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Browsing by Author "Tavener, Simon, advisor"

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    A two-field finite element solver for linear poroelasticity
    (Colorado State University. Libraries, 2020) Wang, Zhuoran, author; Liu, Jiangguo, advisor; Tavener, Simon, advisor; Zhou, Yongcheng, committee member; Ma, Kaka, committee member
    Poroelasticity models the interaction between an elastic porous medium and the fluid flowing in it. It has wide applications in biomechanics, geophysics, and soil mechanics. Due to difficulties of deriving analytical solutions for the poroelasticity equation system, finite element methods are powerful tools for obtaining numerical solutions. In this dissertation, we develop a two-field finite element solver for poroelasticity. The Darcy flow is discretized by a lowest order weak Galerkin (WG) finite element method for fluid pressure. The linear elasticity is discretized by enriched Lagrangian ($EQ_1$) elements for solid displacement. First order backward Euler time discretization is implemented to solve the coupled time-dependent system on quadrilateral meshes. This poroelasticity solver has some attractive features. There is no stabilization added to the system and it is free of Poisson locking and pressure oscillations. Poroelasticity locking is avoided through an appropriate coupling of finite element spaces for the displacement and pressure. In the equation governing the flow in pores, the dilation is calculated by taking the average over the element so that the dilation and the pressure are both approximated by constants. A rigorous error estimate is presented to show that our method has optimal convergence rates for the displacement and the fluid flow. Numerical experiments are presented to illustrate theoretical results. The implementation of this poroelasticity solver in deal.II couples the Darcy solver and the linear elasticity solver. We present the implementation of the Darcy solver and review the linear elasticity solver. Possible directions for future work are discussed.
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    An adaptive algorithm for an elliptic optimization problem, and stochastic-deterministic coupling: a mathematical framework
    (Colorado State University. Libraries, 2008) Lee, Sheldon, author; Estep, Donald, advisor; Tavener, Simon, advisor
    This dissertation consists of two parts. In the first part, we study optimization of a quantity of interest of a solution of an elliptic problem, with respect to parameters in the data using a gradient search algorithm. We use the generalized Green's function as an efficient way to compute the gradient. We analyze the effect of numerical error on a gradient search, and develop an efficient way to control these errors using a posteriori error analysis. Specifically, we devise an adaptive algorithm to refine and unrefine the finite element mesh at each step in the descent search algorithm. We give basic examples and apply this technique to a model of a healing wound. In the second part, we construct a mathematical framework for coupling atomistic models with continuum models. We first study the case of coupling two deterministic diffusive regions with a common interface. We construct a fixed point map by repeatedly solving the problems, while passing the flux in one direction and the concentration in the other direction. We examine criteria for the fixed point iteration to converge, and offer remedies such as reversing the direction of the coupling, or relaxation, for the case it does not. We then study the one dimensional case where the particles undergo a random walk on a lattice, next to a continuum region. As the atomistic region is random, this technique yields a fixed point iteration of distributions. We run numerical tests to study the long term behavior of such an iteration, and compare the results with the deterministic case. We also discuss a probability transition matrix approach, in which we assume that the boundary conditions at each iterations follow a Markov chain.
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    An analysis of domain decomposition methods using deal.II
    (Colorado State University. Libraries, 2021) Rigsby, Christina, author; Tavener, Simon, advisor; Bangerth, Wolfgang, committee member; Heyliger, Paul, committee member; Liu, Jiangguo, committee member
    Iterative solvers have attracted significant attention since the mid-20th century as the computational problems of interest have grown to a size beyond which direct methods are viable. Projection methods, and the two classical iterative schemes, Jacobi and Gauss-Seidel, provide a framework in which many other methods may be understood. Parallel methods or Jacobi-like methods are particularly attractive as Moore's Law and computer architectures transition towards multiple cores on a chip. We implement and explore two such methods, the multiplicative and restricted additive Schwarz algorithms for overlapping domain decomposition. We implement these in deal.II software, which is written in C++ and uses the finite element method. Finally, we point out areas for potential improvement in the implementation and present a possible extension of this work to an agent-based modeling prototype currently being developed by the Air Force Research Laboratory's Autonomy Capability Team (ACT3).
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    HIV-1 Gag trafficking and assembly: mathematical models and numerical simulations
    (Colorado State University. Libraries, 2013) Munoz-Alicea, Roberto, author; Liu, Jiangguo, advisor; Tavener, Simon, advisor; Chen, Chaoping, committee member; Mueller, Jennifer, committee member; Shipman, Patrick, committee member
    AIDS (acquired immune deficiency syndrome) is an infectious disease that takes away many people's lives each year. Group-specific antigen (Gag) polyprotein precursor is the major structural component of HIV, the causing agent of AIDS. Gag is essential and sufficient for the formation of new HIV virus-like particles. The late stages of the HIV-1 life cycle include the transport of Gag proteins towards the cell membrane, the oligomerization of Gag near the cell membrane during the budding process, and core assembly during virion maturation. The mechanisms for Gag protein trafficking and assembly are not yet fully understood. In order to gain further insight into the mechanisms of HIV-1 replication, we develop and analyze mathematical models and numerical algorithms for intracellular Gag protein trafficking, Gag trimerization near the cell membrane, and HIV-1 core assembly. Our preliminary results indicate that active transport plays an important role for Gag trafficking in the cytoplasm. This process can be mathematically modeled by convection-diffusion equations, which can be solved efficiently using characteristic finite element methods. We employ differential dynamical systems to model Gag trimerization and HIV-1 core assembly. For the Gag trimerization model, we estimate relationships between the association and dissociation parameters as well as the Gag arrival and multimerization parameters. We also find expressions for the equilibrium concentrations of the monomer and trimer species, and show that the equilibrium is asymptotically stable. For HIV-1 core assembly, we first consider a model developed by Zlonick and others, which regards assembly as a polymerization reaction. We utilize theoretical and numerical tools to confirm the stability of the equilibrium of CA intermediates. In addition, we propose a cascaded dynamical system model for HIV-1 core assembly. The model consists of two subsystems: one subsystem for nucleation and one for elongation. We perform simulations on the nucleation model, which suggests the existence of an equilibrium of the CA species.
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    Mathematical models for HIV-1 viral capsid structure and assembly
    (Colorado State University. Libraries, 2015) Sadre-Marandi, Farrah, author; Liu, Jiangguo, advisor; Tavener, Simon, advisor; Chen, Chaoping, committee member; Hulpke, Alexander, committee member; Zhou, Yongcheng, committee member
    HIV-1 (human immunodeficiency virus type 1) is a retrovirus that causes the acquired immunodeficiency syndrome (AIDS). This infectious disease has high mortality rates, encouraging HIV-1 to receive extensive research interest from scientists of multiple disciplines. Group-specific antigen (Gag) polyprotein precursor is the major structural component of HIV. This protein has 4 major domains, one of which is called the capsid (CA). These proteins join together to create the peculiar structure of HIV-1 virions. It is known that retrovirus capsid arrangements represent a fullerene-like structure. These caged polyhedral arrangements are built entirely from hexamers (6 joined proteins) and exactly 12 pentamers (5 proteins) by the Euler theorem. Different distributions of these 12 pentamers result in icosahedral, tubular, or the unique HIV-1 conical shaped capsids. In order to gain insight into the distinctive structure of the HIV capsid, we develop and analyze mathematical models to help understand the underlying biological mechanisms in the formation of viral capsids. The pentamer clusters introduce declination and hence curvature on the capsids. The HIV-1 capsid structure follows a (5,7)-cone pattern, with 5 pentamers in the narrow end and 7 in the broad end. We show that the curvature concentration at the narrow end is about five times higher than that at the broad end. This leads to a conclusion that the narrow end is the weakest part on the HIV-1 capsid and a conjecture that “the narrow end closes last during maturation but opens first during entry into a host cell.” Models for icosahedral capsids are established and well-received, but models for tubular and conical capsids need further investigation. We propose new models for the tubular and conical capsid based on an extension of the Caspar-Klug quasi-equivalence theory. In particular, two and three generating vectors are used to characterize respectively the lattice structures of tubular and conical capsids. Comparison with published HIV-1 data demonstrates a good agreement of our modeling results with experimental data. It is known that there are two stages in the viral capsid assembly: nucleation (formation of a nuclei: hexamers) and elongation (building the closed shell). We develop a kinetic model for modeling HIV-1 viral capsid nucleation using a 6-species dynamical system. Numerical simulations of capsid protein (CA) multimer concentrations closely match experimental data. Sensitivity and elasticity analysis of CA multimer concentrations with respect to the association and disassociation rates further reveals the importance of CA dimers in the nucleation stage of viral capsid self-assembly.
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    Numerical solutions of nonlinear systems derived from semilinear elliptic equations
    (Colorado State University. Libraries, 2007) Cruceanu, Stefan-Gicu, author; Allgower, Eugene, advisor; Tavener, Simon, advisor
    The existence and the number of solutions for N-dimensional nonlinear boundary value problems has been studied from a theoretical point of view, but there is no general result that states how many solutions such a problem has or even to determine the existence of a solution. Numerical approximation of all solutions (complex and real) of systems of polynomials can be performed using numerical continuation methods. In this thesis, we adapt numerical continuation methods to compute all solutions of finite difference discretizations of boundary value problems in 2-dimensions involving the Laplacian. Using a homotopy deformation, new solutions on finer meshes are obtained from solutions on coarser meshes. The issue that we have to deal with is that the number of the solutions of the complex polynomial systems grows with the number of mesh points of the discretization. Hence, the need of some filters becomes necessary in this process. We remark that in May 2005, E. Allgower, D. Bates, A. Sommese, and C. Wampler used in [1] a similar strategy for finding all the solutions of two-point boundary value problems in 1-dimension with polynomial nonlinearities on the right hand side. Using exclusion algorithms, we were able to handle general nonlinearities. When tracking solutions sets of complex polynomial systems an issue of bifurcation or near bifurcation of paths arises. One remedy for this is to use the gamma-trick introduced by Sommese and Wampler in [2]. In this thesis we show that bifurcations necessarily occur at turning points of paths and we use this fact to numerically handle the bifurcation, when mappings are analytic.
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    Sensitivity analysis of the basic reproduction number and other quantities for infectious disease models
    (Colorado State University. Libraries, 2012) Mikucki, Michael A., author; Tavener, Simon, advisor; Shipman, Patrick, committee member; Antolin, Michael, committee member
    Performing forward sensitivity analysis has been an integral component of mathematical modeling, yet its implementation becomes increasingly difficult with a model's complexity. For infectious disease models in particular, the sensitivity analysis of a parameter known as the basic reproduction number, or R0, has dominated the attention of ecology modelers. While the biological definition of R0 is well established, its mathematical construction is elusive. An index with a concrete mathematical definition that in many cases matches the biological interpretation of R0 is presented. A software package called Sensai that automatically computes this index and its sensitivity analysis is also presented. Other "quantities of interest" that provide similar information to R0 can also be implemented in Sensai and their sensitivities computed. Finally, some example models are presented and analyzed using Sensai.
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    Weak Galerkin finite element methods for the Darcy equation
    (Colorado State University. Libraries, 2018) Wang, Zhuoran, author; Liu, Jiangguo, advisor; Tavener, Simon, advisor; Donahue, Tammy, committee member
    The Darcy equation models pressure-driven flow in porous media. Because of the importance of ground water flow in oil recovery and waste mitigation, several types of numerical methods have been developed for solving the Darcy equation, such as continuous Galerkin finite element methods (CGFEMs) and mixed finite element methods (MFEMs). This thesis describes the lowest-order weak Galerkin (WG) finite element method to solve the Darcy equation and compares it to those well-known methods. In this method, we approximate the pressure by constants inside elements and on edges. Pressure values in interiors and on edges might be different. The discrete weak gradients specified in the local Raviart-Thomas spaces are used to approximate the classical gradients. The WG finite element method has nice features, e.g., locally mass conservation, continuous normal fluxes and easy implementation. Numerical experiments on quadrilateral and hybrid meshes are presented to demonstrate its good approximation and expected convergence rates. We discuss the extension of WG finite element methods to three-dimensional domains.