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 1152 products
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34/2023 - 04/23/2023
Caldana, M.; Antonietti, P. F.; Dede', L.
A Deep Learning algorithm to accelerate Algebraic Multigrid methods in Finite Element solvers of 3D elliptic PDEs | Abstract | | 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%. |
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32/2023 - 04/15/2023
Gambarini, M.; Ciaramella, G.; Miglio, E.; Vanzan, T.
Robust optimization of control parameters for WEC arrays using stochastic methods | Abstract | | This work presents a new computational optimization framework for the robust control of parks of Wave Energy Converters (WEC) in irregular waves. The power of WEC parks is maximized with respect to the individual control damping and stiffness coefficients of each device. The results are robust with respect to the incident wave direction, which is treated as a random variable. Hydrodynamic properties are computed using the linear potential model, and the dynamics of the system is computed in the frequency domain. A slamming constraint is enforced to ensure that the results are physically realistic. We show that the stochastic optimization problem is well posed. Two optimization approaches for dealing with stochasticity are then considered: stochastic approximation and sample average approximation. The outcomes of the above mentioned methods in terms of accuracy and computational time are presented.
The results of the optimization for complex and realistic array configurations of possible engineering interest are then discussed. Results of extensive numerical experiments demonstrate the efficiency of the proposed computational framework.
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31/2023 - 04/12/2023
Orlando, G.
Assessing ChatGPT for coding finite element methods | Abstract | | ChatGPT is a language model trained by OpenAI to follow an instruction in a prompt and to provide a detailed response. We investigate the capabilities of ChatGPT to generate codes which implement the finite element method. The Finite element method (FEM) is a popular technique for the numerical solution of partial differential equations (PDEs). More specifically, we analyze the codes generated for two open source platforms: deal.II, a C++ software library, and FEniCS, for which we focus on its Python interface. We consider as benchmark problems the Poisson equation and a linear advection problem. The outcomes suggest that ChatGPT can be employed as initial building block to write finite element codes, but certain limitations and failures, which require further improvement of the machine learning model and a human supervision, are still present. |
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30/2023 - 04/03/2023
Antonietti, P. F.; Bonizzoni, F.; Verani, M.
A DG-VEM method for the dissipative wave equation | Abstract | | A novel space-time discretization for the (linear) scalar-valued dissipative wave equation is presented. It is a structured approach, namely, the discretization space is obtained tensorizing the Virtual Element (VE) discretization in space with the Discontinuous Galerkin (DG) method in time. As such, it combines the advantages of both the VE and the DG methods. The proposed scheme is implicit and it is proved to be unconditionally stable and accurate in space and time. |
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29/2023 - 03/24/2023
Carbonaro, D.; Mezzadri, F.; Ferro, N.; De Nisco, G.; Audenino, A.L.; Gallo, D.; Chiastra, C.; Morbiducci, U.; Perotto, S.
Design of innovative self-expandable femoral stents using inverse homogenization topology optimization | Abstract | | In this study, we propose a novel computational framework for designing innovative self-expandable femoral stents. First, a two-dimensional stent unit cell is designed by inverse homogenization topology optimization. In particular, the unit cell is optimized in terms of contact area with the target of matching prescribed mechanical properties. The topology optimization is enriched by an anisotropic mesh adaptation strategy, enabling a time- and cost-effective procedure that promotes original layouts. Successively, the optimized stent unit cell is periodically repeated on a hollow cylindrical surface to construct the corresponding three-dimensional device. Finally, structural mechanics and computational fluid dynamics simulations are carried out to verify the performance of the newly-designed stent.
The proposed workflow is being tested through the design of five proof-of-concept stents. These devices are compared through specific performance evaluations, which include the assessments of the minimum requirement for usability, namely the ability to be crimped into a catheter, and the quantification of the radial force, the foreshortening, the structural integrity and the induced blood flow disturbances. |
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28/2023 - 03/24/2023
Zingaro, A.; Vergara, C.; Dede', L.; Regazzoni, F.; Quarteroni, A.
A comprehensive mathematical model for myocardial perfusion | Abstract | | We present a novel mathematical model that simulates myocardial blood perfusion by embedding multiscale and multiphysics features. Our model incorporates cardiac electrophysiology, active and passive mechanics, hemodynamics, reduced valve modeling, and a multicompartment Darcy model of perfusion . We consider a fully coupled electromechanical model of the left heart that provides input for a fully coupled Navier-Stokes - Darcy Model for myocardial perfusion. The fluid dynamics problem is modeled in a left heart geometry that includes large epicardial coronaries, while the multicompartment Darcy model is set in a biventricular domain. Using a realistic and detailed cardiac geometry, our simulations demonstrate the accuracy of our model in describing cardiac perfusion, including myocardial blood flow maps. Additionally, we investigate the impact of a regurgitant aortic valve on myocardial perfusion, and our results indicate a reduction in myocardial perfusion due to blood flow taken away by the left ventricle during diastole. To the best of our knowledge, our work represents the first instance where electromechanics, hemodynamics, and perfusion are integrated into a single computational framework. |
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27/2023 - 03/22/2023
Beirao da Vega, L.; Canuto, C.; Nochetto, R.H.; Vacca, G.; Verani, M.
Adaptive VEM for variable data: convergence and optimality | Abstract | | We design an adaptive virtual element method (AVEM) of lowest order over triangular meshes with hanging nodes in 2d, which are treated as polygons. AVEM hinges on the stabilization-free a posteriori error estimators recently derived in [8]. The crucial property, that also plays a central role in this paper, is that the stabilization term can be made arbitrarily small relative to the a posteriori error estimators upon increasing the stabilization parameter. Our AVEM concatenates two modules, GALERKIN and DATA. The former deals with piecewise constant data and is shown in [8] to be a contraction between consecutive iterates. The latter approximates general data by piecewise constants to a desired accuracy. AVEM is shown to be convergent and quasi-optimal, in terms of error decay versus degrees of freedom, for solutions and data belonging to appropriate approximation classes. Numerical experiments illustrate the interplay between these two modules and provide computational evidence of optimality |
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26/2023 - 03/18/2023
Artoni, A.; Antonietti, P.F.; Corradi, R.; Mazzieri, I.; Parolini, N.; Rocchi, D.; Schito P.; Semeraro, F.F.;
AeroSPEED: a high order acoustic solver for aeroacoustic applications | Abstract | | We propose AeroSPEED, a solver based on the Spectral Element Method (SEM) that solves the aeroacoustic Lighthill's wave equation. First, the fluid solution is computed employing a cell centered Finite Volume method. Then, AeroSPEED maps the sound source coming from the flow solution onto the acoustic grid, where finally the Lighthill's wave equation is solved.
An ad-hoc projection strategy is adopted to apply the flow source term in the acoustic solver.
A model problem with a manufactured solution and the Noise Box test case are used as benchmark for the acoustic problem. We studied the noise generated by the complex flow field around tandem cylinders as a relevant aeroacoustic application. AeroSPEED is an effective and accurate solver for both acoustics and aeroacoustic problems. |
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