R-3 Repository :: Browsing by Author "Riviere, Beatrice"

Browsing by Author "Riviere, Beatrice"

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A Discontinuous Galerkin Method for Two-phase Flow in Deformable Porous Media
(2020-10-07) Shen, Bo; Riviere, Beatrice
The proposed numerical scheme solves the linear poroelasticity equations, which refers to fluid flow within a deformable porous media under the assumption of relative small deformations. More precisely, we approximate the displacement of solid structure, the wetting phase pressure, and saturation of immiscible two-phase fluid flow in the deformable porous media in three dimensions. The model of linear poroelasticity is becoming increasingly essential in a diverse range of engineering fields such as the reservoir, biomedical, and environmental engineering. Thus predicting the deformation of the solid structure and the evolution of the phases in space and time plays an essential role in the risk-managing and decision-making process. The proposed scheme solves the coupled equations sequentially while keeping each equation implicitly with respect to its unknown. A high-order interior penalty discontinuous Galerkin spatial discretization is combined with a backward Euler discretization in time. With this sequential approach, the equations are fully decoupled, which reduces the computational cost significantly compared to the existing implicit approach. Numerical results show the convergence of the scheme with the expected rates.
A Hybrid Numerical Scheme for Immiscible Two-Phase Flow
(2020-05-05) Doyle, Bryan; Riviere, Beatrice
This thesis proposes a hybrid numerical scheme for immiscible, two-phase flow in porous media, for two separate partial differential equation (PDE) formulations. Discontinuous Galerkin (DG) methods are a commonly used numerical scheme in such applications due to their local mass conservation and ability to handle discontinuous coefficients. Another popular choice are fi nite volume (FV) methods, which are computationally cheaper than their DG counterparts but are only first order accurate and struggle when discontinuous coefficients are introduced. The proposed hybrid numerical scheme uses the DG method in areas of the domain where accuracy is important or around regions where coefficients are discontinuous, and the FV method in all other areas. Preliminary numerical results show that such a hybrid method produces similar results to the standard DG and FV methods in cases of homogeneous and heterogeneous fluid flow, at a fraction of the computational cost. Applications of this work include simulating the quarter- five spot validation test and the channel-flow problem.
A Two-grid Method for Coupled Free Flow with Porous Media Flow
(2010-06) Chidyagwai, Prince; Riviere, Beatrice
This paper presents a two-grid method for solving systems of partial differential equations modelling free flow coupled with porous media flow. This work considers both the coupled Stokes and Darcy as well as the coupled Navier-Stokes and Darcy problems. The numerical schemes proposed are based on combinations of the continuous finite element method and the discontinuous Galerkin method. Numerical errors and convergence rates for solutions obtained from the two-grid method are presented. CPU times for the two-grid algorithm are shown to be significantly less than those obtained by solving the fully coupled problem.
Analysis of discontinuous Galerkin schemes for flow and transport problems
(2022-04-21) Masri, Rami; Riviere, Beatrice
We formulate and theoretically analyze interior penalty discontinuous Galerkin (dG) methods for flow and transport problems. In particular, the analyses of dG formulations for (1) non-linear convection diffusion equations, (2) incompressible Navier– Stokes equations, (3) Cahn–Hilliard–Navier–Stokes equations, and (4) elliptic and parabolic problems with a Dirac line source are presented. First, we formulate a new locally implicit dG method for nonlinear convection diffusion equations, show that this scheme yields a less restrictive constraint on the time step, and prove optimal error estimates. This formulation is motivated by applications to coupled systems of solute transport and blood flow where it is combined with a Runge–Kutta dG scheme to simulate these systems in one dimensional vessel networks. The second scheme we analyze is a pressure correction dG scheme for the incompressible Navier–Stokes equations in two and three dimensional domains. Studying this scheme is motivated by its ability to efficiently simulate flow in large-scale complex computational domains. We show unconditional stability, unique solvability, and convergence of the discrete velocity by obtaining error estimates. The derivation of these error estimates requires the development of several tools including new lifting operators. Further, optimal error estimates in the L2 norm for velocity are obtained via introducing dual Stokes problems. To complete this analysis, we also show convergence of the discrete pressure. The pressure correction dG approach is extended to the Cahn–Hilliard–Navier– Stokes equations within a decoupled splitting framework. We show that the proposed scheme is uniquely solvable. The discrete mass conservation, the energy dissipation, and the L∞ stability of the order parameter, are established. We prove optimal a priori error estimates in the broken gradient norm. Using multiple duality arguments, we obtain an optimal error estimate in the L2 norm. The stability proofs and error analysis are based on induction arguments without any regularization of the potential function. The third class of problems we consider are elliptic and parabolic problems with a Dirac line source. Such problems are used to couple one dimensional flow models in blood vessels to three dimensional models in tissues. The analysis of such problems is challenging since the gradient of the true solution is singular. We propose dG discretizations of such problems and prove convergence in the global L 2 norm. For the elliptic problem, we show convergence in weighted energy norms. In addition, we show almost optimal local error estimates in the L 2 and energy norms in domains excluding the line. For the parabolic problem, we establish global error estimates for the semidiscrete formulation and for the fully discrete backward Euler dG discretization.
Black oil simulation utilizing a central finite volume scheme
(2016-04-25) Chinomona, Rujeko; Riviere, Beatrice
Black-oil simulation is a valuable tool in predicting the multi-phase multi-component flow of fluids in reservoirs. This research validates the use of a central high resolution finite volume scheme developed by Kurganov and Tadmor (KT) to the black-oil model problem. The KT scheme is desirable because of its relative ease of implementation and its ability to generate high resolution solutions at low computational costs. In addition, convergence rates on simple hyperbolic conservation law problems provided in this thesis indicate that the KT scheme is second order. Results obtained from simulations are in alignment with published literature and simulations also accommodate changing variables with predictable outcomes. The KT scheme can be applied to the black-oil model problem with increased confidence
Comparison of Reduced Models for Blood Flow Using Runge–Kutta Discontinuous Galerkin Methods
(2015-11) Puelz, Charles; Riviere, Beatrice; Canic, Suncica; Rusin, Craig G.
Reduced, or one–dimensional blood flow models take the general form of nonlinear hyperbolic systems, but differ greatly in their formulation. One class of models considers the physically conserved quantities of mass and momentum, while another class describes mass and velocity. Further, the averaging process employed in the model derivation requires the specification of the axial velocity profile; this choice differentiates models within each class. Discrepancies among differing models have yet to be investigated. In this paper, we systematically compare several reduced models of blood flow for physiologically relevant vessel parameters, network topology, and boundary data. The models are discretized by a class of Runge–Kutta discontinuous Galerkin methods.
Convergence of a high order method in time and space for the miscible displacement equations
(EDP Sciences, 2015) Li, Jizhou; Riviere, Beatrice; Walkington, Noel
A numerical method is formulated and analyzed for solving the miscible displacement problem under low regularity assumptions. The scheme employs discontinuous Galerkin time stepping with mixed and interior penalty discontinuous Galerkin finite elements in space. The numerical approximations of the pressure, velocity, and concentration converge to the weak solution as the mesh size and time step tend to zero. To pass to the limit a compactness theorem is developed which generalizes the Aubin-Lions theorem to accommodate discontinuous functions both in space and in time.
Coupled Flow and Transport in an Organ and its Vasculature
(2024-08-08) Tzolova, Bilyana; Riviere, Beatrice; Fuentes, David
In contrast to many other types of cancer, the incidence of liver cancer, specifically hepatocellular carcinoma (HCC), is on the rise. For most patients, surgical intervention is not a viable option, leaving them reliant on chemotherapy treatments, particularly transarterial chemoembolization (TACE), for relief. Our study aims to understand how these treatments function within the liver and their impact on tumor growth. Building upon existing research, we model the flow and transport of chemotherapy drugs and embolic agents in the liver using the miscible displacement equations. Utilizing CT images from liver cancer patients, we extract a 1D centerline of the hepatic vascular structures that deliver blood to the tumors, and then construct a 3D mesh from the liver segmentations. We employ the singularity subtraction technique to create a finite element model for the flow of blood in the liver, specifically focusing on areas affected by the TACE treatment. We extend the singularity subtraction technique to the time-dependent advection-diffusion equation to model the concentration of chemotherapy drugs in the liver and tumors. We first solve the time-dependent non-conservative advection-diffusion equation using the finite element method. To address instabilities arising when the model is advection dominated, we then utilize the discontinuous Galerkin method to solve the time-dependent conservative advection-diffusion equation. We couple the models for blood flow following the injection of an embolic agent with the transport of chemotherapy to develop a comprehensive model based on the miscible displacement equations in the liver. We then apply the simulation to data from MD Anderson patients diagnosed with hepatocellular carcinoma who have undergone transarterial chemoembolization treatment. This final model enables us to provide insights into the evolving dynamics of TACE within the liver.
Derivation and Numerical Simulation of Oxygen Transport in Blood Vessels
(2019-09-09) Masri, Rami; Riviere, Beatrice
Modeling and simulating the transport of oxygen in blood provides critical insight on the planning of cardiovascular surgeries. Mathematical simulation provides a quantitative angle on the understanding of changes in hemodynamics. Due to the complexity of the cardio- vascular circulation, this is a computationally challenging task. Further, oxygen transport is coupled to the velocity field of blood. Thus, the numerical solution of the transport equation requires either the specification or the computation of the velocity field of blood. The latter approach is expensive when the three-dimensional Navier Stokes equations are considered, and the a-priori specification of the velocity does not account for changes in the velocity field. To counteract these difficulties, we propose a model reduction of the convection diffusion equation of oxygen in a compliant vessel with varying radius. We ob- tain a one-dimensional equation coupled to the reduced model of blood flow. We employ discontinuous Galerkin methods to efficiently solve the resulting system in one vessel. We show stability of the proposed numerical scheme for a general nonlinear convection diffusion equation. We verify the model using the method of manufactured solutions. We extend the numerical method to a bifurcation of vessels, and we simulate oxygen transport in a three vessel network
Discontinuous Galerkin approximations to elliptic and parabolic problems with a Dirac line source
(EDP Sciences, 2023) Masri, Rami; Shen, Boqian; Riviere, Beatrice
The analyses of interior penalty discontinuous Galerkin methods of any order k for solving elliptic and parabolic problems with Dirac line sources are presented. For the steady state case, we prove convergence of the method by deriving a priori error estimates in the L^{2^{norm and in weighted energy norms. In addition, we prove almost optimal local error estimates in the energy norm for any approximation order. Further, almost optimal local error estimates in the L^{2^{norm are obtained for the case of piecewise linear approximations whereas suboptimal error bounds in the L^{2^{norm are shown for any polynomial degree. For the time-dependent case, convergence of semi-discrete and of backward Euler fully discrete scheme is established by proving error estimates in L^{2^{in time and in space. Numerical results for the elliptic problem are added to support the theoretical results.}}}}}}}}
Discontinuous Galerkin Methods for Pore-scale Multiphase Flow: Theoretical Analysis and Simulation
(2019-04-17) Liu, Chen; Riviere, Beatrice
In this dissertation, we formulate a pressure-correction projection algorithm, in conjunction with the interior penalty discontinuous Galerkin scheme for time and space discretization to build a single-phase incompressible Navier–Stokes simulator and a two-phase Cahn–Hilliard–Navier–Stokes simulator. The method is a decoupled algorithm, which is especially convenient for large-scale 3D numerical simulations in complex geometry, such as in porous structures obtained from microtomography scanning. The simulators we implemented are robust. The numerical experiment results have been validated on a series of realistic physical problems and exhibit the potential for computing effective properties of single/two-phase flow such as permeability and saturation. Theoretical analysis of the numerical methods for solving multiphase flow model will also contribute to the understanding of the complex multi-scale fluid system from the mathematical point of view. In this dissertation, we also analyze a non-symmetric interior penalty discontinuous Galerkin scheme for solving the mixed form of the Cahn–Hilliard equation and a symmetric interior penalty discontinuous Galerkin scheme for solving the Cahn–Hilliard–Navier–Stokes equations. We prove several numerical properties for these numerical schemes, including unique solvability, stability analysis, and error analysis.
Inexact Hierarchical Scale Separation for Linear Systems in Modal Discontinuous Galerkin Discretizations
(2018-04-20) Thiele, Christopher; Riviere, Beatrice
This thesis proposes the inexact hierarchical scale separation (IHSS) method for the solution of linear systems in modal discontinuous Galerkin (DG) discretizations. Like p-multigrid methods, IHSS alternates between discretizations of different polynomial order to improve the computational performance of solving linear systems. IHSS uses two discretizations, which are obtained from a hierarchical splitting of the modal DG basis, resulting in two weakly coupled problems for the low-order and high-order components of the solution (coarse and fine scale). While a global linear system of reduced size is solved for the coarse-scale problem, the fine-scale components are updated locally. IHSS extends the original hierarchical scale separation method, using an iterative solver to approximate the coarse-scale problem and shifting more work to the highly parallel local fine-scale updates. Convergence and computational performance of IHSS are evaluated using example problems from an application in the oil and gas industry, the simulation of the phase separation of binary fluid mixtures in three spatial dimensions. The problem is modeled by the Cahn–Hilliard equation, a fourth-order, nonlinear partial differential equation, which is discretized using the nonsymmetric interior penalty DG method. Numerical experiments demonstrate the applicability of IHSS to the linear systems arising in this problem. It is shown that their solution can be significantly accelerated when common iterative methods are used as coarse-scale solvers within IHSS instead of being applied directly. All parameters of the method are discussed in detail, and their impact on computational performance is evaluated.
Interactive Brain Tumor Segmentation
(2023-06-08) Balsells, Cito; Riviere, Beatrice; Fuentes, David
Machine learning based image segmentation relies on having access to a large dataset of labeled scans. Challenges arise when a sufficient training dataset is not available. To build a labeled dataset, one can manually create segmentations either by hand or assisted by a semi-automatic interactive tool. Here, interaction is given by the user in form of foreground and background clicks on the image. This thesis evaluates semi-automatic interactive image segmentation models applied to 3D brain tumor segmentation from MRI scans under two conditions. The first condition involves training models on the entire dataset. This provides a baseline for the best expected performance for each model. The second condition involves training models on portions of the dataset. This is done in an effort to model a dataset being gradually created from a small dataset. In both conditions, we find that the Wilcoxon signed rank test indicates significant results when comparing some of our interactive models with their fully-automatic counterpart. However, we ultimately deem the difference to be clinically irrelevant.
Iterative Methods and Multiscale Methods for Linear Systems in Modal Discontinuous Galerkin Discretizations
(2021-04-28) Thiele, Christopher; Riviere, Beatrice
Iterative methods for the solution of linear systems are a core component of many scientific software packages, especially of numerical simulations in which the discretization of partial differential equations in two or three dimensions and with high spatial resolution often results in large, sparse linear systems. Since their introduction decades ago, iterative approaches such as the conjugate gradient method, GMRES, and BiCGStab have remained popular and relevant, and they have proven themselves to be scalable tools in eras of exponential growth in computing power and increasing heterogeneity of computing hardware. In this thesis, I evaluate the convergence and computational performance of iterative solvers for linear systems obtained from modal discontinuous Galerkin (DG) discretizations. Specifically, I focus on systems that arise in the simulation of pore-scale multi-phase flow. Besides standard Krylov subspace methods with algebraic preconditioners, the evaluation focuses on multigrid methods, which, in their more algebraic variants, can also be viewed as iterative linear solvers. In particular, I discuss how p-multigrid methods, which use discretizations of different order instead of discretizations on different meshes, assume a simple algebraic structure for modal DG discretizations. I then show how hierarchical scale separation (HSS), a recently proposed multiscale method for modal DG discretizations, can be incorporated into the p-multigrid framework, and I discuss the unified implementation of both methods in highly parallel computing environments. I analyze how the computational performance of these methods is affected by their tunable parameters, and I demonstrate in numerical experiments that properly calibrated p-multigrid methods can accelerate large pore-scale flow simulations significantly. Moreover, I show how the main ideas of HSS techniques can be used to further accelerate p-multigrid methods.
Machine Learning Methods for Vessel Segmentation in Organs
(2022-04-19) Tzolova, Bilyana; Riviere, Beatrice; Fuentes, David
The vascular system plays a crucial role in diagnostics, treatment, and surgical planning in a wide array of diseases. Recently, there has been a growing interest in automating the manual vessel segmentation process to save time. We aim to efficiently and effectively segment the vascular system in the liver organ using deep learning techniques in order to improve on current manual methods. We propose a 3D DenseNet using PocketNet paradigm with binary and ternary classifications that has less parameters to train than the state of the art methods. We explore the impact of various preprocessing techniques on the accuracy of the neural network. We are able to reduce training times and increase accuracy per training parameter in medical imaging segmentation of the liver vessels. Finally, we assess the accuracy of our model predictions using the dice score coefficient. We find that successful preprocessing filters and neural network parameters are necessary for consistently high dice scores.
Mathematical Approaches to Liver and Tumor Medical Image Segmentation
(2021-04-13) Actor, Jonas Albert; Riviere, Beatrice; Fuentes, David T
Hepatocellular carcinoma (HCC), more commonly known as liver cancer, is the most common cause of liver-related deaths in the United States, and a leading cause of cancer deaths worldwide. Diagnosis and treatment methods for HCC are often obtained by CT imaging; necessitating image segmentation to provide pixel-wise labels of what is liver, tumor, or healthy tissue. Such segmentations are costly, in time, effort, and money, to obtain manually. In contrast, automated segmentation methods, such as PDE-based methods or deep learning algorithms, are more efficient, but they too suffer from their own flaws; specifically, deep learning models can achieve state-of-the-art segmentation accuracies, but are viewed as “black boxes” that cannot cope with noise. This thesis describes mathematical approaches to overcome these issues, namely, to provide a mathematical foundation for deep learning segmentation methods that build upon classical applied mathematics techniques. First, we propose various improvements on existing deep learning architectures to perform liver and tumor segmentation. Second, we build upon classical techniques from applied mathematics, such as the discretization of partial differential equations and such as tensor factorization, to make sense of the underlying structure of the operators in convolutional neural networks for image segmentation. Third, we analyze the stability and uncertainty in deep learning segmentation models, and we derive a new bound for lower the Lipschitz constants of deep convolutional neural networks for image segmentation, improving the resilience of these networks to imaging noise.
Numerical Error Quantification of Agent-Based Models as Applied to Oil Reservoir Simulation
(2018-01-05) Doyle, Bryan; Riviere, Beatrice
Agent-based models (ABMs) provide a fast alternative to traditional oil reservoir models by applying localized inexpensive simulations, rather than solving a partial differential equation at every time-step. However, while there have been theoretical and numerical results obtained with ABMs in social science applications, the accuracy of ABMs has not been analyzed in the context of oil reservoir modeling. My project quantifies the accuracy of a specific ABM by comparing its results to a widely accepted reservoir model, based on Darcy's law. I show that while modeling single phase flow with a variety of reservoir scenarios, this ABM matches results given by the traditional simulator with less than 5.4% difference. I propose extensions of my work, including modeling two and three phase flow, and obtaining an accurate correlation between the ABM and traditional simulator parameters; such results would provide significant motivation in the extended use of ABMs in oil reservoir modeling.
Numerical methods and applications for reduced models of blood flow
(2017-04-11) Puelz, Charles; Riviere, Beatrice; Rusin, Craig G.
The human cardiovascular system is a vastly complex collection of interacting components, including vessels, organ systems, valves, regulatory mechanisms, microcirculations, remodeling tissue, and electrophysiological signals. Experimental, mathematical, and computational research efforts have explored various hemodynamic questions; the scope of this literature is a testament to the intricate nature of cardiovascular physiology. In this work, we focus on computational modeling of blood flow in the major vessels of the human body. We consider theoretical questions related to the numerical approximation of reduced models for blood flow, posed as nonlinear hyperbolic systems in one space dimension. Further, we apply this modeling framework to abnormal physiologies resulting from surgical intervention in patients with congenital heart defects. This thesis contains three main parts: (i) a discussion of the implementation and analysis for numerical discretizations of reduced models for blood flow, (ii) an investigation of solutions to different classes of models in the realm of smooth and discontinuous solutions, and (iii) an application of these models within a multiscale framework for simulating flow in patients with hypoplastic left heart syndrome. The two numerical discretizations studied in this thesis are a characteristics-based method for approximating the Riemann-invariants of reduced blood flow models, and a discontinuous Galerkin scheme for approximating solutions to the reduced models directly. A priori error estimates are derived in particular cases for both methods. Further, two classes of hyperbolic systems for blood flow, namely the mass-momentum and the mass-velocity formulations, are systematically compared with each numerical method and physiologically relevant networks of vessels and boundary conditions. Lastly, closed loop vessel network models of various Fontan physiologies are constructed. Arterial and venous trees are built from networks of one-dimensional vessels while the heart, valves, vessel junctions, and organ beds are modeled by systems of algebraic and ordinary differential equations.
Numerical methods for boundary integral equations
(2020-08-13) Zhang, Yabin; Gillman, Adrianna; Chan, Jesse; Riviere, Beatrice; Stanciulescu, Ilinca
The thesis focuses on numerical methods for boundary integral equation (BIE) formulations of partial differential equations (PDEs). The work contains three parts: the first two consider numerical solution methods for boundary integral equations in wave scattering and Stokes flow, respectively. The last part proposes an adaptive discretization technique for BIEs in 2D. The proposed work is based on previous developments in fast direct solution techniques for BIEs. Such methods exploit the rank deficiency in the off-diagonal blocks of the discretized system and build an approximation to the inverse with linear cost for two-dimensional problems. Once the inverse approximation is constructed, applying it to any given vector is very cheap, making the methods ideal for problems with lots of right-hand-sides. The two direct solvers presented in this thesis are driven by real-life applications. The scattering solver is built to assist practitioners in designing acoustic and optic devices to manipulate waves. Its efficiency in handling multiple incident angles and minor modifications in the structure will be handy in an optimal design setting. The Stokes solver is to help with numerical simulation of objects such as bacteria and vesicles in viscous flow. To accurately capture the interaction between the objects and the confining wall, the discretization of the wall often needs to be locally refined in the region approached by the objects. This makes standard fast direct solvers too expensive to be useful, as the linear system changes for each time step. The proposed approach avoids this by pre-constructing a fast direct solver for the wall independently of time and updating the original solver to accommodate any refinements in discretization. The last part of the thesis describes an adaptive discretization technique for two-dimensional BIEs. Standard adaptive discretization method often requires a sequence of global boundary density solves each on a finer grid and terminates with the last grid if the improvements obtained from the next finer level is very small. The global density solves make the cost of the standard approach relatively high. The proposed alternative reduces the cost by replacing global solves with local solves for an approximate of the true density.
Numerical Methods for Two-phase Flow in Rigid and Deformable Porous Media
(2022-04-22) Shen, Boqian; Riviere, Beatrice
The thesis focuses on developing numerical schemes for two-phase flow in rigid and deformable porous media problems. We present a stable and efficient sequential Discontinuous Galerkin (DG) method for solving the linear poroelasticity equations, which characterize two-phase flow within a deformable porous media. More precisely, we approximate the pressure of the wetting phase, the pressure of the non-wetting phase, and the displacement of the solid skeleton in three dimensions by a high-order interior penalty discontinuous Galerkin (IPDG) spatial discretization combined with a backward Euler discretization in time. The proposed work is based on previous developments in single fluid flow in deformable porous media. The numerical scheme solves the coupled equations sequentially while keeping each equation implicitly with respect to its unknown. The equations are fully decoupled in this sequential approach, which significantly reduces the computational cost compared to the implicit and iterative approaches. Numerical experiments show the convergence of the scheme is optimal. Finally, we apply the sequential DG scheme to a variety of physical problems with realistic data including common benchmarks, heterogeneous porous media with discontinuous permeability, porosity and capillary pressure, and porous media subjected to load. The second part of this thesis proposes an adaptive hybrid numerical scheme for solving two-phase flow in rigid porous media problems. The spatial discretization for transport phenomena problem in heterogeneous porous media requires locally mass conservative methods, such as finite volume methods and discontinuous Galerkin methods. The numerical scheme uses discontinuous Galerkin methods in regions of interest where high accuracy is needed and uses finite volume methods in the rest of the domain. The proposed schemes take advantage of the high accuracy of the discontinuous Galerkin method due to its local mesh adaptivity and local choice of polynomial degree. Finite volume methods are only first-order accurate but computationally efficient and robust for general geometries with structured mesh. We develop adaptive indicators to dynamically identify the regions of each method. By using such an adaptive indicator we are able to find the optimal balance between accuracy and computational cost.

Browsing by Author "Riviere, Beatrice"

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