Applied Mathematics

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This is the collection for the University of Waterloo's Department of Applied Mathematics.

Research outputs are organized by type (eg. Master Thesis, Article, Conference Paper).

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Browsing Applied Mathematics by Author "Del Rey Fernández, David"

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    Continuous-Galerkin Summation-by-Parts Discretization of the Khokhlov-Zabolotskaya-Kuznetsov Equation with Application to High-Intensity-Focused Ultrasound
    (University of Waterloo, 2024-09-18) Xie, Zhongyu; Del Rey Fernández, David; Sivaloganathan, Sivabal
    Over the last two decades, High Intensity Focused Ultrasound (HIFU) has emerged as a promising non-invasive medical approach for locally and precisely ablating tissue, offering versatile applications in tumor treatment, drug delivery, and addressing brain disorders such as essential tremor. Its advantages include targeted energy delivery with no affect on skin integrity, low system maintenance costs, minimal impact on normal tissues, and swift recovery. Despite its’ merits, HIFU remains underutilized, primarily employed in specific breast cancer and prostate cancer treatments. To expand its range of applicability, a comprehensive understanding of the interaction between the ultrasound beam and local tissues at the focal point is essential. This thesis focuses on modeling critical nonlinear effects in the thermal modulation of local tissues by numerically solving the Khokhlov- Zabolotskaya-Kuznetsov (KZK) equation—which is an excellent model for the nonlinear acoustic field arising in HIFU. Constructing high-order stable discretizations of the KZK equations poses significant challenges due to the presence of polynomial nonlinear terms and a second derivative of an integral term within in this equation. Employing a continuous Galerkin approach, an operator is formulated to approximate the integral term, facilitating the construction of a modified second derivative operator. This establishes a clear correspondence between continuous and discrete stability proofs. Additionally, a skew-symmetric splitting technique is used to discretize the nonlinear advective term. The resulting semi-discrete scheme is proven to be stable. Numerical experiments using the method of manufactured solutions demonstrate the high-order accuracy and stability of the proposed numerical method. Finally, a HIFU verification test case demonstrates the applicability of the proposed scheme to investigate HIFU.
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    Entropy-Stable Positivity-Preserving Schemes for Multiphase Flows
    (University of Waterloo, 2024-01-22) Simpson, Benjamin Jacob; Del Rey Fernández, David; Sivaloganathan, Sivabal
    High-intensity focused ultrasound is a promising non-invasive medical technology that has been successfully used to ablate tumors, as well as in the treatment of other conditions. Researchers believe high-intensity focused ultrasound could see clinical application in other areas such as disruption of the blood brain barrier and sonoporation. However, such advances in medical technology requires fundamental insight into the physics associated with high-intensity focused ultrasound, such as the phenomena known as acoustic cavitation and the collapse of the ensuing bubble cavity. The multiphase description of flow phenomena is an attractive option for modelling such problems as all fluids in the domain are modelled using a single set of governing equations, as opposed to separate systems of equations for each phase and therefore, separate meshes for each fluid. In this thesis, we are interested in studying the bubble collapse problem numerically, to elucidate the physics behind the collapse of acoustically driven bubbles. We seek to develop high-order numerical methods to solve this problem, due to their potential to increase computational efficiency. However, high-order methods typically have stability issues, especially when considering complex physics. For this reason, high-order entropy-stable summation-by-parts schemes are a popular method used to simulate compressible flow equations. These methods offer provable stability through satisfying a discrete entropy inequality, which is used to prove discrete L2 stability. Such stability proofs rely on the fundamental assumption that the densities and volume (or void) fractions of both phases remain positive. However, we seek numerical schemes that can simulate flows where the densities and volume fractions get arbitrarily close to zero and, as such, could become negative as the simulation progresses. To address this problem, we present a novel high-order entropy-stable positivity-preserving scheme to solve the 1-D isentropic Baer-Nunziato model. The key to our proposed scheme is a novel artificial dissipation operator, which has tuneable dissipation coefficients that allow the scheme to have provable nodewise positivity of the densities and volume fractions. This new scheme is constructed by mixing a high-order entropy-conservative scheme with a first-order entropy-stable positivity-preserving scheme to create a high-order entropy-stable positivity-preserving scheme. Numerical results which demonstrate the convergence, positivity, and shock capturing capabilities of the scheme are presented.