Penalty method with Crouzeix-Raviart approximation for the Stokes equations under slip boundary condition

The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain $\Omega \subset \mathbb R^N \, (N=2,3)$. We propose a finite element scheme based on the nonconforming P1/P0 approximation (Crouzeix-Raviart approximation) combined with a penalty formulation and with reduced-order numerical integration in order to address the essential boundary condition $u \cdot n_{\partial\Omega} = g$ on $\partial\Omega$. Because the original domain $\Omega$ must be approximated by a polygonal (or polyhedral) domain $\Omega_h$ before applying the finite element method, we need to take into account the errors owing to the discrepancy $\Omega \neq \Omega_h$, that is, the issues of domain perturbation. In particular, the approximation of $n_{\partial\Omega}$ by $n_{\partial\Omega_h}$ makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., right-continuous inverse of the normal trace operator $H^1(\Omega)^N \to H^{1/2}(\partial\Omega)$; $u \mapsto u\cdot n_{\partial\Omega}$. In this paper we indeed prove such a discrete lifting theorem, taking advantage of the nonconforming approximation, and consequently we establish the error estimates $O(h^\alpha + \epsilon)$ and $O(h^{2\alpha} + \epsilon)$ for the velocity in the $H^1$- and $L^2$-norms respectively, where $\alpha = 1$ if $N=2$ and $\alpha = 1/2$ if $N=3$. This improves the previous result [T. Kashiwabara et al., Numer. Math. 134 (2016), pp. 705--740] obtained for the conforming approximation in the sense that there appears no reciprocal of the penalty parameter $\epsilon$ in the estimates.Comment: 21 page

A PENALTY METHOD FOR THE TIME-DEPENDENT STOKES PROBLEM WITH THE SLIP BOUNDARY CONDITION AND ITS FINITE ELEMENT APPROXIMATION (Numerical Analysis : New Developments for Elucidating Interdisciplinary Problems II)

We consider the finite element method for the time-dependent Stokes problem with the slip boundary condition in a smooth domain. To avoid a variational crime of numerical computation, a penalty method is applied, which also facilitates the numerical implementation. For the continuous problems, the convergence of the penalty method is investigated. Then, we consider the P1/P1-stabilization or P1b/P1 finite element approximations with penalty and time-discretization. For the penalty term, we propose the reduced and non-reduced integration schemes, and obtain the error estimate for velocity and pressure. The theoretical results are verified by numerical experiments

Sensor Attacks and Resilient Defense on HVAC Systems for Energy Market Signal Tracking

The power flexibility from smart buildings makes them suitable candidates for providing grid services. The building automation system (BAS) that employs model predictive control (MPC) for grid services relies heavily on sensor data gathered from IoT-based HVAC systems through communication networks. However, cyber-attacks that tamper sensor values can compromise the accuracy and flexibility of HVAC system power adjustment. Existing studies on grid-interactive buildings mainly focus on the efficiency and flexibility of buildings' participation in grid operations, while the security aspect is lacking. In this paper, we investigate the effects of cyber-attacks on HVAC systems in grid-interactive buildings, specifically their power-tracking performance. We design a stochastic optimization-based stealthy sensor attack and a corresponding defense strategy using a resilient control framework. The attack and its defense are tested in a physical model of a test building with a single-chiller HVAC system. Simulation results demonstrate that minor falsifications caused by a stealthy sensor attack can significantly alter the power profile, leading to large power tracking errors. However, the resilient control framework can reduce the power tracking error by over 70% under such attacks without filtering out compromised data

Wetting layer evolution and its temperature dependence during self assembly of InAs/GaAs quantum dots

For InAs/GaAs(001) quantum dot (QD) system, the wetting layer (WL) evolution and its temperature dependence were studied using reflectance difference spectroscopy (RDS) and analyzed with a rate equation model. The WL thicknesses showed a monotonic increase at relatively low growth temperatures but a first increase and then decrease at higher temperatures, which were unexpected from the thermodynamic understanding. By adopting a rate equation model, the temperature dependence of QD growth was assigned as the origin of different WL evolutions. A brief discussion on the indium desorption was also given. Those results gave hints of the kinetic aspects of QD self-assembly.Comment: 13 pages, 3 figure

A model-data asymptotic-preserving neural network method based on micro-macro decomposition for gray radiative transfer equations

We propose a model-data asymptotic-preserving neural network(MD-APNN) method to solve the nonlinear gray radiative transfer equations(GRTEs). The system is challenging to be simulated with both the traditional numerical schemes and the vanilla physics-informed neural networks(PINNs) due to the multiscale characteristics. Under the framework of PINNs, we employ a micro-macro decomposition technique to construct a new asymptotic-preserving(AP) loss function, which includes the residual of the governing equations in the micro-macro coupled form, the initial and boundary conditions with additional diffusion limit information, the conservation laws, and a few labeled data. A convergence analysis is performed for the proposed method, and a number of numerical examples are presented to illustrate the efficiency of MD-APNNs, and particularly, the importance of the AP property in the neural networks for the diffusion dominating problems. The numerical results indicate that MD-APNNs lead to a better performance than APNNs or pure data-driven networks in the simulation of the nonlinear non-stationary GRTEs