Prediction of probabilistic settlements via spectral stochastic meshless local Petrov–Galerkin method

This study introduces the prediction of probabilistic settlements with the uncertainty in the spatial variability of Young’s modulus to illustrate the preliminary development of a spectral stochastic meshless local Petrov–Galerkin (SSMLPG) method. Generalized polynomial chaos expansions of Young’s moduli and a two-dimensional meshfree weak–strong formulation in elasticity are combined to derive the SSMLPG formulation. Because of the local and truly meshless nature, the SSMLPG method is more computationally efficient than available stochastic numerical methods. Two examples further show that SSMLPG-based predictions remain sufficiently accurate even in case of scattered nodes. Therefore, the SSMLPG method can be a valuable alternative for solving stochastic boundary-value problems.     

The Lissajous transformation IV. Delaunay and Lissajous variables

Relations are established between the Delaunay variables defined over a phase space E in four dimensions and the Lissajous variables defined over a four-dimensional phase space F when the latter is mapped onto E by a parabolic canonical transformation.     

Decomposition of functions for elliptic orbits

In the algebraA of functions periodic in the mean anomaly we relate the problem of integrating over the mean anomaly with that of decomposing an element ofA as the direct sum of two functions, one in the kernel of the Lie derivative in the Keplerian flow and one in the image of this Lie derivative. We propose recursive rules amenable to general purpose symbolic processors for accomplishing such decomposition in a wide subclass ofA. We introduce the dilogarithmic function to express in exact terms quadratures involving the equation of the center.     

Spatio-temporal filtering using wavelets

 In this paper, a class of spatio-temporal processes with first-order autoregressive temporal structure and functional spatio-temporal interaction is introduced. The spatial second-order regularity is allowed to change over time and is characterized in terms of fractional Sobolev spaces. The associated filtering problem is considered, assuming that observations are defined by spatial linear functionals of the process of interest, being affected by additive noise. Conditions under which a stable solution to this problem is obtained are studied. A functional least-squares linear estimate fusion method is derived to calculate this solution A multiscale finite-dimensional approximation to the problem is obtained from the wavelet-based orthogonal expansions of the time cross-section spatial processes, which allows the numerical inversion of the linear operator involved.     

Spherical harmonics in texture analysis

The objective of this contribution is to emphasize the fundamental role of spherical harmonics in constructive approximation on the sphere in general and in texture analysis in particular. The specific purpose is to present some methods of texture analysis and pole-to-orientation probability density inversion in a unifying approach, i.e. to show that the classic harmonic method, the pole density component fit method initially introduced as a distinct alternative, and the spherical wavelet method for high-resolution texture analysis share a common mathematical basis provided by spherical harmonics. Since pole probability density functions and orientation probability density functions are probability density functions defined on the sphere Ω^{3 3} or hypersphere Ω^{4 4}, respectively, they belong at least to the space of measurable and integrable functions ¹(Ω^d), d=3, 4, respectively.

Therefore, first a basic and simplified method to derive real symmetrized spherical harmonics with the mathematical property of providing a representation of rotations or orientations, respectively, is presented. Then, standard orientation or pole probability density functions, respectively, are introduced by summation processes of harmonic series expansions of ¹(Ω^d) functions, thus avoiding resorting to intuition and heuristics. Eventually, it is shown how a rearrangement of the harmonics leads quite canonically to spherical wavelets, which provide a method for high-resolution texture analysis. This unified point of view clarifies how these methods, e.g. standard functions, apply to texture analysis of EBSD orientation measurements.