arXiv subjects

Suncica Canic

Publications and source records attributed to Suncica Canic.

15 recordsLinked to original sources

arXiv2026

Second order explicit splitting scheme for fluid-poroelastic structure interaction problems

Efficient and provably accurate partitioned methods for fluid-poroelastic structure interaction remain challenging because explicit treatment of the Stokes-Biot interface coupling condition can compromise stability. In this work, we develop and analyze a fully discrete, second-order, explicit splitting scheme for the time-dependent Stokes-Biot problem on fixed domains. The method combines BDF2 time stepping with second-order Adams-Bashforth extrapolation of interface data through a Robin reformulation, yielding a partitioned algorithm in which the Stokes and Biot subproblems are solved independently and in parallel at each time step. The main analytical contribution is a rigorous stability and error analysis for this second-order explicit coupling strategy. Using BDF2 energy identities, a sharp decomposition of the extrapolated interface terms, and discrete trace estimates, we prove a closed stability bound under a parabolic CFL condition. We then derive an a priori error estimate through a projection-based framework using a Fortin projection for the fluid variables and Ritz-type projections for the poroelastic variables. The analysis identifies consistency defects from BDF2 time discretization, Adams-Bashforth interface extrapolation, and the projected kinematic relation. It shows that the total errors in fluid velocity, structure velocity, pore pressure, and elastic displacement are bounded by C times the sum of the kth power of the mesh size and the square of the time step, for k from 1 to 3, in bulk energy norms. Numerical experiments with manufactured solutions confirm second-order temporal convergence and optimal-order spatial convergence. We also include a moving-domain example with Navier-Stokes fluid flow, demonstrating applicability beyond the fixed-domain Stokes-Biot setting analyzed.

math.NA↗

arXiv2026

Error Analysis of the Explicit Splitting Scheme for Fluid-Poroelastic Structure Interaction Problems

We present a priori error analysis for a fully discrete, parallelizable, explicit loosely coupled scheme for the time-dependent Stokes-Biot problem. The method decouples the fluid and poroelastic subproblems in a fully explicit fashion, allowing each problem to be solved independently at each time step, with a consistent treatment of the interface conditions that provides stability and convergence of the scheme. The error analysis is carried out in a discrete energy framework. More specifically, we introduce Ritz-type projections in each subdomain, and subtract the fully discrete scheme from the time-discrete continuous formulation. This yields reduced error equations in which the dominant interpolation contributions cancel. The remaining consistency terms stem primarily from time discretization residuals and lagged interface data inherent to the explicit splitting. The main result of this manuscript is the derivation of a discrete error energy identity, and establishment of unconditional error estimates in a combined energy-dissipation norm via a Gronwall-type argument. These estimates demonstrate first-order accuracy in time and optimal spatial convergence rates, as determined by the degree of the finite element polynomials. Numerical experiments based on a manufactured solution corroborate the theory, confirming first-order temporal convergence for all variables, and spatial convergence orders consistent with the chosen approximation spaces.

math.NA↗

arXiv2024

Extended one-dimensional reduced model for blood flow within a stenotic artery

In this paper, we introduce an adapted one-dimensional (1D) reduced model aimed at analyzing blood flow within stenosed arteries. Differing from the prevailing 1D model \cite{Formaggia2003, Sherwin2003_2, Sherwin2003, Quarteroni2004, 10.1007/978-3-642-56288-4_10}, our approach incorporates the variable radius of the blood vessel. Our methodology begins with the non-dimensionalization of the Navier-Stokes equations for axially symmetric flow in cylindrical coordinates and then derives the extended 1D reduced model, by making additional adjustments to accommodate the effects of variable radii of the vessel along the longitudinal direction. Additionally, we propose a method to extract radial velocity information from the 1D results during post-processing, enabling the generation of two-dimensional (2D) velocity data. We validate our model by conducting numerical simulations of blood flow through stenotic arteries with varying severities, ranging from 23% to 50%. The results were compared to those from the established 1D model and a full three-dimensional (3D) simulation, highlighting the potential and importance of this model for arteries with variable radius. All the code used to generate the results presented in the paper is available at https://github.com/qcutexu/Extended-1D-AQ-system.git.

math.NA↗

arXiv2024

Existence of martingale solutions to a nonlinearly coupled stochastic fluid-structure interaction problem

In this paper we study a nonlinear stochastic fluid-structure interaction problem with a multiplicative, white-in-time noise. The problem consists of the Navier-Stokes equations describing the flow of an incompressible, viscous fluid in a 2D cylinder interacting with an elastic lateral wall whose elastodynamics is described by a membrane/shell equation. The flow is driven by the inlet and outlet data, and by the stochastic forcing. The stochastic noise is applied both to the fluid equations as a volumetric body force, and to the structure as an external forcing to the deformable fluid boundary. The fluid and the structure are nonlinearly coupled via the kinematic and dynamic conditions assumed at the moving interface, which is a random variable not known a priori. The geometric nonlinearity due to the nonlinear coupling requires the development of new techniques to capture martingale solutions for this class of stochastic fluid-structure interaction problems. We introduce a constructive approach based on a Lie splitting scheme and prove the existence of martingale solutions to the system. To the best of our knowledge, this is the first result in the field of stochastic PDEs that addresses existence of solutions on moving fluid domains involving incompressible viscous fluids, where the displacement of the boundary and the fluid domain are random variables that are not known a priori and are parts of the solution itself.

math.AP↗

arXiv2021

A stochastically perturbed fluid-structure interaction problem modeled by a stochastic viscous wave equation

We study well-posedness for fluid-structure interaction driven by stochastic forcing. This is of particular interest in real-life applications where forcing and/or data have a strong stochastic component. The prototype model studied here is a stochastic viscous wave equation, which arises in modeling the interaction between Stokes flow and an elastic membrane. To account for stochastic perturbations, the viscous wave equation is perturbed by spacetime white noise scaled by a nonlinear Lipschitz function, which depends on the solution. We prove the existence of a unique function-valued stochastic mild solution to the corresponding Cauchy problem in spatial dimensions one and two. Additionally, we show that up to a modification, the stochastic mild solution is $α$-Hölder continuous for almost every realization of the solution's sample path, where $α\in[0,1)$ for spatial dimension $n=1$, and $α\in [0,1/2)$ for spatial dimension $n=2$. This result contrasts the known results for the heat and wave equations perturbed by spacetime white noise, including the damped wave equation perturbed by spacetime white noise, for which a function-valued mild solution exists only in spatial dimension one and not higher. Our results show that dissipation due to fluid viscosity, which is in the form of the Dirichlet-to-Neumann operator applied to the time derivative of the membrane displacement, sufficiently regularizes the roughness of white noise in the stochastic viscous wave equation to allow the stochastic mild solution to exist even in dimension two, which is the physical dimension of the problem. To the best of our knowledge, this is the first result on well-posedness for a stochastically perturbed fluid-structure interaction problem.

math.AP↗

arXiv2021

Deterministic ill-posedness and probabilistic well-posedness of the viscous nonlinear wave equation describing fluid-structure interaction

We study low regularity behavior of the nonlinear wave equation in $\mathbb{R}^2$ augmented by the viscous dissipative effects described by the Dirichlet-Neumann operator. Problems of this type arise in fluid-structure interaction where the Dirichlet-Neumann operator models the coupling between a viscous, incompressible fluid and an elastic structure. We show that despite the viscous regularization, the Cauchy problem with initial data $(u,u_t)$ in $H^s(\mathbb{R}^2)\times H^{s-1}(\mathbb{R}^2)$, is ill-posed whenever ${0 < s < s_{cr}}$, where the critical exponent $s_{cr}$ depends on the degree of nonlinearity. In particular, for the quintic nonlinearity $u^5$, the critical exponent in $\mathbb{R}^2$ is $s_{cr} = 1/2$, which is the same as the critical exponent for the associated nonlinear wave equation without the viscous term. We then show that if the initial data is perturbed using a Wiener randomization, which perturbs initial data in the frequency space, then the Cauchy problem for the quintic nonlinear viscous wave equation is well-posed almost surely for the supercritical exponents $s$ such that $-1/6 < s \le s_{cr} = 1/2$. To the best of our knowledge, this is the first result showing ill-posedness and probabilistic well-posedness for the nonlinear viscous wave equation arising in fluid-structure interaction.

math.AP↗

arXiv2016

Comparison of reduced models for blood flow using Runge-Kutta discontinuous Galerkin methods

One-dimensional blood flow models take the general form of nonlinear hyperbolic systems but differ greatly in their formulation. One class of models considers the physically conserved quantities of mass and momentum, while another class describes mass and velocity. Further, the averaging process employed in the model derivation requires the specification of the axial velocity profile; this choice differentiates models within each class. Discrepancies among differing models have yet to be investigated. In this paper, we systematically compare several reduced models of blood flow for physiologically relevant vessel parameters, network topology, and boundary data. The models are discretized by a class of Runge-Kutta discontinuous Galerkin methods.

physics.flu-dyn↗

arXiv2016

Existence of a weak solution to a fluid-elastic structure interaction problem with the Navier slip boundary condition

We study a nonlinear, moving boundary fluid-structure interaction problem between an incompressible, viscous Newtonian fluid, modeled by the 2D Navier-Stokes equations, and an elastic structure modeled by the shell or plate equations. The fluid and structure are coupled via the {\em Navier slip boundary condition} and balance of contact forces at the fluid-structure interface. The slip boundary condition is more realistic than the classical no-slip boundary condition in situations, e.g., when the structure is "rough", and in modeling dynamics near, or at a contact. Cardiovascular tissue and cell-seeded tissue constructs, which consist of grooves in tissue scaffolds that are lined with cells, are examples of "rough" elastic interfaces interacting with and incompressible, viscous fluid. The problem of heart valve closure is an example of a fluid-structure interaction problem with a contact. We prove the existence of a weak solution to this class of problems by designing a constructive proof based on the time discretization via operator splitting. This is the first existence result for fluid-structure interaction problems involving elastic structures satisfying the Navier slip boundary condition

math.AP↗

arXiv2014

Runge-Kutta Discontinuous Galerkin Method for Traffic Flow Model on Networks

We propose a bound-preserving Runge-Kutta (RK) discontinuous Galerkin (DG) method as an efficient, effective and compact numerical approach for numerical simulation of traffic flow problems on networks, with arbitrary high order accuracy. Road networks are modeled by graphs, composed of a finite number of roads that meet at junctions. On each road, a scalar conservation law describes the dynamics, while coupling conditions are specified at junctions to define flow separation or convergence at the points where roads meet. We incorporate such coupling conditions in the RK DG framework, and apply an arbitrary high order bound preserving limiter to the RK DG method to preserve the physical bounds on the network solutions (car density). We showcase the proposed algorithm on several benchmark test cases from the literature, as well as several new challenging examples with rich solution structures. Modeling and simulation of Cauchy problems for traffic flows on networks is notorious for lack of uniqueness or (Lipschitz) continuous dependence. The discontinuous Galerkin method proposed here deals elegantly with these problems, and is perhaps the only realistic and efficient high-order method for network problems.

math.NA↗

arXiv2014

Stability of the kinematically coupled β-scheme for fluid-structure interaction problems in hemodynamics

It is well-known that classical Dirichlet-Neumann loosely coupled partitioned schemes for fluid-structure interaction (FSI) problems are unconditionally unstable for certain combinations of physical and geometric parameters that are relevant in hemodynamics. It was shown in \cite{causin2005added} on a simple test problem, that these instabilities are associated with the so called ``added-mass effect''. By considering the same test problem as in \cite{causin2005added}, the present work shows that a novel, partitioned, loosely coupled scheme, recently introduced in \cite{MarSun}, called the kinematically coupled $β$-scheme, does not suffer from the added mass effect for any $β\in [0,1]$, and is unconditionally stable for all the parameters in the problem. Numerical results showing unconditional stability are presented for a full, nonlinearly coupled benchmark FSI problem, first considered in \cite{formaggia2001coupling}.

math.NA↗

arXiv2014

A partitioned scheme for fluid-composite structure interaction problems

We present a loosely-coupled partitioned scheme for a benchmark problem in fluid-composite structure interaction. The benchmark problem proposed here consists of an incompressible, viscous fluid interacting with a composite structure that consists of two layers: a thin elastic layer which is in contact with the fluid, and a thick elastic layer which sits on top of the thin layer. The motivation comes from fluid-structure interaction (FSI) in hemodyam- ics. The equations of linear elasticity are used to model the thick structural layer, while the Koiter member/shell equations are used to model the thin structural layer which serves as fluid-structure interface with mass. An effi- cient, modular, operator-splitting scheme is proposed to simulate solutions to the coupled, nonlinear FSI problem. The operator splitting scheme sepa- rates the elastodynamics structure problem, from a fluid problem in which the thin structure inertia is included as a Robin-type boundary condition to achieve unconditional stability, without requiring any sub-iterations within time-steps. An energy estimate associated with unconditional stability is derived for the fully nonlinear FSI problem defined on moving domains. Two instructive numerical examples are presented to test the performance of the scheme, where it is shown numerically, that the scheme is at least first-order accurate in time. The second example reveals a new phenomenon in FSI problems: the presence of a thin fluid-structure interface with mass regularizes solutions to the full FSI problem.

math.NA↗

arXiv2013

A Modular, Operator Splitting Scheme for Fluid-Structure Interaction Problems with Thick Structures

We present an operator-splitting scheme for fluid-structure interaction (FSI) problems in hemodynamics, where the thickness of the structural wall is comparable to the radius of the cylindrical fluid domain. The equations of linear elasticity are used to model the structure, while the Navier-Stokes equations for an incompressible viscous fluid are used to model the fluid. The operator splitting scheme, based on Lie splitting, separates the elastodynamics structure problem, from a fluid problem in which structure inertia is included to achieve unconditional stability. We prove energy estimates associated with unconditional stability of this modular scheme for the full nonlinear FSI problem defined on a moving domain, without requiring any sub-iterations within time steps. Two numerical examples are presented, showing excellent agreement with the results of monolithic schemes. First-order convergence in time is shown numerically. Modularity, unconditional stability without temporal sub-iterations, and simple implementation are the features that make this operator-splitting scheme particularly appealing for multi-physics problems involving fluid-structure interaction.

math.NA↗

arXiv2013

Existence of a solution to a fluid-multi-layered-structure interaction problem

We study a nonlinear, unsteady, moving boundary, fluid-structure (FSI) problem in which the structure is composed of two layers: a thin layer which is in contact with the fluid, and a thick layer which sits on top of the thin structural layer. The fluid flow, which is driven by the time-dependent dynamic pressure data, is governed by the 2D Navier-Stokes equations for an incompressible, viscous fluid, defined on a 2D cylinder. The elastodynamics of the cylinder wall is governed by the 1D linear wave equation modeling the thin structural layer, and by the 2D equations of linear elasticity modeling the thick structural layer. The fluid and the structure, as well as the two structural layers, are fully coupled via the kinematic and dynamic coupling conditions describing continuity of velocity and balance of contact forces. The thin structural layer acts as a fluid-structure interface with mass. The resulting FSI problem is a nonlinear moving boundary problem of parabolic-hyperbolic type. This problem is motivated by the flow of blood in elastic arteries whose walls are composed of several layers, each with different mechanical characteristics and thickness. We prove existence of a weak solution to this nonlinear FSI problem as long as the cylinder radius is greater than zero. The proof is based on a novel semi-discrete, operator splitting numerical scheme, known as the kinematically coupled scheme. We effectively prove convergence of that numerical scheme to a solution of the nonlinear fluid-multi-layered-structure interaction problem. The spaces of weak solutions presented in this manuscript reveal a striking new feature: the presence of a thin fluid-structure interface with mass regularizes solutions of the coupled problem.

math.AP↗

arXiv2012

Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls

We study a nonlinear, unsteady, moving boundary, fluid-structure interaction (FSI) problem arising in modeling blood flow through elastic and viscoelastic arteries. The fluid flow, which is driven by the time-dependent pressure data, is governed by 2D incompressible Navier-Stokes equations, while the elastodynamics of the cylindrical wall is modeled by the 1D cylindrical Koiter shell model. Two cases are considered: the linearly viscoelastic and the linearly elastic Koiter shell. The fluid and structure are fully coupled (2-way coupling) via the kinematic and dynamic lateral boundary conditions describing continuity of velocity (the no-slip condition), and balance of contact forces at the fluid-structure interface. We prove existence of weak solutions to the two FSI problems (the viscoelastic and the elastic case) as long as the cylinder radius is greater than zero. The proof is based on a novel semi-discrete, operator splitting numerical scheme, known as the kinematically coupled scheme, introduced in \cite{GioSun} to solve the underlying FSI problems. The backbone of the kinematically coupled scheme is the well-known Marchuk-Yanenko scheme, also known as the Lie splitting scheme. We effectively prove convergence of that numerical scheme to a solution of the corresponding FSI problem.

math.AP↗

arXiv2012

Fluid-structure interaction in blood flow capturing non-zero longitudinal structure displacement

We present a new model and a novel loosely coupled partitioned numerical scheme modeling fluid-structure interaction (FSI) in blood flow allowing non-zero longitudinal displacement. Arterial walls are modeled by a {linearly viscoelastic, cylindrical Koiter shell model capturing both radial and longitudinal displacement}. Fluid flow is modeled by the Navier-Stokes equations for an incompressible, viscous fluid. The two are fully coupled via kinematic and dynamic coupling conditions. Our numerical scheme is based on a new modified Lie operator splitting that decouples the fluid and structure sub-problems in a way that leads to a loosely coupled scheme which is {unconditionally} stable. This was achieved by a clever use of the kinematic coupling condition at the fluid and structure sub-problems, leading to an implicit coupling between the fluid and structure velocities. The proposed scheme is a modification of the recently introduced "kinematically coupled scheme" for which the newly proposed modified Lie splitting significantly increases the accuracy. The performance and accuracy of the scheme were studied on a couple of instructive examples including a comparison with a monolithic scheme. It was shown that the accuracy of our scheme was comparable to that of the monolithic scheme, while our scheme retains all the main advantages of partitioned schemes, such as modularity, simple implementation, and low computational costs.

math.NA↗