A Numerical Method for Computing Double Integrals with Variable Upper Limits

We propose and justify a numerical method for computing the double integral with variable upper limits that leads to the variableness of the region of integration. Imposition of simple variables as functions for upper limits provides the form of triangles of integration region and variable in the external limit of integral leads to a continuous set of similar triangles. A variable grid is overlaid on the integration region. We consider three cases of changes of the grid for the division of the integration region into elementary volumes. The first is only the size of the imposed grid changes with the change of variable of the external upper limit. The second case is the number of division elements changes with the change of the external upper limit variable. In the third case, the grid size and the number of division elements change after fixing their multiplication. In these cases, the formulas for computing double integrals are obtained based on the application of cubatures in the internal region of integration and performing triangulation division along the variable boundary. The error of the method is determined by expanding the double integral into the Taylor series using Barrow’s theorem. Test of efficiency and reliability of the obtained formulas of the numerical method for three cases of ways of the division of integration region is carried out on examples of the double integration of sufficiently simple functions. Analysis of the obtained results shows that the smallest absolute and relative errors are obtained in the case of an increase of the number of division elements changes when the increase of variable of the external upper limit and the grid size is fixed.

A Model-Based FDD Approach for an EHA Using Updated Interactive Multiple Model SVSF

Ubiquitous applications, especially in harsh environments and with strict safety requirements, make Fault Detection and Diagnosis (FDD) in hydraulic actuators an imperative concern for the industry. Model-based FDD uses estimation strategies, including observers and filters as estimation tools. In these methods, observability is a limiting factor in information extraction and parameter estimation for most applications such as in fluid power systems. To address the observability problem, adaptive strategies like Interactive Multiple Model (IMM) estimation have proven to be effective. In this paper a computationally efficient form of IMM referred to as the Updated IMM (UIMM) is used and applied to an Electro-Hydrostatic Actuator (EHA) for FDD. The UIMM is suited to fault conditions that are irreversible, meaning that if a fault happens it will persist in the system. In essence the UIMM follows through a progression of models that in line with the progression of the fault condition in lieu of having all models being considered at the same time (as is the case for IMM). Hence, UIMM significantly reduces the number of models running in parallel and at the same time. This has two major advantages which are higher computational efficiency and avoiding combinatorial explosion. The state and parameters estimation strategies that is used in conjunction with UIMM is the Variable Boundary Layer Smooth Variable Structure Filter (VBL-SVSF). The VBL SVSF is a robust optimal estimation strategy that is more stable than the Kalman Filter in relation to system and modeling uncertainties. The UIMM method is validated by simulation of fault conditions on an EHA.

Slip Gas Flow and Heat Transfer in Confined Porous Media With Different Shape Cylinders

With the recent advances in micro devices, an accurate gas flow and heat transfer analysis become more relevant considering the slip effect. A micro-scale, multiple-relaxation-time (MRT) lattice Boltzmann method with double distribution function approach is used to simulate flow and heat transfer through circular- and diamond-shaped cylinders at the porescale level. The velocity slip and temperature jump are captured at the boundaries using a non-equilibrium extrapolation scheme with the counter-extrapolation method. A pore-scale domain of micro-cylinders comprised of circle and diamond shape are studied. It is found that the permeability increases linearly with an increase in Knudsen number for both circular- and diamond-shaped cylinders. However, the permeability increase for circular obstacle is larger than that of the diamond one. A larger surface area for diamond cylinder will offer more resistance to flow, hence resulting in lower values. For heat transfer, the Nusselt number shows an increase with increasing Reynolds number, however, it decreases with the increase in porosity. Nusselt number values are found to be higher for a circular obstacle. A variable boundary gradient for circular obstacle could be a possible explanation for this difference.

In Situ Stress Inversion and Distribution Characteristics of Tunnel Based on Numerical Simulation and Neural Network Technology

According to the geological conditions of the study area, the measured data of in situ stress was analyzed and the influence degree of buried depth was obtained. A numerical simulation research model with full consideration of fault structure and surface characteristics is established, and boundary condition functions with variables are used. The neural network optimized by genetic algorithm is used to establish the nonlinear relationship between the measured value and the simulated value of the variable boundary condition, and the optimal boundary condition function is obtained. Finally, the in situ stress in the study area was predicted. Through the analysis of the in situ stress field in the research target area, the stress boundary conditions are provided for the follow-up study, and the practical basis for the division of the dangerous area of the surrounding rock of the deep and long tunnel is provided.

Internal water storage capacity of terrestrial planets and the effect of hydration on the M-R relation

<p>The discovery of low density exoplanets in the super-Earth mass regime suggests that ocean planets could be abundant in the galaxy. Understanding the chemical interactions between water and Mg-silicates or iron is essential for constraining the interiors of water-rich planets. Hydration effects have, however, been mostly neglected by the astrophysics community so far. As such effects are unlikely to have major impacts on theoretical mass-radius relations, this is justified as long as the measurement uncertainties are large. However, upcoming missions, such as the PLATO mission (scheduled launch 2026), are envisaged to reach a precision of up to ≈ 3% and ≈ 10% for radii and masses, respectively. As a result, we may soon enter an area in exoplanetary research where various physical and chemical effects such as hydration can no longer be ignored. We have constructed interior models for planets that include reliable prescriptions for hydration of the cores and mantles. These models can be used to refine previous results for which hydration has been neglected and to guide future characterization of observed exoplanets. We have developed numerical tools to solve for the structure of multi-layered planets with variable boundary conditions and compositions. Here we have considered three types of planets: dry interiors, hydrated interiors, and dry interiors plus surface ocean, where the ocean mass fraction corresponds to the mass fraction of the H<sub>2</sub>O equivalent in the hydrated case. We find H and OH storage capacities in the hydrated planets equivalent to 0 - 6 wt% H<sub>2</sub>O corresponding to up to ≈800 km deep ocean layers. In the mass range 0.1 ≤ M/M<sub>⊕</sub> ≤ 3, the effect of hydration on the total radius is found to be ≤ 2.5%, whereas the effect of separation into an isolated surface ocean is ≤ 5 %. Furthermore, we find that our results are very sensitive to the bulk composition.</p>

Internal water storage capacity of terrestrial planets and the effect of hydration on the M-R relation

Context. The discovery of low density exoplanets in the super-Earth mass regime suggests that ocean planets could be abundant in the galaxy. Understanding the chemical interactions between water and Mg-silicates or iron is essential for constraining the interiors of water-rich planets. Hydration effects have, however, been mostly neglected by the astrophysics community so far. As such effects are unlikely to have major impacts on theoretical mass-radius relations, this is justified as long as the measurement uncertainties are large. However, upcoming missions, such as the PLATO mission (scheduled launch 2026), are envisaged to reach a precision of up to ≈3 and ≈10% for radii and masses, respectively. As a result, we may soon enter an area in exoplanetary research where various physical and chemical effects such as hydration can no longer be ignored. Aims. Our goal is to construct interior models for planets that include reliable prescriptions for hydration of the cores and mantles. These models can be used to refine previous results for which hydration has been neglected and to guide future characterization of observed exoplanets. Methods. We have developed numerical tools to solve for the structure of multi-layered planets with variable boundary conditions and compositions. Here we consider three types of planets: dry interiors, hydrated interiors, and dry interiors plus surface ocean, where the ocean mass fraction corresponds to the mass fraction of the H2O equivalent in the hydrated case. Results. We find H and OH storage capacities in the hydrated planets equivalent to 0−6 wt% H2O corresponding to up to ≈800 km deep ocean layers. In the mass range 0.1 ≤ M∕M⊕≤ 3, the effect of hydration on the total radius is found to be ≤2.5%, whereas the effect of separation into an isolated surface ocean is ≤5%. Furthermore, we find that our results are very sensitive to the bulk composition.