13th June – Jean-luc Guermond: Invariant-domain preserving IMEX time stepping methods

Chérif Amrouche Thursday 13th June at 11:00   ABSTRACT: I will present high-order time discretizations of a Cauchy problem where the evolution operator comprises a hyperbolic part and a parabolic part (say diffusion and stiff relaxation terms). The said problem is assumed to possess an invariant domain. I will propose a technique that makes every implicit-explicit (IMEX) time stepping scheme invariant domain preserving and mass conservative. The IMEX scheme is written in incremental form and, at each stage of the scheme, we first compute low-order hyperbolic and parabolic updates, followed by their high-order counterparts. The proposed technique, which is agnostic to the space discretization, allows to optimize the time step restrictions induced by the hyperbolic sub-step. To illustrate the proposed methodology, we derive three novel IMEX schemes with optimal efficiency and for which the implicit scheme is singly-diagonal and L-stable: a third-order, four-stage scheme; and two fourth-order schemes, one with five stages and one with six stages. The novel IMEX schemes are evaluated numerically on a stiff ODE system. We also apply these schemes to nonlinear convection-diffusion problems with stiff reaction and to compressible viscous flows possibly including grey radiation.

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16th June – ‪Chérif Amrouche: Elliptic Problems in Lipschitz and in $C^{1,1}$ Domains

Chérif Amrouche Thursday 16th June at 11:30   ABSTRACT: We are interested here in questions related to the maximal regularity of solutions to elliptic problems with Dirichlet or Neumann boundary conditions (see ([1]). For the last 40 years, many works have been concerned with questions when Ω is a Lipschitz domain. Some of them contain incorrect results that are corrected in the present work. We give here new proofs and some complements for the case of the Laplacian (see [3]), the Bilaplacian ([2] and [6]) and the operator div (A∇) (see ([5]) when A is a matrix or a function. And we extend this study to obtain other regularity results for domains having an adequate regularity. We give also new results for the Dirichlet-to-Neumann operator for Laplacian and Bilaplacian. Using the duality method, we can then revisit the work of Lions-Magenes [4], concerning the so-called very weak solutions, when the data are less regular. References : [1]  C. Amrouche and M. Moussaoui. Laplace equation in smooth or non smooth do- mains. Work in Progress. [2]  B.E.J. Dahlberg, C.E. Kenig, J. Pipher and G.C. Verchota. Area integral estimates for higher-order elliptic equations and systems. Ann. Inst. Fourier, 47-5, 1425– 1461, (1997). [3]  D. Jerison and C.E. Kenig. The Inhomogeneous Dirichlet Problem in Lipschitz Domains, J. Funct. Anal. 130, 161–219, (1995). [4]  J.L. Lions and E. Magenes. Probl`emes aux limites non-homog`enes et applications, Vol. 1, Dunod, Paris, (1969). [5]  J. Necas. Direct methods in the theory of elliptic equations. Springer Monographs in Mathematics. Springer, Heidelberg, (2012). [6]  G.C. Verchota. The biharmonic Neumann problem in Lipschitz domains. Acta Math. 194-2, 217–279, (2005).

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20 January – ‪Isabelle Ramiere‬ : Automatic multigrid adaptive mesh refinement with controlled accuracy for quasi-static nonlinear solid mechanics

Isabelle Ramiere‬ Thursday 20th January at 11:00   ABSTRACT: Many real industrial problems involve localized effects (nonlinearity, contact, heterogenity,…). Adaptive Mesh Refinement (AMR) approaches are well-suited numerical techniques to take into account mesoscale phenomena in simulation processes. For implicit solvers (such as for quasi-static mechanics problems), classical h and/or p-adaptive refinement strategies consisting in generating a unique global mesh locally refined (in mesh step and/or in degree of basis function) are limited by the resulting size of problems to be solved (cf. number of DoFs). Hence, we were interested in local multigrid methods, consisting in adding local refined nested meshes in zones of interest without modifying the initial computation mesh. An iterative process (similar to standard multigrid solvers) enables to correct to various levels solutions. We have extended the multigrid Local Defect Correction (LDC) method (Hackbusch, 1984), initially introduced in Computational Fluid Dynamics, to elastostaticity (Barbié et al., 2014) with a multilevel generalization of the algorithm. In order to automatically detect the zone of interest and hence to avoid the pollution error, the LDC method has been coupled with an a posteriori error estimate of Zienckiewicz-Zhu type (Barbié et al., 2014; Barbié et al., 2015; Liu et al., 2017). We also proposed an original stopping criterion in case of local singularity (Ramière et al., 2019). We have compared in (Koliesnikova et al.,2021) within a unified AMR framework the efficiency of the LDC method with respect to conforming and nonconforming h-adaptive strategies. We have also extended the LDC method to structural mechanics nonlinearities. In (Liu et al., 2017), an efficient algorithm has been developed in order to deal with frictional contact via the LDC method. For nonlinear material behaviours, a one time step algorithm has been first introduced in (Barbié et al., 2015) while a fully automatic algorithm in time with…

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25 November – Pierre Gosselet: Asynchronous Global/Local coupling

Pierre Gosselet: Thursday 25th November at 11:00   ABSTRACT: Non-intrusive global/local coupling can be seen as an exact iterative version of the submodeling (structural zoom) technique widely used by industry in their simulations. A global model, coarse but capable of identifying general trends in the structure, is locally patched by fine models with refined geometries, materials and meshes. The coupling is achieved by alternating Dirichlet resolutions on the patches and global resolutions with a well-chosen immersed Neumann condition. After the preliminary work of (Whitcomb, 1991), the method has been rediscovered by many authors. Our work starts with (Gendre et al., 2009). From a theoretical point of view, the method is related to the optimized Schwarz domain decomposition methods (Gosselet et al., 2018). It has been applied in many contexts (localized or generalized (visco)plasticity, stochastic calculations, cracking, damage, fatigue…). In the ANR project ADOM, we are working on the implementation of an asynchronous version of the method. The expected benefits of asynchronism (Magoulès et al., 2018; Glusa et al., 2020) are to reach the solution faster, to adapt to many computational hardware by being more resilient in case of poor load balancing, network latencies or even outages. During the presentation, I will show how to adapt the global/local coupling to asynchronism and will illustrate its performance on thermal and linear elasticity calculations. This work is realized with the support of National Research Agency, project [ANR-18-CE46-0008]. [1] Gendre, Lionel et al. (2009). “Non-intrusive and exact global/local techniques for structural problems with local plasticity”. In: Computational Mechanics 44.2, pp. 233–245. [2] Glusa, Christian et al. (2020). “Scalable Asynchronous Domain Decomposition Solvers”. In: SIAM Journal on Scientific Computing 42.6, pp. C384–C409. doi: 10.1137/19M1291303. [3] Gosselet, Pierre et al. (2018). “Non-invasive global-local coupling as a Schwarz domain decomposition method: acceleration and generalization”. In: Advanced Modeling and…

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24 November – Grégory Etangsale: A primal hybridizable discontinuous Galerkin method for modelling flows in fractured porous media

Grégory Etangsale: Wednesday 24th November at 10:30   ABSTRACT: Modeling fluid flow in fractured porous media has received tremendous attention from engineering, geophysical, and other research fields over the past decades. We focus here on large fractures described individually in the porous medium, which act as preferential paths or barriers to the flow. Two different approaches are available from a computational aspect: The first one, and definitively the oldest, consists of meshing inside the fracture. In this case, the flow is governed by a single Darcy equation characterized by a large scale of variation of the permeability coefficient within the matrix region and the fracture, respectively. However, this description becomes quite challenging since it requires a considerable amount of memory storage, severely increasing the CPU time. A more recent approach differs by considering the fracture as an encapsulated object of lower dimension, i.e., (d − 1)-dimension. As a result, the flow process is now governed by distinctive equations in the matrix region and fractures, respectively. Thus, coupling conditions are added to close the problem. This mathematical description of the fractured porous media has been initially introduced by Martin et al. in [4] and is referred to as the Discrete Fracture-Matrix (DFM) model. The DFM description is particularly attractive since it significantly simplifies the meshing of fractures and allows the coupling of distinctive discretizations such as Discontinuous and Continuous Galerkin methods inside the bulk region and the fracture network, respectively. For instance, we refer the reader to the recent works of Antonietti et al. [1] (and references therein), where the authors coupled the Interior Penalty DG method with the (standard) H1-Conforming finite element method to solve the DFM problem (see e.g., [3]). However, it is well-known that DG methods are generally more expensive than most other numerical methods due to their high…

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06 September – Rolf Stenberg: Nitsche’s Method for Elastic Contact Problems

Rolf Stenberg: Monday 06th June at 15:00   ABSTRACT: In this talk, we present a priori and a posteriori error estimates for the frictionless contact problem between two elastic bodies. The analysis is built upon interpreting Nitsche’s method as a stabilised finite element method for which the error estimates can be derived with minimal regularity assumptions and without a saturation assumption. The stabilising term corresponds to a master-slave mortaring technique on the contact boundary. The numerical experiments show the robustness of Nitsche’s method and corroborate the efficiency of the a posteriori error estimators. [1] T. Gustafsson, R. Stenberg, J. Videman. On Nitsche’s method for elastic contact problems. SIAM Journal of Scientific Computing. 42 (2020) B425–B446 [2] T. Gustafsson, R. Stenberg, J. Videman. The masters-slave Nitsche method for elastic contact problems. Numerical Mathematics and Advanced Applications – ENUMATH 2019. J.F. Vermolen, C. Vuik, M. Moller (Eds.). Springer Lecture Notes in Computational Science and Engineering. 2021

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17 June – Elyes Ahmed: Adaptive fully-implicit solvers and a posteriori error control for multiphase flow with wells

Elyes Ahmed: Thursday 17 June at 11:00 am   ABSTRACT: Flow is driven by the wells in most reservoir simulation workflows. From a numerical point of view,  wells can be seen as singular source-terms due to their small-scale relative to grid blocks used in field-scale simulation. Near-well models, such as Peaceman model, are used to account for the highly non-linear flow field in the vicinity of the wellbore. The singularities that wells introduce in the solution create difficulties for the gridding strategy and usually result in a less flexible time-stepping strategy to ensure convergence of the nonlinear solver. We present in this work a-posteriori error estimators for multiphase flow with singular well sources. The estimators are fully and locally computable and target the singular effects of wells.  The error estimate uses the appropriate weighted norms, where the weight weakens the norm only around the wells, letting it behave like the usual H^{1} -norm far from the near-well region. The error estimators are used to modify a fully implicit solver in the MATLAB Reservoir Simulation Toolbox (MRST). We demonstrate the benefits of the adaptive implicit solver through a range of test cases.

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March 16 – Bochra Mejri: Topological sensitivity analysis for identification of voids under Navier’s boundary conditions in linear elasticity

Bochra Mejri: Monday 16 March at 15:00, A415 Inria Paris. This talk is concerned with a geometric inverse problem related to the two-dimensional linear elasticity system. Thereby, voids under Navier’s boundary conditions are reconstructed from the knowledge of partially over-determined boundary data. The proposed approach is based on the so-called energy-like error functional combined with the topological sensitivity method. The topological derivative of the energy-like misfit functional is computed through the topological-shape sensitivity method. Firstly, the shape derivative of the corresponding misfit function is presented briefly from previous work. Then, an explicit solution of the fundamental boundary-value problem in the infinite plane with a circular hole is calculated by the Muskhelishvili formulae. Finally, the asymptotic expansion of the topological gradient is derived explicitly with respect to the nucleation of a void. Numerical tests are performed in order to point out the efficiency of the developed approach.

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February 25 – Jakub Both: Robust iterative solvers for thermo-poro-visco-elasticity via gradient flows

Jakub Both: Tuesday 25 February at 15:00, A415 Inria Paris. Coupled flow and mechanical deformation of porous media has been of increased interest in the recent past with applications ranging from geotechnical to biomedical engineering. With increased model complexity a high demand in numerical solvers arises. In this context, physically-based iterative splitting solvers, sequentially solving the physical subproblems, have been widely popular due to their simple implementation and the possibility of reusing existing solver technologies. For unconditional stability however suitable and model-dependent stabilization is typically required. In the previous literature, the main motivation for specific choices has mostly been based on physical intuition. In this talk, a systematic development of such solvers is presented based on mathematical justification. A gradient flow framework is presented for the modeling, analysis, and development of numerical solvers for coupled processes in poroelastic media. Various existing poroelasticity models fall into the framework, e.g., the linear Biot equations but also extensions involving viscoelastic, thermal, and/or nonlinear material laws. Besides of enabling abstract tools for the well-posedness analysis, the approach naturally leads to robust physically-based iterative splitting solvers. Gradient flow formulations are naturally discretized in time using a series of (convex) optimization problems. In the spirit of splitting solvers, we propose applying the fundamental alternating minimization for a systematic and robust decoupling of the physical subproblems. By this we re-discover popular solvers as the undrained and fixed-stress splits for the linear Biot equations, and we also provide novel iterative splittings for more advanced models. A priori convergence is established in a unified fashion utilizing abstract convergence theory for alternating minimization. This is joint work with Kundan Kumar, Jan M. Nordbotten, and Florin A. Radu (all UiB).

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