Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

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Numerical solution of saddle point problems

We present first results of the H2O Southern Galactic Plane Survey (HOPS), using the Mopra Radio Telescope with a broad-band backend and a beam size of about 2 arcmin. We have observed 100 deg2 of the southern Galactic plane at 12 mm (19.5-27.5 GHz), including spectral line emission from H2O masers, multiple metastable transitions of ammonia, cyanoacetylene, methanol and radio recombination lines. In this paper, we report on the characteristics of the survey and H2O maser emission. We find 540 H2O masers, of which 334 are new detections. The strongest maser is 3933 Jy and the weakest is 0.7 Jy, with 62 masers over 100 Jy. In 14 maser sites, the spread in the velocity of the H2O maser emission exceeds 100 km s-1. In one region, the H2O maser velocities are separated by 351.3 km s-1. The rms noise levels are typically between 1 and 2 Jy, with 95 per cent of the survey under 2 Jy. We estimate completeness limits of 98 per cent at around 8.4 Jy and 50 per cent at around 5.5 Jy. We estimate that there are between 800 and 1500 H2O masers in the Galaxy that are detectable in a survey with similar completeness limits to HOPS. We report possible masers in NH3 (11,9) and (8,6) emission towards G19.61-0.23 and in the NH3 (3,3) line towards G23.33-0.30.

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The H2O Southern Galactic Plane Survey (HOPS) – I. Techniques and H2O maser data

http://www.iust.ac.ir/files/cefsse/pg.cef/Articles/Prof._Kaveh/sizing,_geometry_and_topology_optimization.pdf

Sizing, geometry and topology optimization of trusses via force method and genetic algorithm

It is known that the matrix force method has certain advantages over the displacement method for a class of structural problems. It is also known that the force method, when carried out by the conventional Gauss-Jordan procedure, tends to fill in the problem data, making the method unattractive for large size, sparse problems. This poor fill-in property, however, is not necessarily inherent to the method, and the sparsity may be maintained if one uses what we call the Turn-Back LU Procedure. The purpose of this paper is two-fold. First, it is shown that there exist some close relationships between the force method and the least squares problem, and that many existing algebraic procedures to perform the force method can be regarded as applications/extensions of certain well-known matrix factorization schemes for the least squares problem. Secondly, it is demonstrated that these algebraic procedures for the force method can be unified form the matrix factorization viewpoint. Included in this unification is the Turn-Back LU Procedure, which was originally proposed by Topcu in his thesis.8 It is explained why this procedure tends to produce sparse and banded ‘self-stress’ and flexibility matrices with small band width. Some computational results are presented to demonstrate the superiority of the Turn-Back LU Procedure over the other schemes considered in this paper.

On computational procedures for the force method

Improved cycle bases for the flexibility analysis of structures

Two methods are presented for the automatic selection of a cycle basis leading to a sparse flexibility matrix for the analysis of rigid-jointed skeletal structures.

The first method having a local approach forms a maximal set of admissible minimal cycles, while the second having a global approach constructs admissible minimal cycles on the ordered chords of a shortest route tree. A cycle ordering algorithm is also given to reduce the band width of the corresponding flexibility matrix.

Cycle bases for the flexibility analysis of structures

The development of the flexibility method of analysis of skeletal structures has been hindered by the difficulty of determining a suitable statical basis on which to form the flexibility matrix. A combinatorial approach reduces the difficulty to one of selecting a minimal basis of the cycle vector space. After an introduction to flexibility analysis and a brief review of earlier work using combinatorics, the paper presents a procedure to construct a finite sub-set of the cycle vector space containing the elements of all minimal bases. This makes the generation of the required basis feasible by a finite procedure, such as Welsh's generalization of the Kruskal algorithm. It is thus possible to have an automatic method for the analysis of skeletal structures which uses an optimal combinatorial approach.

Cycle bases of minimal measure for the structural analysis of skeletal structures by the flexibility method

The selection of a system of cut releases which would render a continuous skeletal structure statically determinate is governed principally by the topological and geometric characteristics of the structure, and to a lesser extent by its material properties. Only the topological aspects will be considered here, the other factors being purposely excluded. It is shown how the knowledge of the topology of the structure can be exploited to improve the conditioning of the associated equations of compatability. Owing to difficulties inherent in the design of any automatic procedure for the reduction to a canonical form, a method of condensation has been evolved for complex multi‐member structures. In this way the large scale features of the structure are taken into account without neglecting the effect of local characteristics.

Topological Aspects of Structural Linear Analysis: Improving the Conditioning of the Equations of Compatibility of a Multi‐Member Skeletal Structure by Use of the Knowledge of Topology

J. C. de C. Henderson

Papers

Cycle bases for the flexibility analysis of structures

Cycle bases of minimal measure for the structural analysis of skeletal structures by the flexibility method

Topological Aspects of Structural Linear Analysis: Improving the Conditioning of the Equations of Compatibility of a Multi‐Member Skeletal Structure by Use of the Knowledge of Topology