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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21/2014 - 06/09/2014
Artina, M.; Fornasier, M.; Micheletti, S.; Perotto, S
The benefits of anisotropic mesh adaptation for brittle fractures under plane-strain conditions | Abstract | | We develop a reliable a posteriori anisotropic first order estimator for the numerical simulation of the Francfort and Marigo model of brittle fracture, after its approximation by means of the Ambrosio-Tortorelli variational model. We show that an adaptive algorithm based on this estimator reproduces all the previously obtained well-known benchmarks on fracture development with particular attention to the fracture directionality. Additionally, we explain why our method, based on an extremely careful tuning of the anisotropic adaptation, has the potential of outperforming significantly in terms of numerical complexity the ones used to achieve similar degrees of accuracy in previous studies. |
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20/2014 - 05/26/2014
Artina, M.; Fornasier, M.; Micheletti, S.; Perotto, S.
Anisotropic mesh adaptation for crack detection in brittle materials | Abstract | | The quasistatic brittle fracture model proposed by G. Francfort and J.-J. Marigo can be Gamma-approximated at each time evolution step by the Ambrosio-Tortorelli functional. In this paper, we focus on a modification of this functional which includes additional constraints via penalty terms to enforce the irreversibility of the fracture as well as the applied displacement field. Secondly, we build on this variational model an adapted discretization to numerically compute the time-evolving minimizing solution. We present the derivation of a novel a posteriori error estimator driving the anisotropic adaptive procedure. The main properties of these automatically generated meshes are to be very fine and strongly anisotropic in a very thin neighborhood of the crack, but only far away from the crack tip, while they show a highly isotropic behavior in a neighborhood of the crack tip instead. As a consequence of these properties, the resulting discretizations follow very closely the propagation of the fracture, which is not significantly influenced by the discretization itself, delivering a physically sound prediction of the crack path, with a reasonable computational effort. In fact, we provide numerical tests which assess the balance between accuracy and complexity of the algorithm. We compare our results with isotropic mesh adaptation and we highlight the remarkable improvements both in terms of accuracy and computational cost with respect to simulations in the pertinent most recent literature. |
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19/2014 - 05/22/2014
Bonaventura, L.; Ferretti, R.
Semi-Lagrangian methods for parabolic problems in divergence form | Abstract | | Semi-Lagrangian methods have traditionally been developed in the framework of hyperbolic equations, but several extensions of the Semi-Lagrangian approach to diffusion and advection-diffusion problems have been proposed recently. These extensions are mostly based on probabilistic arguments and share the common feature of treating second-order operators in trace form, which makes them unsuitable for mass conservative models like the classical formulations of turbulent diffusion employed in computational fluid dynamics. We propose here some basic ideas for treating second-order operators in divergence form. A general framework for constructing consistent schemes in one space dimension is presented, and a specific case of nonconservative discretization is discussed in detail and analysed. Finally, an extension to (possibly nonlinear) problems in an arbitrary number of dimensions is proposed. Although the resulting discretization approach is only of first order in time, numerical results in a number of test cases highlight the advantages of these methods for applications to computational fluid dynamics and their superiority over to more standard low order time discretization approaches. |
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18/2014 - 05/21/2014
Tumolo, G.; Bonaventura, L.
An accurate and efficient numerical framework for adaptive numerical weather prediction | Abstract | | We present an accurate and efficient discretization approach for the adaptive discretization of typical model equations employed in numerical weather prediction. A semi-Lagrangian approach is combined with the TR-BDF2 semi-implicit time discretization method and with a spatial discretization based on adaptive discontinuous finite elements. The resulting method has full second order accuracy in time and can employ polynomial bases of arbitrarily high degree in space, is unconditionally stable and can effectively adapt the number of degrees of freedom employed in each element, in order to balance accuracy and computational cost. The p-adaptivity approach employed does not require remeshing, therefore it is especially suitable for applications, such as numerical weather prediction, in which a large number of physical quantities are associated with a given mesh. Furthermore, although the proposed method can be implemented on arbitrary unstructured and nonconforming meshes, even its application on simple Cartesian meshes in spherical coordinates can cure effectively the pole problem by reducing the polynomial degree used in the polar elements. Numerical simulations of classical benchmarks for the shallow water and for the fully compressible Euler equations validate the method and demonstrate its capability to achieve accurate results also at large Courant numbers, with time steps up to 100 times larger than those of typical explicit discretizations of the same problems, while reducing the computational cost thanks to the adaptivity algorithm. |
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17/2014 - 05/06/2014
Discacciati, M.; Gervasio, P.; Quarteroni, A.
Interface Control Domain Decomposition (ICDD) Method for Stokes-Darcy coupling | Abstract | | We propose the ICDD method to solve the coupling between Stokes and Darcy equations. According to this approach, the problem is formulated as an optimal control problem whose control variables are the traces of the velocity and the pressure on the internal boundaries of the subdomains that provide an overlapping decomposition of the original computational domain. A theoretical analysis is carried out, an efficient solution algorithm is proposed, and several numerical tests are implemented. Our results show the accuracy of the ICDD methos, its computational efficiency and robustness with respect to the different parameters involved (grid-size, polynomial degrees, permeability of the porous domain, thickness of the overlapping region). The ICDD approach is more versatile and easier to implement than the model based on the Beavers, Joseph and Saffman coupling conditions. |
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16/2014 - 05/05/2014
Dede, L.; Jaggli, C.; Quarteroni, A.
Isogeometric numerical dispersion analysis for elastic wave propagation | Abstract | | In this paper, we carry out a numerical dispersion analysis for the linear elastodynamics equations approximated by means of NURBS-based Isogeometric Analysis in the framwork of the Galerkin method; specifically, we consider the analysis of harminic plane waves in an isotropic and homogeneous elastic medium. We compare and discuss the errors associated to the compressional and shear wave velocities and we provide the anisotropic curves for numerical approximations obtained by considerin B-splines and NURBS basis functions of different regularity, namely globally C^0- and C^(p-1)- continuous, being p the polynomial degree. We conclude our analysis by numerically simulating the seismic wave propagation in a sinusoidal shaped valley with discontinuous elastic parameters across an internal interface. |
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15/2014 - 05/02/2014
Esfandiar, B.; Porta, G.; Perotto, S.; Guadagnini, A;
Anisotropic mesh and time step adaptivity for solute transport modeling in porous media | Abstract | | We assess the impact of space-time mesh adaptivity on the modeling of solute transport in porous media. This approach allows an automatic selection of both the spatial mesh and the time step on the basis of a suitable recovery-based error estimator. In particular, we deal with an anisotropic control of the spatial mesh. The solver coupled with the adaptive module deals with an advection-dispersion equation to model the transport of dissolved species, which are assumed to be convected by a Darcy flow field. The whole solution-adaptation procedure is assessed through two-dimensional numerical tests. A numerical convergence analysis of the spatial mesh adaptivity is first performed by considering a test-case with analytical solution. Then, we validate the space-time adaptive procedure by reproducing a set of experimental observations associated with solute transport in a homogeneous sand pack. The accuracy and the efficiency of the methodology are discussed and numerical results are compared with those associated with fixed uniform space-time discretizations. This assessment shows that the proposed approach is robust and reliable. In particular, it allows us to obtain a significant improvement of the simulation quality of the early solute arrivals times at the outlet of the medium. |
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14/2014 - 04/23/2014
Dassi, F.; Formaggia, L.; Zonca, S.
Degenerate Tetrahedra Recovering | Abstract | | Many mesh generation or optimization algorithms could produce a low quality tetrahedral mesh, i.e. a mesh where the tetrahedra have very small solid or dihedral angles. In this paper, we propose a series of operations to recover these degenerate tetrahedra. In particular, we will focus on the standard shapes of these undesired mesh elements (sliver, cap, wedge and spade) and, for each of these configurations, we apply a suitable sequence of classical mesh modification procedures to get a higher quality mesh. The reliability of the proposed mesh optimization algorithm is numerically proved with some examples. |
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