DAE Solver Research Articles

Application of TUSSIM with a variable Tromp curve for predicting optimal operation of multi-compartment mills with various ball size distributions

A true unsteady-state simulator (TUSSIM) for ball milling was integrated with a variable Tromp curve for classification to simulate and optimize closed-circuit, multi-compartment cement ball milling. Using representative model–operational parameters from available literature, we first investigated the system dynamics for a two-compartment mill. Then, various simulations examined the impacts of closed-circuit vs. open-circuit operation, number of compartments, and various ball size distributions. Our results suggest that integrating an air classifier into an open-circuit ball mill can increase the production rate by 15% or increase the cement-specific surface area by 13%. A single-compartment mill entails a pre-milled feed for proper operation, whereas a two-compartment mill yields a finer cement product than a three-compartment mill. Uniform mass distribution of balls led to slightly finer product than uniform surface area or number distributions, while the impact of a classifying liner was negligibly small. Finally, we identified optimal ball mixtures in a two-compartment mill using a combined global optimizer–DAE solver, which suggests 14% capacity increase with desirable cement quality. Overall, TUSSIM’s results are not only in line with limited, full-scale experimental studies and industry best practices, but also provide fundamental process insights, while enabling process optimization with tailored ball mixtures in different compartments.

ODES: a high level interface to ODE and DAE solvers

ODES: a high level interface to ODE and DAE solvers

DAE-EKF-Based Nonlinear Predictive Control of Reactive Distillation Systems Exhibiting Input and Output Multiplicities

Reactive distillation (RD) has become one of the most important hybrid separation processes in recent times because of its economic and operational advantages. Reactive distillation systems, however, can exhibit complex input and output multiplicity behavior simultaneously in the desired operating region. Moreover, such a system is generally stiff and is typically modeled as a set of differential-algebraic equations. These two factors render the control of RD systems a challenging problem, particularly when the desirable operating point is unstable. In this work, an observer error feedback-based NMPC scheme has been developed for achieving offset-free control of RD systems modeled as DAEs. A recently developed version of the EKF for DAE systems (Mandela et al. Chem. Eng. Sci. 2010, 65, 4548−4556) was used to carry out state estimation. Because direct use of a DAE solver in the NMPC formulation can prove to be prohibitively computationally intensive and unsuitable for online implementation, a successive-linearization-based NMPC scheme (SLNMPC) was also developed. The effectiveness of the proposed control schemes is demonstrated by simulating servo and regulatory control problems associated with a hypothetical ideal RD column that exhibits input and output multiplicity behavior simultaneously at an unstable but economically desirable operating point. The servo and regulatory performances of the proposed SLNMPC scheme were also studied by simulating an industrial RD system involving MTBE synthesis. Analysis of the simulation results indicates that the proposed SLNMPC formulation provides an effective approach for handling control problems involving moderately large-magnitude servo and regulatory changes in the operation of RD systems. Moreover, the average computation time for the SLNMPC formulation was found to be quite small when compared with the sampling interval, which establishes the feasibility of implementing the SLNMPC scheme in real time.

무인 국방 로봇의 효과적인 다물체 동역학 해석을 위한 부분시스템 합성방법 기반 DAE 해석 기법 비교 연구

무인 국방 로봇의 실시간 해석을 위해서는 효과적인 해석기법이 필수적인 요소이다. 이러한 효과적인 해석을 위하여 부분시스템 합성방법이 개발되었다. 부분시스템 합성방법은 기준 물체의 운동방정식과 각 부분시스템들의 운동방정식을 독립적으로 계산함으로써 계산량의 이득을 볼 수 있다. 운동방정식은 미분방정식과 대수방정식이 혼합된 미분대수방정식으로 표현된다. 이러한 미분대수방정식의 정확하고 효과적인 해석을 위해서 직접 적분방법, 구속조건식 안정화기법, 일반 좌표 분할기법 등이 개발되었다. 본 논문에서는 무인 국방 로봇의 효과적인 해석을 위하여 부분시스템 합성방법을 적용하고 위에서 기술한 세 가지의 미분대수방정식 해석기법을 비교하는 연구를 수행하였다.

Time Efficient Solution of Phase Equilibria in Dynamic and Distributed Systems with Differential Algebraic Equation Solvers

Solution of phase equilibria with flash calculations is central in many processes. During the integrating of the conservation equations in these systems, flash calculations are traditionally solved in inner loops at each integration step. Some of these systems can be solved more efficiently using modern DAE solvers, where the differential equations and the algebraic equations describing the phase equilibrium are solved simultaneously. In this paper we present a framework, called the Thermodynamic Differential Algebraic Equation (TDAE) method, which handles most two-phase flash variants. The phase boundary is tracked, enabling a robust solution also with phase changes. The time consumption of the TDAE method has been compared to the traditional approach in several examples implemented in both Fortran and Matlab. In some cases, the TDAE method is more than 800 times faster, and in other cases the traditional methodology should be used. We will give insight into how and when DAE solvers can be used to speed up phase equilibrium calculations.

Strategies for solving index one DAE with non-negative constraints: Application to liquid–liquid extraction

Liquid–liquid extraction modeling leads to solve an index one DAE system. For the sake of robustness, it is desirable to account for non-negative constraints. Based on the DASSL architecture (a classical index one DAE solver) we propose and compare three different strategies to implement these bound constraints. Each of these strategies corresponds to a different Newton modification: clipping, damping, or interior point method. The comparisons are made on two test cases: the Robertson ODE problem, and an example from liquid–liquid extraction modeling.

Pryce pre-analysis adapted to some DAE solvers

The ODE solver HBT(12)5 of order 12 (T. Nguyen-Ba, H. Hao, H. Yagoub, R. Vaillancourt, One-step 5-stage Hermite–Birkho–Taylor ODE solver of order 12, Appl. Math. Comput. 211 (2009) 313–328. doi:10.1016/j.amc.2009.01.043), which combines a Taylor series method of order 9 with a Runge–Kutta method of order 4, is expanded into the DAE solver HBT(12)5DAE of order 12. Dormand–Prince’s DP(8, 7)13M is also expanded into the DAE solver DP(8, 7)DAE. Pryce structural pre-analysis, extended ODEs and ODE first-order forms are adapted to these DAE solvers with a stepsize control based on local error estimators and a modified Pryce algorithm to advance integration. HBT(12)5DAE uses only the first nine derivatives of the unknown variables as opposed to the first 12 derivatives used by the Taylor series method T12DAE of order 12. Numerical results show the advantage of HBT(12)5DAE over T12DAE, DP(8, 7)DAE and other known DAE solvers.

Efficient Solution and Sensitivity Analysis of Partial Differential-Algebraic Equation Systems: Application to Corrosion Pit Initiation

A mathematical model based on transport and reaction equations describing initiation of pitting corrosion at a MnS sulfide inclusion in NaCl electrolyte was used as a test problem for development of efficient numerical procedures for corrosion simulation and sensitivity analysis. The equations associated with the corrosion model were formulated by using a reaction invariant approach, which resulted in an index-2 partial differential-algebraic equation (PDAE) system. The original PDAE system was regularized to yield a lower index PDAE system, for which we were able to formulate an effective preconditioner for use in the DASPK 3.0 DAE solver. This approach was found to improve the efficiency of the Krylov iterative solver as well as to provide the parameter sensitivities with little additional effort in comparison with another method reported recently. In agreement with previous findings, the results obtained with this approach indicate that the model is most sensitive to the parameters associated with the heterogeneous dissolution of the sulfide inclusion.

BACOL

BACOL is a new, high quality, robust software package in Fortran 77 for solving one-dimensional parabolic PDEs, which has been shown to be significantly more efficient than any other widely available software package of the same class (to our knowledge), especially for problems with solutions exhibiting rapid spatial variation. A novel feature of this package is that it employs high order, adaptive methods in both time and space, controlling and balancing both spatial and temporal error estimates. The software implements a spline collocation method at Gaussian points, with a B-spline basis, for the spatial discretization. The time integration is performed using a modification of the popular DAE solver, DASSL. Based on the computation of a second, higher order, global solution, a high quality a posteriori spatial error estimate is obtained after each successful time step. The spatial error is controlled by a sophisticated new mesh selection algorithm based on an equidistribution principle. In this article we describe the overall structure of the BACOL package, and in particular the modifications to the DASSL package that improve its performance within BACOL. An example is provided in the online Appendix to illustrate the use of the package.

A high-order global spatially adaptive collocation method for 1-D parabolic PDEs*1

In this paper, we describe a high-order solver, adaptive in space and time, for the efficient numerical solution of one-dimensional parabolic PDEs. Collocation at Gaussian points is employed for the spatial discretization, using a B-spline basis. A modification of the well-known DAE solver, DASSL is used for the time integration. An a posteriori spatial error estimate is calculated at each successful time step. A new mesh selection strategy based on an equidistribution principle is presented for controlling the spatial error which is balanced with respect to the temporal error. This new mesh adaptation algorithm is shown to be robust, and particularly efficient for problems having solutions with rapid variation.

A high-order global spatially adaptive collocation method for 1-D parabolic PDEs

In this paper, we describe a high-order solver, adaptive in space and time, for the efficient numerical solution of one-dimensional parabolic PDEs. Collocation at Gaussian points is employed for the spatial discretization, using a B-spline basis. A modification of the well-known DAE solver, DASSL is used for the time integration. An a posteriori spatial error estimate is calculated at each successful time step. A new mesh selection strategy based on an equidistribution principle is presented for controlling the spatial error which is balanced with respect to the temporal error. This new mesh adaptation algorithm is shown to be robust, and particularly efficient for problems having solutions with rapid variation.

DAE 해법과 PPTA(Partial Periodic Trimming Algorithm)를 이용한 헬리콥터 트림해석 및 비행 시뮬레이션

Lever II 헬리콥터 비행운동 방정식으로부터 주기성 트림해를 구하기 위해 DAE 해법에 근거한 PPTA(Partial Periodic Trimming Algorithm)를 제안하였다. PPTA의 반복계산으로 수정된 상태변수는 적합한 초기조건이 요구되는 DAE해법에서 수치불안정을 일으킬 수 있다. 간단하게 DAE 차수를 조절함으로써 정확한 주기성 트림을 얻을 수 있었다. 수치해법을 CBM(Common Baseline Model) 헬리콥터에 적용하여 harmonic balance 방법과 동일한 트림해를 얻었으며 시뮬레이션 초기의 과도응답을 효과적으로 제거할 수 있음을 밝혔다. To get a periodic trim solution from Level II helicopter flight dynamic equations, DAE based PPT A (partial Periodic Trimming Algorithm) has been proposed. Iterative update of state variables from PPT A can cause a numerical instability in DAE solver which needs compatible initial conditions. By simply adjusting the order of DAE solver a periodic trim can be obtained with good accuracy. Application for CBM (Common Baseline Model) helicopter showed the same trim result as harmonic balance method and the effective elimination of unrealistic initial responses at the start of flight simulation.

Force Coupling Versus Differential Algebraic Description of Constrained Multibody Systems

The force coupling and the differential algebraic descriptionof constrained multibody systems are compared and applied to two testmodels used in vehicle dynamics. Based on the Newtonian and Eulerianequations the equations of motion for local subsystems are presentedusing explicitly and implicitly formulated constraints. In the nextstage the coupling of local subsystems using implicit algebraicconstraints and an approximative description of the constraints byapplied coupling forces is explained. The two coupling methods areapplied to test models used in vehicle dynamics to examine theircharacteristics by numerical integration with different methods. Themost efficient among the tested DAE solvers is identified.

Dynamic simulation of Brazed Plate-Fin Heat Exchangers

In this paper a dynamic simulator of Brazed Plate-Fin Heat Exchangers (PFHEs) ProSec \ m is presented. A rigorous model is used which allows one to represent the very broad range of configurations for this kind of equipment. A DAE solver (DISCo) is used with a sparse direct solver (MA38) to handle the large number of equations. Applications for control and design are highlighted through an industrial case study involving shutdown.

PDEFIT: A Fortran code for data fitting in partial differential equations

PDEFIT is a computer program to estimate parameters in a system of one-dimensional differential equations and coupled ordinary differential equations. Equations without time-dependent derivatives are permitted. The model allows arbitrary transition conditions between separate integration areas for functions and derivatives, switching conditions and dynamic constraints. Proceeding from given experimental data, e.g. observation times and measurements, the distance of these measured data from the solution of a system of differential equations at designated spatial values is to be minimized in the L 2-L 1-, or L ∞-norm. The method of lines is used to discretize the partial differential equation with respect to polynomial approximation, difference formulae and several special upwind and related formulae for hyperbolic PDE's. The original system is transformed into a set of ordinary differential equations or, alternatively, into a set of differential algebraic equations, that is solved then by standard ODE or DAE solvers. We describe program organization and outline the usage of the corresponding FORTRAN code. A few examples are added to prove the feasibility of the proposed approach.

Towards on-scale crystalliser design using compartmental models

In this paper the importance of on-scale crystalliser design is outlined. An on-scale approach is specifically required for the analysis and optimisation tasks in design. The need for this approach is a direct consequence of the nonlinear dependency of most physical processes in crystallisation on the degree of saturation, the energy dissipation, the crystal size, and its distribution. The hydrodynamics in a crystalliser vessel are typically such, that these process variables are distributed non-uniformly throughout the vessel. The conventional, geometrically lumped description of the physical process inside a crystalliser vessel, i.e nucleation, growth, dissolution, attrition, breakage agglomeration and particle segregation, has therefore never proven to be reliable for scale-up purposes. Furthermore, as the interactions between these processes lead to an intricate dynamic behaviour, models describing the effect of changes in time of process variables on the product quality are essential. Compartmental modelling, a well known technique in reactor engineering and applied within crystallisation since a number of years, facilitates on-scale design since it allows a natural separation of kinetic and hydrodynamic mechanisms. The resulting dynamic models (order of 10 4 equations) can be easily tackled with standard DAE solvers. Here we will focus upon the need for a proper physical description of the aforementioned crystallisation mechanisms. First of all, a brief description of the dependencies of these mechanisms upon local supersaturation or undersaturation, local energy dissipation and crystal size is given. Depending on the type of crystallisation process, suspension crystallisation or precipitation, the dependencies necessary to be included in the compartmental model, in order to describe their overall effect are discussed. The next step is deriving the geometric structure of a compartmental model for a certain scale crystalliser and material, for which two methodologies will be presented. Finally, the approach will be illustrated for evaporative crystallisation of ammonium sulphate from water in 0.15 and 18.5 m 3 FC (Forced Circulation) and 0.022 and 1.1 m 3 DTB (Draft Tube Baffle) crystallisers, using size dependent nucleation, growth, dissolution, attrition and segregation models.

Simulating discontinuous phenomena affecting robot motion

The paper deals with simulation of mechanical systems affected by discontinuous phenomena. These phenomena involve impulsive events and/or models whose structure changes depending on the values of some system variables. The models of three kinds of these discontinuities (joint with static friction, collisions with rigid environment, bifurcation behavior near kinematic singularities) are given, and a simulation environment, based on the DAE solver DASSL is presented, that also allows efficient simulation of sample-data systems. Some simulation results achieved with the proposed environment are finally presented.

Dynamic Simulation in Concurrent Control Engineering

Concurrent Control Engineering aims at multidisciplinary integration of control engineering methods with the data and procedures of complementary engineering disciplines in view of holistic system optimization. The basic view must be that of an engineering data-and methodbase with an application dependent configurability of functional analysis and synthesis modules, open to the modelling methodologies of different engineering disciplines. In this view, dynamic (time-) simulation, although a key interdisciplinary analysis tool, is just one functional module among others. If based on the features of a general-purpose engineering database system, however, an “open” simulation module is of wider applicability than a selfcontained “closed” simulation package. We describe the features of our simulation module DSSIM. This simulation module operates on blocks which can also refer to a data structure on an engineering database. A block is defined by a set of FORTRAN subroutines. It describes a subclass of systems described by general nonlinear differentialalgebraic equations, as for instance linear state space control systems or nonlinear multibody mechan-ical systems. Any number of models can be kept in the loaded program module which allows online switching between design alternatives or system representations of different complexity. Also the numerics interface is specifically designed to make best use of the features of modern ODE and DAE solvers.

Dynamic simulation of multispecies reaction/diffusion in nonisothermal porous spheres

This is the last in a series of papers on the numerical dynamic simulation of a single, spherically symmetric, porous pellet in which diffusive transport and chemical reactions are taking place. It describes the extension of the technique to multiple fluid and solid reactants and products, underoing multiple reactions. As shown previously, the particular form of the model in which explicit references to individual fluid species fluxes have been eliminated offers favorable numerical characteristics. This paper describes the considerable additional effort, by comparison to the binary fluid system of the previous paper, required to accomplish the flux elimination. The fluxes can be replaced by functions of the fluid species mole fractions and the total molar flux if the transport equations can be inverted. An algorithm for obtaining a unique solution to the Stefan—Maxwell equations is described in detail with references to other approaches in the literature. This permits formulation of the model as an index one set of differential/algebraic equations suitable for numerical solution by a general-purpose DAE solver. We briefly describe a code, MSRS (Multispecies Shale Retorting Simulator), which uses the DAE solver DASSL to simulate nonisothermal reacting/diffusing pellet dynamics. Two example problems are presented: a smaller, three fluid/two solid/one reaction simplified model of oil shale combustion, and a larger, nine fluid/10 solid/15 reaction simulation of in situ oil shale retorting with time-dependent boundary conditions. Both models were run with coarse and fine spatial solution meshes. Simulator performance is discussed and several modifications to the models made to speed up execution.