on bond lengths and bond angles. This allows us to extend the length of the time interval and therefore the range of physical phenomena that we can afford to simulate. We examine the existing algorithms and software for solving nonlinear constraint equations in parallel and we explain why it is necessary to advance the state-of-the-art. We present ILVES-PC, a new algorithm for imposing bond constraints on proteins accurately and efficiently. It solves the same system of differential algebraic equations as the celebrated SHAKE algorithm, but ILVES-PC solves the nonlinear constraint equations using Newton's method rather than the nonlinear Gauss-Seidel method. Moreover, ILVES-PC solves the necessary linear systems using a specialized linear solver that exploits the structure of the protein. ILVES-PC can rapidly solve constraint equations as accurately as the hardware will allow. The run-time of ILVES-PC is proportional to the number of constraints. We have integrated ILVES-PC into GROMACS and simulated proteins of different sizes. Compared with SHAKE, we have achieved speedups of up to 4.9× in single-threaded executions and up to 76× in shared-memory multi-threaded executions. Moreover, ILVES-PC is more accurate than P-LINCS algorithm. Our work is a proof-of-concept of the utility of software designed specifically for the simulation of polymers.

A space of filters for the consistent initialization of differential-algebraic equations in optimization tasks

In this article, a new approach to determining consistent initial conditions for an optimization task with differential-algebraic equations (DAEs) is presented. The work is motivated by the fact that initial conditions have a significant impact on both the obtained state trajectories, as well as the progress of the optimization procedure. In the first step of the designed approach, an α-parametrization of the DAE model takes place. This gives the possibility of influencing the dynamics of the system, and in particular its total reduction. When the initial values of the differential state variables are given, then the solution of the DAE system depends only on the algebraic state variables and the input function, which should have the character of the consistent initial conditions. To obtain the consistent initial conditions and, at the same time, to optimize the considered process, in the second step of the procedure a filter space is defined and applied. The filter space is an innovative extension of Fletcher's filter method. It is built from a finite number of filters, and the dimension of the filter space depends on the input function parametrization approach. As a result of using the filter space, it is possible to obtain a reasonable initial solution to the whole optimization problem, which can be improved by available exact local algorithms. The operation of the new method is tested in the heat and mass transfer optimization task, which is described by a system of nonlinear differential-algebraic equations.

Thermo-elasticity displacement formulation for constrained articulated mechanical systems

This paper proposes an approach for thermal analysis of articulated systems subject to boundary and motion constraints (BMC). The solution framework is designed to capture large temperature fluctuations, significant change in geometry due to reference configuration and deformations, and geometric nonlinearity due to articulated mechanical systems (AMS) large displacements and spinning motion. Thermal-expansion displacements, which do not contribute to rigid-body translations, are determined from thermal stretch of position-gradient vectors using a new sweeping matrix technique designed to eliminate dependence on translational rigid-body modes. A new quadratic thermal-energy kinetic form is defined and used to formulate a dynamic force vector that accounts for thermal transient and inertia effects. Nodal thermal displacements are used in formulating AMS differential/algebraic equations (DAEs), and consequently, thermal stresses due to BMC equations are automatically accounted for based on integration of thermal analysis and Lagrange-D’Alembert principle, which is the foundation of computational multibody system (MBS) algorithms. The approach used in this study for large-displacement constrained and unconstrained thermal expansions is based on multiplicative decomposition of position-gradient matrix, instead of strain additive decomposition, for solution of thermo-elasticity problems. Four configurations are used to define continuum geometry and displacements: straight configuration, reference configuration, thermal-expansion configuration, and current configuration. The proposed approach allows applying thermal loads during constrained large displacements, does not impose restrictions on the choice of thermal coefficients, captures reference-configuration geometry and change in inertia forces due to temperature fluctuations, accounts for thermal displacement in formulating AMS nonlinear constraint equations, and allows for integration with MBS algorithms for the study of a wide range of thermo-elasticity problems.

Multistep Methods for Second-Order Differential-Algebraic Equations

We consider the initial value problem for linear systems of second-order ordinary differential equations with an identically singular matrix multiplying the principal part. Sufficient conditions for the existence of a unique solution are given in terms of matrix polynomials. For such problems, multistep difference schemes are proposed. An analysis of their stability and calculations of a model example are carried out.

An integrated enabling technology interfacing multiple space/time methods/algorithms/domains with model reduction for first-order systems

PurposeThe purpose of this study is to further advance the multiple space/time subdomain framework with model reduction. Existing linear multistep (LMS) methods that are second-order time accurate, and useful for practical applications, have a significant limitation. They do not account for separable controllable numerical dissipation of the primary variables. Furthermore, they have little or no significant choices of altogether different algorithms that can be integrated in a single analysis to mitigate numerical oscillations that may occur. In lieu of such limitations, under the generalized single-step single-solve (GS4) umbrella, several of the deficiencies are circumvented.Design/methodology/approachThe GS4 framework encompasses a wide variety of LMS schemes that are all second-order time accurate and offers controllable numerical dissipation. Unlike existing state-of-art, the present framework permits implicit–implicit and implicit–explicit coupling of algorithms via differential algebraic equations (DAE). As further advancement, this study embeds proper orthogonal decomposition (POD) to further reduce model sizes. This study also uses an iterative convergence check in acquiring sufficient snapshot data to adequately capture the physics to prescribed accuracy requirements. Simple linear/nonlinear transient numerical examples are presented to provide proof of concept.FindingsThe present DAE-GS4-POD framework has the flexibility of using different spatial methods and different time integration algorithms in altogether different subdomains in conjunction with the POD to advance and improve the computational efficiency.Originality/valueThe novelty of this paper is the addition of reduced order modeling features, how it applies to the previous DAE-GS4 framework and the improvement of the computational efficiency. The proposed framework/tool kit provides all the needed flexibility, robustness and adaptability for engineering computations.

Semiexplicit time integration of a reduced magnetic vector potential magneto‐quasistatic field formulation

AbstractIn this article, a reduced vector potential formulation for transient magneto‐quasistatic field problems is presented. The finite element method for spatial discretization yields a system of differential algebraic equations (DAE). Applying a generalized Schur complement the DAE system decomposes into a system of ordinary differential equations (ODE) and an algebraic equation. In order to avoid computationally expensive linearization schemes to solve nonlinear eddy current problems the ODE system is integrated in time using a semiexplicit Euler scheme. The vector potential is split into a source vector potential and a reduced vector potential which describes the eddy current and magnetization effects. The reduced formulation is compared with the underlying nonreduced formulation regarding the overall performance. In addition, the proper orthogonal decomposition method is applied to the iterative preconditioned conjugate gradient method to minimize the number of iterations per time step.

Differential–algebraic systems with dissipative Hamiltonian structure

Different representations of linear dissipative Hamiltonian and port-Hamiltonian differential–algebraic equations (DAE) systems are presented and compared. Using global geometric and algebraic points of view, translations between different representations are presented. Characterizations are also derived when a general DAE system can be transformed into one of these structured representations. Approaches for computing the structural information and the described transformations are derived that can be directly implemented as numerical methods. The results are demonstrated with a large number of examples.