+publisher:"University of Texas – Austin" +contributor:("Hughes, Thomas J. R.")

University of Texas – Austin

1. Liu, Ju. Thermodynamically consistent modeling and simulation of multiphase flows.

Degree: PhD, Computational Science, Engineering, and Mathematics, 2014, University of Texas – Austin

► Multiphase flow is a familiar phenomenon from daily life and occupies an important role in physics, engineering, and medicine. The understanding of multiphase flows relies… (more)

▼ Multiphase flow is a familiar phenomenon from daily life and occupies an important role in physics, engineering, and medicine. The understanding of multiphase flows relies largely on the theory of interfaces, which is not well understood in many cases. To date, the Navier-Stokes-Korteweg equations and the Cahn-Hilliard equation have represented two major branches of phase-field modeling. The Navier-Stokes-Korteweg equations describe a single component fluid material with multiple states of matter, e.g., water and water vapor; the Cahn-Hilliard type models describe multi-component materials with immiscible interfaces, e.g., air and water. In this dissertation, a unified multiphase fluid modeling framework is developed based on rigorous mathematical and thermodynamic principles. This framework does not assume any ad hoc modeling procedures and is capable of formulating meaningful new models with an arbitrary number of different types of interfaces. In addition to the modeling, novel numerical technologies are developed in this dissertation focusing on the Navier-Stokes-Korteweg equations. First, the notion of entropy variables is properly generalized to the functional setting, which results in an entropy-dissipative semi-discrete formulation. Second, a family of quadrature rules is developed and applied to generate fully discrete schemes. The resulting schemes are featured with two main properties: they are provably dissipative in entropy and second-order accurate in time. In the presence of complex geometries and high-order differential terms, isogeometric analysis is invoked to provide accurate representations of computational geometries and robust numerical tools. A novel periodic transformation operator technology is also developed within the isogeometric context. It significantly simplifies the procedure of the strong imposition of periodic boundary conditions. These attributes make the proposed technologies an ideal candidate for credible numerical simulation of multiphase flows. A general-purpose parallel computing software, named PERIGEE, is developed in this work to provide an implementation framework for the above numerical methods. A comprehensive set of numerical examples has been studied to corroborate the aforementioned theories. Additionally, a variety of application examples have been investigated, culminating with the boiling simulation. Importantly, the boiling model overcomes several challenges for traditional boiling models, owing to its thermodynamically consistent nature. The numerical results indicate the promising potential of the proposed methodology for a wide range of multiphase flow problems. Advisors/Committee Members: Hughes, Thomas J. R. (advisor).

Subjects/Keywords: Multiphase flows; Phase-field model; Non-convex entropy; Van der Waals theory; Coleman-Noll approach; Isogeometric analysis; Entropy variable formulation; Temporal scheme; Parallel computing; Thermocapillarity; Boiling

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APA (6^th Edition):

Liu, J. (2014). Thermodynamically consistent modeling and simulation of multiphase flows. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/28353

Chicago Manual of Style (16^th Edition):

Liu, Ju. “Thermodynamically consistent modeling and simulation of multiphase flows.” 2014. Doctoral Dissertation, University of Texas – Austin. Accessed April 14, 2021. http://hdl.handle.net/2152/28353.

MLA Handbook (7^th Edition):

Liu, Ju. “Thermodynamically consistent modeling and simulation of multiphase flows.” 2014. Web. 14 Apr 2021.

Vancouver:

Liu J. Thermodynamically consistent modeling and simulation of multiphase flows. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2014. [cited 2021 Apr 14]. Available from: http://hdl.handle.net/2152/28353.

Council of Science Editors:

Liu J. Thermodynamically consistent modeling and simulation of multiphase flows. [Doctoral Dissertation]. University of Texas – Austin; 2014. Available from: http://hdl.handle.net/2152/28353

University of Texas – Austin

2. -6969-6857. New ideas in adjoint methods for PDEs : a saddle-point paradigm for finite element analysis and its role in the DPG methodology.

Degree: PhD, Computational Science, Engineering, and Mathematics, 2018, University of Texas – Austin

URL: http://hdl.handle.net/2152/68919

► This dissertation presents a novel framework for the construction and analysis of finite element methods with trial and test spaces of unequal dimension. At the… (more)

▼ This dissertation presents a novel framework for the construction and analysis of finite element methods with trial and test spaces of unequal dimension. At the heart of this work is a new duality theory suitable for variational formulations with non-symmetric functional settings. The primary application of this theory, in this dissertation, is the development and analysis of discontinuous Petrov–Galerkin (DPG) finite element methods. This dissertation introduces the DPG* finite element method: the dual to the DPG method. DPG, as a methodology, can be viewed as a practical means to solve overdetermined discretizations of boundary value problems. In a similar way, DPG* delivers a methodology for underdetermined discretizations. Supporting this new finite element method are new results on a priori error estimation and a posteriori error control. Notably, it is demonstrated that the convergence of a DPG* method is controlled, in part, by a Lagrange multiplier variable which plays the role of the solution variable in DPG methods. An important new result on a posteriori error control for DPG methods and comparisons with other related methods are also featured. The theory developed here is applied to two representative problems coming from linear and nonlinear partial differential equation (PDE) models. To facilitate a thorough mathematical analysis, Poisson's equation is considered. To demonstrate the utility of the approach in less tractable scenarios, the Oldroyd-B fluid model is also considered. Taken together, the combined analysis of these two models effectively demonstrates the utility of the newly developed paradigm. Extensive computational experiments support the theoretical work presented in this dissertation. In these experiments, h- and hp-adaptive mesh refinement play a central role. For standard solution-oriented adaptive mesh refinement, local error contributions coming from a global a posteriori error estimate are selected to mark individual elements for refinement. For goal-oriented adaptive mesh refinement, local contributions coming from both a primal (DPG) and a dual (or adjoint; DPG*) problem are combined to deliver effective refinement strategies for linear output functionals, also known as quantities of interest. Advisors/Committee Members: Demkowicz, Leszek (advisor), Biros, George (committee member), Hughes, Thomas J. R. (committee member), Oden, J. Tinsley (committee member), Roberts, Nathan V. (committee member).

Subjects/Keywords: DPG method; DPG* method; Mixed method; Adjoint method; Finite element method

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APA (6^th Edition):

-6969-6857. (2018). New ideas in adjoint methods for PDEs : a saddle-point paradigm for finite element analysis and its role in the DPG methodology. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/68919

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Chicago Manual of Style (16^th Edition):

-6969-6857. “New ideas in adjoint methods for PDEs : a saddle-point paradigm for finite element analysis and its role in the DPG methodology.” 2018. Doctoral Dissertation, University of Texas – Austin. Accessed April 14, 2021. http://hdl.handle.net/2152/68919.

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Author name may be incomplete

MLA Handbook (7^th Edition):

-6969-6857. “New ideas in adjoint methods for PDEs : a saddle-point paradigm for finite element analysis and its role in the DPG methodology.” 2018. Web. 14 Apr 2021.

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Author name may be incomplete

Vancouver:

-6969-6857. New ideas in adjoint methods for PDEs : a saddle-point paradigm for finite element analysis and its role in the DPG methodology. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2018. [cited 2021 Apr 14]. Available from: http://hdl.handle.net/2152/68919.

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Author name may be incomplete

Council of Science Editors:

-6969-6857. New ideas in adjoint methods for PDEs : a saddle-point paradigm for finite element analysis and its role in the DPG methodology. [Doctoral Dissertation]. University of Texas – Austin; 2018. Available from: http://hdl.handle.net/2152/68919

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Author name may be incomplete

University of Texas – Austin

3. Kamensky, David Michael. Immersogeometric fluid–structure interaction analysis of bioprosthetic heart valves.

Degree: PhD, Computational science, engineering, and mathematics, 2016, University of Texas – Austin

URL: http://hdl.handle.net/2152/46874

► The purpose of this dissertation is to develop numerical methods for fluid–structure interaction (FSI) analysis that are suitable for modeling and simulating bioprosthetic heart valves… (more)

▼ The purpose of this dissertation is to develop numerical methods for fluid–structure interaction (FSI) analysis that are suitable for modeling and simulating bioprosthetic heart valves (BHVs). BHVs are prosthetic replacements for the valves that regulate blood flow through the heart. BHVs reproduce natural hemodynamic conditions by mimicking the structure of native heart valves: they consist of thin flexible leaflets, passively driven by interaction with surrounding fluid. Current designs frequently require replacement 10–15 years after implantation. Computer simulation may help identify causes of and solutions to durability issues. Despite much previous research into computer simulation of heart valve FSI, inconvenience or inaccuracy of readily available numerical methods have prevented widespread incorporation of FSI into models of heart valve mechanics. Challenges associated with heart valve FSI simulation include large deformations of the region occupied by fluid, with changes of topology as the valve opens and closes, and low mass of the structure relative to the fluid, which necessitates careful treatment of fluid–structure coupling. The presence of large pressure gradients also requires special attention to the treatment of fluid mass conservation. Further, a useful numerical method for studying and improving designs of BHVs should be able to capture variations of valve geometry without requiring major effort to construct geometry-specific discretizations. To meet these challenges, I develop a new numerical approach, combining the immersed boundary concept of capturing fluid–structure interfaces on unfitted discretizations with recent developments in isogeometric analysis (IGA), which directly uses geometrical designs of engineered systems as discrete analysis meshes. In this work, I immerse an isogeometric structure discretization into an unfitted analysis mesh of the fluid subproblem. I refer to the immersion of design geometries into unfitted analysis meshes as immersogeometric analysis. To reliably couple unfitted discretizations of the fluid and structure subproblems, I introduce a new semi-implicit time integration procedure and analyze its stability and convergence in the context of linear model problems. I verify that this analysis extrapolates to the nonlinear setting through numerical experiments and explore the validity of my modeling assumptions by comparing computer simulations with observations from an in vitro experiment. Advisors/Committee Members: Sacks, Michael S. (advisor), Hughes, Thomas J. R. (advisor), Ghattas, Omar (committee member), Moser, Robert D (committee member), Hsu, Ming-Chen (committee member).

Subjects/Keywords: Fluid–structure interaction; Isogeometric analysis; Immersogeometric analysis

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APA (6^th Edition):

Kamensky, D. M. (2016). Immersogeometric fluid–structure interaction analysis of bioprosthetic heart valves. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/46874

Chicago Manual of Style (16^th Edition):

Kamensky, David Michael. “Immersogeometric fluid–structure interaction analysis of bioprosthetic heart valves.” 2016. Doctoral Dissertation, University of Texas – Austin. Accessed April 14, 2021. http://hdl.handle.net/2152/46874.

MLA Handbook (7^th Edition):

Kamensky, David Michael. “Immersogeometric fluid–structure interaction analysis of bioprosthetic heart valves.” 2016. Web. 14 Apr 2021.

Vancouver:

Kamensky DM. Immersogeometric fluid–structure interaction analysis of bioprosthetic heart valves. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2016. [cited 2021 Apr 14]. Available from: http://hdl.handle.net/2152/46874.

Council of Science Editors:

Kamensky DM. Immersogeometric fluid–structure interaction analysis of bioprosthetic heart valves. [Doctoral Dissertation]. University of Texas – Austin; 2016. Available from: http://hdl.handle.net/2152/46874

4. -4190-6493. Dynamically adaptive data-driven simulation of extreme hydrological flows.

Degree: PhD, Engineering Mechanics, 2019, University of Texas – Austin

URL: http://dx.doi.org/10.26153/tsw/3153

► Hydrological hazards such as storm surges, tsunamis, and rainfall-induced flooding are physically complex events that are costly in loss of human life and economic productivity.… (more)

▼ Hydrological hazards such as storm surges, tsunamis, and rainfall-induced flooding are physically complex events that are costly in loss of human life and economic productivity. Many such disasters could be mitigated through improved emergency evacuation in real-time and through the development of resilient infrastructure based on knowledge of how systems respond to extreme events. Datadriven computational modeling is a critical technology underpinning these efforts. This investigation focuses on the novel combination of methodologies in forward simulation and data assimilation. The forward geophysical model utilizes adaptive mesh refinement (AMR), a process by which a computational mesh can adapt in time and space based on the current state of a simulation. The forward solution is combined with ensemble based data assimilation methods, whereby observations from an event are assimilated into the forward simulation to improve the veracity of the solution, or used to invert for uncertain physical parameters. The novelty in our approach is the tight two-way coupling of AMR and ensemble filtering techniques. The data assimilation system is implemented on various test cases that delve into the aspects of ensemble based assimilation filters. Additionally, data assimilation on tsunami models is analyzed and a methodology to map the uncertainties in the seabed deformation due to the associated earthquake to the water surface elevation forecast has been presented. Further, using other simulated environments such as the Chile tsunami event of February 2010, a systematic way to calibrate the assimilation system is presented. Finally, the technology is tested by assimilating actual gauge data from the Tohoku tsunami event. These advances offer the promise of significantly transforming data-driven, real-time modeling of hydrological hazards, with potentially broader applications in other science domains. Advisors/Committee Members: Dawson, Clinton N. (advisor), Hughes, Thomas J. R. (committee member), Ghattas, Omar (committee member), Bui-Thanh, Tan (committee member).

Subjects/Keywords: Data assimilation; Ensemble Kalman filter; Adaptive mesh refinement; Tsunami; Okada model; Shallow water equations; Uncertainty quantification

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APA (6^th Edition):

-4190-6493. (2019). Dynamically adaptive data-driven simulation of extreme hydrological flows. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://dx.doi.org/10.26153/tsw/3153

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Chicago Manual of Style (16^th Edition):

-4190-6493. “Dynamically adaptive data-driven simulation of extreme hydrological flows.” 2019. Doctoral Dissertation, University of Texas – Austin. Accessed April 14, 2021. http://dx.doi.org/10.26153/tsw/3153.

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MLA Handbook (7^th Edition):

-4190-6493. “Dynamically adaptive data-driven simulation of extreme hydrological flows.” 2019. Web. 14 Apr 2021.

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Author name may be incomplete

Vancouver:

-4190-6493. Dynamically adaptive data-driven simulation of extreme hydrological flows. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2019. [cited 2021 Apr 14]. Available from: http://dx.doi.org/10.26153/tsw/3153.

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Council of Science Editors:

-4190-6493. Dynamically adaptive data-driven simulation of extreme hydrological flows. [Doctoral Dissertation]. University of Texas – Austin; 2019. Available from: http://dx.doi.org/10.26153/tsw/3153

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Author name may be incomplete

University of Texas – Austin

5. Toshniwal, Deepesh. Isogeometric Analysis : study of non-uniform degree and unstructured splines, and application to phase field modeling of corrosion.

Degree: PhD, Computational Science, Engineering, and Mathematics, 2019, University of Texas – Austin

URL: http://dx.doi.org/10.26153/tsw/4799

► Isogeometric Analysis or IGA was introduced by Hughes et al. (2005) to facilitate efficient design-through-analysis cycles for engineered objects. The goal of this technology is… (more)

▼ Isogeometric Analysis or IGA was introduced by Hughes et al. (2005) to facilitate efficient design-through-analysis cycles for engineered objects. The goal of this technology is the unification of geometric modeling and engineering analysis, and this is realized by exploiting smooth spline spaces used for the former as finite element spaces required for the latter. As intended, this allows the use of geometrically exact representations for the purpose of analysis. Several new spline constructions have been devised on grid-like meshes since IGA’s inception. The excellent approximation and robustness offered by them has rejuvenated the study of high order methods, and IGA has been successfully applied to myriad problems. However, an unintended consequence of adopting a splinebased design-through-analysis paradigm has been the inheritance of open problems that lie at the intersection of the fields of modeling and approximation using splines. The first two parts of this dissertation focus on two such problems: splines of non-uniform degree and splines on unstructured meshes. The last part of the dissertation is focused on phase field modeling of corrosion using splines. The development of non-uniform degree splines is driven by the observation that relaxing the requirement for a spline’s polynomial pieces to have the same degree would be very powerful in the context of both geometric modeling and IGA. This dissertation provides a complete solution in the univariate setting. A mathematically sound foundation for an efficient algorithmic evaluation of univariate non-uniform degree splines is derived. It is shown that the algorithm outputs a nonuniform degree B-spline basis and that, furthermore, it can be applied to create C¹ piecewise-NURBS of non-uniform degree with B-spline-like properties. In the bivariate setting, a theoretical study of the dimension of non-uniform degree splines on planar T-meshes and triangulations is carried out. Combinatorial lower and upper bounds on the spline space dimension are presented. For T-meshes, sufficient conditions for the bounds to coincide are provided, while for triangulations it is shown that the spline space dimension is stable in sufficiently high degree. Modeling complex geometries using only quadrilaterals leads, in general, to unstructured meshes. In locally structured regions of the mesh, smooth splines can be built following standard procedures. However, there is no canonical way of constructing smooth splines on an unstructured arrangement of quadrilateral elements. This dissertation proposes new spline constructions for the two types of unstructuredness that can be encountered – polar points (i.e., mesh vertices that are collapsed edges) and extraordinary points (i.e., mesh vertices shared by µ ≠ 4 quadrilaterals). On meshes containing polar points, smooth spline basis functions that form a convex partition of unity are built. Numerical tests presented to benchmark the construction indicate optimal approximation behavior. On meshes containing extraordinary points, two… Advisors/Committee Members: Hughes, Thomas J. R. (advisor), Speleers, Hendrik (committee member), Landis, Chad M. (committee member), Demkowicz, Leszek F. (committee member), Ghattas, Omar (committee member).

Subjects/Keywords: Isogeometric Analysis; Finite elements; Smooth splines; Unstructured meshes; Non-uniform degree splines; Dimension formula; Corrosion modeling

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APA (6^th Edition):

Toshniwal, D. (2019). Isogeometric Analysis : study of non-uniform degree and unstructured splines, and application to phase field modeling of corrosion. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://dx.doi.org/10.26153/tsw/4799

Chicago Manual of Style (16^th Edition):

Toshniwal, Deepesh. “Isogeometric Analysis : study of non-uniform degree and unstructured splines, and application to phase field modeling of corrosion.” 2019. Doctoral Dissertation, University of Texas – Austin. Accessed April 14, 2021. http://dx.doi.org/10.26153/tsw/4799.

MLA Handbook (7^th Edition):

Toshniwal, Deepesh. “Isogeometric Analysis : study of non-uniform degree and unstructured splines, and application to phase field modeling of corrosion.” 2019. Web. 14 Apr 2021.

Vancouver:

Toshniwal D. Isogeometric Analysis : study of non-uniform degree and unstructured splines, and application to phase field modeling of corrosion. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2019. [cited 2021 Apr 14]. Available from: http://dx.doi.org/10.26153/tsw/4799.

Council of Science Editors:

Toshniwal D. Isogeometric Analysis : study of non-uniform degree and unstructured splines, and application to phase field modeling of corrosion. [Doctoral Dissertation]. University of Texas – Austin; 2019. Available from: http://dx.doi.org/10.26153/tsw/4799

6. -1325-4005. Inverse problems for basal properties in a thermomechanically coupled ice sheet model.

Degree: PhD, Computational Science, Engineering, and Mathematics, 2017, University of Texas – Austin

URL: http://hdl.handle.net/2152/61822

► Polar ice sheets are losing mass at a growing rate, and are likely to make a dominant contribution to 21st century sea-level rise. Thus, modeling… (more)

▼ Polar ice sheets are losing mass at a growing rate, and are likely to make a dominant contribution to 21st century sea-level rise. Thus, modeling the dynamics of polar ice sheets is critical for projections of future sea level rise. Yet, this is difficult due to the complexity of accurately modeling ice sheet dynamics and because of the unobservable boundary conditions at the base of the ice sheet. In this work, we address the inverse problem of inferring basal properties – in particular, the geothermal heat flux – from surface velocity observations and a forward model in the form of thermomechanically coupled nonlinear Stokes and energy equations with complementarity conditions representing melting of basal ice. This inverse problem is made even more challenging due to the severely nonlinear and non-smooth nature of the forward problem. The inverse problem is formulated as a nonlinear least squares optimization that minimizes the misfit between the model prediction and the observation. A Tikhonov regularization term is added to render the problem well-posed. To solve the inverse problem for large-scale ice sheets, the use of adjoint-based Newton methods is critical, which requires a smoothly differentiable forward problem. Thus, we regularize the complementarity conditions of the forward problem by a penalty-like method, such that the solution of the regularized problem approaches that of the original forward problem as the penalty approaches infinity. As a consequence of the Petrov-Galerkin discretization of the energy equation, discretization and differentiation do not commute, i.e., the order in which we discretize the cost functional and differentiate it affects the correctness of the gradient. Here, we employ the discretize-then-optimize approach to guarantee the consistency between the discrete cost function and its gradient. Using two- and three-dimensional model problems, we study the prospects for and limitations of the inference of the geothermal heat flux field from surface velocity observations. The results show that we can approximately locate the melting region of the ice sheet and reconstruct the geothermal heat flux in the cold region. The reconstruction improves as the noise level in the observations decreases but short-wavelength variations in the geothermal heat flux are difficult to recover. We analyze the ill-posedness of the inverse problem as a function of the number of observations by examining the spectrum of the Hessian of the cost functional. Motivated by the popularity of operator-split or staggered solvers for forward multiphysics problems – i.e., those that drop two-way coupling terms to yield a one-way coupled forward Jacobian – we study the effect on the inversion of a one-way coupling of the adjoint energy and Stokes equations. We show that taking such a one-way coupled approach for the adjoint equations leads to an incorrect gradient and premature termination of optimization iterations. This is due to loss of a descent direction stemming from inconsistency of the gradient with the… Advisors/Committee Members: Ghattas, Omar N. (advisor), Hughes, Thomas J. R. (advisor), Stadler, Georg (committee member), Dawson, Clint (committee member), Heimbach, Patrick (committee member).

Subjects/Keywords: Inverse problem; Thermomechanically coupled model; Complementarity conditions; Ice sheet

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APA (6^th Edition):

-1325-4005. (2017). Inverse problems for basal properties in a thermomechanically coupled ice sheet model. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/61822

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Chicago Manual of Style (16^th Edition):

-1325-4005. “Inverse problems for basal properties in a thermomechanically coupled ice sheet model.” 2017. Doctoral Dissertation, University of Texas – Austin. Accessed April 14, 2021. http://hdl.handle.net/2152/61822.

Note: this citation may be lacking information needed for this citation format:
Author name may be incomplete

MLA Handbook (7^th Edition):

-1325-4005. “Inverse problems for basal properties in a thermomechanically coupled ice sheet model.” 2017. Web. 14 Apr 2021.

Note: this citation may be lacking information needed for this citation format:
Author name may be incomplete

Vancouver:

-1325-4005. Inverse problems for basal properties in a thermomechanically coupled ice sheet model. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2017. [cited 2021 Apr 14]. Available from: http://hdl.handle.net/2152/61822.

Note: this citation may be lacking information needed for this citation format:
Author name may be incomplete

Council of Science Editors:

-1325-4005. Inverse problems for basal properties in a thermomechanically coupled ice sheet model. [Doctoral Dissertation]. University of Texas – Austin; 2017. Available from: http://hdl.handle.net/2152/61822

Note: this citation may be lacking information needed for this citation format:
Author name may be incomplete

University of Texas – Austin

7. Evans, John Andrews. Divergence-free B-spline discretizations for viscous incompressible flows.

Degree: PhD, Computational and Applied Mathematics, 2011, University of Texas – Austin

URL: http://hdl.handle.net/2152/ETD-UT-2011-12-4506

► The incompressible Navier-Stokes equations are among the most important partial differential systems arising from classical physics. They are utilized to model a wide range of… (more)

▼ The incompressible Navier-Stokes equations are among the most important partial differential systems arising from classical physics. They are utilized to model a wide range of fluids, from water moving around a naval vessel to blood flowing through the arteries of the cardiovascular system. Furthermore, the secrets of turbulence are widely believed to be locked within the Navier-Stokes equations. Despite the enormous applicability of the Navier-Stokes equations, the underlying behavior of solutions to the partial differential system remains little understood. Indeed, one of the Clay Mathematics Institute's famed Millenium Prize Problems involves the establishment of existence and smoothness results for Navier-Stokes solutions, and turbulence is considered, in the words of famous physicist Richard Feynman, to be "the last great unsolved problem of classical physics." Numerical simulation has proven to be a very useful tool in the analysis of the Navier-Stokes equations. Simulation of incompressible flows now plays a major role in the industrial design of automobiles and naval ships, and simulation has even been utilized to study the Navier-Stokes existence and smoothness problem. In spite of these successes, state-of-the-art incompressible flow solvers are not without their drawbacks. For example, standard turbulence models which rely on the existence of an energy spectrum often fail in non-trivial settings such as rotating flows. More concerning is the fact that most numerical methods do not respect the fundamental geometric properties of the Navier-Stokes equations. These methods only satisfy the incompressibility constraint in an approximate sense. While this may seem practically harmless, conservative semi-discretizations are typically guaranteed to balance energy if and only if incompressibility is satisfied pointwise. This is especially alarming as both momentum conservation and energy balance play a critical role in flow structure development. Moreover, energy balance is inherently linked to the numerical stability of a method. In this dissertation, novel B-spline discretizations for the generalized Stokes and Navier-Stokes equations are developed. The cornerstone of this development is the construction of smooth generalizations of Raviart-Thomas-Nedelec elements based on the new theory of isogeometric discrete differential forms. The discretizations are (at least) patch-wise continuous and hence can be directly utilized in the Galerkin solution of viscous flows for single-patch configurations. When applied to incompressible flows, the discretizations produce pointwise divergence-free velocity fields. This results in methods which properly balance both momentum and energy at the semi-discrete level. In the presence of multi-patch geometries or no-slip walls, the discontinuous Galerkin framework can be invoked to enforce tangential continuity without upsetting the conservation and stability properties of the method across patch boundaries. This also allows our method to default to a… Advisors/Committee Members: Hughes, Thomas J. R. (advisor), Babuska, Ivo (committee member), Demkowicz, Leszek (committee member), Ghattas, Omar (committee member), Moser, Robert D. (committee member), Bazilevs, Yuri (committee member).

Subjects/Keywords: Incompressible Navier-Stokes equations; B-splines; Mixed discretizations

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APA (6^th Edition):

Evans, J. A. (2011). Divergence-free B-spline discretizations for viscous incompressible flows. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/ETD-UT-2011-12-4506

Chicago Manual of Style (16^th Edition):

Evans, John Andrews. “Divergence-free B-spline discretizations for viscous incompressible flows.” 2011. Doctoral Dissertation, University of Texas – Austin. Accessed April 14, 2021. http://hdl.handle.net/2152/ETD-UT-2011-12-4506.

MLA Handbook (7^th Edition):

Evans, John Andrews. “Divergence-free B-spline discretizations for viscous incompressible flows.” 2011. Web. 14 Apr 2021.

Vancouver:

Evans JA. Divergence-free B-spline discretizations for viscous incompressible flows. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2011. [cited 2021 Apr 14]. Available from: http://hdl.handle.net/2152/ETD-UT-2011-12-4506.

Council of Science Editors:

Evans JA. Divergence-free B-spline discretizations for viscous incompressible flows. [Doctoral Dissertation]. University of Texas – Austin; 2011. Available from: http://hdl.handle.net/2152/ETD-UT-2011-12-4506

University of Texas – Austin

8. -4039-082X. Various applications of discontinuous Petrov-Galerkin (DPG) finite element methods.

Degree: PhD, Computational Science, Engineering, and Mathematics, 2018, University of Texas – Austin

URL: http://hdl.handle.net/2152/65710

► Discontinuous Petrov-Galerkin (DPG) finite element methods have garnered significant attention since they were originally introduced. They discretize variational formulations with broken (discontinuous) test spaces and… (more)

▼ Discontinuous Petrov-Galerkin (DPG) finite element methods have garnered significant attention since they were originally introduced. They discretize variational formulations with broken (discontinuous) test spaces and are crafted to be numerically stable by implicitly computing a near-optimal discrete test space as a function of a discrete trial space. Moreover, they are completely general in the sense that they can be applied to a variety of variational formulations, including non-conventional ones that involve non-symmetric functional settings, such as ultraweak variational formulations. In most cases, these properties have been harnessed to develop numerical methods that provide robust control of relevant equation parameters, like in convection-diffusion problems and other singularly perturbed problems. In this work, other features of DPG methods are systematically exploited and applied to different problems. More specifically, the versatility of DPG methods is elucidated by utilizing the underlying methodology to discretize four distinct variational formulations of the equations of linear elasticity. By taking advantage of interface variables inherent to DPG discretizations, an approach to coupling different variational formulations within the same domain is described and used to solve interesting problems. Moreover, the convenient algebraic structure in DPG methods is harnessed to develop a new family of numerical methods called discrete least-squares (DLS) finite element methods. These involve solving, with improved conditioning properties, a discrete least-squares problem associated with an overdetermined rectangular system of equations, instead of directly solving the usual square systems. Their utility is demonstrated with illustrative examples. Additionally, high-order polygonal DPG (PolyDPG) methods are devised by using the intrinsic discontinuities present in ultraweak formulations. The resulting methods can handle heavily distorted non-convex polygonal elements and discontinuous material properties. A polygonal adaptive strategy was also proposed and compared with standard techniques. Lastly, the natural high-order residual-based a posteriori error estimator ingrained within DPG methods was further applied to problems of physical relevance, like the validation of dynamic mechanical analysis (DMA) calibration experiments of viscoelastic materials, and the modeling of form-wound medium-voltage stator coils sitting inside large electric machinery. Advisors/Committee Members: Demkowicz, Leszek (advisor), Babuska, Ivo M. (committee member), Caffarelli, Luis A. (committee member), Hughes, Thomas J. R. (committee member), Oden, J. Tinsley (committee member), Wilder, Aleta (committee member).

Subjects/Keywords: Finite element methods; Numerical analysis; Computational mathematics; PDEs; DPG methods; DLS methods; PolyDPG methods; Linear elasticity; Viscoelasticity; Thermoviscoelasticity; DMA experiments; Form-wound coils

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APA (6^th Edition):

-4039-082X. (2018). Various applications of discontinuous Petrov-Galerkin (DPG) finite element methods. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/65710

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Chicago Manual of Style (16^th Edition):

-4039-082X. “Various applications of discontinuous Petrov-Galerkin (DPG) finite element methods.” 2018. Doctoral Dissertation, University of Texas – Austin. Accessed April 14, 2021. http://hdl.handle.net/2152/65710.

Note: this citation may be lacking information needed for this citation format:
Author name may be incomplete

MLA Handbook (7^th Edition):

-4039-082X. “Various applications of discontinuous Petrov-Galerkin (DPG) finite element methods.” 2018. Web. 14 Apr 2021.

Note: this citation may be lacking information needed for this citation format:
Author name may be incomplete

Vancouver:

-4039-082X. Various applications of discontinuous Petrov-Galerkin (DPG) finite element methods. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2018. [cited 2021 Apr 14]. Available from: http://hdl.handle.net/2152/65710.

Note: this citation may be lacking information needed for this citation format:
Author name may be incomplete

Council of Science Editors:

-4039-082X. Various applications of discontinuous Petrov-Galerkin (DPG) finite element methods. [Doctoral Dissertation]. University of Texas – Austin; 2018. Available from: http://hdl.handle.net/2152/65710

Note: this citation may be lacking information needed for this citation format:
Author name may be incomplete

University of Texas – Austin

9. Nugen, Frederick Theodore. Advances in saccular aneurysm biomechanics : enlargement via rate-sensitive inelastic growth, bio-mathematical stages of aneurysm disease, and initiation profiles.

Degree: PhD, Mechanical engineering, 2016, University of Texas – Austin

URL: http://hdl.handle.net/2152/45553

► I have created the first simulation of saccular aneurysm initiation and development from a healthy artery geometry. It is capable of growing saccular aneurysm geometries… (more)

▼ I have created the first simulation of saccular aneurysm initiation and development from a healthy artery geometry. It is capable of growing saccular aneurysm geometries from patient-specific data. My model describes aneurysm behavior in a way that bridges fields. I assume arteries are made of a rate-sensitive inelastic material which produces irreversible deformation when it is overstressed. The material is assumed to consist of a 3D hyperelastic background material embedded with 1D transversely-isotropic fibers. I optionally use a Winkler foundation term to model support of external organs and distinguish healthy tissue from diseased tissue. Lesions are defined as a local degradation of artery wall structure. My work suggests passive mechanisms of growth are insufficient for predicting saccular aneurysms. Furthermore, I identify a new concept of stages of aneurysm disease. The stages connect mathematical descriptions of the simulation with clinically-relevant changes in the modeled aneurysm. They provide an evocative framework through which clinical descriptions of arteries can be neatly matched with mathematical features of the model. The framework gives a common language of concepts – e.g., collagen fiber, pseudoelastic limit, inelastic strain, and subclinical lesion – through which researchers in different fields, with different terminologies, can engage in an ongoing dialog: under the model, questions in medicine can be translated into equivalent questions in mathematics. A new stage of “subclinical lesion” has been identified, with a suggested direction for future biomechanics research into early detection and treatment of aneurysms. This stage defines a preclinical aneurysm-producing lesion which occurs before any artery dilatation. It is a stage of aneurysm development involving microstructural changes in artery wall makeup. Under the model, this stage can be identified by its reduced strength: its structural support is still within normal limits, but presumably would perform more poorly in ex vivo failure testing than healthy tissue from the same individual. I encourage clinicians and biomechanicians to measure elastin degradation, and to build detailed multiscale models of elastin degradation profiles as functions of aging and tortuosity; and similarly for basal tone. I hope such measurements will to lead to early detection and treatment of aneurysms. I give specific suggestions of biological tissue experiments to be performed for improving and reinforming constitutive modeling techniques. Advisors/Committee Members: Hughes, Thomas J. R. (advisor), Moser, Robert deLancey (advisor), Sacks, Michael S (committee member), Barr, Ronald (committee member), Gonzalez, Oscar (committee member), Kemper, Craig (committee member), Beasley, Haley K (committee member).

Subjects/Keywords: Saccular aneurysm initiation; Saccular aneurysm; Cranial aneurysm; Aneurysm; Arterial modeling; Inelasticity; Rate-sensitive hyperelasticity; Collagen fibers; Aneurysm stages; Isogeometric analysis; Numerical simulation; Cardiovascular engineering; Computational medicine

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APA (6^th Edition):

Nugen, F. T. (2016). Advances in saccular aneurysm biomechanics : enlargement via rate-sensitive inelastic growth, bio-mathematical stages of aneurysm disease, and initiation profiles. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/45553

Chicago Manual of Style (16^th Edition):

Nugen, Frederick Theodore. “Advances in saccular aneurysm biomechanics : enlargement via rate-sensitive inelastic growth, bio-mathematical stages of aneurysm disease, and initiation profiles.” 2016. Doctoral Dissertation, University of Texas – Austin. Accessed April 14, 2021. http://hdl.handle.net/2152/45553.

MLA Handbook (7^th Edition):

Nugen, Frederick Theodore. “Advances in saccular aneurysm biomechanics : enlargement via rate-sensitive inelastic growth, bio-mathematical stages of aneurysm disease, and initiation profiles.” 2016. Web. 14 Apr 2021.

Vancouver:

Nugen FT. Advances in saccular aneurysm biomechanics : enlargement via rate-sensitive inelastic growth, bio-mathematical stages of aneurysm disease, and initiation profiles. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2016. [cited 2021 Apr 14]. Available from: http://hdl.handle.net/2152/45553.

Council of Science Editors:

Nugen FT. Advances in saccular aneurysm biomechanics : enlargement via rate-sensitive inelastic growth, bio-mathematical stages of aneurysm disease, and initiation profiles. [Doctoral Dissertation]. University of Texas – Austin; 2016. Available from: http://hdl.handle.net/2152/45553

10. -9567-1322. Fast integral equation solver for variable coefficient elliptic PDEs in complex geometries.

Degree: PhD, Computational Science, Engineering, and Mathematics, 2017, University of Texas – Austin

URL: http://hdl.handle.net/2152/63349

► This dissertation presents new numerical algorithms and related software for the numerical solution of elliptic boundary value problems with variable coefficients on certain classes of… (more)

▼ This dissertation presents new numerical algorithms and related software for the numerical solution of elliptic boundary value problems with variable coefficients on certain classes of geometries. The target applications are problems in electrostatics, fluid mechanics, low-frequency electromagnetic and acoustic scattering. We present discretizations based on integral equation formulations which are founded in potential theory and Green's functions. Advantages of our methods include high-order discretization, optimal algorithmic complexity, mesh-independent convergence rate, high-performance and parallel scalability. First, we present a parallel software framework based on kernel independent fast multipole method (FMM) for computing particle and volume potentials in 3D. Our software is applicable to a wide range of elliptic problems such as Poisson, Stokes and low-frequency Helmholtz. It includes new parallel algorithms and performance optimizations which make our volume FMM one of the fastest constant-coefficient elliptic PDE solver on cubic domains. We show that our method is orders of magnitude faster than other N-body codes and PDE solvers. We have scaled our method to half-trillion unknowns on 229K CPU cores. Second, we develop a high-order, adaptive and scalable solver for volume integral equation (VIE) formulations of variable coefficient elliptic PDEs on cubic domains. We use our volume FMM to compute integrals and use GMRES to solve the discretized linear system. We apply our method to compute incompressible Stokes flow in porous media geometries using a penalty function to enforce no-slip boundary conditions on the solid walls. In our largest run, we achieved 0.66 PFLOP/s on 2K compute nodes of the Stampede system (TACC). Third, we develop novel VIE formulations for problems on geometries that can be smoothly mapped to a cube. We convert problems on non-regular geometries to variable coefficient problems on cubic domains which are then solved efficiently using our volume FMM and GMRES. We show that our solver converges quickly even for highly irregular geometries and that the convergence rates are independent of mesh refinement. Fourth, we present a parallel boundary integral equation solver for simulating the flow of concentrated vesicle suspensions in 3D. Such simulations provide useful insights on the dynamics of blood flow and other complex fluids. We present new algorithmic improvements and performance optimizations which allow us to efficiently simulate highly concentrated vesicle suspensions in parallel. Advisors/Committee Members: Biros, George (advisor), Engquist, Bjorn (committee member), van de Geijn, Robert A (committee member), Hughes, Thomas J R (committee member), Kloeckner, Andreas (committee member).

Subjects/Keywords: Elliptic boundary value problems; Fast multipole method; Integral equations

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APA (6^th Edition):

-9567-1322. (2017). Fast integral equation solver for variable coefficient elliptic PDEs in complex geometries. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/63349

Note: this citation may be lacking information needed for this citation format:
Author name may be incomplete

Chicago Manual of Style (16^th Edition):

-9567-1322. “Fast integral equation solver for variable coefficient elliptic PDEs in complex geometries.” 2017. Doctoral Dissertation, University of Texas – Austin. Accessed April 14, 2021. http://hdl.handle.net/2152/63349.

Note: this citation may be lacking information needed for this citation format:
Author name may be incomplete

MLA Handbook (7^th Edition):

-9567-1322. “Fast integral equation solver for variable coefficient elliptic PDEs in complex geometries.” 2017. Web. 14 Apr 2021.

Note: this citation may be lacking information needed for this citation format:
Author name may be incomplete

Vancouver:

-9567-1322. Fast integral equation solver for variable coefficient elliptic PDEs in complex geometries. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2017. [cited 2021 Apr 14]. Available from: http://hdl.handle.net/2152/63349.

Note: this citation may be lacking information needed for this citation format:
Author name may be incomplete

Council of Science Editors:

-9567-1322. Fast integral equation solver for variable coefficient elliptic PDEs in complex geometries. [Doctoral Dissertation]. University of Texas – Austin; 2017. Available from: http://hdl.handle.net/2152/63349

Note: this citation may be lacking information needed for this citation format:
Author name may be incomplete

University of Texas – Austin

11. Taus, Matthias Franz. Isogeometric Analysis for boundary integral equations.

Degree: PhD, Computational Science, Engineering, and Mathematics, 2015, University of Texas – Austin

URL: http://hdl.handle.net/2152/32824

► Since its emergence, Isogeometric Analysis (IgA) has initiated a revolution within the field of Finite Element Methods (FEMs) for two reasons: (i) geometry descriptions originating… (more)

▼ Since its emergence, Isogeometric Analysis (IgA) has initiated a revolution within the field of Finite Element Methods (FEMs) for two reasons: (i) geometry descriptions originating from Computer Aided Design (CAD) can be used directly for analysis purposes, and (ii) the availability of smooth exact geometry descriptions and smooth basis functions can be used to develop new, highly accurate and highly efficient numerical methods. Whereas in FEMs the first issue is still open, it has already been shown that Isogeometric BEMs (IBEMs) provide a complete design-through-analysis framework. However, in contrast to FEMs, the effect of smoothness provided by IgA has not yet been explored in IBEMs. In this dissertation, we address this aspect of IgA. We show that the smoothness and exactness properties provided by the IgA framework can be used to design highly accurate and highly efficient BEMs which are not accessible with conventional BEMs. We develop Collocation IBEMs on piecewise smooth geometries. This allows us to show that IBEMs converge in the expected rates and result in system matrices with mesh-independent condition numbers. The latter property is particularly beneficial for large-scale problems that require iterative linear solvers. However, using conventional Collocation BEMs, this approach is not accessible because hyper-singular integrals have to be evaluated. In contrast, using Collocation IBEMs, the smoothness properties of the IgA framework can be used to regularize the hyper-singular integrals and reduce them to weakly singular integrals which can be evaluated using well-known techniques. We perform several numerical examples on canonical shapes to show these results. In addition, we use well-known mathematical results to develop a sound theoretical foundation to some of our methods, a result that is very rare for Collocation discretizations. Finally, using the exactness of IgA geometry descriptions, we design Patch Tests that allow one to rigorously test IBEM implementations. We subject our implementation to these Patch Tests which not only shows the reliability of our method but also shows that IBEMs can be as accurate as machine precision. We apply our IBEMs to Laplace's equation and the equations of linear elasticity. In addition, input files for our implementation can be automatically obtained from commercial CAD packages. These practical aspects allow us to apply IBEMs to analyze a propeller under a wind load. Advisors/Committee Members: Rodin, G. J. (Gregory J.) (advisor), Hughes, Thomas J. R. (advisor), Demkowicz, Leszek F. (committee member), Biros, George (committee member), Sayas, Francisco Javier (committee member).

Subjects/Keywords: Boundary integral equations; Isogeometric Analysis; Boundary element methods; Isogeometric boundary element methods; Collocation

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APA (6^th Edition):

Taus, M. F. (2015). Isogeometric Analysis for boundary integral equations. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/32824

Chicago Manual of Style (16^th Edition):

Taus, Matthias Franz. “Isogeometric Analysis for boundary integral equations.” 2015. Doctoral Dissertation, University of Texas – Austin. Accessed April 14, 2021. http://hdl.handle.net/2152/32824.

MLA Handbook (7^th Edition):

Taus, Matthias Franz. “Isogeometric Analysis for boundary integral equations.” 2015. Web. 14 Apr 2021.

Vancouver:

Taus MF. Isogeometric Analysis for boundary integral equations. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2015. [cited 2021 Apr 14]. Available from: http://hdl.handle.net/2152/32824.

Council of Science Editors:

Taus MF. Isogeometric Analysis for boundary integral equations. [Doctoral Dissertation]. University of Texas – Austin; 2015. Available from: http://hdl.handle.net/2152/32824

University of Texas – Austin

12. Hossain, Shaolie Samira. Mathematical modeling of coupled drug and drug-encapsulated nanoparticle transport in patient-specific coronary artery walls.

Degree: PhD, Mechanical Engineering, 2009, University of Texas – Austin

URL: http://hdl.handle.net/2152/7861

► A vast majority of heart attacks occur due to rapid progression of plaque buildup in the coronary arteries that supply blood to the heart muscles.… (more)

▼ A vast majority of heart attacks occur due to rapid progression of plaque buildup in the coronary arteries that supply blood to the heart muscles. The diseased arteries can be treated with drugs delivered locally to vulnerable plaques—ones that may rupture and release emboli, resulting in the formation of thrombus, or blood clot that can cause blockage of the arterial lumen. In designing these local drug delivery devices, important issues regarding drug distribution and targeting need to be addressed to ensure device design optimization as physiological forces can cause the local concentration to be very different from mean drug tissue concentration estimated from in vitro experiments and animal studies. Therefore, the main objective of this work was to develop a computational tool-set to support the design of a catheter-based local drug delivery system that uses nanoparticles as drug carriers by simulating drug transport and quantifying local drug distribution in coronary artery walls. Toward this end, a three dimensional mathematical model of coupled transport of drug and drug-encapsulated nanoparticles was developed and solved numerically by applying finite element based isogeometric analysis that uses NURBS-based techniques to describe the artery wall geometry. To gain insight into the parametric sensitivity of drug distribution, a study of the effect of Damkohler number and Peclet number was carried out. The tool was then applied to a three-dimensional idealized multilayered model of the coronary artery wall under healthy and diseased condition. Preliminary results indicated that use of realistic geometry is essential in creating physiological flow features and transport forces necessary for developing catheter-based drug delivery design procedures. Hence, simulations were run on a patient-specific coronary artery wall segment with a typical atherosclerotic plaque characterized by a lipid pool encased by a thin fibrous cap. Results show that plaque heterogeneity and artery wall inhomogeneity have a considerable effect on drug distribution. The computational tool-set developed was able to successfully capture trends observed in local drug delivery by incorporating a multitude of relevant physiological phenomena, and thus demonstrated its potential utility in optimizing drug design parameters including delivery location, nanoparticle surface properties and drug release rate. Advisors/Committee Members: Hughes, Thomas J. R. (advisor).

Subjects/Keywords: Nanoparticles; Drug transport; Coupled transport; Drug-encapsulated nanoparticles; Arteries; Heart disease; Atherosclerotic plaque; Drug delivery

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APA (6^th Edition):

Hossain, S. S. (2009). Mathematical modeling of coupled drug and drug-encapsulated nanoparticle transport in patient-specific coronary artery walls. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/7861

Chicago Manual of Style (16^th Edition):

Hossain, Shaolie Samira. “Mathematical modeling of coupled drug and drug-encapsulated nanoparticle transport in patient-specific coronary artery walls.” 2009. Doctoral Dissertation, University of Texas – Austin. Accessed April 14, 2021. http://hdl.handle.net/2152/7861.

MLA Handbook (7^th Edition):

Hossain, Shaolie Samira. “Mathematical modeling of coupled drug and drug-encapsulated nanoparticle transport in patient-specific coronary artery walls.” 2009. Web. 14 Apr 2021.

Vancouver:

Hossain SS. Mathematical modeling of coupled drug and drug-encapsulated nanoparticle transport in patient-specific coronary artery walls. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2009. [cited 2021 Apr 14]. Available from: http://hdl.handle.net/2152/7861.

Council of Science Editors:

Hossain SS. Mathematical modeling of coupled drug and drug-encapsulated nanoparticle transport in patient-specific coronary artery walls. [Doctoral Dissertation]. University of Texas – Austin; 2009. Available from: http://hdl.handle.net/2152/7861

University of Texas – Austin

13. Cottrell, John Austin, 1980-. Isogeometric analysis and numerical modeling of the fine scales within the variational multiscale method.

Degree: PhD, Computational and Applied Mathematics, 2007, University of Texas – Austin

URL: http://hdl.handle.net/2152/3190

► This work discusses isogeometric analysis as a promising alternative to standard finite element analysis. Isogeometric analysis has emerged from the idea that the act of… (more)

Subjects/Keywords: Numerical analysis

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APA (6^th Edition):

Cottrell, John Austin, 1. (2007). Isogeometric analysis and numerical modeling of the fine scales within the variational multiscale method. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/3190

Chicago Manual of Style (16^th Edition):

Cottrell, John Austin, 1980-. “Isogeometric analysis and numerical modeling of the fine scales within the variational multiscale method.” 2007. Doctoral Dissertation, University of Texas – Austin. Accessed April 14, 2021. http://hdl.handle.net/2152/3190.

MLA Handbook (7^th Edition):

Cottrell, John Austin, 1980-. “Isogeometric analysis and numerical modeling of the fine scales within the variational multiscale method.” 2007. Web. 14 Apr 2021.

Vancouver:

Cottrell, John Austin 1. Isogeometric analysis and numerical modeling of the fine scales within the variational multiscale method. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2007. [cited 2021 Apr 14]. Available from: http://hdl.handle.net/2152/3190.

Council of Science Editors:

Cottrell, John Austin 1. Isogeometric analysis and numerical modeling of the fine scales within the variational multiscale method. [Doctoral Dissertation]. University of Texas – Austin; 2007. Available from: http://hdl.handle.net/2152/3190

University of Texas – Austin

14. Bazilevs, Jurijs. Isogeometric analysis of turbulence and fluid-structure interaction.

Degree: PhD, Computational and Applied Mathematics, 2006, University of Texas – Austin

URL: http://hdl.handle.net/2152/2677

► This work puts Isogeometric Analysis, a new analysis framework for computational engineering and sciences, on a firm mathematical foundation. FEM-like theory is developed in which… (more)

Subjects/Keywords: Finite element method; Fluid-structure interaction – Mathematical models; Turbulence – Mathematical models

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APA (6^th Edition):

Bazilevs, J. (2006). Isogeometric analysis of turbulence and fluid-structure interaction. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/2677

Chicago Manual of Style (16^th Edition):

Bazilevs, Jurijs. “Isogeometric analysis of turbulence and fluid-structure interaction.” 2006. Doctoral Dissertation, University of Texas – Austin. Accessed April 14, 2021. http://hdl.handle.net/2152/2677.

MLA Handbook (7^th Edition):

Bazilevs, Jurijs. “Isogeometric analysis of turbulence and fluid-structure interaction.” 2006. Web. 14 Apr 2021.

Vancouver:

Bazilevs J. Isogeometric analysis of turbulence and fluid-structure interaction. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2006. [cited 2021 Apr 14]. Available from: http://hdl.handle.net/2152/2677.

Council of Science Editors:

Bazilevs J. Isogeometric analysis of turbulence and fluid-structure interaction. [Doctoral Dissertation]. University of Texas – Austin; 2006. Available from: http://hdl.handle.net/2152/2677

15. Scott, Michael Andrew. T-splines as a design-through-analysis technology.

Degree: PhD, Computational and Applied Mathematics, 2011, University of Texas – Austin

URL: http://hdl.handle.net/2152/ETD-UT-2011-08-3795

► To simulate increasingly complex physical phenomena and systems, tightly integrated design-through-analysis (DTA) tools are essential. In this dissertation, the complementary strengths of isogeometric analysis and… (more)

▼ To simulate increasingly complex physical phenomena and systems, tightly integrated design-through-analysis (DTA) tools are essential. In this dissertation, the complementary strengths of isogeometric analysis and T-splines are coupled and enhanced to create a seamless DTA framework. In all cases, the technology de- veloped meets the demands of both design and analysis. In isogeometric analysis, the smooth geometric basis is used as the basis for analysis. It has been demonstrated that smoothness offers important computational advantages over standard finite elements. T-splines are a superior alternative to NURBS, the current geometry standard in computer-aided design systems. T-splines can be locally refined and can represent complicated designs as a single watertight geometry. These properties make T-splines an ideal discretization technology for isogeometric analysis and, on a higher level, a foundation upon which unified DTA technologies can be built. We characterize analysis-suitable T-splines and develop corresponding finite element technology, including the appropriate treatment of extraordinary points (i.e., unstructured meshing). Analysis-suitable T-splines form a practically useful subset of T-splines. They maintain the design flexibility of T-splines, including an efficient and highly localized refinement capability, while preserving the important analysis-suitable mathematical properties of the NURBS basis. We identify Bézier extraction as a unifying paradigm underlying all isogeometric element technology. Bézier extraction provides a finite element representation of NURBS or T-splines, and facilitates the incorporation of T-splines into existing finite element programs. Only the shape function subroutine needs to be modified. Additionally, Bézier extraction is automatic and can be applied to any T-spline regardless of topological complexity or polynomial degree. In particular, it represents an elegant treatment of T-junctions, referred to as "hanging nodes" in finite element analysis We then detail a highly localized analysis-suitable h-refinement algorithm. This algorithm introduces a minimal number of superfluous control points and preserves the properties of an analysis-suitable space. Importantly, our local refinement algorithm does not introduce a complex hierarchy of meshes. In other words, all local refinement is done on one control mesh on a single hierarchical “level” and all control points have similar influence on the shape of the surface. This feature is critical for its adoption and usefulness as a design tool. Finally, we explore the behavior of T-splines in finite element analysis. It is demonstrated that T-splines possess similar convergence properties to NURBS with far fewer degrees of freedom. We develop an adaptive isogeometric analysis framework which couples analysis-suitable T-splines, local refinement, and Bézier extraction and apply it to the modeling of damage and fracture processes. These examples demonstrate the feasibility of applying T-spline element technology to… Advisors/Committee Members: Hughes, Thomas J. R. (advisor), Sederberg, Thomas W. (advisor), Taylor, Robert L. (committee member), Ghattas, Omar (committee member), Landis, Chad M. (committee member), Ying, Lexing (committee member).

Subjects/Keywords: Isogeometric analysis; T-splines; Design-through-analysis; Local refinement; Fracture; Extraordinary points; Bézier extraction

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APA (6^th Edition):

Scott, M. A. (2011). T-splines as a design-through-analysis technology. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/ETD-UT-2011-08-3795

Chicago Manual of Style (16^th Edition):

Scott, Michael Andrew. “T-splines as a design-through-analysis technology.” 2011. Doctoral Dissertation, University of Texas – Austin. Accessed April 14, 2021. http://hdl.handle.net/2152/ETD-UT-2011-08-3795.

MLA Handbook (7^th Edition):

Scott, Michael Andrew. “T-splines as a design-through-analysis technology.” 2011. Web. 14 Apr 2021.

Vancouver:

Scott MA. T-splines as a design-through-analysis technology. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2011. [cited 2021 Apr 14]. Available from: http://hdl.handle.net/2152/ETD-UT-2011-08-3795.

Council of Science Editors:

Scott MA. T-splines as a design-through-analysis technology. [Doctoral Dissertation]. University of Texas – Austin; 2011. Available from: http://hdl.handle.net/2152/ETD-UT-2011-08-3795

16. Borden, Michael Johns. Isogeometric analysis of phase-field models for dynamic brittle and ductile fracture.

Degree: PhD, Computational and Applied Mathematics, 2012, University of Texas – Austin

URL: http://hdl.handle.net/2152/ETD-UT-2012-08-6113

► To date, efforts to model fracture and crack propagation have focused on two broad approaches: discrete and continuum damage descriptions. The discrete approach incorporates a… (more)

▼ To date, efforts to model fracture and crack propagation have focused on two broad approaches: discrete and continuum damage descriptions. The discrete approach incorporates a discontinuity into the displacement field that must be tracked and updated. Examples of this approach include XFEM, element deletion, and cohesive zone models. The continuum damage, or smeared crack, approach incorporates a damage parameter into the model that controls the strength of the material. An advantage of this approach is that it does not require interface tracking since the damage parameter varies continuously over the domain. An alternative approach is to use a phase-field to describe crack propagation. In the phase-field approach to modeling fracture the problem is reformulated in terms of a coupled system of partial differential equations. A continuous scalar-valued phase-field is introduced into the model to indicate whether the material is in the unfractured or fractured ''phase''. The evolution of the phase-field is governed by a partial differential equation that includes a driving force that is a function of the strain energy of the body in question. This leads to a coupling between the momentum equation and the phase-field equation. The phase-field model also includes a length scale parameter that controls the width of the smooth approximation to the discrete crack. This allows discrete cracks to be modeled down to any desired length scale. Thus, this approach incorporates the strengths of both the discrete and continuum damage models, i.e., accurate modeling of individual cracks with no interface tracking. The research presented in this dissertation focuses on developing phase-field models for dynamic fracture. A general formulation in terms of the usual balance laws supplemented by a microforce balance law governing the evolution of the phase-field is derived. From this formulation, small-strain brittle and large-deformation ductile models are then derived. Additionally, a fourth-order theory for the phase-field approximation of the crack path is postulated. Convergence and approximation results are obtained for the proposed theories. In this work, isogeometric analysis, and particularly T-splines, plays an important role by providing a smooth basis that allows local refinement. Several numerical simulations have been performed to evaluate the proposed theories. These results show that phase-field models are a powerful tool for predicting fracture. Advisors/Committee Members: Hughes, Thomas J. R. (advisor), Ghattas, Omar (committee member), Landis, Chad M. (committee member), Ravi-Chandar, Krshnaswamy (committee member), Wheeler, Mary F. (committee member).

Subjects/Keywords: Fracture; Phase-field; Brittle fracture; Ductile fracture; Isogeometric analysis; Bezier extraction

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APA (6^th Edition):

Borden, M. J. (2012). Isogeometric analysis of phase-field models for dynamic brittle and ductile fracture. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/ETD-UT-2012-08-6113

Chicago Manual of Style (16^th Edition):

Borden, Michael Johns. “Isogeometric analysis of phase-field models for dynamic brittle and ductile fracture.” 2012. Doctoral Dissertation, University of Texas – Austin. Accessed April 14, 2021. http://hdl.handle.net/2152/ETD-UT-2012-08-6113.

MLA Handbook (7^th Edition):

Borden, Michael Johns. “Isogeometric analysis of phase-field models for dynamic brittle and ductile fracture.” 2012. Web. 14 Apr 2021.

Vancouver:

Borden MJ. Isogeometric analysis of phase-field models for dynamic brittle and ductile fracture. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2012. [cited 2021 Apr 14]. Available from: http://hdl.handle.net/2152/ETD-UT-2012-08-6113.

Council of Science Editors:

Borden MJ. Isogeometric analysis of phase-field models for dynamic brittle and ductile fracture. [Doctoral Dissertation]. University of Texas – Austin; 2012. Available from: http://hdl.handle.net/2152/ETD-UT-2012-08-6113

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