116 research outputs found
A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs
Traditional reduced order modeling techniques such as the reduced basis (RB)
method (relying, e.g., on proper orthogonal decomposition (POD)) suffer from
severe limitations when dealing with nonlinear time-dependent parametrized
PDEs, because of the fundamental assumption of linear superimposition of modes
they are based on. For this reason, in the case of problems featuring coherent
structures that propagate over time such as transport, wave, or
convection-dominated phenomena, the RB method usually yields inefficient
reduced order models (ROMs) if one aims at obtaining reduced order
approximations sufficiently accurate compared to the high-fidelity, full order
model (FOM) solution. To overcome these limitations, in this work, we propose a
new nonlinear approach to set reduced order models by exploiting deep learning
(DL) algorithms. In the resulting nonlinear ROM, which we refer to as DL-ROM,
both the nonlinear trial manifold (corresponding to the set of basis functions
in a linear ROM) as well as the nonlinear reduced dynamics (corresponding to
the projection stage in a linear ROM) are learned in a non-intrusive way by
relying on DL algorithms; the latter are trained on a set of FOM solutions
obtained for different parameter values. In this paper, we show how to
construct a DL-ROM for both linear and nonlinear time-dependent parametrized
PDEs; moreover, we assess its accuracy on test cases featuring different
parametrized PDE problems. Numerical results indicate that DL-ROMs whose
dimension is equal to the intrinsic dimensionality of the PDE solutions
manifold are able to approximate the solution of parametrized PDEs in
situations where a huge number of POD modes would be necessary to achieve the
same degree of accuracy.Comment: 28 page
A Deep Learning algorithm to accelerate Algebraic Multigrid methods in Finite Element solvers of 3D elliptic PDEs
Algebraic multigrid (AMG) methods are among the most efficient solvers for
linear systems of equations and they are widely used for the solution of
problems stemming from the discretization of Partial Differential Equations
(PDEs). The most severe limitation of AMG methods is the dependence on
parameters that require to be fine-tuned. In particular, the strong threshold
parameter is the most relevant since it stands at the basis of the construction
of successively coarser grids needed by the AMG methods. We introduce a novel
Deep Learning algorithm that minimizes the computational cost of the AMG method
when used as a finite element solver. We show that our algorithm requires
minimal changes to any existing code. The proposed Artificial Neural Network
(ANN) tunes the value of the strong threshold parameter by interpreting the
sparse matrix of the linear system as a black-and-white image and exploiting a
pooling operator to transform it into a small multi-channel image. We
experimentally prove that the pooling successfully reduces the computational
cost of processing a large sparse matrix and preserves the features needed for
the regression task at hand. We train the proposed algorithm on a large dataset
containing problems with a highly heterogeneous diffusion coefficient defined
in different three-dimensional geometries and discretized with unstructured
grids and linear elasticity problems with a highly heterogeneous Young's
modulus. When tested on problems with coefficients or geometries not present in
the training dataset, our approach reduces the computational time by up to 30%
Accelerating Algebraic Multigrid Methods via Artificial Neural Networks
We present a novel deep learning-based algorithm to accelerate—through the use of Arti- ficial Neural Networks (ANNs)—the convergence of Algebraic Multigrid (AMG) methods for the iterative solution of the linear systems of equations stemming from finite element discretizations of Partial Differential Equations (PDE). We show that ANNs can be success- fully used to predict the strong connection parameter that enters in the construction of the sequence of increasingly smaller matrix problems standing at the basis of the AMG algo- rithm, so as to maximize the corresponding convergence factor of the AMG scheme. To demonstrate the practical capabilities of the proposed algorithm, which we call AMG-ANN, we consider the iterative solution of the algebraic system of equations stemming from finite element discretizations of two-dimensional model problems. First, we consider an ellip- tic equation with a highly heterogeneous diffusion coefficient and then a stationary Stokes problem. We train (off-line) our ANN with a rich dataset and present an in-depth analy- sis of the effects of tuning the strong threshold parameter on the convergence factor of the resulting AMG iterative scheme
Preserving the positivity of the deformation gradient determinant in intergrid interpolation by combining RBFs and SVD: application to cardiac electromechanics
The accurate robust and efficient transfer of the deformation gradient tensor
between meshes of different resolution is crucial in cardiac electromechanics
simulations. We present a novel method that combines rescaled localized Radial
Basis Function (RBF) interpolation with Singular Value Decomposition (SVD) to
preserve the positivity of the determinant of the deformation gradient tensor.
The method involves decomposing the evaluations of the tensor at the quadrature
nodes of the source mesh into rotation matrices and diagonal matrices of
singular values; computing the RBF interpolation of the quaternion
representation of rotation matrices and the singular value logarithms;
reassembling the deformation gradient tensors at quadrature nodes of the
destination mesh, to be used in the assembly of the electrophysiology model
equations. The proposed method overcomes limitations of existing interpolation
methods, including nested intergrid interpolation and RBF interpolation of the
displacement field, that may lead to the loss of physical meaningfulness of the
mathematical formulation and then to solver failures at the algebraic level,
due to negative determinant values. The proposed method enables the transfer of
solution variables between finite element spaces of different degrees and
shapes and without stringent conformity requirements between different meshes,
enhancing the flexibility and accuracy of electromechanical simulations.
Numerical results confirm that the proposed method enables the transfer of the
deformation gradient tensor, allowing to successfully run simulations in cases
where existing methods fail. This work provides an efficient and robust method
for the intergrid transfer of the deformation gradient tensor, enabling
independent tailoring of mesh discretizations to the particular characteristics
of the physical components concurring to the of the multiphysics model.Comment: 24 pages; 11 figure
Hemodynamics of the heart's left atrium based on a Variational Multiscale-LES numerical model
In this paper, we investigate the hemodynamics of a left atrium (LA) by
proposing a computational model suitable to provide physically meaningful fluid
dynamics indications and detailed blood flow characterization. In particular,
we consider the incompressible Navier-Stokes equations in Arbitrary Lagrangian
Eulerian (ALE) formulation to deal with the LA domain under prescribed motion.
A Variational Multiscale (VMS) method is adopted to obtain a stable formulation
of the Navier-Stokes equations discretized by means of the Finite Element
method and to account for turbulence modeling based on Large Eddy Simulation
(LES). The aim of this paper is twofold: on one hand to improve the general
understanding of blood flow in the human LA in normal conditions; on the other,
to analyse the effects of the turbulence VMS-LES method on a situation of blood
flow which is neither laminar, nor fully turbulent, but rather transitional as
in LA. Our results suggest that if relatively coarse meshes are adopted, the
additional stabilization terms introduced by the VMS-LES method allow to better
predict transitional effects and cycle-to-cycle blood flow variations than the
standard SUPG stabilization method
Numerical Modelling of the Brain Poromechanics by High-Order Discontinuous Galerkin Methods
We introduce and analyze a discontinuous Galerkin method for the numerical
modelling of the equations of Multiple-Network Poroelastic Theory (MPET) in the
dynamic formulation. The MPET model can comprehensively describe functional
changes in the brain considering multiple scales of fluids. Concerning the
spatial discretization, we employ a high-order discontinuous Galerkin method on
polygonal and polyhedral grids and we derive stability and a priori error
estimates. The temporal discretization is based on a coupling between a Newmark
-method for the momentum equation and a -method for the pressure
equations. After the presentation of some verification numerical tests, we
perform a convergence analysis using an agglomerated mesh of a geometry of a
brain slice. Finally we present a simulation in a three dimensional
patient-specific brain reconstructed from magnetic resonance images. The model
presented in this paper can be regarded as a preliminary attempt to model the
perfusion in the brain
Modeling isovolumetric phases in cardiac flows by an Augmented Resistive Immersed Implicit Surface method
A major challenge in the computational fluid dynamics modeling of the heart function is the simulation of isovolumetric phases when the hemodynamics problem is driven by a prescribed boundary displacement. During such phases, both atrioventricular and semilunar valves are closed: consequently, the ventricular pressure may not be uniquely defined, and spurious oscillations may arise in numerical simulations. These oscillations can strongly affect valve dynamics models driven by the blood flow, making unlikely to recovering physiological dynamics. Hence, prescribed opening and closing times are usually employed, or the isovolumetric phases are neglected altogether. In this article, we propose a suitable modification of the Resistive Immersed Implicit Surface (RIIS) method (Fedele et al., Biomech Model Mechanobiol 2017, 16, 1779-1803) by introducing a reaction term to correctly capture the pressure transients during isovolumetric phases. The method, that we call Augmented RIIS (ARIIS) method, extends the previously proposed ARIS method (This et al., Int J Numer Methods Biomed Eng 2020, 36, e3223) to the case of a mesh which is not body-fitted to the valves. We test the proposed method on two different benchmark problems, including a new simplified problem that retains all the characteristics of a heart cycle. We apply the ARIIS method to a fluid dynamics simulation of a realistic left heart geometry, and we show that ARIIS allows to correctly simulate isovolumetric phases, differently from standard RIIS method. Finally, we demonstrate that by the new method the cardiac valves can open and close without prescribing any opening/closing times
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