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Large-scale dynamic process simulation

Collaborator: J. Borchardt , F. Grund , D. Horn  

Cooperation with: S. Kurz (Robert Bosch GmbH, Stuttgart), A. Rogowski (EMS Inventa-Fischer, Berlin), D. Zeitz (Alstom (Switzerland) AG, Baden)

Supported by: Robert Bosch GmbH, Stuttgart

Description:

Over the last three decades, dynamic process simulation has become an indispensable tool for process design, analysis, and operation in chemical industry. Due particularly to an improved accuracy of mathematical process models and an increasing degree of integration in process modeling, the sizes of the problems which have to be solved numerically have grown considerably within this time. As a result, the plantwide dynamic process simulation has become a challenging field of application for parallel numerical methods.

For a dynamic process simulation of complex, highly interconnected plants, initial value problems for large-scale systems of coupled differential and algebraic equations (DAEs) have to be solved. Generally, the differential equations arise from balances of energy, mass, and momentum, while the algebraic relations result from constitutive relations for phenomenological quantities or different constraints. These DAE systems are highly nonlinear and can involve several tens of thousands of equations or even more. Usually their solution components process at different time scales. The process models depend on numerous parameters and are often characterized by discontinuities.

Our simulation concept is based on divide-and-conquer techniques and exploits the hierarchical, modular structure of chemical plants for large-scale dynamic process simulation on parallel computers. For that, a plant is considered as a network of connected process units like reactors, pumps, heat exchangers, or trays of distillation columns (see Fig. 1). In an equation-based flowsheet technique, a parameter-dependent mathematical model, describing the unit operations, is assigned to each unit type, and the units are connected, e.g., by mass and energy streams. With it, the corresponding system of DAEs is structured into subsystems according to the units and can be appropriately partitioned into blocks, which can be treated almost concurrently. The approach has been implemented in the Block Oriented Process simulator BOP   that uses its own compiler to generate a hierarchically structured data interface from a process description with our modeling language.

In the period under report, we have mainly worked on improving the homogeneous simulation approach within BOP, in which the entire plant is modeled by a hierarchically structured, block-partitioned system of DAEs

$$\begin{eqnarray*} F_j(t,Y_j(t),\dot{Y}_j(t),U_j(t),\dot{U}_j(t),u(t)) = 0, ~~j=1(1)p,\end{eqnarray*}$$

$$F_j : \mathbb{R}\times\mathbb{R}^{m_j}\times\mathbb{R}^{m_j... ...thbb{R}^{m_j}, ~~~\sum_{i=1}^{p}m_j=n, ~~~t \in [t_0,t_{end}],$$

where the vectors Y_j(t) and U_j(t) denote the unknown and coupling variables of the blocks, respectively, and u(t) the parameter functions. Within this approach, we use efficiently parallelizable block-structured Newton-type methods ([1])   to get a widely applicable parallel approach, covering ``exact'' methods as well as relaxation techniques. This approach has been improved concerning block partitioning, an adaptive relaxation decoupling, as well as the treatment of analytical derivatives for procedures, and a continuation method has been included into the simulator. With it, the parallel performance of BOP has been further improved ([2]).

The numerical solution within BOP requires a repeated solution of linear systems with the same pattern structure of sparse, unsymmetric coefficient matrices, and with multiple right-hand sides. Usually the linear solver requires between 50 and 80 percent of the total amount of computing time in large-scale applications. Our direct solver GSPAR is used to solve the linear systemAx = b. It uses advanced direct methods based on the Gaussian elimination method

$$\begin{eqnarray*} PAQ &=& LU, \ Ly = Pb,&~& UQ^{-1}x = y.\end{eqnarray*}$$

The row permutation matrix P is used to provide numerical stability, and the column permutation matrix Q is chosen to control sparsity. In a new approach, at each step of the elimination, the algorithm is searching for columns with a minimal number of nonzero elements, and a partial pivoting technique is used to maintain numerical stability.

  

Table 1: CPU times (in seconds) for factorization: GSPAR2 compared with UMFPACK V3

GSPAR2 UMFPACK V3

First Second Symb.+Num. Redo 

Matrix N NNZ Factorization Factorization Factorization Factorization

Our new linear solver GSPAR2 has proven to be successful in simulating several real-life problems. For linear systems with matrices arising from different technical problems, it has been compared with the solver UMFPACK V3 (T.A. Davis, University of Florida, USA) concerning computing time. The results ([5]) are given in Table 1. Here N denotes the order and NNZ the number of nonzeros of the matrices. The first two matrices are from chemical process simulation, while the last two are from circuit simulation. The computations have been performed on a Compaq AlphaServer GS80 6/731 with 9 GB memory and alpha EV6.7 (21264A) processors, which operate at 731 MHz.

The generation of the data interface for the simulator BOP consists of three phases that can be controlled separately. To improve the usage of the compiler, a graphical user interface (GUI) has been implemented in JAVA. The GUI (see Fig. 2) enables the selection of examples, the editing of the model and process description, the selection of different compiler phases, the setting of compiler parameters, and the evaluation of the resulting data interface. The compiler itself has been improved regarding the analysis and treatment of modeling errors.

 

Fig. 2: GUI of the modeling language compiler for the simulator BOP 

In the period under report, a continuing interest of industrial partners in our simulation concept has been maintained. The work concerning the realization of a continuation method within BOP for solving a special industrial problem has been successfully finished and was fully financed by the cooperation partner. Due to the request of another industrial partner, a PC version of BOP running under WindowsNT is in preparation. Finally we have started work to adapt a SPICE-like description language in the area of electric circuit simulation to our process description language, so that in the future, the simulator BOP will be applicable to circuit simulation problems as well.

First results concerning an inhomogeneous simulation approach have been reported in [3], [4].

References:

1.   J. BORCHARDT, Newton-type decomposition methods in large-scale dynamic process simulation , Comput. Chem. Engng., 25 (2001), pp. 951-961.
2.   , Newton-type decomposition methods in parallel process simulation , in: Proceedings of ICheaP-5, the 5th Italian Conference on Chemical and Process Engineering, S. Pierucci, ed., vol. 1, AIDIC, Florence, 2001, pp. 459-464.
3.   K. EHRHARDT, J. BORCHARDT, F. GRUND, D. HORN, Distributed dynamic process simulation , Z. Angew. Math. Mech., 81 (2001), pp. S715-S716.
4.   F. GRUND, K. EHRHARDT, J. BORCHARDT, D. HORN, Heterogeneous dynamic process flowsheet simulation of chemical plants , to appear in: Mathematics -- Key Technology for the Future II, K.-H. Hoffmann, W. Jäger, T. Lohmann, H. Schunck, eds., Springer, Berlin.
5.   F. GRUND, Solution of linear systems with sparse matrices, talk at the workshop ``Modellierung, Simulation und Optimierung integrierter Schaltkreise'', November 2001, Mathematisches Forschungsinstitut Oberwolfach,
   (see http://www.wias-berlin.de/~grund/oberwolfach_01.pdf ).

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