An extension of Purcell's vector method with applications to panel element equations

The classical Purcell's vector method, for the construction of solutions to dense systems of linear equations is extended to a flexible orthogonalisation procedure. Some properties are revealed of the orthogonalisation procedure in relation to the classical Gauss-Jordan elimination with or without pivoting. Additional properties that are not shared by the classical Gauss-Jordan elimination are exploited. Further properties related to distributed computing are discussed with applications to panel element equations in subsonic compressible aerodynamics. Using an orthogonalisation procedure within panel methods enables a functional decomposition of the sequential panel methods and leads to a two-level parallelism.

On the coupling of Navier–Stokes and linearised Euler equations for aeroacoustic simulation

Aerodynamic generation of sound is governed by the Navier–Stokes equations while acoustic propagation in a non-uniform medium is effectively described by the linearised Euler equations. Different numerical schemes are required for the efficient solution of these two sets of equations, and therefore, coupling techniques become an essential issue. Two types of one-way coupling between the flow solver and the acoustic solver are discussed: (a) for aerodynamic sound generated at solid surfaces, and (b) in the free stream. Test results indicate how the coupling achieves the necessary accuracy so that Computational Fluid Dynamics codes can be used in aeroacoustic simulations.

An improved result of exponential stability of stochastic semilinear evolution equations

Sufficient conditions for the exponential stability of a class ofnonlinear, non-autonomous stochastic differential equations in infinitedimensions are studied. The analysis consists of introducing a suitableapproximating solution systems and using a limiting argument to pass onstability of strong solutions to mild ones. As a consequence, the classicalcriteriaof stability in A. Ichikawa [8] are improved and extended to cover a class ofnon-autonomous stochastic evolution equations.Two examples are investigated to illustrate our theory.

Applications of Feller-Reuter-Riley transition functions

A Feller–Reuter–Riley function is a Markov transition function whose corresponding semigroup maps the set of the real-valued continuous functions vanishing at infinity into itself. The aim of this paper is to investigate applications of such functions in the dual problem, Markov branching processes, and the Williams-matrix. The remarkable property of a Feller–Reuter–Riley function is that it is a Feller minimal transition function with a stable q-matrix. By using this property we are able to prove that, in the theory of branching processes, the branching property is equivalent to the requirement that the corresponding transition function satisfies the Kolmogorov forward equations associated with a stable q-matrix. It follows that the probabilistic definition and the analytic definition for Markov branching processes are actually equivalent. Also, by using this property, together with the Resolvent Decomposition Theorem, a simple analytical proof of the Williams' existence theorem with respect to the Williams-matrix is obtained. The close link between the dual problem and the Feller–Reuter–Riley transition functions is revealed. It enables us to prove that a dual transition function must satisfy the Kolmogorov forward equations. A necessary and sufficient condition for a dual transition function satisfying the Kolmogorov backward equations is also provided.

Moment decay rates of solutions of stochastic differential equations

The objective of this paper is to investigate the p-ίh moment asymptotic stability decay rates for certain finite-dimensional Itό stochastic differential equations. Motivated by some practical examples, the point of our analysis is a special consideration of general decay speeds, which contain as a special case the usual exponential or polynomial type one, to meet various situations. Sufficient conditions for stochastic differential equations (with variable delays or not) are obtained to ensure their asymptotic properties. Several examples are studied to illustrate our theory.

On a flexible elimination algorithm with applications to panel element equations

A flexible elimination algorithm is presented and is applied to the solution of dense systems of linear equations. Properties of the algorithm are exploited in relation to panel element methods for potential flow and subsonic compressible flow. Further properties in relation to distributed computing are also discussed.

Condition assessment of a 63-year old reinforced concrete structure exposed to the North Sea

This paper describes the condition of a reinforced concrete balustrade consisting of some 1000 individual beam elements all exposed similarly to the hostile marine environment of the North Sea at Arbroath, Scotland since 1943. A comparison is made of the condition of the original construction with the condition of repairs carried out in 1968 and in 1993. It is shown that the 1943 construction shows very little corrosion-induced cracking and little rust staining even though it does not appear to be of high construction quality. Only a very low percentage of the balustrade beams have been replaced. In contrast the beam installed in 1968 and later in 1993 show very considerable and large concrete cracks directly attributable to the corrosion of the longitudinal reinforcement, even though the concrete is of a higher quality and density. A detailed condition survey and statistics of crack sizes are presented in the paper. It is found that the higher corrosion resistance of the 1943 concrete is generally consistent with the concrete electrical resistivity measurements but the degree of corrosion of the reinforcing bars is inconsistent with chloride penetration measurements. The results are compared with the very few observations available in the literature for ageing concrete structures in marine environments. The results cast doubt on the conventional wisdom that chloride content at the reinforcement correlates well with reinforcement corrosion.