Discontinuous Galerkin methods for the Ostrovsky-Vakhnenko equation

In this paper, we develop discontinuous Galerkin (DG) methods for the Ostrovsky-Vakhnenko (OV) equation, which yields the shock solutions and singular soliton solutions, such as peakon, cuspon and loop solitons. The OV equation has also been shown to have a bi-Hamiltonian structure. We directly develop the energy stable or Hamiltonian conservative discontinuous Galerkin (DG) schemes for the OV equation. Error estimates for the two energy stable schemes are also proved. For some singular solutions, including cuspon and loop soliton solutions, the hodograph transformation is adopted to transform the OV equation or the generalized OV system to the coupled dispersionless (CD) system. Subsequently, two DG schemes are constructed for the transformed CD system. Numerical experiments are provided to demonstrate the accuracy and capability of the DG schemes, including shock solution and, peakon, cuspon and loop soliton solutions

Ultrasonic and image-based characterization of progressive damage processes in intact rocks

Includes bibliographical references.2019 Fall.Stress-driven deformation behavior of rocks is relevant in many practical scenarios, such as damage development around underground structures, the failure of mine pillars, stability of rock slopes and rock-based bridge or dam abutments, hydraulic fracturing operations, rock burst dynamics, earthquake physics, etc. In order to model and predict rock behavior in these practical situations, it is necessary to develop techniques through which the mechanisms associated with rock damage evolution can be accurately monitored and interpreted in detail, which is the primary goal of this research. In this research, the ultrasonic testing approach (both linear ultrasonic testing (LUT) and non-linear ultrasonic testing (NLUT)) has been implemented in conjunction with the non-contact 2-dimensional Digital Image Correlation (2D-DIC) procedure to develop an improved interpretation of the rock damage evolution processes in progressively stressed intact rock specimens. Intact rock specimens of Lyons sandstone, Gosford sandstone, Granodiorite, Barre granite and Stanstead granite were damaged under monotonically increasing uniaxial load for this purpose. Prismatic specimens of these rock types were prepared to ensure a planar surface as necessary for accurate in-plane 2D-DIC measurements. Compressional ultrasonic wave pulses were directly transmitted (T-mode) through and reflected (R-mode) off the damage features in the rock specimens while digital images of the specimen surface were continuously acquired during the test. It was established that the changes in the directly transmitted linear ultrasonic wave (T-mode LUT) parameters (amplitude and mean frequency) are illustrative of the state of damage in progressive loaded intact rock specimens, and that the damage-induced changes in these parameters can be utilized for the estimation of the crack initiation (CI) and crack damage (CD) thresholds of rocks. The non-linear indicator θ, which is indicative of the non-linear elasto-dynamic response of rocks and evaluated through the NLUT-SSM (Scaling Subtraction Method) approach was more sensitive, in comparison to the traditional linear LUT wave attributes, in the illumination of early stages of damage evolution in intact rocks (e.g. close to CI). The analysis of the reflected ultrasonic waves (R-mode LUT) with the progression of damage in the rock specimens showed that R-mode LUT, similar to T-mode LUT, is also capable of illuminating the extent of damage in the rock specimens. This is perhaps the only study that tracks the changes in the reflected ultrasonic waves in progressively damaged intact rock specimens. From the 2D-DIC measured deformation fields, evolution of the tensile damage and shear damage in progressively damaged rock specimens was quantitatively evaluated in real-time. Through this damage quantification, it was explicitly proven that the changes in the energy (amplitude) of the ultrasonic waves is in-fact dominated by the evolution of damage in the rock specimens. The 2D-DIC measurements also facilitated in the quantitative analysis of the evolution of strain-field heterogeneity with increasing levels of load on the rock specimens. The analysis validated that the magnitude of strain-field heterogeneity is representative of the state of damage in the rock volume, and that, due to the multi-scale nature of damage progression in rocks, the smaller scales of measurement are more representative of the active state of damage in the rock specimens. These findings have numerous potential practical applications. They verify that the ultrasonic testing approach can be used for the assessment of the state of damage and the associated damage mechanism in rocks in a near-continuous manner, and that it can also be employed for prediction of damage related failures in structures