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Organizing Committee
University of Notre Dame
University of Colorado Boulder
Clemson University
Southern Methodist University
University of Houston
University of Florida
Università di Bologna
Abstract
It is practically rare that a natural phenomenon or engineering problem can be accurately described by a single law of physics. The striking diversity of rules of life forces scientists to continuously increase the complexity of models to address the ever-growing requirements for their prediction capabilities. It remains a formidable challenge to derive and analyze numerical methods which are universal enough to handle complex multiphysics problems with the same ease and efficiency as traditional methods do for textbook PDEs.
The workshop will focus on recent trends in the field of numerical methods for multiphysics problems that include the development of monolithic approaches, structure preserving discretizations, geometrically unfitted methods, data-driven techniques, and modern algebraic methods for the resulting linear and nonlinear discrete systems. The topics of interest include models and discretizations for fluid - elastic structure interaction, non-Newtonian fluids, phase field models for fluid mixtures, bulk-surface coupled problems, and biological flows. The workshop will also address emerging topics in scientific computing such as randomized algorithms, tensor methods, and structured numerical linear algebra methods, with the goal to better understand advances they offer for multiphysics problems.
The workshop will focus on recent trends in the field of numerical methods for multiphysics problems that include the development of monolithic approaches, structure preserving discretizations, geometrically unfitted methods, data-driven techniques, and modern algebraic methods for the resulting linear and nonlinear discrete systems. The topics of interest include models and discretizations for fluid - elastic structure interaction, non-Newtonian fluids, phase field models for fluid mixtures, bulk-surface coupled problems, and biological flows. The workshop will also address emerging topics in scientific computing such as randomized algorithms, tensor methods, and structured numerical linear algebra methods, with the goal to better understand advances they offer for multiphysics problems.
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Workshop Schedule
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.
