CABARET scheme on moving grids for two-dimensional equations of gas dynamics and dynamic elasticity

Схема КАБАРЕ, являющаяся представителем семейства балансно-характеристических методов, широко используется при решении многих задач для систем дифференциальных уравнений гиперболического типа в эйлеровых переменных. Возрастающая актуальность задач взаимодействия деформируемых тел с потоками жидкости и газа требует адаптации этого метода на лагранжевы и смешанные эйлерово-лагранжевы переменные. Ранее схема КАБАРЕ была построена для одномерных уравнений газовой динамики в массовых лагранжевых переменных, а также для трехмерных уравнений динамической упругости. В первом случае построенную схему не удалось обобщить на многомерные задачи, а во втором — использовался необратимый по времени алгоритм передвижения сетки. В данной работе представлено обобщение метода КАБАРЕ на двумерные уравнения газовой динамики и динамической упругости в смешанных эйлерово-лагранжевых и лагранжевых переменных. Построенный метод является явным, легко масштабируемым и обладает свойством временн´ой обратимости. Метод тестируется на различных одномерных и двумерных задачах для обеих систем уравнений (соударение упругих тел, поперечные колебания упругой балки, движение свободной границы идеального газа). The conservative-characteristic CABARET scheme is widely used in solving many problems for systems of differential equations of hyperbolic type in Euler variables. The increasing urgency of the problems of interaction of deformable bodies with liquid and gas flows requires the adaptation of this method to Lagrangian and arbitrary Lagrangian-Eulerian variables. Earlier, the CABARET scheme was constructed for one-dimensional equations of gas dynamics in mass Lagrangian variables, as well as for three-dimensional equations of dynamic elasticity. In the first case, the constructed scheme could not be generalized to multidimensional problems, and in the second, a time-irreversible grid movement algorithm was used. This paper presents a generalization of the CABARET method to two-dimensional equations of gas dynamics and dynamic elasticity in arbitrary Lagrangian-Eulerian and Lagrangian variables. The constructed method is explicit, easily scalable, and has the property of temporal reversibility. The method is tested on various one-dimensional and two-dimensional problems for both systems of equations (collision of elastic bodies, transverse vibrations of an elastic beam, motion of the free boundary of an ideal gas).

Non-Destructive Assessment of the Dynamic Elasticity Modulus of Eucalyptus nitens Timber Boards

Eucalyptus nitens is a fast-growing wood species with a relevant presence in countries like Australia and Chile. The sustainable construction goals have driven the search of structural applications for Eucalyptus nitens; however, this process has been complicated due to the defects usually presented in these timber boards. This study aims to evaluate the dynamic elasticity modulus (Exd) of Eucalyptus nitens timber boards through non-destructive vibration-based tests. Thirty-six timber boards with different levels of knots and cracks were instrumented and tested in a simply supported condition by measuring longitudinal and transverse vibrations. In the first stage, the Exd was calculated globally through simplified normative formulas. Then, in a second stage, the local variability of the Exd was estimated using operational modal analysis (OMA), finite element numerical simulations (FEM), and regional sensitivity analysis (RSA). The positive correlation found between the global static modulus of elasticity and Exd suggests that non-destructive techniques could be used as a reliable and fast alternative for the assessment of bending stiffness. Finally, the proposed method to estimate the local variability of Exdt based on the combination of OMA, FEM, and RSA techniques was useful to improve the structural selection process of timber boards for lightweight social housing floors.

On a Method in Dynamic Elasticity Problems for Heterogeneous Wedge-Shaped Medium

The method of analysis of steady oscillations arising in the piecewise homogeneous wedge-shaped medium composed by two homogeneous elastic wedges with different mechanical and geometric characteristics is presented. Method is based on the distributions’ integral transform technique and allows reconstructing the wave field in the whole medium by displacement oscillations given in the domain on the boundary of the medium. The problem in question is reduced to a boundary integral equation (BIA). Solvability problems of the BIA are examined and the structure of its solution is established.