Deterministic Spectral Method Research Articles

Towards an H-theorem for granular gases

.The H-theorem, originally derived at the level of the Boltzmann nonlinear kinetic equation for a dilute gas undergoing elastic collisions, strongly constrains the velocity distribution of the gas to evolve irreversibly towards equilibrium. As such, the theorem could not be generalized to account for dissipative systems: the conservative nature of collisions is an essential ingredient in the standard derivation. For a dissipative gas of grains, we construct here a simple functional related to the original H, that can be qualified as a Lyapunov functional. It is positive, and results backed by three independent simulation approaches (a deterministic spectral method, the stochastic direct simulation Monte Carlo technique, and molecular dynamics) indicate that it is also nonincreasing. Both driven and unforced cases are investigated.

Spectral Fatigue Analysis of Shallow Water Jacket Platforms

Abstract The spectral analysis approach is a very elegant and computationally efficient method of analyzing the fatigue life of offshore jacket platforms. The primary limitation of the approach is that it assumes linearity of both the structural system and the wave-loading mechanism. The approach is now widely used for the analysis of deepwater, dynamically responsive platforms where nonlinearities are usually not serious. There are also advantages associated with using the approach for shallow water platforms although nonlinearities then become significant, particularly the wave-loading mechanism. Various methods have been proposed to enable the spectral method to be used for some nonlinear situations, including a new approach which uses the Longuet-Higgins wave height-period joint probability density function in order to obtain a better linearization technique. This linearization process is associated with the particular wave heights chosen for producing the transfer functions. The new approach provides a better method for choosing the appropriate height of each so-called base wave case. In order to verify the new approach, a time series analysis, including wave-loading nonlinearities, has been adopted to obtain a reference fatigue life. The sea surface elevation spectrum has been decomposed into a set of equivalent harmonic components. The water particle velocities and accelerations were then individually evaluated and the appropriate (Morison’s) wave loading was computed for each time step in the sea surface time history. The structural stress response time history was then calculated, from which a fatigue life estimate was obtained. This paper presents the results obtained using this new approach, as well as comparative results obtained using the deterministic, spectral, and time domain approaches applied with a representative sea state. The results show that the deterministic-spectral method has a considerable amount of potential, especially for new design work where weight savings and/or increased confidence levels may be achieved.