Parallelepiped

Abstract: In this paper, a geometric and electromagnetic model of a typical element of urban structure is presented, in order to analytically evaluate in closed form its electromagnetic return to an active microwave sensor. This model can be used to understand what information on geometric and dielectric properties of a building can be extracted from microwave remote sensing data. The geometrical model consists of a rectangular parallelepiped whose vertical walls form a generic angle with respect to the sensor line of flight. The parallelepiped is placed on a rough surface. The radar return from such a structure can be decomposed into single-scattering contributions from the (rough) ground, the building roof (a plane surface in our model), and vertical walls and multiple scattering contributions from dihedral structures formed by vertical walls and ground. In our model, single-scattering contributions are evaluated by using either physical optics (PO) or geometrical optics (GO), depending on surface roughness. In order to account for multiple scattering between buildings and terrain, we use GO to evaluate the field reflected by the smooth wall toward the ground (first bounce) or the sensor (second or third bounce) and GO or PO (according to ground surface roughness) to evaluate the field scattered by the ground toward the wall (first or second bounce) or the sensor (second bounce). Finally, the above model is used to analyze the field backscattered from a building as a function of the main scene parameters; in particular, the angle between vertical walls and sensor line of night and the dependence on the look angle are analyzed.

Abstract: Summary Non-probabilistic convex models need to be provided only the changing boundary of parameters rather than their exact probability distributions; thus, such models can be applied to uncertainty analysis of complex structures when experimental information is lacking. The interval and the ellipsoidal models are the two most commonly used modeling methods in the field of non-probabilistic convex modeling. However, the former can only deal with independent variables, while the latter can only deal with dependent variables. This paper presents a more general non-probabilistic convex model, the multidimensional parallelepiped model. This model can include the independent and dependent uncertain variables in a unified framework and can effectively deal with complex ‘multi-source uncertainty’ problems in which dependent variables and independent variables coexist. For any two parameters, the concepts of the correlation angle and the correlation coefficient are defined. Through the marginal intervals of all the parameters and also their correlation coefficients, a multidimensional parallelepiped can easily be built as the uncertainty domain for parameters. Through the introduction of affine coordinates, the parallelepiped model in the original parameter space is converted to an interval model in the affine space, thus greatly facilitating subsequent structural uncertainty analysis. The parallelepiped model is applied to structural uncertainty propagation analysis, and the response interval of the structure is obtained in the case of uncertain initial parameters. Finally, the method described in this paper was applied to several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.

Abstract: A solution is presented for the three‐dimensional problem of determining the free vibration frequencies and mode shapes for a rectangular parallelepiped which is completely fixed on one face and free on the other five faces. This problem apparently is previously unsolved in the published literature. The Ritz method is used, with displacements assumed in the form of algebraic polynomials. Convergence is studied. Numerical results are given for the first five frequencies of each of the four symmetry classes of vibration, for five thick parallelepiped configurations, including the cube. Contour plots are exhibited for the modal displacements of the cube. The effects of varying Poisson’s ratio are also observed.