Large-scale Nonlinear Problems Research Articles

Opportunities for research on electric power systems problems

Long-term high-risk investigations into topics related to future planning and operation functions of large-scale electric utilities encompass a wide range of issues. These issues include computer applications, modeling, security assessment, control under uncertainty, distribution systems, protection, reliability, economics/pricing, and knowledge-based systems. The paper discusses the origin of these topics and their relation to future large-scale nonlinear systems problems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Seismic Velocity Field Estimation — Strategies for Large-scale Nonlinear Inverse Problems

The estimation of the seismic velocity field in two or three dimensions, by modelling the travel times of particular seismic phases or by matching observed and computed seismograms, represents a large-scale nonlinear inverse problem. The solution can be obtained by determining the minimum of a misfit function between observations and theoretical predictions, subject to some regularisation conditions on the behaviour of the model parameters. The minimisation can be achieved without the inversion of large matrices by using a search scheme based on the local properties of the misfit function. At each step in the iterative process, a subspace of a small number of directions is constructed in model space and then the minimum sought in a quadratic approximation on this set. At least two directions are required for rapid convergence. This approach is very suitable when the model parameters are of different types, since partitioning by parameter class avoids dependence on scaling. If the model is to remain close to a reference then the regularisation term is particularly important and different types of a priori information (e.g. geological) can be introduced via the character of this term. When fit-to-data is emphasised there is the chance of finding features suppressed in a more conservative approach, but at the risk of introducing spurious detail.

Ta πanta pei

The paper presents a survey of recent work on flow phenomena performed in the closing days of the traditional Institute for Statics and Dynamics of the University of Stuttgart. It serves as a swan song to the latter institution and as a clarion call initiating the work of the new Institute for Computer Applications. The paper develops, in particular, a rigorous natural methodology for the analysis of flow problems when described by the Navier-Stokes equations. The authors consider, apart from the continuous and the finite element formulations, also the solution of the associated large-scale nonlinear problems on present-day supercomputers. The theory is illustrated on a number of selected applications encompassing three-dimensional flow, thermally coupled phenomena, fluid-solid interaction and Stokesian motion for solid materials.

Design of array processor software for nonlinear structural analysis

This paper presents an investigation of the solution of large-scale nonlinear structural problems using a 32-bit minicomputer with an attached 64-bit array processor. The two processors communicate via a common memory interface. A user-oriented software package has been designed to allow the use of the given computer configuration by a typical engineer or a scientist. It is possible to use the system without a detailed knowledge of the operation of the hardware and the complex data handling necessary to manipulate the data associated with a large problem. Several test examples are considered using finite element 3-D frame models and the Newton-Raphson solution scheme. The array processor could not be utilized as yet, due to the lack of the proper vendor software. Hence, a simulator was designed to predict the system performance.

Potential of minicomputer-array processor system for nonlinear finite-element analysis

A study is made of the potential of using a minicomputer-array processor system for efficient solution of large-scale nonlinear finite-element problems. A PRIME 750 is used as the host computer, and a software simulator residing on the PRIME is employed to assess the performance of the Floating Point Systems AP-120B array processor. Major hardware characteristics of the system such as virtual memory, parallel and pipeline processing are reviewed and the interplay between various hardware components is examined. Effective use of the minicomputer-array processor system for nonlinear analysis requires the following: (a) proper selection of the computational procedure and the capability to vectorize the numerical algorithms; (b) reduction of I/O operations; and (c) overlapping host and array-processor operations. A detailed discussion is given of techniques to accomplish each of these tasks. Two benchmark problems with 1715 and 3230 degrees of freedom, respectively, are selected to measure the anticipated gain in speed obtained by using the proposed algorithms on the array processor. Results of the study of the two benchmarks indicate that these two problems would run faster on a PRIME 750 coupled with the AP-120B than on the PRIME 750 alone. The 1715 degree-of-freedom problem would run about five times faster, and the 3230 degree-of-freedom problem would run about ten times faster. New advances in array-processor hardware are outlined, and possible improvements in the computational algorithms are discussed. The combination of the two can significantly enhance the effectiveness of the minicomputer-array processor system for large-scale nonlinear analysis.

On making large nonlinear problems small

A number of aspects of reduction methods for solution of large-scale nonlinear problems are discussed including: (a) selection of basis vectors for steady-state problems; (b) identification and determination of bifurcation and limit points, and tracing post-limit-point and post-bifurcation point paths using reduction methods; (c) application of reduction methods to nonlinear problems with prescribed nonzero values of the field variable; and (d) use of reduction methods in conjunction with multifield (mixed) finite element models. Four numerical examples are presented to demonstrate the effectiveness of using reduction methods for the solution of nonlinear thermal and structural problems. Also, a number of research areas which have high potential for application of reduction methods are identified.

Two approaches to nonlinear decomposition

Unlike the situation for the linear case, nonlinear programming problems rapidly become numerically very difficult to solve, as the number of variables grows. For this reason, the solution of large-scale nonlinear problems requires some form of decomposition or partitioning. This involves breaking the problem down into a master problem and one or more subproblems, which are all solved iteratively, exchanging information between the subproblems on one hand and the master on the other hand, yielding approximate solutions nearer and nearer to the true optimum. In this paper, we consider two basically different approaches to nonlinear decomposition, which we shall call the relaxedmaster problem approach and the directmaster problem approach. After transforming the problem into two-stage form, and considering some related properties, we briefly review the Kbonsjo (1969) relaxed master decomposition algorithm. A direct master problem decomposition algorithm is then presented. Some material on subdifferentiable ...

Multidimensional curve-fitting with self-organizing automata

In large-scale linear problems a one-dimensional electrical network often serves as a model, at discrete points, for a function of k independent variables. Such a discrete animated model has been generalized by the author to represent a class of large-scale nonlinear problems by introducing a sequence of linear-, planar-, cubic-, etc., up to k-dimensional networks, all interconnected into a single polyhedral structure. However, a hierarchy of multidimensional networks can no longer be energized by mere currents and voltages. A sequence of multidimensional electromagnetic waves must be propagated across the polyhedron (and its dual polyhedron), in order that the waves may satisfy Stokes' theorem, as they step across the boundaries between two different-dimensional networks. Such an animated polyhedral model can represent not only a function of k independent variables, but also all its divided differences of higher order (estimated directional derivatives along the lines, planes, cubes, etc.) all simultaneously. Furthermore, if the polyhedron and its dual are immersed into a k-dimensional region filled with stationary or moving magnetohydrodynamic plasma, the amorphous field crystallizes into a sequence of 2 k sets of transmission networks, coupled by and energized with a large number of k-dimensional magnetohydrodynamic generators (“generalized” rotating electrical machines). Even in the absence of motion (velocity terms), the crystallized field-structure may assume a self-adaptive “oscillatory” state, in which it can represent not one, but any number of arbitrarily picked functions of k independent variables, as well as their higher order divided differences—all simultaneously and automatically—without needing any adjustment or interference by the analyst. The resultant oscillatory polyhedron (or self-organizing “automaton”) is applied in the present paper to model (curve-fit) simultaneously any number of functions of a set of nonuniformly-spaced variables. In particular, simple numerical examples are shown of estimating—by regression theory and a least-square criterion—several arbitrary functions of two independent variables at four nonuniformly-spaced points on a plane. The same oscillatory model (automaton) is used, without any change, for the highly satisfactory estimation of not one but six different arbitrarily picked functions, plus their divided differences. Numerical examples with two nonoscillatory polyhedra show that the latter can satisfactorily fit only one, or at most, only a small class of functions plus their divided differences.