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 1249 products
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47/2015 - 09/25/2015
Colombo, M. C.; Giverso, C.; Faggiano, E.; Boffano,C.; Acerbi, F.; Ciarletta, P.
Towards the personalized treatment of glioblastoma: integrating patient-specific clinical data in a continuous mechanical model | Abstract | | Glioblastoma multiforme (GBM) is the most aggressive and malignant among brain tumors. In addition to uncontrolled proliferation and genetic instability, GBM is characterized by a diffuse infiltration, developing long protrusions that penetrate deeply along the fibers of the white matter. These features, combined with the underestimation of the invading GBM area by available imaging techniques, make a definitive treatment of GBM particularly difficult.
A multidisciplinary approach combining mathematical, clinical and radiological data has the potential to foster our understanding of GBM evolution in every single patient throughout his/her oncological history, in order to target therapeutic weapons in a patient-specific manner.
In this work, we propose a continuous mechanical model and we perform numerical simulations of GBM invasion combining the main mechano-biological characteristics of GBM with the micro-structural information extracted from radiological images, i.e. by elaborating patient-specific Diffusion Tensor Imaging (DTI) data.
The numerical simulations highlight the influence of the different biological parameters on tumor progression and they demonstrate the fundamental importance of including anisotropic and heterogeneous patient-specific DTI data in order to obtain a more accurate prediction of GBM evolution. The results of the proposed mathematical model have the potential to provide a relevant benefit for clinicians involved in the treatment of this particularly aggressive disease and, more importantly, they might drive progress towards improving tumor control and patient's prognosis. |
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46/2015 - 09/24/2015
Giverso, C.; Verani, M.; Ciarletta P.
Emerging morphologies in round bacterial colonies: comparing volumetric versus chemotactic expansion | Abstract | | Biological experiments performed on living bacterial colonies have demonstrated
the microbial capability to develop finger-like shapes and highly irregular contours,
even starting from an homogeneous inoculum. In this work, we study from
the continuum mechanics viewpoint the emergence of such branched morphologies
in an initially circular colony expanding on the top of a Petri dish coated with
agar. The bacterial colony expansion, based on either a source term, representing
volumetric mitotic processes, or a non-convective mass flux, describing chemotactic
expansion, is modelled at the continuum scale. We demonstrate that the
front of the colony is always linearly unstable, having similar dispersion curves
to the ones characterizing branching instabilities. We also perform finite element
simulations, which not only prove the emergence of branching, but also highlight
dramatic differences between the two mechanisms of colony expansion in the nonlinear
regime. Furthermore, the proposed combination of analytical and numerical
analysis allowed studying the influence of different model parameters on the
selection of specific patterns. A very good agreement has been found between
the resulting simulations and the typical structures observed in biological assays.
Finally, this work provides a new interpretation of the emergence of branched patterns
in living aggregates, depicted as the results of a complex interplay among
chemical, mechanical and size effects. |
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45/2015 - 09/24/2015
Lange, M.; Palamara, S.; Lassila, T.; Vergara, C.; Quarteroni, A.; Frangi, A.F.
Improved hybrid/GPU algorithm for solving cardiac electrophysiology problems on Purkinje networks | Abstract | | The cardiac Purkinje fibres provide an important stimulus to the coordinated
contraction of the heart. We present a numerical algorithm for the
solution of electrophysiology problems on the Purkinje network that is efficient enough to be used on realistic networks with physiologically detailed
membrane models. The algorithm is based on operator splitting and is provided
with three different implementations: pure CPU, hybrid CPU/GPU,
and pure GPU. Compared to our previous work based on the model of
Vigmond et al., we modify the explicit gap junction term at network bifurcations
in order to improve its mathematical consistency. Due to this
improved consistency of the model, we are able to perform a convergence
study against analytical solutions and verify that all three implementations
produce equivalent convergence rates. Finally, we compare the efficiency of
all three implementations on Purkinje networks of increasing spatial resolution
using membrane models of increasing complexity. Both hybrid and
pure-GPU implementations outperform the pure-CPU implementation, but
their relative performance difference depends on the size of the Purkinje
network and the complexity of the membrane model used. |
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44/2015 - 09/20/2015
Antonietti, P.F.; Houston, P.; Smears, I.
A note on optimal spectral bounds for nonoverlapping domain decomposition preconditioners for hp-Version Discontinuous Galerkin methods | Abstract | | In this article, we consider the derivation of hp-optimal spectral bounds for a class of domain decomposition preconditioners based on the Schwarz framework for discontinuous Galerkin finite element approximations of second-order elliptic partial differential equations. In particular, we improve the bounds derived in our earlier article [P.F. Antonietti and P. Houston, J. Sci. Comput., 46(1):124–149, 2011] in the sense that the resulting bound on the condition number of the preconditioned system is not only explicit with respect to the coarse and fine mesh sizes H and h, respectively, and the fine mesh polynomial degree p, but now also explicit with respect to the polynomial degree q employed for the coarse grid solver. More precisely, we show that the resulting spectral bounds are of order p2H/(qh) for the hp-version of the discontinuous Galerkin method. |
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43/2015 - 09/09/2015
Deparis, S.; Forti, D.; Gervasio, P.; Quarteroni, A.
INTERNODES: an accurate interpolation-based method for coupling the Galerkin solutions of PDEs on subdomains featuring non-conforming interfaces | Abstract | | We are interested in the approximation of partial differential equations on domains decomposed into two (or several) subdomains featuring non-conforming interfaces. The non-conformity may be due to different meshes and/or different polynomial degrees used from the two sides, or even to a geometrical mismatch. Across each interface, one subdomain is identified as master and the other as slave. We consider Galerkin methods for the discretization (such as finite element or spectral element methods) that make use of two interpolants for transferring information across the interface: one from master to slave and another one from slave to master. The former is used to ensure continuity of the primal variable (the problem solution), while the latter for the dual variable (the normal flux). In particular, since the dual variable is expressed in weak form, we first compute a strong representation of the dual variable from the slave side, interpolate it, transform the interpolated quantity back into weak form and assign it to the master side. In case of slightly non-matching geometries, we use a radial-basis function interpolant instead of Lagrange interpolant.
We name the proposed method INTERNODES (INTERpolation for NOnconforming DEcompositionS). It can be regarded as an alternative to the mortar element method and it is much simpler to implement in a numerical code. We show on two dimensional problems that by using the Lagrange interpolation we obtain at least as good convergence results as with the mortar element method with any order of polynomials. When using low order polynomials, the radial-basis interpolant leads to the same convergence properties as the Lagrange interpolant. We conclude with a comparison between INTERNODES and a standard conforming approximation in a three dimensional case. |
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42/2015 - 09/07/2015
Brugiapaglia, S.; Nobile, F.; Micheletti, S.; Perotto, S.
A theoretical study of COmpRessed SolvING for advection-diffusion-reaction problems | Abstract | | We present a theoretical analysis of the CORSING (COmpRessed SolvING) method for the numerical approximation of partial differential equations based on compressed sensing. In particular, we show that the best s-term approximation of the weak solution of a PDE with respect to an orthonormal system of N trial functions, can be recovered via a Petrov-Galerkin approach using m ? N orthonormal test functions. This recovery is guaranteed if the local a-coherence associated with the bilinear form and the selected trial and test bases fulfills suitable decay properties. The fundamental tool of this analysis is the restricted inf-sup property, i.e., a combination of the classical inf-sup condition and the well-known restricted isometry property of compressed sensing. |
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41/2015 - 08/28/2015
Quarteroni, A.; Veneziani, A.; Vergara, C.
Geometric multiscale modeling of the cardiovascular system, between theory and practice | Abstract | | This review paper addresses the so called geometric multiscale approach for the numerical simulation of blood flow problems, from its origin (that we can identify in 1997) to our days. By this approach the blood fluid-dynamics in the whole circulatory system is described mathematically by means of heterogeneous featuring different degree of detail and different geometric dimension that interact together through appropriate interface coupling conditions.
Our review starts with the introduction of the stand-alone problems, namely the 3D fluid-structure interaction problem,
its reduced representation by means of 1D models, and the so-called lumped parameters (aka 0D) models, where only the dependence on time survives.
We then address specific methods for stand-alone models when the available boundary data are not enough to ensure the mathematical well posedness. These so-called ``defective problems'' naturally arise in practical applications of clinical relevance but also because of the interface coupling of heterogeneous problems that are generated by the geometric multiscale process.
We also describe specific issues related to th boundary treatment of reduced models, particularly relevant to the multiscale coupling.
Next, we detail the most popular numerical
algorithms for the solution of these problems.
Finally, we review some of the most representative works - from different research groups - which addressed the geometric multiscale approach in the past years.
A proper treatment of the different scales relevant to the hemodynamics and their interplay is essential for the accuracy of numerical simulations and eventually for their clinical impact. This paper aims at providing a state-of-the-art picture of these topics, where the gap between theory and practice demands rigorous mathematical models to be reliably filled. |
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40/2015 - 08/24/2015
Patelli, A.S.; Dedè, L.; Lassila, T.; Bartezzaghi, A.; Quarteroni, A.
Isogeometric approximation of cardiac electrophysiology models on surfaces: an accuracy study with application to the human left atrium | Abstract | | We consider Isogeometric Analysis in the framework of the Galerkin method for the spatial approximation of cardiac electrophysiology models defined on NURBS surfaces; specifically, we perform a numerical comparison between basis functions of degree p ? 1 and globally C^k-continuous, with k = 0 or p ? 1, to find the most accurate approximation of a propagating front with the minimal number of degrees of freedom.We show that B-spline basis functions of degree p ? 1, which are C^(p?1)-continuous capture accurately the front velocity of the transmembrane potential even with moderately refined meshes; similarly, we show that,for accurate tracking of curved fronts, high-order continuous B-spline basis functions should be used. Finally, we apply Isogeometric Analysis to an idealized human left atrial geometry described by NURBS with physiologically sound fiber directions and anisotropic conductivity tensor to demonstrate that the numerical scheme retains its favorable approximation properties also in a more realistic setting. Keywords: Isogeometric Analysis, cardiac electrophysiology, surface PDEs, high-order approximation |
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