Numerical Analysis of Multiphysics Problems

Institute for Computational and Experimental Research in Mathematics (ICERM)

February 12, 2024 - February 16, 2024
Monday, February 12, 2024
  • 8:30 - 8:50 am EST
    Check In
    11th Floor Collaborative Space
  • 8:50 - 9:00 am EST
    Welcome
    11th Floor Lecture Hall
    • Brendan Hassett, ICERM/Brown University
  • 9:00 - 9:45 am EST
    Numerical modeling of fluid-structure interaction
    11th Floor Lecture Hall
    • Speaker
    • Martina Bukač, University of Notre Dame
    • Session Chair
    • Sara Pollock, University of Florida
    Abstract
    Fluid-structure interaction problems arise in many applications. In biomedicine, such models are used to describe the interaction between blood and arterial walls. Other applications include geomechanics and aerodynamics. When a deformable structure is porous and allows flow through it, poroelastic models are commonly used to describe its behavior. The numerical simulation of fluid-elastic/poroelastic structure interaction problems has received considerable attention, but still remains a significant challenge in the mathematical and computational sciences. Main difficulties stem from the the intricate multiphysics nature of the problem, and strong nonlinearities. In this talk, we will present some recent advances in numerical modeling of fluid-structure interaction problems, including adaptive, partitioned methods where the domain movement is handled using an Arbitrary Lagragian-Eulerian approach, and a fixed mesh scheme based on the diffuse interface method. We will also present an application of solvers for fluid-structure interaction in the design of a bioartifical pancreas.
  • 10:00 - 10:30 am EST
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EST
    On the design and analysis of property-preserving finite element schemes for hyperbolic problems
    11th Floor Lecture Hall
    • Speaker
    • Dmitri Kuzmin, Technische Universität Dortmund
    • Session Chair
    • Sara Pollock, University of Florida
    Abstract
    This talk presents a family of algebraically constrained finite element schemes for hyperbolic conservation laws. The validity of (generalized) discrete maximum principles is enforced using monolithic convex limiting (MCL), a new flux correction procedure based on representation of spatial semi-discretizations in terms of admissible intermediate states. Semi-discrete entropy stability is enforced using a limiter-based fix. Time integration is performed using explicit or implicit Runge-Kutta methods, which can also be equipped with property-preserving flux limiters. In MCL schemes for nonlinear systems, problem-dependent inequality constraints are imposed on scalar functions of conserved variables to ensure physical and numerical admissibility of approximate solutions. After explaining the design philosophy behind our flux-corrected finite element approximations and showing some numerical examples, we turn to the analysis of consistency and convergence. In particular, we prove a Lax-Wendroff-type theorem for the inequality-constrained semi-discrete problem. A key component of our analysis is the use of a weak estimate on bounded variation, which follows from the semi-discrete entropy stability property of the method under investigation. For the Euler equations of gas dynamics, we prove weak convergence to a dissipative weak solution. The convergence analysis to be presented in this talk is joint work with Maria Lukáčová-Medvid’ová and Philipp Öffner.
  • 11:30 am - 12:15 pm EST
    Coupling mechanics with biochemistry to understand single and collective cell migration: A geometric bulk-surface partial differential equation approach
    11th Floor Lecture Hall
    • Speaker
    • Anotida Madzvamuse, The University of British Columbia
    • Session Chair
    • Sara Pollock, University of Florida
    Abstract
    In this talk I will present a geometric bulk-surface partial differential equations (GBS-PDEs) approach for coupling mechanics with biochemistry to understand mechanisms for single and collective cell migration. The GBS-PDEs are solved efficiently using tailor-made numerical methods depending on the properties of the mathematical models; two novel numerical methods will be presented: (i) the evolving bulk-surface finite element method suitable for solving GBS-PDEs on evolving domains and manifolds for sharp-interface formulations and (ii) the geometric multigrid method suitable for solving PDEs on evolving domains and manifolds using diffuse interface formulations. Experimentally driven inspired applications will be presented, demonstrating the novelty, applicability and generality of this mechanobiochemical modelling approach to studying single and collective cell migration. Cell migration is essential for many physiological and pathological processes. It plays a central role in the development and maintenance of multicellular organisms. Tissue formation during embryonic development, wound healing and immune responses as well as the formation of cancer, all require the orchestrated movement of cells in particular directions to specific locations. Hence, understanding single cell dynamics during movement is critically important in development, biomedicine and biomedical engineering.
  • 12:30 - 2:30 pm EST
    Lunch/Free Time
  • 2:30 - 3:15 pm EST
    Convergence of numerical methods for coupled Cahn-Hilliard and Navier-Stokes equations
    11th Floor Lecture Hall
    • Speaker
    • Beatrice Riviere, Rice University
    • Session Chair
    • Sara Pollock, University of Florida
    Abstract
    Modeling multicomponent flows in porous media is important for many applications relevant to energy and environment. Advances in pore-scale imaging, increasing availability of computational resources, and developments in numerical algorithms have started rendering direct pore-scale numerical simulations of multiphase flow in pore structures feasible. This talk presents recent advances in the discretization of phase-field models for systems of two-phase flows. Spatial discretization is based on the interior penalty discontinuous Galerkin methods. Time discretization utilizes a decoupled splitting approach. Both theory and application of the proposed methods to model flows in porous structures are discussed.
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 4:45 pm EST
    New optimized Robin-Robin domain decomposition methods using Krylov solvers for the Stokes-Darcy system
    11th Floor Lecture Hall
    • Speaker
    • Xiaoming He, Missouri University of Science and Technology
    • Session Chair
    • Sara Pollock, University of Florida
    Abstract
    In this presentation, we design optimized Schwarz domain decomposition algorithms to accelerate the Krylov type solution for the Stokes-Darcy system. We use particular solutions of this system on a circular geometry to analyze the iteration operator mode by mode. We introduce a new optimization strategy of the so-called Robin parameters based on a specific linear relation between these parameters, using the min-max and the expectation minimization approaches. Moreover, we use a Krylov solver to deal with the iteration operator and accelerate this new optimized domain decomposition algorithm. Several numerical experiments are provided to validate the effectiveness of this new method.
  • 5:00 - 6:30 pm EST
    Welcome Reception
    Reception - 11th Floor Collaborative Space
Tuesday, February 13, 2024
  • 9:00 - 9:45 am EST
    Immersed methods for fluid-structure interaction
    Virtual
    • Virtual Speaker
    • Boyce Griffith, University of North Carolina at Chapel Hill
    • Session Chair
    • Amnon Meir, Southern Methodist University
    Abstract
    The immersed boundary (IB) method is a framework for modeling systems in which an elastic structure interacts with a viscous incompressible fluid. The fundamental feature of the IB approach to such fluid-structure interaction (FSI) problems is its combination of an Eulerian formulation of the momentum equation and incompressibility constraint with a Lagrangian description of the structural deformations and resultant forces. In conventional IB methods, Eulerian and Lagrangian variables are linked through integral equations with Dirac delta function kernels, and these singular kernels are replaced by regularized delta functions when the equations are discretized for computer simulation. This talk will focus on three related extensions of the IB method. I first detail an IB approach to structural models that use the framework of large-deformation nonlinear elasticity. I will focus on efficient numerical methods that enable finite element structural models in large-scale simulations, with examples focusing on models of the heart and its valves. Next, I will describe an extension of the IB framework to simulate soft material failure using peridynamics, which is a nonlocal structural mechanics formulation. Numerical examples demonstrate constitutive correspondence with classical mechanics for non-failure cases along with essentially grid-independent predictions of fluid-driven soft material failure. Finally, I will introduce a reformulation of the IB large-deformation elasticity framework that enables accurate and efficient fluid-structure coupling through a version of the immersed interface method, which is a sharp-interface IB-type method. Computational examples demonstrate the ability of this methodology to simulate a broad range of fluid-structure mass density ratios without suffering from artificial added mass instabilities, and to facilitate subgrid contact models. I will also present biomedical applications of the methodology, including models of clot capture by inferior vena cava filters.
  • 10:00 - 10:30 am EST
    Coffee Break
    Virtual
  • 10:30 - 11:15 am EST
    AAPicard-Newton solver for Navier-Stokes and related problems
    Virtual
    • Virtual Speaker
    • Leo Rebholz, Clemson University
    • Session Chair
    • Amnon Meir, Southern Methodist University
    Abstract
    We consider the composition of AA-Picard fixed point iteration with Newton iteration, to create a more robust and stable yet still quadratically convergent solver. We analyze for Navier-Stokes in cases of small data (sufficient for uniqueness) and large data. Tests for NSE and other PDEs with this solver show a remarkable ability to converge for larger Reynolds number (NSE), Rayleigh number (Boussinesq), Kerr coefficient and refraction index (nonlinear Helmholtz), and so on.
  • 11:30 am - 12:15 pm EST
    On Reliability and Economics
    Virtual
    • Virtual Speaker
    • Jed Brown, University of Colorado Boulder
    • Session Chair
    • Amnon Meir, Southern Methodist University
    Abstract
    Data structures and algorithms have changed the relative costs, but few production pipelines have internalized the new economics of simulation. For example, matrix-free invert the marginal cost of high order discretizations, enabling large speed-ups for engineering workflows. Meanwhile, there is frequent dispute among practitioners of when to apply linearizations and assumptions on physical regime, when to use structured vs unstructured meshes, and many other important design choices in a simulation tool. For single-physics problems, it is perhaps tractable for analysts to determine (usually a posteriori) when these approximations are valid, but that is more difficult and error-prone for multi-physics problems. We reflect on the computational cost, robustness, and user interface consequences of "simplifying" this decision landscape by embracing fully nonlinear formulations with unstructured meshes and implicit solvers. Via case studies in fluid and structural mechanics, we observe that solvers for more general regimes can have low overhead relative to regime-specific solvers, yet come equipped with diagnostics that provide effective a posteriori indicators of physical regime.
  • 12:30 - 2:30 pm EST
    Lunch/Free Time
  • 2:30 - 3:15 pm EST
    Fluid-poroelastic structure interaction
    Virtual
    • Speaker
    • Suncica Canic, University of California, Berkeley
    • Session Chair
    • Amnon Meir, Southern Methodist University
    Abstract
    We will discuss some recent results on fluid-poroelastic structrure interaction between multilayered poroelastic media and viscous, incompressible fluids.
Wednesday, February 14, 2024
  • 9:00 - 9:45 am EST
    Multiscale/Multiphysics Modeling and Simulation in Ophthalmology
    11th Floor Lecture Hall
    • Virtual Speaker
    • Riccardo Sacco, Politecnico di Milano
    • Session Chair
    • Hyesuk Lee, Clemson University
    Abstract
    Glaucoma is a multifactorial ocular neuropathology representing the second major cause of irreversible blindness. Elevated intraocular pressure (IOP) is an established risk factor of glaucoma determined by aqueous humor dynamics (AHDyn), the balance among production (Pr), diffusion (Diff) and drainage (Dr) of aqueous humor (AH), a watery transparent fluid including electrolytes and low protein concentration. Reducing AH-Pr and/or increasing AH-Dr are possible approaches to reduce IOP. In this talk we illustrate a multiscale/multiphysics approach to develop a computational virtual laboratory (CVL) for the simulation of AHDyn based on a 3D-to-0D reduction of (1) 3D Velocity-Extended Poisson-Nernst-Planck PDE system to model AH-Pr; (2) 3D diffusion bulk flow to model AH-Diff; (3) 3D poroelastic PDE system to model AH-Dr. Model variables are the compartment values of electric potential, ion molar densities and fluid pressure and are numerically determined by a fixed-point iteration which transforms AHDyn simulation into the successive solution of two nonlinear systems of algebraic equations, representing mass balance in AH-Pr and in AH-Diff + AH-Dr, respectively. Computational tests suggest that Na+/K+ pump and TM/Uv hydraulic facilities are the main biomarkers of a pathological increase in IOP. These results support the potential use of a CVL to assist and optimize the design of IOP lowering medications.
  • 10:00 - 10:30 am EST
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EST
    Multiphysics at multiple scales for coupled [TpHM] processes in permafrost soils
    11th Floor Lecture Hall
    • Speaker
    • Malgorzata Peszynska, Oregon State University
    • Session Chair
    • Hyesuk Lee, Clemson University
    Abstract
    In this joint work with many students and collaborators we describe robust and conservative computational schemes for coupled process of flow, deformation, and energy with phase change (TpHM) in the soils in permafrost regions. The models present challenges due to the free boundary of freezing/thawing, strong dependence of constitutive parameters on the micro-physics of TpHM, disparate time scales, and micro- and macro heterogeneity. We also discuss how to get the data for the Darcy scale [TpH] ad [HM] models from the models at the pore-scale by computational upscaling.
  • 11:30 am - 12:15 pm EST
    Multiphysics problems related to brain clearance, sleep and dementia
    11th Floor Lecture Hall
    • Speaker
    • Kent-Andre Mardal, University of Oslo
    • Session Chair
    • Hyesuk Lee, Clemson University
    Abstract
    Recent theories suggest that a fundamental reason for sleep is simply clearance of metabolic waste produced during the activities of the day. In this talk we will present multi-physics problems and numerical schemes that target these applications. In particular, we will be lead from basic applications of neuroscience into multi-physics problems involving Stokes, Biot and fractional solvers at the brain-fluid interface.
  • 12:25 - 12:30 pm EST
    Group Photo (Immediately After Talk)
    11th Floor Lecture Hall
  • 12:30 - 2:30 pm EST
    Poster Session Lunch
    Poster Session - 10th Floor Collaborative Space
  • 2:30 - 3:15 pm EST
    Direct van der Waals Simulation (DVS): Towards Predictive Simulations of Cavitation and Boiling
    11th Floor Lecture Hall
    • Speaker
    • Hector Gomez, Purdue University
    • Session Chair
    • Hyesuk Lee, Clemson University
    Abstract
    Cavitating flows are ubiquitous in engineering and science. Despite their significance, a number of fundamental problems remain open; and our ability to make quantitative predictions is very limited. The Navier-Stokes-Korteweg equations constitute a fundamental model of cavitation, which has potential for predictive computations of liquid-vapor flows, including cavitation inception —one of the most elusive aspects of cavitation. However, numerical simulation of the Navier-Stokes-Korteweg equations is very challenging, and state of the art simulations are limited to very small Reynolds numbers, open flows (no walls), and in most cases, micrometer length scales. The computational challenges emerge from, at least, (a) the dispersive nature of the solutions to the equations, (b) a complicated eigenstructure of the isentropic form of the equations, which limits the use of standard CFD techniques, and (c) the need to resolve the liquid-vapor interface, which without special treatment, has a thickness in the order of nanometers. Here, we present Direct van der Waals simulation (DVS), a new approach that permits, for the first time as far as we are aware, large-scale simulations of wall-bounded flows with large Reynolds numbers. The proposed discretization scheme is a residual-based approach that emanates from the dispersive nature of the equations and outperforms standard stabilization schemes for advection-dominated problems. We feel that this work opens possibilities for predictive simulations of cavitation.
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 4:45 pm EST
    A data-driven exterior calculus
    11th Floor Lecture Hall
    • Speaker
    • Nathaniel Trask, Sandia National Laboratory
    • Session Chair
    • Hyesuk Lee, Clemson University
    Abstract
    Despite the recent flurry of work employing machine learning to develop surrogate models to accelerate scientific computation, the "black-box" underpinnings of current techniques fail to provide the verification and validation guarantees provided by modern finite element methods. In this talk we present a data-driven finite element exterior calculus for developing reduced-order models of multiphysics systems when the governing equations are either unknown or require closure. The framework employs deep learning architectures typically used for logistic classification to construct a trainable partition of unity which provides notions of control volumes with associated boundary operators. This alternative to a traditional finite element mesh is fully differentiable and allows construction of a discrete de Rham complex with a corresponding Hodge theory. We demonstrate how models may be obtained with the same robustness guarantees as traditional mixed finite element discretization, with deep connections to contemporary techniques in graph neural networks. For applications developing digital twins where surrogates are intended to support real time data assimilation and optimal control, we further develop the framework to support Bayesian optimization of unknown physics on the underlying adjacency matrices of the chain complex. By framing the learning of fluxes via an optimal recovery problem with a computationally tractable posterior distribution, we are able to develop models with intrinsic representations of epistemic uncertainty.
Thursday, February 15, 2024
  • 9:00 - 9:45 am EST
    The Role of Multiphysics Modeling in the Design of Coronary Stents
    11th Floor Lecture Hall
    • Speaker
    • Alessandro Veneziani, Emory University
    • Session Chair
    • Noel Walkington, Carnegie Mellon University
    Abstract
    Since their introduction in the Eighties, coronary stents have undergone significant design improvements, making them a critical tool for treating severe obstructions. From original Bare-Metal Stents (BMS) to Drug Eluting Stents (DES) to the most recent experience of Bioresorbable Stents, the design of these scaffolds was minimally supported by mathematical tools. The patient-specific quantitative analysis of stented coronaries is a difficult task for the variety of complex morphologies left by the stent deployment. Therefore, this type of analysis was limited to a minimal number of patients, not compatible with clinical trials. On the other hand, the development and the failure of Brioresorbable Stents clearly pointed out the importance of rigorous quantitative tools in the design of next-generation scaffolds. In this talk, we will present recent results in investigating coronary stents based on Applied Mathematics (as opposed to traditional animal models). We will consider in detail (i) the modeling of the elution in a multidomain problem solved by iterative substructuring methods involving simultaneously the lumen, the wall, and the struts of the stents; (2) the impact of the struts on the wall shear stress of a significant number of patients; (3) the consequent role of shape optimization and model order reduction in the design of scaffolds. This journey through a sophisticated combination of data and models will pinpoint the critical role of applied mathematics and scientific computing not only for a basic understanding of the biomechanics of stents but also for the clinical routine and the design of more performing prostheses.
  • 10:00 - 10:30 am EST
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EST
    Domain Agnostic Neural Operators for Multiphysics Problems
    11th Floor Lecture Hall
    • Speaker
    • Yue Yu, Lehigh University
    • Session Chair
    • Noel Walkington, Carnegie Mellon University
    Abstract
    Over the past several decades, physics-based Partial Differential Equations (PDEs) have been the cornerstone for modeling multiphysics problems. Traditional numerical methods have been employed to solve these PDEs and various approaches have been proposed to capture the multiphysics interfaces. However, their accuracy and computational feasibility can be compromised when dealing with unknown governing laws or complex interface geometries, such as in the crack propagation, fluid—structure interaction, and heterogeneous material design problems. In this talk, we develop to use data-driven modeling approaches to learn the hidden physics, capture irregular geometries, and provide accelerated predictions. In particular, we introduce domain agnostic Fourier neural operator (DAFNO), which learns the surrogate mapping between loading conditions and the corresponding physical responses with irregular geometries and evolving domains. The key idea is to incorporate a smoothed characteristic function in the integral layer architecture of neural operators, and leverage FFT to achieve rapid computations for evaluating these integrals, in such a way that the geometric information is explicitly encoded in the architecture. Once trained, DAFNO can provide efficient predictions for physical problems under unseen loading scenarios and evolving domain geometries, which makes it especially suitable to handle the complex interfacial problems in multiphysics modeling. To illustrate the applicability of DAFNO in multiphysics problems, we show three examples. Firstly, we consider a brittle material crack propagation problem which features complex domains with topology changes. Then, in the second example we further consider the corrosion induced cracking in reinforced concrete, which is a multiphysics system involving the interactions between diffusion, chemical reaction, mechanical strain, and crack fields. Last but not least, we show that DAFNO can act as an efficient surrogate for the inverse microstructure design of multifunctional metamaterials. These examples highlight the features of DAFNOs in its generalizability, flexibility, and efficiency.
  • 11:30 am - 12:15 pm EST
    The Shifted Boundary Method: An embedded approach for computational mechanics
    11th Floor Lecture Hall
    • Speaker
    • Guglielmo Scovazzi, Duke University
    • Session Chair
    • Noel Walkington, Carnegie Mellon University
    Abstract
    Embedded/immersed/unfitted boundary methods obviate the need for continual re-meshing in many applications involving rapid prototyping and design. Unfortunately, many finite element embedded boundary methods are also difficult to implement due to the need to perform complex cell cutting operations at boundaries, and the consequences that these operations may have on the overall conditioning of the ensuing algebraic problems. We present a new, stable, and simple embedded boundary method, named Shifted Boundary Method (SBM), which eliminates the need to perform cell cutting. Boundary conditions are imposed on a surrogate discrete boundary, lying on the interior of the true boundary interface. We then construct appropriate field extension operators by way of Taylor expansions, with the purpose of preserving accuracy when imposing the boundary conditions. We demonstrate the SBM on large-scale solid and fracture mechanics problems; thermoelasticity problems; porous media flow problems; incompressible flow problems governed by the Navier-Stokes equations (also including free surfaces); and problems governed by hyperbolic conservation laws.
  • 12:30 - 2:30 pm EST
    Lunch: Classical theory vs machine learning in education
    Working Lunch - 11th Floor Collaborative Space
  • 2:30 - 3:15 pm EST
    A FEM for a phase-field model of two-phase incompressible surface flow with electrostatic interaction
    11th Floor Lecture Hall
    • Speaker
    • Annalisa Quaini, University of Houston
    • Session Chair
    • Noel Walkington, Carnegie Mellon University
    Abstract
    We consider a thermodynamically consistent phase-field model of a two-phase flow of incompressible viscous fluids with electrostatic interaction. The model allows for a nonlinear dependence of the fluid density on the phase-field order parameter. Driven by applications in biomembrane studies, the model is written for tangential flows of fluids constrained to a surface and consists of (surface) Navier–Stokes–Cahn–Hilliard type equations. We apply an unfitted finite element method to discretize the system and introduce a fully discrete time-stepping scheme with the following properties: (i) the scheme decouples the fluid and phase-field equation solvers at each time step, (ii) the resulting two algebraic systems are linear, and (iii) the numerical solution satisfies the same stability bound as the solution of the original system under some restrictions on the discretization parameters. We provide numerical examples to demonstrate the stability, accuracy, and overall efficiency of the approach and provide validation against experimental data.
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 4:45 pm EST
    Positivity-preserving discretisations in general meshes
    11th Floor Lecture Hall
    • Speaker
    • Gabriel Barrenechea, University of Strathclyde
    • Session Chair
    • Noel Walkington, Carnegie Mellon University
    Abstract
    In this talk I will present a method that enforces bound-preservation (at the degrees of freedom) of the discrete solution (recently presented in [1]). The method is built by first defining an algebraic projection onto the convex closed set of finite element functions that satisfy the bounds given by the solution of the PDE. Then, this projection is hardwired into the definition of the method by writing a discrete problem posed for this projected part of the solution. Since this process is done independently of the shape of the basis functions, and no result on the resulting finite element matrix is used, this process guarantees bound-preservation independently of the underlying mesh. The core of the talk will be devoted to explaining the main idea in the context of linear (and nonlinear) reaction-diffusion equations. Then, I will explain the main difficulties encountered when extending this method to convection-diffusion equations, and to a finite element method defined in polytopal meshes. The results in this talk have been carried out in collaboration with Abdolreza Amiri (Strathclyde, UK), Emmanuil Geourgoulis (Heriot-Watt, UK and Athens, Greece), Tristan Pryer (Bath, UK), and Andreas Veeser (Milan, Italy). References 1. G.R. Barrenechea, E. Georgoulis, T. Pryer, and A. Veeser, A nodally bound-preserving finite element method. arXiv:2304.01067, IMA Journal on Numerical Analysis, to appear.
Friday, February 16, 2024
  • 9:00 - 9:45 am EST
    Quantum Digital Twins - a numerical methodist’s adventure in the land of quantum computers
    11th Floor Lecture Hall
    • Speaker
    • Daniel Appelö, Virginia Tech
    • Session Chair
    • Valeria Simoncini, Università di Bologna
    Abstract
    In this talk I will introducing the most basic concepts in quantum computing and describe one type of quantum computing hardware (a transmon) and how it is modeled. We will then outline the computationally challenging tasks that are needed for making a quantum computer run and introduce numerical methods tailored especially for these tasks. Time permitting I will take you on a comprehensive journey through a real-world example involving characterization, control, and experimental validation, showcasing our experiences with a qutrit device within the Lawrence Livermore QUDIT testbed.
  • 10:00 - 10:30 am EST
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EST
    Monolithic and Partitioned FEM for FSI: ALE divergence-free HDG fluid solver + TDNNS structural solver
    11th Floor Lecture Hall
    • Speaker
    • Guosheng Fu, University of Notre Dame
    • Session Chair
    • Valeria Simoncini, Università di Bologna
    Abstract
    We present novel (high-order) finite element schemes for the fluid-structure interaction (FSI) problem based on an arbitrary Lagrangian-Eulerian divergence-free hybridizable discontinuous Gakerkin (ALE divergence-free HDG) incompressible flow solver, a Tangential-DisplacementNormal-Normal-Stress (TDNNS) nonlinear elasticity solver, and a generalized Robin interface condition treatment. Temporal discretization is performed using the high-order backward difference formulas (BDFs). Both monolithic and strongly coupled partitioned fully discrete schemes are obtained. Numerical convergence studies are performed for the flow and elasticity solvers, and the coupled FSI solver, which verify the high-order space-time convergence of the proposed schemes. Numerical results on classical two dimensional benchmark problems also showed good performance of our proposed methods.
  • 11:30 am - 12:15 pm EST
    Mixed methods for the coupled Stokes/Poisson-Nernst-Planck equations in Banach spaces
    11th Floor Lecture Hall
    • Speaker
    • Ricardo Ruiz-Baier, Monash University
    • Session Chair
    • Valeria Simoncini, Università di Bologna
    Abstract
    I will discuss a Banach spaces-based framework and new mixed finite element methods for the numerical solution of the coupled Stokes and Poisson-Nerns-Planck equations (a nonlinear model describing the dynamics of electrically charged incompressible fluids). The pseudostress tensor, the electric field (rescaled gradient of the potential) and total ionic fluxes are used as new mixed unknowns. The resulting fully mixed variational formulation consists of two saddle-point problems, each one with nonlinear source terms depending on the remaining unknowns, and a perturbed saddle-point problem with linear source terms, which is in turn additionally perturbed by a bilinear form. The well-posedness of the continuous formulation is a consequence of a fixed-point strategy in combination with the Banach theorem, the Babuska-Brezzi theory, the solvability of abstract perturbed saddle-point problems, and the Banach-Necas-Babuska theorem. An analogous approach (but using now both the Brouwer and Banach theorems and stability conditions on arbitrary FE subspaces) is employed at the discrete level. A priori error estimates are derived, and examples of discrete spaces that fit the theory, include, e.g., Raviart--Thomas elements along with piecewise polynomials. Finally, several numerical experiments confirm the theoretical error bounds and illustrate the balance-preserving properties and applicability of the proposed family of methods. This talk is based on joint work with Claudio I. Correa and Gabriel N. Gatica (from CI2MA, Concepcion).
  • 12:30 - 2:30 pm EST
    Lunch: Networking
    Working Lunch - 11th Floor Collaborative Space
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space

All event times are listed in ICERM local time in Providence, RI (Eastern Standard Time / UTC-5).

All event times are listed in .