Efficient Numerical Solver for Partially Structured Differential and Algebraic Equation Systems

Abstract

Given a sparse set of differential and algebraic equations (DAEs), it is always recommended to exploit the structure of the system’s sparsity (e.g., tridiagonal blocks matrix, band matrix, and staircase matrix, etc.), thus to use tailored numerical solvers in order to reduce the computation time. Very frequently, though, while highly structured, a couple of elements enter the description which make it difficult for the solvers to reach a solution. They are common in process control applications, where the states added to the plant description by the integral parts of the controllers introduce unstructured elements in the otherwise very structured Jacobian of the mathematical model. Such systems are characterized by a partially structured Jacobian, which inhibits the use of the solvers tailored to fit problems with fully structured matrices. In such cases, one can either use a solver with lower performance, resulting in larger computation times, or alternatively one seeks an approximation for the unstructured points. A solution to the handling of “dirty” Jacobians is presented, which is implemented in a DAE solver package available freely on the Internet. This novel DAE solver fully exploits the overall structure of the system’s sparsity, without compromising CPU computation time and precision of the results. A numerical comparison with different approaches is given by solving a DAE model representing an existing nonequilibrium distillation column.

Reduced Kinetic Schemes of Complex Reaction Systems: Fossil and Biomass‐Derived Transportation Fuels

“…All simulations were performed with the OpenSMOKE code , with an extensive use of the BzzMath numerical library .…”

Model predictive control and dynamic real-time optimization of steam cracking units

“…• DynSim™ (Schneider-Electric): widespread commercial dynamic simulation suite, used by major engineering, procurement and construction (EPC) societies for purposes of design, validation and control of chemical processes; • ROMeo™ (Schneider-Electric): package for dynamic simulation and real-time optimization, which represents the industrial state of the art in this field; • RobOpt™ and BzzMath (Politecnico di Milano): comprehensive set of numerical tools, including robust derivative-free optimizers and numerical solvers for systems of differential and differentialalgebraic equations (Buzzi-Ferraris and Manenti, 2012Manenti, , 2013Manenti et al, 2009); • SPYRO ® (Pyrotec-Technip): well-established program for the simulation of the coils of steam cracking furnaces. The integration of these four tools into a full-fledged DRTO platform is facilitated by object-oriented programming, which allows embedding both the efficient ODE/DAE solvers of the BzzMath library and the commercial package SPYRO® into DynSim™.…”

First-principles models and sensitivity analysis for the lignocellulosic biomass-to-methanol conversion process

“…The resulting differential-algebraic system obtained in the GasDS Suite is solved using the BzzDae solver of the BzzMath library, a freely downloadable tool for universities and research centres (Buzzi-Ferraris and . The use and potential of the differential-algebraic solvers are broached elsewhere for theoretical aspects (Manenti et al, 2009) and different applications are collected (Buzzi-Ferraris and . The GasDS Suite is then applied to simulate a countercurrent fluidized bed gasifier with steam and oxygen fed at the bottom of the unit (Badeau and Levi, 2009).…”