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A high performance interrogation method for ultra-weak FBG (UWFBG) using double-pulse and heterodyne detection method is proposed. The perturbation along the UWFBG array is located quickly through the use of double-pulsed optical input waveform. Then the perturbation of fiber is quantified precisely by demodulating the phase of differential signals from a heterodyne configuration. The efficiency of measuring the perturbation is improved by more than 20 times than that of using single probe pulse. In comparison with conventional Rayleigh scattering based approach, the proposed method is supreme in signal-to-noise ratio (SNR), approximately 18 dB higher. The use of differential signaling method can effectively remove the influence from frequency drift of the laser source, making this proposed method capable of measuring low frequency vibration. In our experiment, perturbations with both sinusoidal and triangle waveform were generated to quantitatively evaluate the performance of the proposed method. The minimum detectable fiber length variation is 14.85 nm, and the sensing frequency can be as low as 0.2 Hz.

, "Phase sensitive distributed vibration sensing based on ultraweak fiber Bragg grating array using double-pulse," Opt. Abstract. A distributed vibration sensing technique using double-optical-pulse based on phase-sensitive optical time-domain reflectometry (Φ-OTDR) and an ultraweak fiber Bragg grating (UWFBG) array is proposed for the first time. The single-mode sensing fiber is integrated with the UWFBG array that has uniform spatial interval and ultraweak reflectivity. The relatively high reflectivity of the UWFBG, compared with the Rayleigh scattering, gains a high signal-to-noise ratio for the signal, which can make the system achieve the maximum detectable frequency limited by the round-trip time of the probe pulse in fiber. A corresponding experimental Φ-OTDR system with a 4.5 km sensing fiber integrated with the UWFBG array was setup for the evaluation of the system performance. Distributed vibration sensing is successfully realized with spatial resolution of 50 m. The sensing range of the vibration frequency can cover from 3 Hz to 9 kHz.

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In this paper, an important discovery has been found for nonconforming immersed finite element (IFE) methods using integral-value degrees of freedom for solving elliptic interface problems. We show that those IFE methods can only achieve suboptimal convergence rates (i.e., O(h 1/2 ) in the H 1 norm and O(h) in the L 2 norm) if the tangential derivative of the exact solution and the jump of the coefficient are not zero on the interface. A nontrivial counter example is also provided to support our theoretical analysis. To recover the optimal convergence rates, we develop a new nonconforming IFE method with additional terms locally on interface edges. The unisolvence of IFE basis functions is proved on arbitrary triangles. Furthermore, we derive the optimal approximation capabilities of both the Crouzeix-Raviart and the rotated-Q1 IFE spaces for interface problems with variable coefficients via a unified approach different from multipoint Taylor expansions. Finally, optimal error estimates in both H 1 -and L 2 -norms are proved and confirmed with numerical experiments.

In this article, we apply a modified weak Galerkin method to solve variational inequality of the first kind which includes Signorini and obstacle problems. Optimal order a priori error estimates in the energy norm are derived. We also provide some numerical experiments to validate the theoretical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1459–1474, 2017