Explicit Ordinary Differential Equations Research Articles

Probabilistic evolution theory for explicit autonomous ordinary differential equations: recursion of squarified telescope matrices and optimal space extension

Probabilistic evolution theory facilitates the solution of initial value problem of explicit autonomous ordinary differential equations with second degree multinomial right hand side functions. Its formulation has components we call telescope matrices. The matrices grow in size very rapidly and has many zeroes and repeating structures. In order to avoid the computational complexity coming from telescope matrices, squarified telescope matrices are utilized. Their calculation is through a recursion. This recursion has been used in several works by the authors and their colleagues but its proof was not given. This work gives the proof of the recursion and all the surrounding details. A second purpose of this work is to provide a method for most facilitative (optimal) space extension. Space extension is needed for using probabilistic evolution theory when degree of multinomiality of the right hand side functions is more than two. For this purpose, an approach using method of exhaustion (brute-force) is proposed.

Couples and pairs formation—thermodynamic and kinetic modelling applied to Al–Mg–Si

Kinetics of formation of couples (B–C) and pairs (B–B, C–C) is studied in an alloy with B-atoms and C-atoms as dilute solutes and A-atoms as solvent. The trapping concept, allowing division of the system into subsystems, the Bragg–Williams approximation used for the configurational entropy of each subsystem, and the thermodynamic extremal principle are applied in the model. The derived equilibrium conditions provide an extended Oriani-type equation. The kinetics of couples and pairs formation is given by explicit ordinary differential equations for the concentrations of couples and pairs allowing calculation of all relevant state variables. The concept is applied to an Al–Mg–Si system with the bonding energy terms EBB, ECC, EBC taken from ab-initio calculations. The presence of excess vacancies that are eventually frozen in by instant quenching as well as their subsequent annihilation influence the formation kinetics in a decisive way.

New answers to an old question in the theory of differential–algebraic equations: Essential underlying ODE versus inherent ODE

In the context of linear differential–algebraic equations (DAEs) one finds different associated explicit ordinary differential equations (ODEs), among them essential underlying and inherent explicit regular ones, abbreviated: EUODEs and IERODEs. EUODEs have been introduced in 1991 for index-2 DAEs in Hessenberg form by means of special transformations. IERODEs result within the framework of the projector based decoupling. Each such explicit ODE is occasionally considered to rule the flow of the DAE.The question to which extend EUODEs and IERODEs are related to each other has been asked promptly after 1991. For index-2 Hessenberg-form DAEs, answers have been given in 2005, saying that EUODEs represent somehow condensed IERODEs. Recently, EUODEs have been indicated for general arbitrary-index DAEs and it has been proved that they are condensed IERODEs. The understanding of the relation between the IERODE and the EUODEs enables to uncover the stability behavior of the DAE flow.In the present paper we show that both, the IERODEs and EUODEs of a DAE with arbitrary high index do not at all depend on derivatives of the right-hand side. We consider adjoint pairs of DAEs and provide generalizations of the classical Lagrange identity. Furthermore, we address Lyapunov spectra and Lyapunov regularity.

Probabilistic evolution theory for the solution of explicit autonomous ordinary differential equations: squarified telescope matrices

Probabilistic evolution theory (PREVTH) is used for the solution of initial value problems of first order explicit autonomous ordinary differential equation sets with second degree multinomial right hand side functions. It is an approximation method based on Kronecker power series: a rewriting of multivariate Taylor series using matrices having certain flexible parameters. Kronecker power series have matrices which are called telescope matrices: \(n \times n^{j+1}\) matrices where j is the index of summation. The additive terms of each telescope matrix is formed through Kronecker product from both sides by Kronecker powers of identity matrices. Recently, squarification is proposed in order to avoid the growing of the matrices in size at each additive term of the series. This paper explains the squarification procedure: the procedure used in order to avoid Kronecker multiplications within PREVTH so that the sizes of the matrices do not grow and so that the amount of necessary computation is reduced. The recursion between squarified matrices is also given. As a numerical application, the solution of a Henon–Heiles system is provided.

Modeling and simulation aspects of AC machines

Abstract In the field of power and drive systems, electrical AC machines are mostly modeled using a set of explicit ordinary differential equations in a state space representation. It is shown, that by using other equation types for simulation, algebraic constraints arising from aggregating several machines to a more complex system can directly be considered. The effects of different model variants on numerical ODE/DAE solvers are investigated in the focus of this work in order perform efficient simulations of larger systems possessing electrical AC machines.

Analytical design of refractive optical elements generating one-parameter intensity distributions.

We calculate a refractive surface to generate a one-parameter intensity distribution described by a vector function of one argument from an incident beam with a plane wavefront. The calculation of the refractive surface is reduced to the solution of two explicit ordinary differential equations of the first order. An approach to solving the problem of the existence of solutions for various one-parameter intensity distributions is proposed. Optical elements that form a variety of intensity distributions (a line, a circular arc, and a double-arc) are designed. The results of numerical simulations of the designed optical elements demonstrate the high quality of the generated line-shaped patterns.

Converting DAE models to ODE models: application to reactive Rayleigh distillation

This paper illustrates the application of an index reduction method to some differential algebraic equations (DAE) modelling the reactive Rayleigh distillation. After two deflation steps, this DAE is converted to an equivalent first-order explicit ordinary differential equation (ODE). This ODE involves a reduced number of dependent variables, and some evaluations of implicit functions defined, either from the original algebraic constraints, or from the hidden ones. Consistent initial conditions are no longer to be computed; at the opposite of some other index reduction methods, which generate a drift-off effect, the algebraic constraints remain satisfied at any time; and, finally, the computational effort to solve the ODE may be less than the one associated to the original DAE.

Analytic design of refractive optical elements generating one-parameter directivity diagram

Design of refractive surface generating a one-parameter directivity diagram is considered for the case of collimated incident beam. Calculation of optical element generating a directivity diagram with an arbitrary shape is reduced to sequential solution of two explicit ordinary differential equations. The optical elements generating one-parameter directivity diagrams in the form of a line-segment, an arc, and a combination of two arcs are designed. The results of simulations demonstrate high performances of the designed optical elements.

Design of an optical element forming an axial line segment for efficient LED lighting systems

An LED optical element is proposed as an alternative to cold-cathode fluorescent lamps. The optical element generates two symmetric uniformly illuminated line segments on the diffuse reflector. The illuminated segments then act as secondary linear light sources. The calculation of the optical element is reduced to the integration of the system of two explicit ordinary differential equations. The results of the simulation of an illumination system module consisting of a set of optical elements generating a set of line segments on the surface of the diffuse reflector are presented. The elements are located directly on the surface of the reflector. The simulation results demonstrate the uniform illumination of a rectangular area at a distance of 30-40 mm from the light source plane. The lighting efficiency of the designed system exceeds 83%.

Dynamic Modelling and Trajectory Tracking of Parallel Manipulator with Flexible Link

This paper mainly focuses on dynamic modelling and real-time control for a parallel manipulator with flexible link. The Lagrange principle and assumed modes method (AMM) substructure technique is presented to formulate the dynamic modelling of a two-degrees-of-freedom (DOF) parallel manipulator with flexible links. Then, the singular perturbation technique (SPT) is used to decompose the nonlinear dynamic system into slow time-scale and fast time-scale subsystems. Furthermore, the SPT is employed to transform the differential algebraic equations (DAEs) for kinematic constraints into explicit ordinary differential equations (ODEs), which makes real-time control possible. In addition, a novel composite control scheme is presented; the computed torque control is applied for a slow subsystem and the H∞ technique for the fast subsystem, taking account of the model uncertainty and outside disturbance. The simulation results show the composite control can effectively achieve fast and accurate tracking control.

Simulation of Constrained Mechanical Systems—Part II: Explicit Numerical Integration

This paper presents an explicit to integrate differential algebraic equations (DAEs) method for simulations of constrained mechanical systems modeled with holonomic and nonholonomic constraints. The proposed DAE integrator is based on the equation of constrained motion developed in Part I of this work, which is discretized here using explicit ordinary differential equation schemes and applied to solve two nontrivial examples. The obtained results show that this integrator allows one to precisely solve constrained mechanical systems through long time periods. Unlike many other implicit DAE solvers which utilize iterative constraint correction, the presented DAE integrator is explicit, and it does not use any iteration. As a direct consequence, the present formulation is simple to implement, and is also well suited for real-time applications.

Cheap global error estimation in some Runge-Kutta pairs

The global error estimation and control problem is studied in embedded Runge–Kutta methods applied to ordinary differential equations (ODEs). We consider both nonstiff equations and stiff ones. Such problems arise in many areas of application and their accurate numerical solution is an important issue of applied mathematics. Usually, global error estimation and control techniques implemented in ODE solvers considerably extend execution time and they are rarely used in practical computations, especially, in implicit methods applied to stiff ODEs. In this paper we show that useful information can be obtained via simple summation of local error estimates calculated cheaply in the course of integration by some embedded Runge–Kutta pairs. Thus, since the embedded method local error estimation is a standard option of any practical solver for differential equations, our technique allows the global error estimation to be implemented naturally and without extra computations. Numerical tests presented here confirm this conclusion in practice. A comparison with explicit and implicit ODE solvers implemented with only local error control in MATLAB is also conducted.

Accelerating multi-dimensional combustion simulations using GPU and hybrid explicit/implicit ODE integration

Simulating multi-dimensional combustion with detailed kinetics often requires solving a large number of ordinary differential equation (ODE) problems at each global time step. In many cases, the ODE integrations account for the bulk of the total wall-clock time for the simulation. This paper introduces CHEMEQ2-GPU – a new explicit stiff ODE solver (based on the existing CHEMEQ2 solver) that exploits the parallel architecture of the modern graphics processing unit (GPU) to accelerate ODE integration in multi-dimensional combustion simulations. We also demonstrate efficient application of the CPU and GPU as co-processors, for further speedup. We describe a hybrid explicit/implicit ODE solver approach that combines the strengths of both solver types running simultaneously on the GPU and CPU, respectively. A dynamic load balancing scheme was used to assign the kinetics ODE integrations over all grid points to either the CPU-based implicit solver DVODE (which is the more efficient solver for highly stiff grid points) or CHEMEQ2-GPU (more efficient for moderately stiff or non-stiff grid points). We demonstrate CHEMEQ2-GPU and the hybrid approach in 3-D simulations of homogeneous charge compression ignition (HCCI) engines. The test cases applied two different n-heptane reaction mechanisms (a large detailed model and a small skeletal model) and three different mesh sizes. Engine simulations were performed using KIVA-CHEMKIN. CHEMEQ2 was about 2–3 times faster than DVODE, with similar prediction accuracy. The CHEMEQ2-GPU speedup relative to CHEMEQ2 increased linearly with the number of grid points for the range of meshes tested in this work. Assuming ideal linear scaling of simulation time with number of processors, the speed of CHEMEQ2-GPU on the Tesla C2050 GPU was equivalent to CHEMEQ2 running on approximately 13 parallel 2.8GHz CPU processors for the finest mesh; and the hybrid solver approach was equivalent to CHEMEQ2 on ∼15 such CPU processors. In summary, CHEMEQ2-GPU provided the additional computing power of 14 parallel CPU processors (for the finest mesh tested) and the hybrid solver approach demonstrated a method to efficiently apply these additional co-processors with existing CPU cores for combustion simulations. CHEMEQ2-GPU scales favorably with the number of grid points and is available by request to the authors. This work presents opportunities for further development, particularly in CPU/GPU load balancing algorithms.

Numerical methods in vehicle system dynamics: state of the art and current developments

Robust and efficient numerical methods are an essential prerequisite for the computer-based dynamical analysis of engineering systems. In vehicle system dynamics, the methods and software tools from multibody system dynamics provide the integration platform for the analysis, simulation and optimisation of the complex dynamical behaviour of vehicles and vehicle components and their interaction with hydraulic components, electronical devices and control structures. Based on the principles of classical mechanics, the modelling of vehicles and their components results in nonlinear systems of ordinary differential equations (ODEs) or differential-algebraic equations (DAEs) of moderate dimension that describe the dynamical behaviour in the frequency range required and with a level of detail being characteristic of vehicle system dynamics. Most practical problems in this field may be transformed to generic problems of numerical mathematics like systems of nonlinear equations in the (quasi-)static analysis and explicit ODEs or DAEs with a typical semi-explicit structure in the dynamical analysis. This transformation to mathematical standard problems allows to use sophisticated, freely available numerical software that is based on well approved numerical methods like the Newton–Raphson iteration for nonlinear equations or Runge–Kutta and linear multistep methods for ODE/DAE time integration. Substantial speed-ups of these numerical standard methods may be achieved exploiting some specific structure of the mathematical models in vehicle system dynamics. In the present paper, we follow this framework and start with some modelling aspects being relevant from the numerical viewpoint. The focus of the paper is on numerical methods for static and dynamic problems, including software issues and a discussion which method fits best for which class of problems. Adaptive components in state-of-the-art numerical software like stepsize and order control in time integration are introduced and illustrated by a well-known benchmark problem from rail vehicle simulation. Over the last few decades, the complexity of high-end applications in vehicle system dynamics has frequently given a fresh impetus for substantial improvements of numerical methods and for the development of novel methods for new problem classes. In the present paper, we address three of these challenging problems of current interest that are today still beyond the mainstream of numerical mathematics: (i) modelling and simulation of contact problems in multibody dynamics, (ii) real-time capable numerical simulation techniques in vehicle system dynamics and (iii) modelling and time integration of multidisciplinary problems in system dynamics including co-simulation techniques.

Explicit formulation of titration models for isothermal titration calorimetry

Isothermal titration calorimetry (ITC) produces a differential heat signal with respect to the total titrant concentration. This feature gives ITC excellent sensitivity for studying the thermodynamics of complex biomolecular interactions in solution. Currently, numerical methods for data fitting are based primarily on indirect approaches rooted in the usual practice of formulating biochemical models in terms of integrated variables. Here, a direct approach is presented wherein ITC models are formulated and solved as numerical initial value problems for data fitting and simulation purposes. To do so, the ITC signal is cast explicitly as a first-order ordinary differential equation (ODE) with total titrant concentration as independent variable and the concentration of a bound or free ligand species as dependent variable. This approach was applied to four ligand–receptor binding and homotropic dissociation models. Qualitative analysis of the explicit ODEs offers insights into the behavior of the models that would be inaccessible to indirect methods of analysis. Numerical ODEs are also highly compatible with regression analysis. Since solutions to numerical initial value problems are straightforward to implement on common computing platforms in the biochemical laboratory, this method is expected to facilitate the development of ITC models tailored to any experimental system of interest.

A Novel Time Integration Method for Solving A Large System of Non-Linear Algebraic Equations

Iterative algorithms for solving a nonlinear system of algebraic equations of the type: Fi(xj )= 0, i, j = 1,...,n date back to the seminal work of Issac Newton. Nowadays a Newton-like algorithm is still the most popu- lar one due to its easy numerical implementa- tion. However, this type of algorithm is sensitive to the initial guess of the solution and is expen- sive in the computations of the Jacobian matrix ∂Fi/∂xj and its inverse at each iterative step. In a time-integration of a system of nonlinear Ordi- nary Differential Equations (ODEs) of the type Bij˙ xj +Fi = 0w hereBij are nonlinear functions of xj, the methods which involve an inverse of theJacobain matrix Bij= ∂Fi/∂xj are called Im- plicit, while those that do not involve an inverse of ∂Fi/∂xj are called Explicit. In this paper a natural system of explicit ODEs is derived from the given system of nonlinear algebraic equations (NAEs), by introducinga fictitioustime, such that it is a mathematically equivalent system in the n+1-dimensional space as the original algebraic equations system is in the n-dimensional space. The iterative equations are obtained by apply- ing numerical integrations on the resultant ODEs, which do not need the information of ∂Fi/∂xj and its inverse. The computational cost is thus greatly reduced. Numerical examples given con- firm that this fictitious time integration method (FTIM) ishighlyefficienttofind thetruesolutions withresidual errors beingmuch smaller. Also,the FTIM is used to study the attracting sets of fixed points, when multiple roots exist.

Explicit stochastic ODE solution methods applied to high-temperature combustion

New stochastic algorithms for the numerical solution of systems of ordinary differential equations (ODEs) are proposed. Furthermore, a correspondence principle is established between these algorithms, which are based on the theory of Markov jump processes, and deterministic schemes. For one of the proposed stochastic algorithms, a detailed numerical study of some of its properties is carried out using examples from high-temperature homogeneous gas-phase combustion. One deterministic method yielded by our correspondence principle is used to shed light on various aspects of the considered stochastic algorithm. In addition, we use the widespread stiff ODE-solver package DASSL for comparison. Advantages of our methods include among others their exceptional simplicity of implementation and negligible start-up costs. For a large system at moderate accuracy requirements, the proposed stochastic algorithms exhibit computational efficiency in the same order of magnitude as implicit solvers, assuming multiple runs. In view of the stiffness of the considered systems and the explicit nature of our algorithms, this is rather surprising. Limitations of our methods concerning the choice of system, initial conditions, and accuracy requirements are also addressed.

State-Space Representation of Vortex Wakes by the Method of Lines

The method of lines (MOL) is applied to the equations of helicopter rotor vortex wakes, and converts the governing partial differential equations into a system of ordinary differential equations (ODE). These ODE can then be coupled to other ODE modeling helicopter dynamics, for time-marching simulations or to extract linearized models. The MOL is applied to a simplified set of wake equations that has an analytical solution. Because these simplified equations neglect key wake physics, the study is only a first step toward applying MOL to realistic models. Therefore, the conclusions only apply to the simplified problem considered. The results show that the MOL is a suitable method to formulate vortex wake models in state-space form. The solutions are accurate and numerically stable. Refining the space discretization increases the stiffness of the ODE, but explicit solvers can still be used. Computational efficiency increases when the accuracies of space and time discretizations are matched, that is, the local error tolerance set for the time solver is matched with the order of accuracy of the space discretization. Formulas of several orders were used in the space discretization. In all cases, the explicit ODE solver used (DE/STEP) was much more computationally efficient than another, implicit ODE solver (DASSL). Linearized state-space wake models can be easily obtained. The MOL could also provide a systematic methodology to extract state-space models from computational fluid dynamics formulations, and therefore to increase the accuracy of helicopter simulation models.

The index of linear differential algebraic equations with properly stated leading terms

For linear differential-algebraic equations with properly stated leading terms and coefficients that are just continuous an index notion is introduced. The index criteria are given in terms of the original coefficients. The index is shown to be invariant under regular transformations of the unknown function, but also under refactorizations of the leading term. Inherent regular explicit ordinary differential equations are described in detail.

Recognizing the symmetry type of O.D.E.s

An algorithm is presented to determine the symmetry type of an explicit ordinary differential equation of order at least 2. From a Janet basis for the determining system, the Lie algebra of point symmetries can be obtained abstractly, as well as the isotropy subalgebra at the origin. This pair is the input of a decision procedure that determines whether the Lie algebra is transitive and, if so, outputs the associated entry on Lie's list of transitive Lie algebras.