An algebraic least squares reduced basis method for the solution of nonaffinely parametrized Stokes equations

Santo, Niccolò Dal; Deparis, Simone; Manzoni, Andrea; Quarteroni, Alfio

doi:10.1016/j.cma.2018.06.035

Cited by 18 publications

(13 citation statements)

References 28 publications

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“…Furthermore, the global system obtained from the nonconforming method introduced in Section 4.2 is also a saddle-point problem where the velocity and the Lagrange multipliers play the role of the primal and dual fields, respectively. Among the ways to deal with the loss of stability in the reduced system are the use of least squares Petrov–Galerkin approaches for the solution of the minimization problem associated to the nonlinear residual of the reduced equations [ 57 , 58 ] and the supremizers enrichment [ 59 – 61 ]. Here we follow the latter approach.…”

Section: The Reduced Basis Element Methods For Flow In Arteriesmentioning

confidence: 99%

Model order reduction of flow based on a modular geometrical approximation of blood vessels

Pegolotti

Pfaller

Marsden

et al. 2021

Computer Methods in Applied Mechanics and Engineering

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We are interested in a reduced order method for the efficient simulation of blood flow in arteries. The blood dynamics is modeled by means of the incompressible Navier–Stokes equations. Our algorithm is based on an approximated domain-decomposition of the target geometry into a number of subdomains obtained from the parametrized deformation of geometrical building blocks (e.g., straight tubes and model bifurcations). On each of these building blocks, we build a set of spectral functions by Proper Orthogonal Decomposition of a large number of snapshots of finite element solutions (offline phase). The global solution of the Navier–Stokes equations on a target geometry is then found by coupling linear combinations of these local basis functions by means of spectral Lagrange multipliers (online phase). Being that the number of reduced degrees of freedom is considerably smaller than their finite element counterpart, this approach allows us to significantly decrease the size of the linear system to be solved in each iteration of the Newton–Raphson algorithm. We achieve large speedups with respect to the full order simulation (in our numerical experiments, the gain is at least of one order of magnitude and grows inversely with respect to the reduced basis size), whilst still retaining satisfactory accuracy for most cardiovascular simulations.

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Section: The Reduced Basis Element Methods For Flow In Arteriesmentioning

confidence: 99%

Model order reduction of flow based on a modular geometrical approximation of blood vessels

Pegolotti

Pfaller

Marsden

et al. 2021

Computer Methods in Applied Mechanics and Engineering

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“…This strategy leads to an RB problem which is inf-sup stable in practice, but whose wellposedness is not rigorously proven [56,7]. Another option to enforce the inf-sup stability would rely on a coarse algebraic least-squares RB method; however, this strategy has only been investigated in the case of steady Stokes problems so far; see [23].…”

Section: Navier-stokes Equationsmentioning

confidence: 99%

8 Reduced-order modeling for applications to the cardiovascular system

2020

Applications

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“…The presence of FSI, coupling the fluid model with a model describing the structural displacement of the non-rigid domain where the fluid flows, makes the problem even more involved. Several strategies have been proposed to address these issues: for instance, hyper-reduction techniques have been devised in a purely algebraic way to treat the nonaffine and nonlinear convective terms appearing in the NS equations [8]; suitable enrichment of the velocity space can be considered to ensure the inf-sup stability of the ROM [7,18] as well as alternative, more effective, stabilization techniques for the ROM [19]; mesh-moving techniques have been exploited to efficiently parameterize domain shapes to address geometric variability in fluid flow simulations [8], and either monolithic or segregated strategies have been considered as first attempts to handle fluid-structure interactions in the RB method for parameterized fluid flows [20,21].…”

Section: Introductionmentioning

confidence: 99%

Real-Time Simulation of Parameter-Dependent Fluid Flows through Deep Learning-Based Reduced Order Models

Fresca

Manzoni

2021

Fluids

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Simulating fluid flows in different virtual scenarios is of key importance in engineering applications. However, high-fidelity, full-order models relying, e.g., on the finite element method, are unaffordable whenever fluid flows must be simulated in almost real-time. Reduced order models (ROMs) relying, e.g., on proper orthogonal decomposition (POD) provide reliable approximations to parameter-dependent fluid dynamics problems in rapid times. However, they might require expensive hyper-reduction strategies for handling parameterized nonlinear terms, and enriched reduced spaces (or Petrov–Galerkin projections) if a mixed velocity–pressure formulation is considered, possibly hampering the evaluation of reliable solutions in real-time. Dealing with fluid–structure interactions entails even greater difficulties. The proposed deep learning (DL)-based ROMs overcome all these limitations by learning, in a nonintrusive way, both the nonlinear trial manifold and the reduced dynamics. To do so, they rely on deep neural networks, after performing a former dimensionality reduction through POD, enhancing their training times substantially. The resulting POD-DL-ROMs are shown to provide accurate results in almost real-time for the flow around a cylinder benchmark, the fluid–structure interaction between an elastic beam attached to a fixed, rigid block and a laminar incompressible flow, and the blood flow in a cerebral aneurysm.

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An algebraic least squares reduced basis method for the solution of nonaffinely parametrized Stokes equations

Cited by 18 publications

References 28 publications

Model order reduction of flow based on a modular geometrical approximation of blood vessels

Model order reduction of flow based on a modular geometrical approximation of blood vessels

8 Reduced-order modeling for applications to the cardiovascular system

Real-Time Simulation of Parameter-Dependent Fluid Flows through Deep Learning-Based Reduced Order Models

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