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 1239 products
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105/2023 - 12/16/2023
Cicci, L.; Fresca, S.; Guo, M.; Manzoni, A.; Zunino, P.
Uncertainty quantification for nonlinear solid mechanics using reduced order models with Gaussian process regression | Abstract | | Uncertainty quantification (UQ) tasks, such as sensitivity analysis and parameter estimation, entail a huge computational complexity when dealing with input-output maps involving the solution of nonlinear differential problems, because of the need to query expensive numerical solvers repeatedly. Projection-based reduced order models (ROMs), such as the Galerkin-reduced basis (RB) method, have been extensively developed in the last decades to overcome the computational complexity of high fidelity full order models (FOMs), providing remarkable speed-ups when addressing UQ tasks related with parameterized differential problems. Nonetheless, constructing a projection-based ROM that can be efficiently queried usually requires extensive modifications to the original code, a task which is non-trivial for nonlinear problems, or even not possible at all when proprietary software is used. Non-intrusive ROMs – which rely on the FOM as a black box – have been recently developed to overcome this issue. In this work, we consider ROMs exploiting proper orthogonal decomposition to construct a reduced basis from a set of FOM snapshots, and Gaussian process regression (GPR) to approximate the RB projection coefficients. Two different approaches, namely a global GPR and a tensor-decomposition-based GPR, are explored on a set of 3D time-dependent solid mechanics examples. Finally, the non-intrusive ROM is exploited to perform global sensitivity analysis (relying on both screening and variance-based methods) and parameter estimation (through Markov chain Monte Carlo methods), showing remarkable computational speed-ups and very good accuracy compared to high-fidelity FOMs. |
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101/2023 - 12/15/2023
Formaggia, L.; Zunino, P.
Hybrid dimensional models for blood flow and mass transport | Abstract | | Mathematical models accounting of several space scales have proved to be very effective tools in the description and simulation of the cardiovascular system. In this chapter, we review the family of models that are based on partial differential equations defined on domains with hybrid dimension. Referring to the vascular applications, the most prominent example consists of coupling three-dimensional (3D) with one-dimensional (1D) mathematical models for blood flow and mass transport. On the basis of their coupling conditions these models can be subdivided into two main categories: the ones based on sequential coupling and those arising from the embedded coupling. We organize this work in two main sections, reflecting this subdivision. |
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104/2023 - 12/15/2023
Possenti, L.; Gallo, A.; Vitullo, P.; Cicchetti, A.; Rancati, T.; Costantino, M.L.; Zunino, P.
A computational model of the tumor microenvironment applied to fractionated radiotherapy | Abstract | | Radiotherapy consists in delivering a precise radiation dose to a specific tumor target in order to eradicate tumor cells and to achieve local tumor control. The definition of the most suitable radiotherapy treatment schedule is not trivial due to the large tumor heterogeneity reported in clinical practice. The ultimate goal is to prescribe a specific treatment pattern for each patient, considering all the different radiobiological properties of the tumor / normal tissues to achieve the best final result possible. The model presented in this work goes in this direction, analyzing oxygen dependency and the role of the vascular network in the tumor microenvironment, since the efficacy of radiation therapy also depends on local oxygen availability. The main purpose of this work is to develop a mathematical model that describes the interaction between microvascular oxygen transfer and the efficacy of fractionated radiotherapy. |
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103/2023 - 12/15/2023
Dimola N.; Kuchta M.; Mardal K.A.; Zunino P.
Robust Preconditioning of Mixed-Dimensional PDEs on 3d-1d domains coupled with Lagrange Multipliers | Abstract | | In the context of micro-circulation, the coexistence of two distinct length scales - the vascular radius and the tissue/organ scale - with a substantial difference in magnitude, poses significant challenges. To handle slender inclusions and simplify
the geometry involved, a technique called topological dimensionality reduction is used, which suppresses the manifold dimensions associated with the smaller characteristic length. However, the algebraic structure of the resulting discretized system presents a challenge in constructing efficient solution algorithms. This chapter addresses this challenge by developing a robust preconditioner for the 3d-1d problem using the operator preconditioning technique. The robustness of the preconditioner is demonstrated with respect to the problem parameters, except for the vascular radius. The vascular radius, as demonstrated, plays a fundamental role in the mathematical well-posedness of the problem and the effectiveness of the preconditioner. |
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98/2023 - 12/14/2023
Lespagnol, F.; Grandmont, C.; Zunino, P.; Fernandez, M.A.
A mixed-dimensional formulation for the simulation of slender structures immersed in an incompressible flow | Abstract | | We consider the simulation of slender structures immersed in a three-dimensional (3D) flow. By exploiting the special geometric configuration of the slender structures, this particular problem can be modeled by mixed-dimensional coupled equations (3D for the fluid and 1D for the solid).
Several challenges must be faced when dealing with this type of problems. From a mathematical point of view, these include defining wellposed trace operators of codimension two. On the computational standpoint, the non-standard mathematical formulation makes it difficult to ensure the accuracy of the solutions obtained with the mixed-dimensional discrete formulation as compared to a fully resolved one.
We establish the continuous formulation using the Navier-Stokes equations for the fluid and a Timoshenko beam model for the structure. We complement these models with a mixed-dimensional version of the fluid-structure interface conditions, based on the projection of kinematic coupling conditions on a finite-dimensional Fourier space. Furthermore, we develop a discrete formulation within the framework of the finite element method, establish the energy stability of the scheme, provide extensive numerical evidence of the accuracy of the discrete formulation, notably with respect to a fully resolved (ALE based) model and a standard reduced modeling approach. |
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100/2023 - 12/14/2023
Vitullo, P.; Cicci, L.; Possenti, L.; Coclite, A.; Costantino, M.L.; Zunino, P.
Sensitivity analysis of a multi-physics model for the vascular microenvironment | Abstract | | The vascular microenvironment is the scale at which microvascular transport, interstitial tissue properties and cell metabolism interact. The vascular microenvironment has been widely studied by means of quantitative approaches, including multi-physics mathematical models as it is a central system for the pathophysiology of many diseases, such as cancer.
The microvascular architecture is a key factor for the fluid balance and mass transfer in the vascular microenvironment, together with the physical parameters characterizing the vascular wall and the interstitial tissue. The scientific literature of this field has witnessed a long debate about which factor of this multifaceted system is the most relevant.
The purpose of this work is to combine the interpretative power of an advanced multi-physics model of the vascular microenvironment with state of the art, robust sensitivity analysis methods, in order to determine what factors affect the most some quantity of interest, related in particular to cancer treatment. We are particularly interested in comparing the factors related to the microvascular architecture with the ones affecting the physics of microvascular transport.
Ultimately, this work will provide further insight of how the vascular microenvironment affects cancer therapies, such as chemotherapy, radiotherapy or immunotherapy. |
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96/2023 - 11/28/2023
Bonetti, S.; Botti, M.; Antonietti, P.F.
Robust discontinuous Galerkin-based scheme for the fully-coupled non-linear thermo-hydro-mechanical problem | Abstract | | We present and analyze a discontinuous Galerkin method for the numerical modeling of the non-linear fully-coupled thermo-hydro-mechanic problem. We propose a high-order symmetric weighted interior penalty scheme that supports general polytopal grids and is robust with respect to strong heteorgeneities in the model coefficients. We focus on the treatment of the non-linear convective transport term in the energy conservation equation and we propose suitable stabilization techniques that make the scheme robust for advection-dominated regimes. The stability analysis of the problem and the convergence of the fixed-point linearization strategy are addressed theoretically under mild requirements on the problem's data. A complete set of numerical simulations is presented in order to assess the convergence and robustness properties of the proposed method. |
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95/2023 - 11/28/2023
Barnafi, N. A.; Regazzoni, F.; Riccobelli, D.
Reconstructing relaxed configurations in elastic bodies: mathematical formulation and numerical methods for cardiac modeling | Abstract | | Modeling the behavior of biological tissues and organs often necessitates the knowledge of their shape in the absence of external loads. However, when their geometry is acquired in-vivo through imaging techniques, bodies are typically subject to mechanical deformation due to the presence of external forces, and the load-free configuration needs to be reconstructed. This paper addresses this crucial and frequently overlooked topic, known as the inverse elasticity problem (IEP), by delving into both theoretical and numerical aspects, with a particular focus on cardiac mechanics. In this work, we extend Shield's seminal work to determine the structure of the IEP with arbitrary material inhomogeneities and in the presence of both body and active forces. These aspects are fundamental in computational cardiology, and we show that they may break the variational structure of the inverse problem. In addition, we show that the inverse problem might be ill-posed, even in the presence of constant Neumann boundary conditions and a polyconvex strain energy functional. We then present the results of extensive numerical tests to validate our theoretical framework, and to characterize the computational challenges associated with a direct numerical approximation of the IEP. Specifically, we show that this framework outperforms existing approaches both in terms of robustness and optimality, such as Sellier's iterative procedure, even when the latter is improved with acceleration techniques. A notable discovery is that multigrid preconditioners are, in contrast to standard elasticity, not efficient, and domain decomposition methods provide a much more reliable alternative. Finally, we successfully address the IEP for a full-heart geometry, demonstrating that the IEP formulation can compute the stress-free configuration in real-life scenarios where Sellier's algorithm proves inadequate. |
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