Browsing by Author "Liu, Kun"

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    Compact sound-speed sensor for quartz enhanced photoacoustic spectroscopy based applications
    (AIP Publishing LLC, 2015) Liu, Kun; Dong, Lei; Tittel, Frank K.
    A compact sound-speed sensor based on a phase difference method was developed. The sensor employs a U-shaped stainless steel tube with two holes located on its front and back ends, which serves as a sound wave guide. The phase difference between the two holes was measured using two mini-microphones by means of a phase-sensitive detection technique. This method offers the advantage of eliminating the influence of signal fluctuations. The frequency of a sound source offered by a loudspeaker can be scanned between 1 kHz and 50 kHz. The slope of the phase difference as a function of frequency was obtained by scanning the frequency of the sound source. The speed of sound was retrieved from the rate of change of the phase difference. The performance of the sensor was evaluated over a wide range of speeds of sound from 260 m/s to 1010 m/s in different gas mixtures. The measured speed of sound was found to be in good agreement with the theoretical value for the sound-speed sensor.
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    Discontinuous Galerkin Methods for Elliptic Partial Differential Equations with Random Coefficients
    (2011) Liu, Kun; Riviere, Beatrice M.
    This thesis proposes and analyses two numerical methods for solving elliptic partial differential equations with random coefficients. The stochastic problem is first transformed into a parametrized one by the use of the Karhunen--Loève expansion. This new problem is then discretized by the discontinuous Galerkin (DG) method. A priori error estimate in the energy norm for the stochastic discontinuous Galerkin solution is derived. In addition, the expected value of the numerical error is theoretically bounded in the energy norm and the L2 norm. In the second approach, the Monte Carlo method is used to generate independent identically distributed realizations of the stochastic coefficients. The resulting deterministic problems are solved by the DG method. Next, estimates are obtained for the error between the average of these approximate solutions and the expected value of the exact solution. The Monte Carlo discontinuous Galerkin method is tested numerically on several examples. Results show that the nonsymmetric DG method is stable independently of meshes and the value of penalty parameter. Symmetric and incomplete DG methods are stable only when the penalty parameter is large enough. Finally, comparisons with the Monte Carlo finite element method and the Monte Carlo discontinuous Galerkin method are presented for several cases.
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    Discontinuous Galerkin Methods for Parabolic Partial Differential Equations with Random Input Data
    (2013-09-16) Liu, Kun; Riviere, Beatrice M.; Heinkenschloss, Matthias; Symes, William W.; Vannucci, Marina
    This thesis discusses and develops one approach to solve parabolic partial differential equations with random input data. The stochastic problem is firstly transformed into a parametrized one by using finite dimensional noise assumption and the truncated Karhunen-Loeve expansion. The approach, Monte Carlo discontinuous Galerkin (MCDG) method, randomly generates $M$ realizations of uncertain coefficients and approximates the expected value of the solution by averaging M numerical solutions. This approach is applied to two numerical examples. The first example is a two-dimensional parabolic partial differential equation with random convection term and the second example is a benchmark problem coupling flow and transport equations. I first apply polynomial kernel principal component analysis of second order to generate M realizations of random permeability fields. They are used to obtain M realizations of random convection term computed from solving the flow equation. Using this approach, I solve the transport equation M times corresponding to M velocity realizations. The MCDG solution spreads toward the whole domain from the initial location and the contaminant does not leave the initial location completely as time elapses. The results show that MCDG solution is realistic, because it takes the uncertainty in velocity fields into consideration. Besides, in order to correct overshoot and undershoot solutions caused by the high level of oscillation in random velocity realizations, I solve the transport equation on meshes of finer resolution than of the permeability, and use a slope limiter as well as lower and upper bound constraints to address this difficulty. Finally, future work is proposed.