Christopher L. Cox

Backward error analysis for a pole assignment algorithm

Of the six or so pole assignment algorithms currently available, several have been claimed to be numerically stable, but no proofs have been published to date. It is shown, by performing a backward error analysis, that one of these algorithms, due to Petkov, Christov, and Konstantinov [IEEE Trans. Automat. Control, AC–29 (1984), pp. 1045–10481 is numerically stable.

One‐dimensional infiltration with moving finite elements and improved soil water diffusivity

A problem of significant interest to environmental scientists is the flow of water and solutes through the vadose zone. The partial differential equations which govern this flow are typically time-dependent and nonlinear. Valid solutions to these equations require (1) accurate relationships between various coefficients and variables on which they depend (e.g., coefficient of diffusivity and water content) and (2) sophisticated numerical methods which can handle complexities such as sharp moving fronts. In cases where coefficients are not known explicitly, curve-fitting techniques are needed to smooth out scattered experimental data. Nonlinear coefficients can then be calculated. A constrained least squares spline fit is compared to empirical function fits which have appeared recently. Then, a state-of-the-art numerical technique is used to accurately model transient flow through unsaturated homogeneous soils. The moving finite element method of Miller and colleagues is an adaptive approach in the sense that the grid moves so that nodes are concentrated where they are most needed. As a result, better accuracy is achieved with fewer nodes than are required for standard fixed-grid methods. Petzolds robust Gear-type solver DASSL is used for time-integration. Numerical results are compared to experimental data. Mass balance errors are neglible, and accurate solutions are obtained at all time steps. Though only one-dimensional problems are considered here, the numerical approach generalizes to heterogeneous media and problems in higher dimensions.

A tridiagonal system solver for distributed memory parallel processors with vector nodes

Abstract A variant of the odd-even cyclic reduction algorithm for solving tridiagonal linear systems is presented. Of particular interest is the case where the number of equations is much larger than the number of processors. The target architecture for this scheme is a distributed-memory parallel computer with nodes which are vector processors. The partitioning of the matrix system is governed by a parameter which determines the amount of redundancy in computations and the amount of communication. One feature of the method is that computations are well balanced, as each processor executes an identical algebraic routine. A brief description of the standard cyclic reduction algorithm is given. Then a divide and conquer strategy is presented along with some estimates of speedup and efficiency. Finally, FORTRAN programs for this algorithm which run on the Intel iPSC/2-VX and Floating Point Systems FPS T-20 computer are discussed along with experimental results. Of particular interest is the performance evaluation of this algorithm according to Gustafsons concept of scaled speedup.

Optimal control of non-isothermal viscous fluid flow

For flow inside a four-to-one contraction domain, we minimize the vortex that occurs in the corner region by controlling the heat flux along the corner boundary. The problem of matching a desired temperature along the outflow boundary is also considered. The energy equation is coupled with the mass, momentum, and constitutive equations through the assumption that viscosity depends on temperature. The latter three equations are a non-isothermal version of the three-field Stokes-Oldroyd model, formulated to have the same dependent variable set as the equations governing viscoelastic flow. The state and adjoint equations are solved using the finite element method. Previous efforts in optimal control of fluid flows assume a temperature-dependent Newtonian viscosity when describing the model equations, but make the simplifying assumption of a constant Newtonian viscosity when carrying out computations. This assumption is not made in the current work.

FISIM: an integrated model for simulation of industrial fibre and film processes

Abstract The present study covers the structure and some applications of FISIM (FIber and film SIMulation), a versatile polymer process modelling package which is being developed by the Center for Advanced Engineering Fibers and Films. Generative programming techniques are used to overcome degradation in performance and storage which is often associated with the C++ language. The process stages are integrated using a component-based design so that configurations representing all or part of a process stage can be easily swapped in and out. The prototype code incorporates filtration, forming, external flow under quench conditions, and drawing, applied to fibre melt-spinning. Visualisation is playing a prominent role, providing a userfriendly means for initialising, monitoring, and post-processing simulation output. Process models, solution strategies, and software tools which are implemented in FISIM are presented, along with comparisons between simulation results and experimental measurements. These ingredients comprise a unique modelling environment which will substantially reduce the need for trial-anderror experiments in the development of new fibre and film processes.

Design Analysis of Polymer Filtration using a Multi‐Objective Genetic Algorithm

Abstract Filtration of particle debris is an important component of the polymer fiber melt‐spinning process. The filter lifespan is determined by the pressure drop across the filter, which increases as debris accumulates inside the filtration medium. The cost of filter replacement is high, as is the cost of a loss of the finished fiber product due to debris inclusion in the spun fiber. We use a multiobjective genetic algorithm to examine the trade‐off curve that evolves from these competing goals. A “blackbox” simulator models the debris deposition, and we choose filter porosity and pore diameter as the design variables. We provide numerical results and analysis for two sets of competing objectives.

Backward error analysis for a pole assignment algorithm II: the complex case

In a previous paper [SIAM J. Matrix Anal. Appl., 10 (1989), pp. 446–456], Cox and Moss proved that the pole assignment algorithm of Petkov, Christov, and Konstantinov [IEEE Trans. Automat. Control, AC-29 (1984), pp. 1045–1048] is numerically stable for the real case. In this paper, a modified version of the algorithm of Petkov, Christov, and Konstantinov for the complex case is analyzed and the full algorithm (real and complex) is shown to be numerically stable.

Simulation of TLCP Deformation During Isothermal Melt Spinning of In Situ Composite Fibers

Abstract The evolution of thermotropic liquid‐crystalline polymer (TLCP) morphology during isothermal melt spinning of in situ composites is predicted using the Lee and Park theory. According to the theory, the blend morphology results from the competition between the external flow, which extends droplets into fibrils, and the interfacial tension that retracts droplets into spheres. In this study, the three parameters (λ, μ, and ν) in the Lee and Park theory describing the characteristics of the blend morphology are treated as curve‐fitting parameters; an algorithm was developed to obtain their correct values. The relationship between blend morphology and rheology is also established and verified using data from online measurement during isothermal spinning.

Simulation of filtration in shaped fiber media

A model for filtration of aerosol particles based on the Stokes equations for air velocity and pressure and the Langevin equation for particle paths is presented. The simulation is developed for a filter media consisting of fibers of various shapes and sizes, and validated through comparison to theoretical results for round fibers. When the simulation is applied to filters composed of capillary-channel polymer (C-CP) fibers and filters composed of round fibers having the same cross-sectional area as the C-CP fibers, single-fiber efficiency and figure of merit results indicate that filters made from C-CP fibers have a performance advantage over round fiber filters.

An adaptive-grid least squares finite element solution for flow in layered soils

Groundwater flow in unsaturated soil is governed by Richards equation, a nonlinear convection-diffusion equation. The process is normally convection-dominated, and steep fronts are common in solution profiles. The problem is further complicated if the medium is heterogeneous, for example when there are two or more different soil layers. In this paper, the least squares finite element method is used to solve for flow through 5 layers with differing hydraulic properties. Solution-dependent coefficients are constructed from smooth fits of experimental data. The least squares finite element approach is developed, along with the method for building an optimized, nonuniform grid. Numerical results are presented for the 1D problem. Generalization to higher dimensions is also discussed.