Index-1 Differential-algebraic Equations Research Articles

Reducing the Constrained Multibody Dynamics Problem to the Solution of a System of Ordinary Differential Equations Via Velocity Partitioning and Lie Group Integration

Abstract In multibody dynamics, formulating the equations of motion in absolute Cartesian coordinates results in a set of index-3 differential algebraic equations (DAEs). In this work, we present an approach that bypasses the DAE problem by partitioning the velocities in the system into dependent and independent coordinates, thereby reducing the task of producing the time evolution of the mechanical system to one of solving a set of ordinary differential equations (ODEs). In this approach, the independent coordinates are integrated directly, while the dependent coordinates are recovered through the kinematic constraint equations at the position and velocity levels. Notably, Lie group integration is employed to directly obtain the orientation matrix A at each time-step of the simulation. This eliminates the need to choose generalized coordinates to capture the orientation of a body, as the matrix A is a by-product of the solution algorithm. Herein, we outline the new approach and demonstrate it in conjunction with four mechanisms: a single pendulum, a double pendulum, a four-link mechanism, and a slider crank. We report on the convergence order behavior of the proposed method and compare its performance with an established method that combines coordinate partitioning with an Euler parameter formulation. The Python code developed to generate the reported results is open-source and available in a public repository for reproducibility studies.

Effective numerical simulations of synchronous generator system

Synchronous generator system is a complicated dynamic system for energy transmission, which plays an important role in modern industrial production. In this article, we propose some predictor-corrector methods and structure-preserving methods for a generator system based on the first benchmark model of subsynchronous resonance, among which the structure-preserving methods preserve a Dirac structure associated with the so-called port-Hamiltonian descriptor systems. To illustrate this, the simplified generator system in the form of index-1 differential-algebraic equations has been derived. Our analyses provide the global error estimates for a special class of structure-preserving methods called Gauss methods, which guarantee their superior performance over the PSCAD/EMTDC and the predictor-corrector methods in terms of computational stability. Numerical simulations are implemented to verify the effectiveness and advantages of our methods.

A Sparse Differential Algebraic Equation (DAE) and Stiff Ordinary Differential Equation (ODE) Solver in Maple

This paper implements efficient numerical methods in Maple to solve index-1 nonlinear Differential Algebraic Equations (DAEs) and stiff Ordinary Differential Equations (ODEs) systems. Single-step methods (like Trapezoid (TR), Implicit-mid point (IMP), Euler-backward (EB), Radau IIA (Rad) methods, TRBDF2, TRX2) and backward-difference formula of order 2 are implemented with adaptive time-stepping methods in Maple to solve index-1 nonlinear DAEs. Maple’s robust and efficient ability to search within a list/set is exploited to identify the sparsity pattern and automatically calculate the analytic Jacobian. The algorithm and implementation are robust and efficient for index-1 DAE problems and scale well for finite difference/finite element discretization of two-dimensional models with system size up to 10,000 nonlinear DAEs and solve the same in a few seconds. The computational efficiency of the proposed algorithm (provided as an open-access code) compares favorably with the commercial solver gPROMs, one of the most commonly used sparse DAE solvers in the industry.

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.

The Differential Transform Method as an Effective Tool to Solve Implicit Hessenberg Index-3 Differential-Algebraic Equations

Differential-algebraic equations (DAEs) are important tools to model complex problems in various application fields easily. Those DAEs with an index-3, even the linear ones, are known to cause problems when solving them numerically. The present article proposes a new algorithm together with its multistage form to efficiently solve a class of nonlinear implicit Hessenberg index-3 DAEs. This algorithm is based on the idea of applying the differential transform method (DTM) directly to the DAE without applying the traditional index reduction methods, which can be complex and often result in violations of the DAE constraints. Also, to deal with the nonlinear terms in the DAE, we approximate them using the Adomian polynomials. This new idea has given us a simple and efficient algorithm, which involves the solution of linear algebraic systems except for the initial recursion terms. This algorithm is easy to implement in Maple or Mathematica. Furthermore, to enlarge the interval of convergence of the power series solution obtained from the DTM, an algorithm for the multistage DTM is also given. Both algorithms are applied to solve two examples of highly nonlinear implicit index-3 Hessenberg DAEs. Numerical results show that the DTM can determine the exact solution in convergent power series, while the multistage DTM can compute accurate numerical solutions over large intervals.

Upper central exponent of linear stochastic differential algebraic equations of index 1

The current study was carried out to assess the antioxidant expression of the seed’s black-testa peanut cultivar namely CNC1 in the artifical drought condition. The oxidative stress with a burst of O2 .- and H2O2 in peanut leaves resulted the obviously cellular membrane damage that was illustrated by an increase of lipid peroxidation. A high induced activity of enzymatic antioxidants such as SOD, CAT and POX would detoxify the excess of O2 .- and H2O2. An increase in content of amino acid proline protects the stressed cells by adjusting intercellular osmotic potential. The antioxidative expression of CNC1 plants was similar to that in L20 cutivar - a drought tolerant cutivar of peanut.

An implicit function formulation for optimization of discretized index-1 differential algebraic systems

A formulation for the optimization of index-1 differential algebraic equation systems (DAEs) is described that uses implicit functions to remove algebraic variables and equations from the optimization problem. The formulation uses the implicit function theorem to calculate derivatives of functions that remain in the optimization problem in terms of a reduced space of variables, allowing it to be solved with second-order nonlinear optimization algorithms. The formulation is shown to lead to more reliable solver convergence when compared with a full-space simultaneous formulation in challenging case studies involving a chemical looping combustion reactor. In a steady state simulation problem, the implicit function formulation solves 82 out of 100 instances, while the full space formulation solves only 42 out of 100 instances. In a dynamic optimization problem, the implicit function formulation solves 189 out of 200 instances, while the full space formulation solves only 152 out of 200 instances.

Trajectory sensitivity analysis of hybrid systems with sliding motion

This paper concerns hybrid control systems exhibiting the sliding motion. It is assumed that the system’s motion on the switching surface is described by index-2 differential–algebraic equations (DAEs), which guarantee the accurate tracking of the sliding motion surface. For those systems the sensitivity analysis is performed with the help of solutions to system’s linearized equations. The paper states conditions under which the solutions to the linearized equations for original DAEs and the solutions to linearized equations for underlying ordinary differential equations (ODEs) exhibit similar properties. Due to the presence of sliding motion, we restrict the class of admissible control functions to piecewise differentiable functions. The presented sensitivity analysis might be useful in deriving the weak maximum principle for optimal control problems with hybrid systems exhibiting sliding motion and in establishing the global convergence of algorithms for solving those problems.

Numerical event location techniques in discontinuous differential algebraic equations

In this paper, we consider numerical methods for the location of events of differential algebraic equations of index one. These events correspond to cross a discontinuity surface, beyond which another differential algebraic equation holds. Convergence theorems of the numerical event time and event point to the true event time and event point are given. It is proved that, for integrations by semi-implicit methods or Rosenbrock methods, the order of convergence of the numerical event location is the order of convergence of the continuous extension and, for integration by implicit Runge-Kutta methods, the order of convergence is the order of convergence of the implicit RK method.

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.

Multiscale modeling and nonlinear model predictive control for flue gas desulfurization

The primary source of sulfur dioxide (SO2) emissions is flue gas from fossil fuels-based power plants. SO2 emissions are known to not only cause health issues, but also have an adverse effect on the environment in various ways. Several Flue Gas Desulfurization (FGD) technologies have been incorporated in power plants. The most popular technology is Wet FGD, where a limestone slurry is used to absorb SO2 from the flue gas. A detailed droplet scale model describing the instantaneous and finite rate chemistry is developed. The ill-posed Differential-Algebraic Equation (DAE) droplet model is reformulated to a well-posed index-1 DAE through index reduction. The droplet model is integrated with the bulk phase by incorporating gas-liquid mass transfer, and an oxidation reactor model to simulate the dynamic operation of the counter-current spray scrubber. As a result, the model has a well-conditioned Jacobian and overcomes the modeling challenges of previous works and enables numerical solution without requiring carefully selected initialization or specialized solution procedures. The model is successfully validated using power plant measurements, and nonlinear model predictive control (NMPC) studies are demonstrated to optimize recycle stream flowrates to minimize pumping costs.

Towards a reliable implementation of least-squares collocation for higher index differential-algebraic equations\u2014Part 2: the discrete least-squares problem

In the two parts of the present note we discuss questions concerning the implementation of overdetermined least-squares collocation methods for higher index differential-algebraic equations (DAEs). Since higher index DAEs lead to ill-posed problems in natural settings, the discrete counterparts are expected to be very sensitive, which attaches particular importance to their implementation. We provide in Part 1 a robust selection of basis functions and collocation points to design the discrete problem whereas we analyze the discrete least-squares problem and substantiate a procedure for its numerical solution in Part 2.

Towards a reliable implementation of least-squares collocation for higher index differential-algebraic equations\u2014Part 1: basics and ansatz function choices

In the two parts of the present note we discuss several questions concerning the implementation of overdetermined least-squares collocation methods for higher index differential-algebraic equations (DAEs). Since higher index DAEs lead to ill-posed problems in natural settings, the discrete counterparts are expected to be very sensitive, which attaches particular importance to their implementation. In the present Part 1, we provide a robust selection of basis functions and collocation points to design the discrete problem. We substantiate a procedure for its numerical solution later in Part 2. Additionally, in Part 1, a number of new error estimates are proven that support some of the design decisions.

Iterative Rational Krylov Algorithms for model reduction of a class of constrained structural dynamic system with Engineering applications

<p style='text-indent:20px;'>This paper discusses model order reduction of large sparse second-order index-3 differential algebraic equations (DAEs) by applying Iterative Rational Krylov Algorithm (IRKA). In general, such DAEs arise in constraint mechanics, multibody dynamics, mechatronics and many other branches of sciences and technologies. By deflecting the algebraic equations the second-order index-3 system can be altered into an equivalent standard second-order system. This can be done by projecting the system onto the null space of the constraint matrix. However, creating the projector is computationally expensive and it yields huge bottleneck during the implementation. This paper shows how to find a reduce order model without projecting the system onto the null space of the constraint matrix explicitly. To show the efficiency of the theoretical works we apply them to several data of second-order index-3 models and experimental resultants are discussed in the paper.</p>

Linearly implicit GARK schemes

Systems driven by multiple physical processes are central to many areas of science and engineering. Time discretization of multiphysics systems is challenging, since different processes have different levels of stiffness and characteristic time scales. The multimethod approach discretizes each physical process with an appropriate numerical method; the methods are coupled appropriately such that the overall solution has the desired accuracy and stability properties. The authors developed the general-structure additive Runge–Kutta (GARK) framework, which constructs multimethods based on Runge–Kutta schemes.This paper constructs the new GARK-ROS/GARK-ROW families of multimethods based on linearly implicit Rosenbrock/Rosenbrock-W schemes. For ordinary differential equation models, we develop a general order condition theory for linearly implicit methods with any number of partitions, using exact or approximate Jacobians. We generalize the order condition theory to two-way partitioned index-1 differential-algebraic equations. Applications of the framework include decoupled linearly implicit, linearly implicit/explicit, and linearly implicit/implicit methods. Practical GARK-ROS and GARK-ROW schemes of order up to four are constructed.

A port-Hamiltonian approach to modeling the structural dynamics of complex systems

With this contribution, we give a complete and comprehensive framework for modeling the dynamics of complex mechanical structures as port-Hamiltonian systems. This is motivated by research on the potential of lightweight construction using active load-bearing elements integrated into the structure. Such adaptive structures are of high complexity and very heterogeneous in nature. Port-Hamiltonian systems theory provides a promising approach for their modeling and control. Subsystem dynamics can be formulated in a domain-independent way and interconnected by means of power flows. The modular approach is also suitable for robust decentralized control schemes.Starting from a distributed-parameter port-Hamiltonian formulation of beam dynamics, we show the application of an existing structure-preserving mixed finite element method to arrive at finite-dimensional approximations. In contrast to the modeling of single bodies with a single boundary, we consider complex structures composed of many simple elements interconnected at the boundary. This is analogous to the usual way of modeling civil engineering structures which has not been transferred to port-Hamiltonian systems before. A block diagram representation of the interconnected systems is used to generate coupling constraints which leads to differential algebraic equations of index one. After the elimination of algebraic constraints, systems in input-state-output (ISO) port-Hamiltonian form are obtained. Port-Hamiltonian system models for the considered class of systems can also be constructed from the mass and stiffness matrices obtained via conventional finite element methods. We show how this relates to the presented approach and discuss the differences, promoting a better understanding across engineering disciplines. A Matlab framework is available on http://github.com/awarsewa/ph_fem/ to facilitate the application of the methods to different problems.

Convergence results for implicit–explicit general linear methods

This paper studies fixed-step convergence of implicit-explicit general linear methods. We focus on a subclass of schemes that is internally consistent, has high stage order, and favorable stability properties. Classical, index-1 differential algebraic equation, and singular perturbation convergence analyses results are given. For all these problems IMEX GLMs from the class of interest converge with the full theoretical orders under general assumptions. The convergence results require the time steps to be sufficiently small, with upper bounds that are independent on the stiffness of the problem.

The mathematical foundations of physical systems modeling languages

Modern modeling languages for general physical systems, such as Modelica, Amesim, or Simscape, rely on Differential Algebraic Equations (DAEs), i.e., constraints of the form f(x′,x,u)=0. This drastically facilitates modeling from first principles of the physics, as well as the reuse of models. In this paper, we develop the mathematical theory needed to establish the development of compilers and tools for DAE-based physical modeling languages on solid mathematical bases.Unlike Ordinary Differential Equations (ODEs, of the form x′=g(x,u)), DAEs exhibit subtle issues because of the notion of differentiation index and related latent equations—ODEs are DAEs of index zero, for which no latent equation needs to be considered. Prior to generating execution code and calling solvers, the compilation of such languages requires a nontrivial structural analysis step that reduces the differentiation index to a level acceptable by DAE solvers.The models supported by tools of the Modelica class involve multiple modes, with mode-dependent DAE-based dynamics and state-dependent mode switching. However, multimode DAEs are much more difficult to handle than DAEs, especially because of the events of mode change. Unfortunately, the large literature devoted to the mathematical analysis of DAEs does not cover the multimode case, typically saying nothing about mode changes. This lack of foundations causes numerous difficulties to the existing modeling tools. Some models are well handled, others are not, with no clear boundary between the two classes.In this paper, we develop a comprehensive mathematical approach supporting compilation and code generation for this class of languages. Its core is the structural analysis of multimode DAE systems. As a byproduct of this structural analysis, we propose sound criteria for accepting or rejecting multimode models. Our mathematical development relies on nonstandard analysis, which allows us to cast hybrid system dynamics to discrete-time dynamics with infinitesimal step size, thus providing a uniform framework for handling both continuous dynamics and mode change events.

Offline motion simulation framework: Optimizing motion simulator trajectories and parameters

This paper presents a method to simultaneously compute optimal simulator motions and simulator parameters for a predefined set of vehicle motions. The optimization can be performed with a model of human motion perception or sensory dynamics taken into account. The simulator dynamics, sensory dynamics, and optimality criterion are provided by the user. The dynamical models are defined by implicit index-1 differential-algebraic equations (DAE). The direct collocation method is used to find the numerical solution of the optimization problem. The possible applications of the method include calculating optimal simulator motion for scenarios when the future motion is perfectly known (e.g., comfort studies with autonomous vehicles), optimizing simulator design, and evaluating the maximum possible cueing fidelity for a given simulator. To demonstrate the method, we calculated optimal trajectories for a set of typical car maneuvers for the CyberMotion Simulator at the Max-Planck Institute for Biological Cybernetics. We also optimize the configurable cabin position of the simulator and assess the corresponding motion fidelity improvement. The software implementation of the method is publicly available.

A stable reentry trajectory for flexible manipulators

Trajectory tracking of fast-lightweight manipulators is an emerging topic. These systems often possess an unstable internal dynamics that is related to the inverse dynamics problem. The main contributions of this work are (i) a direct relation between internal dynamics variables and trajectory constraints as state equations in an observer canonical form; (ii) the entry/reentry trajectory defined as input control in order to stabilise the unstable internal dynamics; (iii) the methodology is well-suited for systems modelled in a finite element approach, related to an index-3 differential algebraic equation; (iv) a linearised design is established, but the results are suitable to consider a nonlinear model predictive control of underactuated/flexible manipulators, since a reentry trajectory is often required to minimise the output error recursively. For the application, (a) a planar underactuated manipulator with one passive joint and (b) a planar flexible manipulator on a cart were considered.