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Browsing by Author "Liu, Jiangguo, 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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    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 modeling of groundwater anomaly detection
    (Colorado State University. Libraries, 2016) Gu, Jianli, author; Liu, Jiangguo, advisor; Carlson, Kenneth H., committee member; Zhou, Yongcheng, committee member
    Public concerns about groundwater quality have increased in recent years due to the massive exploitation of shale gas through hydraulic fracturing which raises the risk of groundwater contamination. Groundwater monitoring can fill the gap between the public fears and the industrial production. However, the studies of groundwater anomaly detection are still insufficient. The complicated sequential data patterns generated from subsurface water environment bring many challenges that need comprehensive modeling techniques in mathematics, statistics and machine learning for effective solutions. In this reseach, Multivariate State Estimation Technique (MSET) and One-class Support Vector Machine (1-SVM) methods are utilized and improved for real-time groundwater anomaly detection. The effectiveness of the two methods are validated based upon different data patterns coming from the historic data of Colorado Water Watch (CWW) program. Meanwhile, to ensure the real-time responsiveness of these methods, a groundwater event with contaminant transport was simulated by means of finite difference methods (FDMs). The numerical results indicate the change of contaminant concentration of chloride with groundwater flow over time. By coupling the transport simulation and groundwater monitoring, the reliability of these methods for detecting groundwater contamination event is tested. This research resolves issues encountered when conducting real-time groundwater monitoring, and the implementation of these methods based on Python can be easily transfered and extended to engineering practices.
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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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    Modeling of atmospherically important vapor-to-particle reactions
    (Colorado State University. Libraries, 2014) Hashmi, Bahaudin, author; Shipman, Patrick, advisor; Liu, Jiangguo, advisor; Thompson, Stephen, committee member
    Liesegang ring formation is a special type of chemical pattern formation in which a spatial order is formed by density fluctuations of a weakly soluble salt. The Vapor-to-Particle nucleation process that is believed to produce these Liesegang rings is theorized to be the cause of mini-tornadoes and mini-hurricanes developed in a lab. In this thesis, we develop a one-dimensional finite element scheme for modeling laboratory experiments in which ammonia and hydrogen chloride vapor sources are presented to either end of the tubes. In these experiments, a reaction zone develops and propagates along the tube. Both numerical simulations and the laboratory experiments find an increasing amplitude of oscillations at the reaction front.
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    Pattern formation in reaction diffusion systems and ion bombardment of surfaces
    (Colorado State University. Libraries, 2017) Hashmi, Bahaudin A., author; Shipman, Patrick D., advisor; Liu, Jiangguo, advisor; Thompson, Stephen, committee member; Bates, Daniel J., committee member
    We have analyzed pattern formation in two different systems: (1) Vapor-to-particle reaction diffusion systems and (2) Highly ordered square arrays in ion bombardment. The vapor-to-particle reaction exhibits oscillatory behavior and produces a spatial pattern called Liesegang rings. In this thesis, we develop a finite element scheme to model the hydrogen chloride and ammonia vapor-to-particle reaction. In our simulations, we develop parametric regions for different patterns and corroborate data obtained from experiments of this reaction. For the ion bombardment of a planar surface, we add the Ehrlich-Schwoebel barrier terms to the Bradley-Shipman equations of motion and see the impact of ion bombardment at normal incidence on a binary crystalline material. A weakly nonlinear stability analysis was conducted and regions were determined where highly ordered square pyramids formed.
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    Weak Galerkin finite element methods for elasticity and coupled flow problems
    (Colorado State University. Libraries, 2020) Harper, Graham Bennett, author; Liu, Jiangguo, advisor; Bangerth, Wolfgang, committee member; Guzik, Stephen, committee member; Tavener, Simon, committee member; Zhou, Yongcheng, committee member
    We present novel stabilizer-free weak Galerkin finite element methods for linear elasticity and coupled Stokes-Darcy flow with a comprehensive treatment of theoretical results and the numerical methods for each. Weak Galerkin finite element methods take a discontinuous approximation space and bind degrees of freedom together through the discrete weak gradient, which involves solving a small symmetric positive-definite linear system on every element of the mesh. We introduce notation and analysis using a general framework that highlights properties that unify many existing weak Galerkin methods. This framework makes analysis for the methods much more straightforward. The method for linear elasticity on quadrilateral and hexahedral meshes uses piecewise constant vectors to approximate the displacement on each cell, and it uses the Raviart-Thomas space for the discrete weak gradient. We use the Schur complement to simplify the solution of the global linear system and increase computational efficiency further. We prove first-order convergence in the L2 norm, verify our analysis with numerical experiments, and compare to another weak Galerkin approach for this problem. The method for coupled Stokes-Darcy flow uses an extensible multinumerics approach on quadrilateral meshes. The Darcy flow discretization uses a weak Galerkin finite element method with piecewise constants approximating pressure and the Arbogast-Correa space for the weak gradient. The Stokes domain discretization uses the classical Bernardi-Raugel pair. We prove first-order convergence in the energy norm and verify our analysis with numerical experiments. All algorithms implemented in this dissertation are publicly available as part of James Liu's DarcyLite and Darcy+ packages and as part of the deal.II library.
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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.