[00319] Robust formulations for coupled multiphysics problems – Theory and applications
Session Date & Time :
00319 (1/5) : 1C (Aug.21, 13:20-15:00)
00319 (2/5) : 1D (Aug.21, 15:30-17:10)
00319 (3/5) : 1E (Aug.21, 17:40-19:20)
00319 (4/5) : 2C (Aug.22, 13:20-15:00)
00319 (5/5) : 2D (Aug.22, 15:30-17:10)
Type : Proposal of Minisymposium
Abstract : The proposed minisymposium aims to bring together experts in the construction and analysis of novel discretization techniques for multiphysics models that maintain robustness with respect to model constants of interest. Particular emphasis will be placed on rigorous analysis of stability using saddle-point and perturbed saddle-point theory, a priori and a posteriori error estimation, as well on the design of robust solvers based on tailored domain decomposition techniques or on operator preconditioning. The session will also focus on the application of these new methodologies in the solution of coupled models arising in mechanobiology and similar multiphysics systems. For instance, our minisymposium features submissions involving brain tissue dynamics, cardiac electromechanics, and respiratory system modeling; but it also welcomes talks related to geophysical flows or other types of fluid-structure interaction multiphysics problems.
Organizer(s) : Wietse Boon, Martin Hornkjøl, Miroslav Kuchta, Ricardo Ruiz Baier
[01558] A diffuse interface method for fluid-poroelastic structure interaction
Author(s) :
Martina Bukač (University of Notre Dame)
Boris Muha (University of Zagreb)
Sunčica Čanić (University of California, Berkeley)
Abstract : The interaction between a free flowing fluid and a poroelastic structure, commonly formulated as a Navier-Stokes/Biot coupled system, has been used to describe problems arising in many applications, including environmental sciences, hydrology, geomechanics and biomedical engineering. Many existing numerical for this problem are based on a sharp interface approach, in the sense that the interface between the two regions is parametrized using an exact specification of its geometry and location, and the nodes in the computational mesh align with the interface. However, the exact location is sometimes not known, or the geometry is complicated, making a proper approximation of the integrals error-prone and difficult to automate. Hence, in this talk, we present a diffuse interface method for the coupled fluid-poroelastic structure interaction. The method uses a phase-field function which transitions from 1 in one region to 0 in the other region. We will first present the analysis of convergence of the discrete diffuse interface solution to the continuous sharp interface solution for the Stokes-Darcy problem. Then, we will discuss the extensions to Navier-Stokes/Biot system, and apply the method to study the optimal design of a bioartificial pancreas.
[01781] Parameter-robust methods for the Biot-Stokes interfacial coupling
Author(s) :
Martin Hornkjøl (University of Oslo)
Wietse Boon (KTH Royal Institute of Technology)
Miroslav Kuchta (Simula Research Laboratory)
Kent-Andre Mardal (University of Oslo)
Ricardo Ruiz Baier (Monash University)
Abstract : In this talk I will discuss a fluid-structure interaction model of the monolithic coupling between the free flow of
a viscous Newtonian fluid and a deformable porous medium separated by an interface. I will present a five-field
mixed-primal finite element scheme, with a preconditioner, for solving the Stokes velocity-pressure and Biot displacement-total pressure-fluid pressure. With adequate inf-sup conditions the stability of the formulation is established robustly in all material parameters.
[01845] Conservative and robust methods for the Biot--Brinkman equations in vorticity form
Author(s) :
Ricardo Ruiz Baier (Monash University)
Martin Hornkjøl (University of Oslo)
Alberto Martin (Australian National University)
Santiago Badia (Monash University)
Kent-Andre Mardal (University of Oslo)
Arbaz Khan (IIT Roorkee)
Abstract : In this paper, we propose a new formulation and a suitable finite element method for the steady coupling of viscous flow in deformable porous media using divergence-conforming filtration fluxes. The proposed method is based on the use of parameter-weighted spaces, which allows for a more accurate and robust analysis of the continuous and discrete problems. Furthermore, we conduct a solvability analysis of the proposed method and derive optimal error estimates in appropriate norms. These error estimates are shown to be robust in the case of large Lam'e parameters and small permeability and storativity coefficients. To illustrate the effectiveness of the proposed method, we provide a few representative numerical examples, including convergence verification, poroelastic channel flow simulation, and test the robustness of block-diagonal preconditioners with respect to model parameters.
[02817] Robust parallel solvers in cardiac modeling
Author(s) :
Nicolás Alejandro Barnafi Wittwer (University of Chile)
Abstract : We will see the physics involved in cardiac modeling, and focus on their efficient solution in an HPC infrastructure measured in both CPU time and scalability. As we will see, one of the main tools for improving performance is quasi-Newton methods, which do not sacrifice on scalability when initialized adequately. In some contexts, it will be possible to circumvent the solution of a linear system, which drastically reduces the complexity of the proposed solvers.
[02934] Stochastic Galerkin mixed finite element approximation for poroelasticity with uncertain inputs
Author(s) :
Arbaz Khan (IIT Roorkee)
Catherine E Powell (University of Manchester, UK)
Abstract : Linear poroelasticity models have important applications in biology and geophysics. In particular, the well-known Biot consolidation model describes the coupled interaction between the linear response of a porous elastic medium {saturated with fluid} and a diffusive fluid flow within it, assuming small deformations. This is the starting point for modeling human organs in computational medicine and for modeling the mechanics of permeable rock in geophysics. Finite element methods for Biot's consolidation model have been widely studied over the past four decades. In the first part of talk, we discuss a novel locking-free stochastic Galerkin mixed finite element method for the Biot consolidation model with uncertain Young's modulus and hydraulic conductivity field. After introducing a five-field mixed variational formulation of the standard Biot consolidation model, we discuss stochastic Galerkin mixed finite element approximation, focusing on the issue of well-posedness and efficient linear algebra for the discretized system. We introduce a new preconditioner for use with MINRES and establish eigenvalue bounds. Finally, we present specific numerical examples to illustrate the efficiency of our numerical solution approach. In the second part of the talk, we discuss a posteriori error estimators for SGFEM of Biot's consolidation model with uncertain inputs.
[03195] Finite element analysis for semilinear problems in liquid crystals
Author(s) :
Neela Nataraj (Professor )
Ruma Maity (Postdoc Fellow)
Apala Majumdar (Strathclyde University)
Abstract : A unified framework for the error control of different lowest-order finite element methods for approximating the regular solutions of systems of partial differential equations is established under a set of hypotheses.
The systems involve cubic nonlinearities in lower order terms, non-homogeneous Dirichlet boundary conditions,
and the results are established under minimal regularity assumptions on the exact solution. The results for existence
and local uniqueness of the discrete solutions using Newton-Kantorovich theorem and error control are presented.
The results are applied to conforming, Nitsche, discontinuous Galerkin, and weakly over penalized symmetric interior penalty schemes for variational models of ferronematics liquid crystals.
[03220] Finite element analysis for the Navier-Lamé eigenvalue problem
Author(s) :
Jesus Vellojin (Universidad del Bío-Bío)
Felipe Lepe (Universidad del Bío Bío)
Gonzalo Rivera (Universidad de Los Lagos)
Abstract : In this talk, the author presents the eigenvalue problem for the Navier-Lamé system. The analysis of the spectral problem is based in the compact operators theory. A finite element method based in polynomials of degree $k\geq 1$ is considered. An a posteriori error analysis is performed, where the reliability and efficiency of the proposed estimator is proved. A series of numerical tests are reported in order to assess the performance of the proposed numerical method.
[03598] Virtual element methods for Biot--Kirchhoff poroelasticity
Author(s) :
Rekha Mallappa Khot (Monash University)
David Mora (Universidad del B\'io-B\'io)
Ricardo Ruiz Baier (Monash University)
Abstract : We propose and analyse conforming and nonconforming virtual element formulations for the coupling of solid and fluid phases in deformable porous plates. The governing equations consist of one fourth-order equation for the transverse displacement of the middle surface coupled with a second-order equation for the pressure head relative to the solid. The discretisation supports arbitrary polynomial degrees on general polygonal meshes and we design companion operators with orthogonal properties and best-approximation estimate. We derive both a priori and a posteriori error estimates in appropriate norms, and these error bounds are robust with respect to the main model parameters. A few computational examples illustrate the properties of the numerical methods.
[03951] Unfitted finite element methods for PDEs with dynamic interfaces and boundaries
Author(s) :
Santiago Badia (Monash University)
Hridya Dilip (Monash University)
Francesc Verdugo (Vrije Universiteit Amsterdam)
Pere Antoni Martorell (Universitat Politecnica de Catalunya)
Abstract : In this presentation, we will present recent advances in the numerical approximation of PDEs with moving interfaces/boundaries using unfitted finite element methods. We will describe the numerical discretisation of transient problems using unfitted finite elements that are robust with respect to the small cut cell problems. We will design these algorithms for transient problems, e.g., by defining space-time discrete extension operators.
We will propose two different ways to design space-time unfitted methods. One approach is a pure space-time formulation, in which our geometries are considered in 4D (for a 3D problem in space). This approach is suitable, e.g., for problems in which the geometry is described via level sets. For complex geometrical representations in terms of oriented surface meshes, we propose a geometrical discretisation framework (for 3D in space) that provides all the quadrature rules needed to integrate our numerical methods on unfitted meshes. The extension of this 3D algorithm to 4D is a challenge.
We propose another space-time approach that solves the problem in the time-varying domain by using an extrusion of the 3D problem and a geometrical map. This way, one does not require 4D geometrical algorithms. In time, we consider discontinuous Galerkin spaces. The integration of inter-slab jump terms involves two functions on each side of the interface that are defined on different meshes (the background mesh and the mapped background mesh). In order to exactly compute these integrals, we propose to use intersection algorithms.
[04279] Isogeometric solvers for derived cardiac stem cell reaction-diffusion models
Author(s) :
Sofia Botti (Università della Svizzera Italiana and University of Pavia)
Michele Torre (University of Pavia)
Abstract : Regenerative cardiology is recently employing human induced pluripotent stem cells derived cardiomyocytes to advance in patient-specific medicine. A multiphysics approach to the problem allows to couple the cardiac Monodomain reaction diffusion model with stem cell ionic models to simulate the action potential propagation in the engineered ventricle. The coupled model is then discretized using Isogeometric Analysis in space and finite differences in time to obtain a virtual representation of a derived cardiomyocytes ventricle.
[04578] Conservative and robust methods for the Biot-Brinkman equations in vorticity form
Author(s) :
Alberto Francisco Martin Huertas (Australian National University)
Ruben Caraballo-Diaz (Universidad del Bio-Bio)
Chansophea Wathanak In (Monash University)
Ricardo Ruiz-Baier (Monash University)
Abstract : In this talk we present a new formulation, a suitable finite element method, along with robust/mesh-independent preconditioners, for the steady coupling of viscous flow in deformable porous media using divergence-conforming filtration fluxes. Apart from the well-posedness of the different formulations, and optimal error estimates, our mathematical analysis confirms robustness of the different methods presented in the case of large Lam\'e parameters and small permeability and storativity coefficients. A few representative numerical examples are presented to back up the analysis.
[04697] A five-field mixed formulation for stationary magnetohydrodynamic flows in porous media
Author(s) :
Jessika Camaño (Universidad Católica de la Santísima Concepción)
Abstract : We introduce a new mixed variational formulation for a stationary magnetohydrodynamic flows in porous media problem, whose governing equations are given by the steady Brinkman-Forchheimer equations coupled with the Maxwell equations. Unique solvability of the continuous and discrete systems has been proven.
Stability, convergence, and optimal a priori error estimates for the associated Galerkin scheme are obtained. Numerical tests illustrate the theoretical results.
[04832] Analyzing Multi-Dimensional Time-Dependent Solute Transport Models
Author(s) :
Marius Zeinhofer (Simula Research Laboratory)
Rami Masri (Simula Research Laboratory)
Miroslav Kuchta (Simula Research Laboratory)
Marie Elisabeth Rognes (Simula Research Laboratory)
Abstract : We derive and analyze 3D-1D time dependent solute transport models for convection, diffusion, and exchange in and around pulsating vascular and perivascular networks from their 3D-3D counterpart. These models are applicable e.g.
for transport in vascularized tissue and brain perivascular spaces. We discuss existence and uniqueness questions,
quantify the modelling error, discuss finite element discretizations and present numerical results. Technical key
challenges are controlling constants in classical numerical analysis tools, such as Poincaré's inequality.
[04946] A conforming finite element method for a nonisothermal fluid-membrane interaction
Author(s) :
Ricardo Oyarzúa (Universidad del Bio-Bio)
Abstract : We propose a conforming finite element method for a nonisothermal fluid-membrane interaction problem. The governing equations are given by a Navier-Stokes/Darcy system for the fluid variables and a convection-diffusion model for the temperature. Both systems are coupled through buoyancy terms and a set of transmission conditions on the fluid-membrane interface given by mass conservation, balance of normal forces, the Beavers-Joseph-Saffman law, and the continuity of the heat flux and the fluid temperature.
[04964] Domain decomposition solvers for problems with strong interface perturbations
Author(s) :
Miroslav Kuchta (Simula Research Laboratory)
Abstract : Operators formed by an elliptic part in the bulk domain and a parameter weighted interface perturbation arise in
coupled multiphysics systems as solution operators or as part of their preconditioners. Such systems are often not
amenable to off-the-shelf methods if robustness with respect to the coupling is to be retained. In this talk we develop
robust and scalable solvers for interface-perturbed operators based on domain/subspace decomposition. We
demonstrate performance of the algorithms for single and multiphysics problems such as the EMI and Darcy-Stokes
equations.
[04995] Numerical solution of the Biot/elasticity interface problem using virtual element methods
Author(s) :
Nitesh Verma (Indian Institute of Technology Bombay)
Ricardo Ruiz Baier (Monash University)
David Mora (Universidad del Bio-Bio)
Sarvesh Kumar (Indian Institute of Space Science and Technology)
Abstract : We propose, analyse and implement a virtual element discretisation for an interfacial poroelasticity/elasticity consolidation problem. The formulation of the time-dependent poroelasticity equations uses displacement, fluid pressure and total pressure, and the elasticity equations are written in the displacement-pressure formulation. The construction of the virtual element scheme does not require Lagrange multipliers to impose the transmission conditions (continuity of displacement and total traction, and no flux for the fluid) on the interface. We show the stability and convergence of the virtual element method for different polynomial degrees, and the error bounds are robust with respect to delicate model parameters (such as Lame constants, permeability, and storativity coefficient). Finally, we provide numerical examples that illustrate the properties of the scheme.
[05072] Multipoint mixed finite elements for Biot poroelasticity using a rotation-based formulation
Author(s) :
Wietse Boon (Politecnico di Milano)
Alessio Fumagalli (Politecnico di Milano)
Anna Scotti (Politecnico di Milano)
Abstract : We propose a discretization method for Biot poroelasticity that employs the lowest-order Raviart-Thomas finite element space for the solid displacement and piecewise constants for the fluid pressure. The solid rotation and fluid flux are introduced as auxiliary variables and subsequently removed from the system using a local quadrature rule, leading to a multipoint rotation-flux mixed finite element method. By analyzing the method in terms of weighted norms, we additionally obtain parameter-robust preconditioners.
[05346] A two-way coupled Stokes-Biot-transport model
Author(s) :
Sergio Caucao (Catolica University Concepcion)
Xing Wang (University of Pittsburgh)
Ivan Yotov (University of Pittsburgh)
Abstract : We study mathematical and computational modeling for coupled fluid-poroelastic structure interaction with transport. The model is two-way coupled and nonlinear, with the velocity driving the transport and the concentration affecting the fluid viscosity. We use a Galerkin method, energy estimates, compactness arguments, and fixed point theory to establish well-posedness of the model. We study the finite element approximation of the model and obtain solvability, stability, and error estimates. Computational experiments are conducted to illustrate the theoretical convergence rates and the performance of the method for modeling physical flow and transport phenomena.
