Iowa State University

MiniSymposium-02: Recent Advances in Numerical Methods for Multiphysics Problems

Abstract: Numerical simulation of multiphysics problems is challenging as it involves the coupling of different physical processes that are governed by different PDEs. There are mainly two approaches for the solution of such problems, either a monolithic strategy or a decoupled strategy (the latter is also known as a partitioned procedure). The goal of this minisymposium is to bring together mathematicians and scientists to discuss new developments and novel algorithms for multiphysics problems and their applications. The topics of interest include, but are not limited to, the coupling Darcy/dual-porosity flow and free flow, fluid-structure interaction, and Stokes-poroelastic systems.


Saturday, October 19th, 2019 at 10:20 - 11:40 (204 Carver Hall)
10:20 - 10:40 Xiaoming He,
Missouri University of Science and Technology
An Iterative Immersed Finite Element Method for an Electric Potential Interface Problem based on given Surface Electric Quantity

In plasma simulation, we often only know the total electric quantity on the surface of the object, not the charge density distribution on the surface which appears as the non-homogeneous flux jump condition in the usual interface problems. We propose an iterative method that employs the immersed finite element (IFE) method to solve the 2D interface problem for the potential field according to the given total electric quantity on the surface of the object.

10:40 - 11:00 Aycil Cesmelioglu,
Oakland University
A Hybridized Discontinuous Galerkin Approximation for Stokes-Darcy Flow Coupled to Transport

We propose a hybridized discontinuous Galerkin (HDG) method for the coupling of Stokes-Darcy problem to transport equation. The flow problem is solved monolithically by a mass-conserving HDG method that results in a divergence-conforming velocity approximation. This velocity is then used as the advective velocity in a transport problem. We analyze the semi-discrete HDG scheme for the transport problem and provide optimal a priori error estimates. We also discuss the compatibility of flow and transport discretizations and, illustrate our findings via numerical results.

11:00 - 11:20 A.J. Meir,
Southern Methodist University
On the Equations of Poroelasticity - Recent Advances

Poroelasticity is a complex coupled multiphysics phenomenon. The equations of poroelasticity (a coupled system of pde) model and predict the mechanical behavior of fluid-infiltrated porous media. Their significance comes from the fact that many natural substances, e.g., rocks, soils, clays, shales, biological tissues, and bones, as well as man-made materials, e.g., foams, gels, concrete, water-solute drug carriers, and ceramics, are all elastic porous materials (hence poroelastic). I will give an overview of the equations of poroelasticity and their mathematical analysis, suggest the use of finite element based numerical methods for efficiently and accurately approximating solutions of model problems in poroelasticity, and discuss some observations, recent research directions, and possible extensions.

11:20 - 11:40 Thi-Thao-Phuong Hoang,
Auburn University
Space-Time Domain Decomposition Methods for the Stokes-Darcy Coupling

We study decoupling iterative algorithms based on domain decomposition for the time-dependent nonlinear Stokes-Darcy model, in which different time steps can be used in the flow region and in the porous medium. The coupled system is formulated as a space-time interface problem based on either physical interface conditions or equivalent Robin-Robin interface conditions. The nonlinear interface problem is then solved by a nested iteration approach which involves, at each Newton iteration, the solution of a linearized interface problem and, at each Krylov iteration, parallel solution of time-dependent linearized Stokes and Darcy problems. Consequently, local discretizations in both space and time can be used to efficiently handle multiphysics systems with discontinuous parameters. Numerical results with nonconforming time grids are presented to illustrate the performance of the proposed methods.

Saturday, October 19th, 2019 at 14:40 - 16:00 (204 Carver Hall)
14:40 - 15:00 Martina Bukac,
University of Notre Dame
Partitioned Numerical Methods for Fluid - Hyperelastic/Hyperporoelastic Structure Interaction Problems

Fluid-structure interaction (FSI) problems arise in many applications, such as geomechanics, aerodynamics, and blood flow dynamics (hemodynamics). In hemodynamic applications, mathematical models must capture the non-linear coupling between blood and the elastic structural dynamics of vessel walls, soft tissue, or cardiac muscles. These structural dynamics create “moving domain” FSI problems that are challenging to numerically solve and analyze. We propose partitioned numerical methods for the interaction between a fluid and a hyperelastic structure and for the interaction between a fluid and a hyperporoelastic structure. Both methods are developed and analyzed on fully non-linear, moving domain problems. In the first method, the fluid and solid are discretized using the Backward Euler scheme, and the coupling conditions are imposed by introducing novel, generalized Robin boundary conditions. The solid problem is post-processed using time filtering, increasing the accuracy of the approximation. We show that the method is unconditionally stable. The interaction between a fluid and hyperporoelastic structure features different coupling conditions, which are exploited in the design of a partitioned method based on BDF2 time discretization. Performance of both methods is demonstrated by numerical examples.

15:00 - 15:20 Sibusiso Mabuza,
Clemson University
Iterative Limiters for Continuous Galerkin Discretization of Hyperbolic Conservation Laws

In this talk, gradient based linearity preserving nodal limiters for continuous Galerkin discretization for hyperbolic systems are proposed. The design and application of these limiters follows the algebraic flux correction paradigm. First, the introduction of low-order artificial dissipation and mass lumping is performed on the semi discrete scheme of the Euler equations to get a first order scheme. Then the anti-diffusion which is the difference between the standard Galerkin scheme and the first order scheme is constrained using an element based limiter in a conservative fashion to get a high resolution scheme. Numerical simulations are performed to illustrate the performance of the scheme using explicit Runge-Kutta for time discretization for transient problems and backward Euler pseudo time-stepping for steady problems.

15:20 - 15:40 Yanzhi Zhang,
Missouri University of Science and Technology
Numerical Study of Bose-Einstein Superfluids

In this talk, I will present our recent study on two-component Bose-Einstein condensation. An efficient and accurate split-step Fourier pseudospectral method is proposed to solve the coupled Gross-Pitaevskii equations - the governing equations for the dynamics of Bose-Einstein condensates. Our method has spectral accuracy in all spatial dimensions, and moreover it can be easily implemented in practice. To examine its performance, we compare our method with those reported in the literature. The dynamics of vortex lattices and giant vortices in rotating two-component Bose-Einstein condensations are studied.

15:40 - 16:00 Hyesuk Lee,
Clemson University
Numerical Methods for Non-Newtonian Fluid Structure Interaction

Simulating viscoelastic fluid-structure interactions is challenging not only due to the coupling between solid and fluid substructures, but also the complexity of fluid model in a moving domain. In this talk, both monolithic and decoupling approaches are considered for the numerical study of fluid-structure interaction problems, where the fluid is governed by a viscoelastic model. Monolithic and partitioned algorithms are presented, where the viscoelastic fluid is stabilized by the streamline upwind Petrov-Galerkin (SUPG) approximation. We will discuss some issues with the stress boundary condition on the interface and present simulation results with and without interface stress boundary conditions.

Coordinated by
Iowa State University