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In this paper we report on the numerical analysis of convection patterns - due to changes in humidity - of air inside a two-dimensional square enclosure. The enclosure, with dimensions of 10 cm by 10 cm, and atmospheric air as the working fluid, is placed in a horizontal position with the gravitational force acting directly downward on it. Thermally, the system and its boundaries are at a constant temperature of 20°C, whereas the humidity varies with position inside the cavity. At the top and bottom walls, the relative humidity is set at 0 and 1, respectively, while the vertical walls are considered as impermeable. The mathematical model is based on two-dimensional versions of the conservation equations for mass, momentum and moisture, in Cartesian coordinates, under laminar flow and steady-state conditions. The governing equations were discretized and solved over the computational domain with the Finite Element method for different system conditions. The results, given in terms of velocity, density and humidity fields, show that convection patterns form as a result of buoyancy forces generated by humidity gradients, just as they do in thermal convection. Further comparison to the thermal convection process of dry air on the same system, show that the two are closely related. 

Traditional full-waveform inversion (FWI) methods only render a “best-fit” model that cannot account for uncertainties of the ill-posed inverse problem. Additionally, local optimization-based FWI methods cannot always converge to a geologically meaningful solution unless the inversion starts with an accurate background model. We seek the solution for FWI in the Bayesian inference framework to address those two issues. In Bayesian inference, the model space is directly probed by sampling methods such that we obtain a reliable uncertainty appraisal, determine optimal models, and avoid entrapment in a small local region of the model space. The solution of such a statistical inverse method is completely described by the posterior distribution, which quantifies the distributions for parameters and inversion uncertainties. 

In this work we determine the time-domain dynamics of a complex mechanical network of integer-order components, e.g., springs and dampers, with an overall transfer function described by implicitly defined operators. This type of transfer functions can be used to describe very large scale dynamics of robot formations, multi-agent systems or viscoelastic phenomena. Such large-scale integrated systems are becoming increasingly important in modern engineering systems, and an accurate model of their dynamics is very important to achieve their control. We give a time domain representation for the dynamics of the system by using a complex variable analysis to find its impulse response. Furthermore, we validate how our infinite order model can be used to describe dynamics of finite order networks, which can be useful as a model reduction method. 

This work presents a simple procedure for designing fractional $$PD^\mu$$ controllers for a type of implicit operators, which have recently been studied to describe large-scale systems. The methodology developed proposes a geometrical approach that allows characterizing the parameter-space of the $$PD^\mu$$ controller into stable and unstable regions. Several numerical examples illustrate the effectiveness of the proposed results.