Sparse matrix processing

Different processes like eddy turbulence, bottom current, stagnation of flows, and storm-water events can be simulated, using either laminar or turbulent flow model for simulation. All processes are displayed in real-time graphical mode (history, contour graph, surface, etc.) you can also record them to data files. Thanks to innovative sparse matrix technology, calculation process is fast and stable a large number of layers in vertical and horizontal directions can be used, as well as a small time step. You can hunt for these on the Web. 

The highest level of integration would be to establish one large set of equations and to apply one solution process to both thermal and airflow-related variables. Nevertheless, a very sparse matrix must be solved, and one cannot use the reliable and well-proven solvers of the present codes anymore. Therefore, a separate solution process for thermal and airflow parameters respectively remains the most promising approach. This seems to be appropriate also for the coupling of computational fluid dynamics (CFD) with a thermal model. ... 

Stadherr, M. A., "A New Sparse Matrix Method for Process Design", Paper presented at Miami AIChE Meeting, November 1978. 

Two extremes are encountered in flowsheeting software. At one extreme, the entire set of equations (and inequalities) representing the process is employed. This representation is known as the equation-oriented method of flowsheeting. The equations can be solved in a sequential fashion analogous to the modular representation described below or simultaneously by Newton s method, Broyden s method, or by employing sparse matrix techniques to reduce the extent of matrix manipulations. Refer to the review by Evans and Chapter 5. ... 

Problem size, as most process optimization problems have on the order of lO to 10 constraints and at least 10 to 100 decision variables. Generally, this is done with sparse matrix algorithms or by partitioning the problem to take advantage of the equation structure due to the unit equations and recycles. 

Theoretically the polymerisation (cross-linking) process should not result in a random network of dextran chains but in a heterogeneous gel with crosslink-dense, matrix-rich, water-poor domains intermingled with cross-link-sparse, matrix-poor, water-rich regions (2,20). 

Approximating the coupling waveforms Uj = (Vj,Vj) and treating them as inputs we can generate an iteration process, where at each iteration step the block systems can be solved concurrently on time windows using general known methods (for instance BDF, Newton s method and sparse matrix solver). 

The final macromolecular density matrix P(A") is rather sparse. The index relations described above help to identify the non-zero matrix elements of P(A"), and the actual computations can be restricted to those. Utilizing these restrictions and carrying out a finite number of steps only for the non-zero matrix elements of each fragment density matrix P (< Kk)), an iterative process is used for the assembly of the macromolecular density matrix P(AT) ... 

For process optimization problems, the sparse approach has been further developed in studies by Kumar and Lucia (1987), Lucia and Kumar (1988), and Lucia and Xu (1990). Here they formulated a large-scale approach that incorporates indefinite quasi-Newton updates and can be tailored to specific process optimization problems. In the last study they also develop a sparse quadratic programming approach based on indefinite matrix factorizations due to Bunch and Parlett (1971). Also, a trust region strategy is substituted for the line search step mentioned above. This approach was successfully applied to the optimization of several complex distillation column models with up to 200 variables. 

For practical processes most of the split-fraction coefficients are zero and the matrix is sparse. 

Once the boundary conditions have been applied to the assembled matrix of equations, standard numerical techniques can be used to solve the global system matrix equation for the unknown field variables. The matrix equations generated by the finite element process are often sparse and sometimes also S3mimetrical. 

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