Boundary feedback stabilization of a semilinear model for the flow in star-shaped gas networks

The flow of gas through a pipeline network can be modelled by a coupled system of 1-d quasilinear hyperbolic equations. Often for the solution of control problems it is convenient to replace the quasilinear model by a simpler semilinear model. We analyze the behavior of such a semilinear model on a star-shaped network. The model is derived from the diagonal form of the quasilinear model by replacing the eigenvalues by the sound speed multiplied by 1 or -1 respectively, thus neglecting the influence of the gas velocity which is justified in the applications since it is much smaller than the sound speed. For a star-shaped network of pipes we present boundary feedback laws that stabilize the system state exponentially fast to a position of rest for sufficiently small initial data. We show the exponential decay of the $L^2$-norm for arbitrarily long pipes. This is remarkable since in general even for linear systems, for certain source terms the system can become exponentially unstable if the space interval is too long. Our proofs are based upon an observability inequality and suitably chosen Lyapunov functions. Numerical examples including a comparison of the semilinear and the quasilinear model are presented.

Crack Identification of Curved-Elements Using Wavelet Packet

Wavelet Transform (WT) and Wavelet Packet Transform (WPT) approaches have shown great promise in the field of signal analysis in recent decades. The main merit of these methods is their capability in localization of each signal in distinct time or space interval. However, the frequency resolution of such transformation is relatively poor in high frequency regions. The WPT, which is an extended form of the WT, provides a complete level-by-level signal decomposition. Therefore, a frequency analysis creates an arbitrary time. In this study, dynamic transient analysis is performed employing a finite element software (ANSYS) on a beam and acceleration time history of various points is investigated. Then, the captured signals are decomposed to the wavelet packet components using MATLAB and energy rate index is calculated for each component utilizing a wavelet packet rate index (WPERI). The results indicate that not only are the obtained index values sensitive, but they also are precise for the crack identification.

Verification Test of High Flap Macrofluidic Air Flow Sensor in Wind Tunnel

This air sensor functioning to detect the speed of air surrounding while in motionor a sudden changes in its environment. The effect of fast detection of a security sensor through the high sensitivity of the airflow sensor has enabled the system to identify and analyze the critical condition in higher accuracy compared to the conventional of any security system. Previous studies have developed the macrofluidic air flow sensor that observed the air flow in higher accuracy while the sensor in motion will be verified by detection of high sensitivity in the relative velocity of the airflow sensor compared to a conventional sensor. An experimental investigation was conducted to verify macrofluidic air flow sensor in wind tunnel by control velocity of range (30 to 110 km/h). The result shows the characterization of the changes in voltage reading with respect to the airflow speed in the wind tunnel. Sensor 1 to 4 have been placed at 0 to 360 degree of orientation with respective of 90 degree space interval.

On non-autonomously forced Burgers equation with periodic and Dirichlet boundary conditions

We study the non-autonomously forced Burgers equation $$u_t(x,t) + u(x,t)u_x(x,t)-u_{xx}(x,t) = f(x,t)$$ on the space interval (0, 1) with two sets of the boundary conditions: the Dirichlet and periodic ones. For both situations we prove that there exists the unique H1 bounded trajectory of this equation defined for all t ∈ ℝ. Moreover we demonstrate that this trajectory attracts all trajectories both in pullback and forward sense. We also prove that for the Dirichlet case this attraction is exponential.

Strength nature of two-dimensional woven nanofabrics under biaxial tension

Woven nanostructures have been acknowledged as a platform for solar cells, supercapacitors, and sensors, making them especially of interest in the fields of materials sciences, nanotechnology, and renewable energy. By employing molecular dynamics simulations, the mechanical properties of two-dimensional woven nanofabrics under biaxial tension are evaluated. Two-dimensional woven nanostructures composed of graphene and graphyne nanoribbons are examined. Dynamic failure process of both graphene woven nanofabric and graphyne woven nanofabric with the same woven unit cell initiates at the edge of interlaced ribbons accompanied by the formation of cracks near the crossover location of yarns. Further stress analysis reveals that such failure mode is attributed to the compression between two overlaced ribbons and consequently their deformation under biaxial tension, which is sensitive to the lattice structure of nanoribbon as well as the density of yarns in fabric. Systemic comparisons between nanofabrics with different yarn width and interval show that the strength of nanofabric can be effectively controlled by tuning the space interval between nanoribbons. For nanofabrics with fixed large gap spacing, the strength of fabric does not change with the ribbon width, while the strength of nanofabric with small gap spacing decreases anomalously with the increase in yarn density. Such fabric strength dependency on gap spacing is the result of the stress concentration caused by the interlace compression. The outcomes of simulation suggest that the compacted arrangement of yarns in carbon woven nanofabric structures should be avoided to achieve high strength performance.

L-space intervals for graph manifolds and cables

We present a graph manifold analog of the Jankins–Neumann classification of Seifert fibered spaces over$S^{2}$admitting taut foliations, providing a finite recursive formula to compute the L-space Dehn-filling interval for any graph manifold with torus boundary. As an application of a generalization of this result to Floer simple manifolds, we compute the L-space interval for any cable of a Floer simple knot complement in a closed three-manifold in terms of the original L-space interval, recovering a result of Hedden and Hom as a special case.

Boundary layer problems for the two-dimensional compressible Navier–Stokes equations

We study the well-posedness of the boundary layer problems for compressible Navier–Stokes equations. Under the non-negative assumption on the laminar flow, we investigate the local spatial existence of solution for the steady equations. Meanwhile, we also obtain the solution for the unsteady case with monotonic laminar flow, which exists for either long time small space interval or short time large space interval. Moreover, the limit of these solutions with vanishing Mach number is considered. Our proof is based on the comparison theory for the degenerate parabolic equations obtained by the Crocco transformation or von Mises transformation.

Fractional Sturm-Liouville Problem and 1D Space-Time Fractional Diffusion With Mixed Boundary Conditions

In the paper, we show a connection between a regular fractional Sturm-Liouville problem with left and right Caputo derivatives of order in the range (1/2, 1) and a 1D space-time fractional diffusion problem in a bounded domain. Both problems include mixed boundary conditions in a finite space interval. We prove that in the case of vanishing mixed boundary conditions, the Sturm-Liouville problem can be rewritten in terms of Riesz derivatives. Then, we apply earlier results on its eigenvalues and eigenfunctions to construct a weak solution of the 1D fractional diffusion equation with variable diffusivity. Adding an assumption on the summability of the eigenvalues’ inverses series, we formulate a theorem on a strong solution of the 1D fractional diffusion problem.