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 1238 products
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45/2016 - 11/08/2016
Bukac, M.; Yotov, I.; Zunino, P.
DIMENSIONAL MODEL REDUCTION FOR FLOW THROUGH FRACTURES IN POROELASTIC MEDIA | Abstract | | We study the interaction between a poroelastic medium and a fracture filled with fluid. The flow in the fracture is described by the Brinkman equations for an incompressible fluid and the poroelastic medium by the quasi-static Biot model. The two models are fully coupled via the kinematic and dynamic conditions. The Brinkman equations are then averaged over the cross-sections, giving rise to a reduced flow model on the fracture midline. We derive suitable interface and closure conditions between the Biot system and the dimensionally reduced Brinkman model that guarantee solvability of the resulting coupled problem. We design and analyze a numerical discretization scheme based on finite elements in space and the Backward Euler in time, and perform numerical experiments to compare the behavior of the reduced model to the full-dimensional formulation and study the response of the model with respect to its parameters. |
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44/2016 - 11/08/2016
Notaro, D.; Cattaneo, L.; Formaggia, L.; Scotti, A.; Zunino, P.
A Mixed Finite Element Method for Modeling the Fluid Exchange between Microcirculation and Tissue Interstitium | Abstract | | Thanks to dimensional (or topological) model reduction techniques, small inclusions in a three-dimensional (3D) continuum can be described as one-dimensional (1D) concentrated sources, in order to reduce the computational cost of simulations. However, concentrated sources lead to singular solutions that still require compu- tationally expensive graded meshes to guarantee accurate approximation. The main computational barrier consists in the ill-posedness of restriction operators (such as the trace operator) applied on manifolds with co-dimension larger than one. We overcome the computational challenges of approximating PDEs on manifolds with high dimensionality gap by means of nonlocal restriction operators that combine standard traces with mean values of the solution on low dimensional manifolds. This new approach has the fundamental advantage of enabling the approximation of the problem using Galerkin projections on Hilbert spaces, which could not be otherwise applied because of regularity issues. This approach, previously applied to second order PDEs, is extended here to the mixed formulation of flow problems with applications to microcirculation. In this way we calculate, in the bulk and on the 1D manifold simultaneously, the approximation of velocity and pressure fields that guarantees good accuracy with respect to mass conservation. |
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43/2016 - 11/08/2016
Ambrosi, D.; Ciarletta, P.; De Falco, C.; Taffetani, M.; Zunino, P.
A multiscale modeling approach to transport of nano-constructs in biological tissues | Abstract | | Nanomedicine is the emerging medical research branch which employs nanotechnological de- vices to improve clinical diagnosis and to design more effective therapeutic methodologies. In particular, functionalized nanoparticles have proved their clinical usefulness for cancer therapy, either as vectors for targeted drug delivery or for hyperthermia treatment. The effectiveness of such novel therapeutic strategies in nanomedicine exploits the capability of the nanoparticles to penetrate into the living tissue through the vascular network and to reach the targeted site. Accordingly, their success is tightly related to the control of the the multi-physics and multi-scale phenomena governing the diffusion and transport properties of the nanoparticles, together with the geometrical and chemo-mechanical factors regulating the nanoparticles- tissue interactions. Indeed, the therapeutic effectiveness of earlier approaches was hindered by a limited ability in penetrating within the tumor tissue essentially due to microfluidic effects. Mathematical modeling is often employed in nanomedicine to analyze in silico the key biophysical mechanisms acting at different scales of investigations, providing useful guidelines to foresee and possibily optimize novel experimental techniques. Since these phenomena involve different characteristic time- and length-scales, a multi-scale modeling approach is mandatory. In this work we outline how a multi-scale analysis starts at the smallest scale, and its results are injected in large-scale models. At the microscale, the transport of nanoparticles is modeled either by the stochastic Langevin equation or by its continuous limit; in both cases short distance interaction forces between particles are considered, such as Coulomb and van der Waals interactions, and small disturbances of the fluid velocity field induced by the presence of nanoparticles are assumed. At the macroscopic scale, the living tissue is typically modeled as a homogeneous (homogenized) porous material of varying permeability, where the fluid flow is modeled by Darcy’s equation and nanoparticle transport is described by a continuum Diffusion-Reaction-Advection equation. One of the most significant features of the model is the ability to incorporate information on the microvascular network based on physiological data. The exploitation of the large aspect ratio between the diameter of a capillary and the intercapillary distance makes it possible to adopt an advanced computational scheme as the embedded multiscale method: with this approach the capillaries are represented as one-dimensional (1D) channels embedded and exchanging mass in a porous medium. Special mathematical operators are used to model the interaction of capillaries with the surrounding tissue. In this general context, we illustrate a bottom-up approach to study the transport and the diffusion of nanoparticles in living materials. We determine the permeability as well as the lumped parameters appearing in the nanoparticle transport equation at the tissue level by means of simulations at the microscale, while the macroscale tissue deposition rate is derived from the results of microscale simulations by means of a suitable upscaling technique.
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42/2016 - 11/08/2016
Iannetti, L.; D'Urso, G.; Conoscenti, G.; Cutri, E.; Tuan, R.S.; Raimondi, M.T.; Gottardi, R.; Zunino, P.
Distributed and lumped parameter models for the characterization of high throughput bioreactors | Abstract | | Next generation bioreactors are being developed to generate multiple human cell-based tissue analogs within the same fluidic system, to better recapitulate the complexity and interconnection of human physiology (1, 2). The effective development of these devices requires a solid understanding of their interconnected fluidics, to predict the transport of nutrients and waste through the constructs and improve the design accordingly. In this work, we focus on a specific model of bioreactor, with multiple input/outputs, aimed at generating osteochondral constructs, i.e., a biphasic construct in which one side is cartilaginous in nature, while the other is osseous. We next develop a general computational approach to model the microfluidics of a multi-chamber, interconnected system that may be applied to human-on-chip devices. This objective requires overcoming several challenges at the level of computational modeling. The main one consists of addressing the multi-physics nature of the problem that combines free flow in channels with hindered flow in porous media. Fluid dynamics is also coupled with advection-diffusion-reaction equations that model the transport of biomolecules throughout the system and their interaction with living tissues and C constructs. Ultimately, we aim at providing a predictive approach useful for the general organ-on- chip community. To this end, we have developed a lumped parameter approach that allows us to analyze the behavior of multi-unit bioreactor systems with modest computational effort, provided that the behavior of a single unit can be fully characterized. |
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41/2016 - 11/07/2016
Giovanardi, B.; Scotti, A.; Formaggia, L.
A hybrid XFEM - Phase Field (Xfield) method for crack propagation in brittle materials | Abstract | | The present work proposes a novel method for the simulation of crack propagation
in brittle elastic materials that combines two of the most popular approaches
in literature. A large scale displacement solution is obtained with the
well known extended finite elements method (XFEM), while propagation is governed
by the solution of a local phase field problem at the tip scale. The method,
which we will refer to as Xfield, is here introduced and tested in 2D under mixed
modes I and II loads. The main features and the capability of the Xfield to efficiently
simulate crack propagation are shown in some numerical tests. |
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40/2016 - 11/07/2016
Miglio, E.; Parolini, N.; Penati, M.; Porcù R.
High-order variational time integrators for particle dynamics | Abstract | | The general family of Galerkin variational integrators are analyzed and a complete classification of such methods is proposed. This classification is based upon the type of basis function chosen to approximate the trajectories of material points and the numerical quadrature formula used in time. This approach leads to the definition of arbitrarily high order method in time. The proposed methodology is applied to the simulation of brownout phenomena occurring in helicopter take-off and landing. |
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39/2016 - 11/07/2016
Andrà, C.; Brunetto, D.; Parolini, N.; Verani, M.
Student interactions during class activities: a mathematical model | Abstract | | We propose a mathematical model describing the interactions among students during work group activities aimed at solving a math problem. The model, which hinges upon the pioneering work of Hegselmann and Krause, is able to incorporate: 1) the feelings of each student towards the classmates (building upon the “I can”-“You can” framework); 2) the influence of the correct solution to model the students’ preparation; 3) the presence of the teacher, who may or may not intervene to drive students towards the correct solution of the problem. Several numerical experiments are presented to assess the capability of the model in reproducing typical realistic scenarios. |
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38/2016 - 11/07/2016
Quarteroni, A.; Manzoni, A.; Vergara, C.
The Cardiovascular System: Mathematical Modeling, Numerical Algorithms, Clinical Applications | Abstract | | Mathematical and numerical modeling of the cardiovascular system is a research topic that
has attracted a remarkable interest from the mathematical community because of the intrinsic
mathematical difficulty and due to the increasing impact of cardiovascular diseases worldwide.
In this review article, we will address the two principle components of the cardiovascular
system, the arterial circulation and the heart function. We systematically go through the
complete pipeline from data imaging acquisition, setting the basic physical principles, analyzing
the associated mathematical models that comprise PDEs and ODEs systems, proposing
sound and efficient numerical methods for their approximation, simulating both benchmark
problems and clinically inspired (driven) problems. Mathematical modeling itself features
tremendous challenges, due to the amazing complexity of the cardiocirculatory system, the
multiscale nature of the involved physiological processes, and the need of devising computational
methods that are stable, reliable, and efficient. A critical issue is about filtering
the data, identifying the parameters of mathematical models, devising optimal treatments,
accounting for uncertainties. For this reason, we will devote the last part of the paper to
control and inverse problems, including parameter estimation, uncertainty quantification and
the development of reduced order models that are of paramount importance when solving
problems with high complexity, that would be out of reach otherwise. |
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