arXiv subjects

Annalisa Quaini

Publications and source records attributed to Annalisa Quaini.

60 records · Page 4Linked to original sources

arXiv2018

A Localized Reduced-Order Modeling Approach for PDEs with Bifurcating Solutions

Reduced-order modeling (ROM) commonly refers to the construction, based on a few solutions (referred to as snapshots) of an expensive discretized partial differential equation (PDE), and the subsequent application of low-dimensional discretizations of partial differential equations (PDEs) that can be used to more efficiently treat problems in control and optimization, uncertainty quantification, and other settings that require multiple approximate PDE solutions. In this work, a ROM is developed and tested for the treatment of nonlinear PDEs whose solutions bifurcate as input parameter values change. In such cases, the parameter domain can be subdivided into subregions, each of which corresponds to a different branch of solutions. Popular ROM approaches such as proper orthogonal decomposition (POD), results in a global low-dimensional basis that does no respect not take advantage of the often large differences in the PDE solutions corresponding to different subregions. Instead, in the new method, the k-means algorithm is used to cluster snapshots so that within cluster snapshots are similar to each other and are dissimilar to those in other clusters. This is followed by the construction of local POD bases, one for each cluster. The method also can detect which cluster a new parameter point belongs to, after which the local basis corresponding to that cluster is used to determine a ROM approximation. Numerical experiments show the effectiveness of the method both for problems for which bifurcation cause continuous and discontinuous changes in the solution of the PDE.

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arXiv2018

On the sensitivity to model parameters in a filter stabilization technique for advection dominated advection-diffusion-reaction problems

We consider a filter stabilization technique with a deconvolution-based indicator function for the simulation of advection dominated advection-diffusion-reaction (ADR) problems with under-refined meshes. The proposed technique has been previously applied to the incompressible Navier-Stokes equations and has been successfully validated against experimental data. However, it was found that some key parameters in this approach have a strong impact on the solution. To better understand the role of these parameters, we consider ADR problems, which are simpler than incompressible flow problems. For the implementation of the filter stabilization technique to ADR problems we adopt a three-step algorithm that requires (i) the solution of the given problem on an under-refined mesh, (ii) the application of a filter to the computed solution, and (iii) a relaxation step. We compare our deconvolution-based approach to classical stabilization methods and test its sensitivity to model parameters on a 2D benchmark problem.

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arXiv2018

A finite element method for the surface Stokes problem

We consider a Stokes problem posed on a 2D surface embedded in a 3D domain. The equations describe an equilibrium, area-preserving tangential flow of a viscous surface fluid and serve as a model problem in the dynamics of material interfaces. In this paper, we develop and analyze a Trace finite element method (TraceFEM) for such a surface Stokes problem. TraceFEM relies on finite element spaces defined on a fixed, surface-independent background mesh which consists of shape-regular tetrahedra. Thus, there is no need for surface parametrization or surface fitting with the mesh. The TraceFEM treated here is based on $P_1$ bulk finite elements for both the velocity and the pressure. In order to enforce the velocity vector field to be tangential to the surface we introduce a penalty term. The method is straightforward to implement and has an $O(h^2)$ geometric consistency error, which is of the same order as the approximation error due to the $P_1$--$P_1$ pair for velocity and pressure. We prove stability and optimal order discretization error bounds in the surface $H^1$ and $L^2$ norms. A series of numerical experiments is presented to illustrate certain features of the proposed TraceFEM.

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arXiv2017

Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: applications to Coanda effect in cardiology

We focus on reducing the computational costs associated with the hydrodynamic stability of solutions of the incompressible Navier-Stokes equations for a Newtonian and viscous fluid in contraction-expansion channels. In particular, we are interested in studying steady bifurcations, occurring when non-unique stable solutions appear as physical and/or geometric control parameters are varied. The formulation of the stability problem requires solving an eigenvalue problem for a partial differential operator. An alternative to this approach is the direct simulation of the flow to characterize the asymptotic behavior of the solution. Both approaches can be extremely expensive in terms of computational time. We propose to apply Reduced Order Modeling (ROM) techniques to reduce the demanding computational costs associated with the detection of a type of steady bifurcations in fluid dynamics. The application that motivated the present study is the onset of asymmetries (i.e., symmetry breaking bifurcation) in blood flow through a regurgitant mitral valve, depending on the Reynolds number and the regurgitant mitral valve orifice shape.

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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.

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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.

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