Multiscale modeling

Résumé

Multiscale modeling or multiscale mathematics is the field of solving problems that have important features at multiple scales of time and/or space. Important problems include multiscale modeling of fluids, solids, polymers, proteins, nucleic acids as well as various physical and chemical phenomena (like adsorption, chemical reactions, diffusion). An example of such problems involve the Navier–Stokes equations for incompressible fluid flow. \begin{array}{lcl} \rho_0(\partial_t\mathbf{u}+(\mathbf{u}\cdot\nabla)\mathbf{u})=\nabla\cdot\tau, \ \nabla\cdot\mathbf{u}=0. \end{array} In a wide variety of applications, the stress tensor \tau is given as a linear function of the gradient \nabla u. Such a choice for \tau has been proven to be sufficient for describing the dynamics of a broad range of fluids. However, its use for more complex fluids such as

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https://en.wikipedia.org/wiki/Multiscale_modeling

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Optimization based methods for highly heterogeneous multiscale problems and multiscale methods for elastic waves

Multiscale or multiphysics partial differential equations are used to model a wide range of physical systems with various applications, e.g. from material and natural science to problems in biology or engineering. When the ratio between the smallest scale in the problem and the size of the physical domain (also the size of the solution) is very large, the numerical approximation of the effective behaviour with classical numerical methods, such as the finite element method (FEM), can become computationally prohibitive. Indeed, as the smallest scale in the problem has to be fully resolved, one obtains a discretization of the computational domain with a very large number of degrees of freedom. In the first part of the thesis, we derive a finite element heterogeneous multiscale method (FE-HMM) applied to the wave equation in a linear elastic medium. We state the FE-HMM and give robust a priori error estimates with explicit convergence rates for the macro and micro discretizations. For simplicity, we start with the static highly heterogeneous linear problem and, then, add the time dependency and consider the wave propagation in a highly heteorgeneous linear elastic medium. In the second part of the thesis we are interested in problems in which the scales are well separated only in some regions of the computational domain, with possibly a continuum of scales in the complementary domain. Such problems arise in various situations, for example in heterogeneous composite materials whose effective properties can be well captured by assuming a (locally) periodic microstructure that can however not be valid near defects of the material. In our modeling, the smallest scale is supposed to be still discretized at the continuum level, but for some applications atomistic scale should be considered. Our coupling method is based on a domain decomposition into a family of overlapping domains. Virtual (interface) controls are introduced as boundary conditions, and act as unknown traces or fluxes. Our method is formulated as a minimization problem with states equations as constraints. The optimal boundary controls of two overlapping domains are found by an heterogeneous optimization problem that is based on minimizing the discrepancy between the two models on, at first, the overlapping region, and at second, over the boundary of the overlapping region. The fully discrete optimization based method couples the continuous or discontinuous Galerkin FE-HMM with the FEM. The well-posedness of our method, in continuous and discrete forms, are established and (fully discrete) a priori error estimates are derived.

EPFL2017

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The purpose of this thesis is to investigate methods for the solution of multiscale problems both from the mathematical and numerical point of view, with a particular concern on applications to flows through heterogeneous porous media. After an overview of the recent developments in this vast field, a study of two representative multiscale techniques is carried out. The multiscale finite element method and a conjugate gradient iterative method preconditioned with an overlapping Schwarz domain decomposition preconditioner are compared in the one-dimensional case. Both methods are well suited for parallel environments and have comparable performance and accuracy. Then, we focus our attention on an aggregation-based two-level overlapping Schwarz domain decomposition preconditioner. We study its theoretical properties and show its robustness to mesh refinement as well as to strong variations in the multiscale coefficients of our model problem. We carry out a convergence analysis to give upper bounds for the condition number of the preconditioned linear system arising from the finite element discretization of the problem at hand. We make explicit the relation between the multiscale coefficient and the coarse space basis functions and show that the condition number can be bounded independently of the ratio of the values of the multiscale coefficient even when the discontinuities in the coefficient are not resolved by the coarse mesh. A new aggregation algorithm is proposed according to the suggestions issuing from the theory which builds a coarse space able to cope with the multiscale coefficient. Numerical experiments on various configurations show that the bounds are sharp and that the method is robust with respect to strong variations. Finally, an application of this preconditioning technique to a two-phase flow problem is presented in order to investigate its performance.

EPFL2007

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We develop a discretisation and solution technique for elliptic problems whose solutions may present strong variations, singularities, boundary layers and oscillations in localised regions. We start with a coarse finite element discretisation with a mesh size H, and we superpose to it local patches of finite elements with finer mesh size h < H to capture local behaviours of the solution. The two meshes (coarse and patch) are not necessarily compatible. The algorithm used to compute the finite element solution on the coarse mesh and patch falls in the class of subspace correction methods [50, 48]. This technique has been introduced in [23]. Similarly to mesh adaptation methods, the location of the fine patches is identified by an a posteriori error estimator. Unlike mesh adaptation, no re-meshing is involved. We discuss the implementation and illustrate the method on an industrial problem. Moreover we generalise the algorithm to several patches which leads to further applications in multi-scale problems.

EPFL2007

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Optimization based methods for highly heterogeneous multiscale problems and multiscale methods for elastic waves

EPFL2017