MOX Reports
The preprint collection of the Laboratory for Modeling and Scientific Computation MOX. It mainly contains works on numerical
analysis and mathematical modeling applied to engineering problems. MOX web site is mox.polimi.it
Found 1349 products
-
07/2019 - 02/04/2019
Dal Santo, N.; Manzoni, A.
Hyper-reduced order models for parametrized unsteady Navier-Stokes equations on domains with variable shape | Abstract | | In this work we set up a new, general and computationally efficient way to tackle parametrized fluid flows modeled through unsteady Navier-Stokes equations defined on domains with variable shape, when relying on the reduced basis method. We easily describe a domain by flexible boundary parametrizations, and generate domain (and mesh) deformations by means of a solid extension, obtained by solving an harmonic extension or a linear elasticity problem. The proposed procedure is built over a two-stages reduction: (i) first, we construct a reduced basis approximation for the mesh motion problem, irrespectively of the fluid flow problem we focus on; (ii) then, we generate a reduced basis approximation of the unsteady Navier-Stokes problem, relying on finite element snapshots evaluated over a set of reduced deformed configuration, and approximating both velocity and pressure fields simultaneously. To deal with unavoidable nonaffine parametric dependencies arising in both the mesh motion and the state problem, we apply a matrix version of the discrete empirical interpolation method, allowing to treat geometrical deformations in a non-intrusive, efficient and purely algebraic way. The same strategy is used to per- form hyper-reduction of nonlinear terms. To assess the numerical performances of the proposed technique, we address the solution of parametrized fluid flows where the parameters describe both the shape of the domain, and relevant physical features. Complex flow patterns such as the ones appearing in a patient specific carotid bifurcation are accurately approximated, as well as derived quantities of potential clinical interest. |
-
06/2019 - 02/04/2019
Pagani, S.; Manzoni, A.; Carlberg, K.
Statistical closure modeling for reduced-order models of stationary systems by the ROMES method | Abstract | | This work proposes a technique for constructing a statistical closure model for reduced-order models (ROMs) applied to stationary systems modeled as parameterized systems of algebraic equations. The proposed technique extends the reduced-order-model error surrogates (ROMES) method to closure modeling. The original ROMES method applied Gaussian-process regression to construct a statistical model that maps cheaply computable error indicators (e.g., residual norm, dual-weighted residuals) to a random variable for either (1) the norm of the state error or (2) the error in a scalar-valued quantity of interest. Rather than target these two types of errors, this work proposes to construct a statistical model for the state error itself; it achieves this by constructing statistical models for the generalized coordinates characterizing both the in-plane error (i.e., the error in the trial subspace) and a low-dimensional approximation of the out-of-plane error. The former can be considered a statistical closure model, as it quantifies the error in the ROM generalized coordinates. Because any quantity of interest can be computed as a functional of the state, the proposed approach enables any quantity-of-interest error to be statistically quantified a posteriori, as the state-error model can be propagated through the associated quantity-of-interest functional. Numerical experiments performed on both linear and nonlinear stationary systems illustrate the ability of the technique (1) to improve (expected) ROM prediction accuracy by an order of magnitude, (2) to statistically quantify the error in arbitrary quantities of interest, and (3) to realize a more cost-effective methodology for reducing the error than a ROM-only approach in the case of nonlinear systems. |
-
05/2019 - 02/04/2019
Gasperoni, F.; Ieva, F.; Paganoni, A.M.; Jackson, C.; Sharples, L.
Evaluating the effect of healthcare providers on the clinical path of Heart Failure patients through a novel semi-Markov multi-state model | Abstract | | This paper introduces a novel exploratory statistical tool for investigating healthcare performance through clinical administrative databases. In particular, we propose a Semi-Markov multi state model in which the transition-specific hazards are estimated through a Cox model with a nonparametric discrete frailty term. The proposed model can be interpreted as probabilistic clustering technique. The main goal of this work is to investigate clusters (latent populations) of providers in each specific transition and then to investigate which are the most frequent and most extreme latent populations across all transitions.
Transitions are defined by rates of readmission to, discharge from providers and death in or outside a provider. Rates of transitions are adjusted for patients' characteristics. It is important to notice that this model does not require the selection of providers characteristics to perform the clustering. Finally, we show the impact of the proposed model through a real application on the administrative database related to Heart Failure patients hospitalised in Lombardia, a northern region in Italy. |
-
04/2019 - 02/04/2019
Delpopolo Carciopolo, L.; Formaggia, L.; Scotti, A.; Hajibeygi, H.
Conservative multirate multiscale simulation of multiphase flow in heterogeneous porous media | Abstract | | Accurate and efficient simulation of multiphase flow in heterogeneous porous media motivates the development of space-time multiscale strategies for the coupled nonlinear flow (pressure) and saturation transport equations. The flow equation entails heterogeneous high-resolution (fine-scale) coefficients
and is global (elliptic or parabolic). The time-dependent saturation profile, on the other hand, has local sharp gradients (fronts) where the accuracy of the solution demands for tight time-step sizes. Therefore, accurate flow solvers need to resolve the spatial multiscale challenge, while advanced transport solvers need to also resolve the challenge related to time-step size. In this work, we develop the first integrated multirate multiscale method which implements a space-time conservative multiscale framework for sequentially coupled flow and transport equations. The method solves the pressure equation with a multiscale finite volume method at the spatial coarse scale, the
transport equation is solved by taking different time-step sizes at different locations of the domain. At each time step, a coarse time step is taken, and then based on an adaptive recursive strategy, the front region is sharpened through a local-fine-scale time stepping strategy. The accuracy and efficiency of the method is investigated for a wide range of heterogeneous test cases. The results demonstrate that the proposed method provides a promising strategy to minimise the accuracy-efficiency tradeoff by developing an integrated space-time multiscale simulation strategy. |
-
03/2019 - 01/22/2019
Ratti, L.; Verani, M.
A posteriori error estimates for the monodomain model in cardiac electrophysiology | Abstract | | We consider the monodomain model, a system of a parabolic semilinear reaction-diffusion equation coupled with a nonlinear ordinary differential equation, arising from the (simplified) mathematical description of the electrical activity of the heart. We derive a posteriori error estimators accounting for different sources of error (space/time discretization and linearization). We prove reliability and efficiency (this latter under a suitable assumption) of the error indicators. Finally, numerical experiments assess the validity of the theoretical results. |
-
02/2019 - 01/19/2019
Micheletti, S.; Perotto, S.; Soli, L.
Topology optimization driven by anisotropic mesh adaptation: towards a free-form design | Abstract | | We propose the new adaptive algorithm SIMPATY for topology opti-
mization to design lightweight and stiff structures exhibiting free-form fea-
tures. This is achieved by properly combining the classical SIMP method
with an anisotropic mesh adaptation strategy based on a recovery-based a
posteriori error estimator. Mesh adaptivity allows us to contain (and, for
simple structures, even to eliminate) the filtering usually adopted in a topol-
ogy optmization context. Thus, the final layout is intrinsically smooth, so
that the post-processing phase can be strongly reduced (or even skipped),
and the structure can directly move on to the production manufacturing
phase. A free-form design is also enhanced by anisotropic mesh adaptivity.
An extensive numerical validation supports these remarkable properties. |
-
01/2019 - 01/17/2019
Regazzoni, F.; Dedè, L.; Quarteroni, A.
Machine learning for fast and reliable solution of time-dependent differential equations | Abstract | | We propose a data-driven Model Order Reduction (MOR) technique, based on Artificial Neural Networks (ANNs), applicable to dynamical systems arising from Ordinary Differential Equations (ODEs) or time-dependent Partial Differential Equations (PDEs). Unlike model-based approaches, the proposed approach is non-intrusive since it just requires a collection of input-output pairs generated through the high-fidelity (HF) ODE or PDE model. We formulate our model reduction problem as a maximum-likelihood problem, in which we look for the model that minimizes, in a class of candidate models, the error on the available input-output pairs. Specifically, we represent candidate models by means of ANNs, which we train to learn the dynamics of the HF model from the training input-output data. We prove that ANN models are able to approximate every time-dependent model described by ODEs with any desired level of accuracy. We test the proposed technique on different problems, including the model reduction of two large-scale models. One of the HF systems of ODEs here considered stems from the spatial discretization of a parabolic PDE, which sheds light on a promising field of application of the proposed technique. |
-
66/2018 - 12/24/2018
Riccobelli, D.; Agosti, A.; Ciarletta, P.
On the existence of elastic minimizers for initially stressed materials | Abstract | | A soft solid is said to be initially stressed if it is subjected to a state of internal stress in its unloaded reference configuration.
Developing a sound mathematical framework to model initially stressed solids in nonlinear elasticity is key for many applications in engineering and biology. This work investigates the links between the existence of elastic minimizers and the constitutive restrictions for initially stressed materials subjected to finite deformations. In particular, we consider a subclass of constitutive responses in which the strain energy density is taken as a scalar valued function of both the deformation gradient and the initial stress tensor. The main advantage of this approach is that the initial stress tensor belongs to the group of the divergence-free symmetric tensors satisfying the boundary condition in any given reference configuration. However, it is still unclear which physical restrictions must be imposed for the well-posedness of this elastic problem. Assuming that the constitutive response depends on the choice of the reference configuration only through the initial stress tensor, under given conditions we prove the local existence of a relaxed state given by an implicit tensor function of the initial stress distribution. This tensor function is generally not unique, and can be transformed accordingly to the symmetry group of the material at fixed initial stresses. These results allow to extend Ball's existence theorem of elastic minimizers for the proposed constitutive choice of initially stressed materials. |
|
