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 1268 products
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97/2024 - 11/12/2024
Ferro, N.; Mezzadri, F.; Carbonaro, D.; Galligani, E.; Gallo, D.; Morbiducci, U.; Chiastra, C.; Perotto, S.
Designing novel vascular stents with enhanced mechanical behavior through topology optimization of existing devices | Abstract | | A variety of different vascular stent designs are currently available on the market, featuring different geometries, manufacturing materials, and physical characteristics. Here, we propose a framework for designing innovative stents that replicate and enhance the mechanical properties of existing devices. The framework includes a SIMP-based topology optimization formulation, assisted by the homogenization theory to constrain the mechanical response, along with a minimum length scale requirement to ensure manufacturability to the designed devices. The optimization problem, discretized on a sequence of computational meshes anisotropically adapted, generates a 2D stent unit cell, which can be automatically converted into a 3D digital version of the device. This virtual prototype is validated through in silico testing via a radial crimping simulation to assess the stent insertion into the catheter, prior to implantation. The results prove that the proposed framework can identify stent designs that are competitive with respect to existing devices in terms of relevant clinical requirements, such as foreshortening, radial stiffness and surface contact area. |
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94/2024 - 11/11/2024
Franco, N.R.; Fresca, S.; Tombari, F.; Manzoni, A.
Deep Learning-based surrogate models for parametrized PDEs: handling geometric variability through graph neural networks | Abstract | | Mesh-based simulations play a key role when modeling complex physical systems that, in many disciplines across science and engineering, require the solution of parametrized time-dependent nonlinear partial differential equations (PDEs). In this context, full order models (FOMs), such as those relying on the finite element method, can reach high levels of accuracy, however often yielding intensive simulations to run. For this reason, surrogate models are developed to replace computationally expensive solvers with more efficient ones, which can strike favorable trade-offs between accuracy and efficiency. This work explores the potential usage of graph neural networks (GNNs) for the simulation of time-dependent PDEs in the presence of geometrical variability. In particular, we propose a systematic strategy to build surrogate models based on a data-driven time-stepping scheme where a GNN architecture is used to efficiently evolve the system. With respect to the majority of surrogate models, the proposed approach stands out for its ability of tackling problems with parameter dependent spatial domains, while simultaneously generalizing to different geometries and mesh resolutions. We assess the effectiveness of the proposed approach through a series of numerical experiments, involving both two- and three-dimensional problems, showing that GNNs can provide a valid alternative to traditional surrogate models in terms of computational efficiency and generalization to new scenarios. We also assess, from a numerical standpoint, the importance of using GNNs, rather than classical dense deep neural networks, for the proposed framework. |
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93/2024 - 11/11/2024
Conti, P.; Kneifl, J.; Manzoni, A.; Frangi, A.; Fehr, J.; Brunton, S.L.; Kutz, J.N.
VENI, VINDy, VICI - a variational reduced-order modeling framework with uncertainty quantification | Abstract | | The simulation of many complex phenomena in engineering and science requires solving expensive, high-dimensional systems of partial differential equations (PDEs). To circumvent this, reduced-order models (ROMs) have been developed to speed up computations. However, when governing equations are unknown or partially known, or when access to full order solvers is restricted, typically ROMs lack interpretability and reliability of the predicted solutions. In this work we present a data-driven, non-intrusive framework for building ROMs where the latent variables and dynamics are identified in an interpretable manner and uncertainty is quantified.
Starting from a limited amount of high-dimensional, noisy data the proposed framework constructs an efficient ROM by leveraging variational autoencoders for dimensionality reduction along with a newly introduced, variational version of sparse identification of nonlinear dynamics (SINDy), which we refer to as Variational Identification of Nonlinear Dynamics (VINDy). In detail, the method consists of Variational Encoding of Noisy Inputs (VENI) to identify the distribution of reduced coordinates. Simultaneously, we learn the distribution of the coefficients of a pre-determined set of candidate functions by VINDy. Once trained offline, the identified model can be queried for new parameter instances and/or new initial conditions to compute the corresponding full-time solutions. The probabilistic setup enables uncertainty quantification as the online testing consists of Variational Inference naturally providing Certainty Intervals (VICI). In this work we showcase the effectiveness of the newly proposed VINDy method in identifying interpretable and accurate dynamical system for the Rossler system with different noise intensities and sources. Then the performance of the overall method – named VENI, VINDy, VICI – is tested on PDE benchmarks including structural mechanics and fluid dynamics. |
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95/2024 - 11/11/2024
Zacchei, F.; Rizzini, F.; Gattere, G.; Frangi, A.; Manzoni, A.
Neural networks based surrogate modeling for efficient uncertainty quantification and calibration of MEMS accelerometers | Abstract | | This paper addresses the computational challenges inherent in the stochastic characterization and uncertainty quantification of Micro-Electro-Mechanical Systems (MEMS) capacitive accelerometers. Traditional methods, such as Markov Chain Monte Carlo (MCMC) algorithms, are often constrained by the computational intensity required for high-fidelity (e.g., finite element) simulations. To overcome these limitations, we propose to use supervised learning-based surrogate models, specifically artificial neural networks, to effectively approximate the response of MEMS capacitive accelerometers. Our approach involves training the surrogate models with data derived from initial high-fidelity finite element analyses (FEA), providing rich datasets to be generated in an offline phase. The surrogate models replicate the FEA accuracy in predicting the behavior of the accelerometer under a wide range of fabrication parameters, thereby reducing the online computational cost without compromising accuracy. This enables extensive and efficient stochastic analyses of complex MEMS devices, offering a flexible framework for their characterization. A key application of our framework is demonstrated in estimating the sensitivity of an accelerometer, accounting for unknown mechanical offsets, over-etching, and thickness variations. We employ an MCMC approach to estimate the posterior distribution of the device’s unknown fabrication parameters, informed by its response to transient voltage signals. The integration of surrogate models for mapping fabrication parameters to device responses, and subsequently to sensitivity measures, greatly enhances both backward and forward uncertainty quantification, yielding accurate results while significantly improving the efficiency and effectiveness of the characterization process. This process allows for the reconstruction of device sensitivity using only voltage signals, without the need for direct mechanical acceleration stimuli. |
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96/2024 - 11/11/2024
Brivio, S.; Fresca, S.; Manzoni, A.
PTPI-DL-ROMs: Pre-trained physics-informed deep learning-based reduced order models for nonlinear parametrized PDEs | Abstract | | Among several recently proposed data-driven Reduced Order Models (ROMs), the coupling of Proper Orthogonal Decompositions (POD) and deep learning-based ROMs (DL-ROMs) has proved to be a successful strategy to construct non-intrusive, highly accurate, surrogates for the real time solution of parametric nonlinear time-dependent PDEs. Inexpensive to evaluate, POD-DL-ROMs are also relatively fast to train, thanks to their limited complexity. However, POD-DL-ROMs account for the physical laws governing the problem at hand only through the training data, that are usually obtained through a full order model (FOM) relying on a high-fidelity discretization of the underlying equations. Moreover, the accuracy of POD-DL-ROMs strongly depends on the amount of available data. In this paper, we consider a major extension of POD-DL-ROMs by enforcing the fulfillment of the governing physical laws in the training process – that is, by making them physics-informed – to compensate for possible scarce and/or unavailable data and improve the overall reliability. To do that, we first complement POD-DL-ROMs with a trunk net architecture, endowing them with the ability to compute the problem’s solution at every point in the spatial domain, and ultimately enabling a seamless computation of the physics-based loss by means of the strong continuous formulation. Then, we introduce an efficient training strategy that limits the notorious computational burden entailed by a physics-informed training phase. In particular, we take advantage of the few available data to develop a low-cost pre-training procedure; then, we fine-tune the architecture in order to further improve the prediction reliability. Accuracy and efficiency of the resulting pre-trained physics-informed DL-ROMs (PTPI-DL-ROMs) are then assessed on a set of test cases ranging from non-affinely parametrized advection-diffusion-reaction equations, to nonlinear problems like the Navier-Stokes equations for fluid flows. |
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91/2024 - 11/10/2024
Ciaramella, G.; Kartmann, M.; Mueller, G.
Solving Semi-Linear Elliptic Optimal Control Problems with L1-Cost via Regularization and RAS-Preconditioned Newton Methods | Abstract | | We present a new parallel computational framework for the efficient solution of a class of L2/L1-regularized optimal control problems governed by semi-linear elliptic partial differential equations (PDEs). The main difficulty in solving this type of problem is the nonlinearity and non-smoothness of the L1-term in the cost functional, which we address by employing a combination of several tools. First, we approximate the non-differentiable projection operator appearing in the optimality system by an appropriately chosen regularized operator and establish convergence of the resulting system solutions. Second, we apply a continuation strategy to control the regularization parameter to improve the behavior of (damped) Newton methods. Third, we combine Newton’s method with a domain-decomposition-based nonlinear preconditioning, which improves its robustness properties and allows for parallelization. The efficiency of the proposed numerical framework is demonstrated by extensive numerical experiments. |
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85/2024 - 11/09/2024
Brivio, S.; Franco, Nicola R.; Fresca, S.; Manzoni, A.
Error estimates for POD-DL-ROMs: a deep learning framework for reduced order modeling of nonlinear parametrized PDEs enhanced by proper orthogonal decomposition | Abstract | | POD-DL-ROMs have been recently proposed as an extremely versatile strategy to build accurate and reliable reduced order models (ROMs) for nonlinear parametrized partial differential equations, combining (i) a preliminary dimensionality reduction obtained through proper orthogonal decomposition (POD) for the sake of efficiency, (ii) an autoencoder architecture that further reduces the dimensionality of the POD space to a handful of latent coordinates, and (iii) a dense neural network to learn the map that describes the dynamics of the latent coordinates as a function of the input parameters and the time variable. Within this work, we aim at justifying the outstanding approximation capabilities of POD-DL-ROMs by means of a thorough error analysis, showing how the sampling required to generate training data, the dimension of the POD space, and the complexity of the underlying neural networks, impact on the solution accuracy. This decomposition, combined with the constructive nature of the proofs, allows us to formulate practical criteria to control the relative error in the approximation of the solution field of interest, and derive general error estimates. Furthermore, we show that, from a theoretical point of view, POD-DL-ROMs outperform several deep learning-based techniques in terms of model complexity. Finally, we validate our findings by means of suitable numerical experiments, ranging from parameter-dependent operators analytically defined to several parametrized PDEs. |
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86/2024 - 11/09/2024
Franco, N.R.; Fraulin, D.; Manzoni, A.; Zunino, P.
On the latent dimension of deep autoencoders for reduced order modeling of PDEs parametrized by random fields | Abstract | | Deep Learning is having a remarkable impact on the design of Reduced Order Models (ROMs) for Partial Differential Equations (PDEs), where it is exploited as a powerful tool for tackling complex problems for which classical methods might fail. In this respect, deep autoencoders play a fundamental role, as they provide an extremely flexible tool for reducing the dimensionality of a given problem by leveraging on the nonlinear capabilities of neural networks. Indeed, starting from this paradigm, several successful approaches have already been developed, which are here referred to as Deep Learning-based ROMs (DL-ROMs). Nevertheless, when it comes to stochastic problems parameterized by random fields, the current understanding of DL-ROMs is mostly based on empirical evidence: in fact, their theoretical analysis is currently limited to the case of PDEs depending on a finite number of (deterministic) parameters. The purpose of this work is to extend the existing literature by providing some theoretical insights about the use of DL-ROMs in the presence of stochasticity generated by random fields. In particular, we derive explicit error bounds that can guide domain practitioners when choosing the latent dimension of deep autoencoders. We evaluate the practical usefulness of our theory by means of numerical experiments, showing how our analysis can significantly impact the performance of DL-ROMs. |
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