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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13 January – Koondanibha Mitra : A posteriori estimates for nonlinear degenerate parabolic and elliptic equations

Koondanibha Mitra Thursday 13th January at 11:00   ABSTRACT: Nonlinear advection-diffusion-reaction equations are used to model various complex flow processes in porous media, and in biological systems. They also exhibit parabolic-hyperbolic and parabolic-elliptic kinds of degeneracies resulting in the loss of regularity of the solutions. The nonlinear degenerate nature of the equations makes it challenging to provide sharp error bounds to any numerical solutions of the problem. When discretized in time, such equations result in a sequence of nonlinear degenerate elliptic problems which requires linear iterative schemes to solve. The linear iterates can be used to provide upper/lower bounds to the error, and to separate the error contributions due to linearization and discretization. However, the nonlinearity, as before, impedes the derivation of sharp error bounds in the standard error norm. In the first part of this study, we provide reliable, fully computable, and locally space-time efficient a posteriori error bounds for numerical approximations of such nonlinear degenerate parabolic problems. For showing global reliability, a nonlocal-in-time error estimate is derived individually for the time-integrated $H^1(H^{-1})$, $L^2(L^2)$, and the $L^2(H^1)$ errors. A maximum principle and a degeneracy estimator are employed for the last one. Global and local space-time efficiency error bounds are then obtained in a standard $H^1(H^{-1})\cap L^2(H^1)$ norm. The reliability and efficiency norms employed coincide when there is no nonlinearity. Moreover, error contributors such as flux nonconformity, time discretization, quadrature, and data oscillation are identified and separated. Numerical tests are conducted for nondegenerate and degenerate cases having exact solutions, as well as for realistic cases. It is shown that the estimators correctly identify the errors up to a factor of the order of unity. In the second part, using linear iterative schemes, we derive reliable, fully computable, and efficient error bounds for the finite element solution of the elliptic problem which…

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10 December – Gregor Gantner: Applications of a space-time first-order system least-squares formulation for parabolic PDEs.

Gregor Gantner Friday 10th December at 11:00   ABSTRACT: Currently, there is a growing interest in simultaneous space-time methods for solving parabolic evolution equations. The main reasons are that, compared to classical time-marching methods, space-time methods are much better suited for massively parallel implementation, are guaranteed to give quasi-optimal approximations from the employed trial space, have the potential to drive optimally converging simultaneously space-time adaptive refinement routines, and they provide enhanced possibilities for reduced order modelling of parameter-dependent problems. On the other hand, space-time methods require more storage. This disadvantage however vanishes for problems of optimal control, for which the solution is needed simultaneously over the whole time interval anyway. While the common space-time variational formulation of a parabolic equation results in a bilinear form that is non-coercive, [1] recently proved the well-posedness of a space-time first-order system least-squares formulation of the heat equation. Least-squares formulations always correspond to a symmetric and coercive bilinear form. In particular, the Galerkin approximation from any conforming trial space exists and is a quasi-best approximation. Additionally, the least-squares functional automatically provides a reliable and efficient error estimator. In [2], we have generalized the least-squares method of [1] to general second-order parabolic PDEs with possibly inhomogeneous Dirichlet or Neumann boundary conditions. For homogeneous Dirichlet conditions, we present in this talk convergence of a standard adaptive finite element method driven by the least-squares estimator, which has also been demonstrated in [2]. The convergence analysis is applicable to a wide range of least-squares formulations for other PDEs, answering a long-standing open question in the literature. Moreover, we employ the space-time least-squares method for parameter-dependent problems as well as optimal control problems. In both cases, the coercivity of the corresponding bilinear form plays a crucial role. [1] T. Führer and M. Karkulik. Space–time least-squares finite elements for parabolic equations.…

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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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3 June – Oliver Sutton: High order, mesh-based multigroup discrete ordinates schemes for the linear Boltzmann transport problem.

Oliver Sutton: Thursday 3 June at 11:00 am   ABSTRACT: The linear Boltzmann problem is a widely used model for the transport of particles through a scattering medium, such as neutrons in a nuclear reactor, or photons during radiotherapy or in the atmosphere of a star. The key challenge in simulating such phenomena using this model lies in the fact that the problem is an integro-partial-differential equation in 6 independent variables: three position variables and three momentum variables (7 if time is also included). Despite this, there is a long history of this model being successfully applied in practice. A well-studied class of numerical schemes for simulating phenomena governed by this model couples a discontinuous Galerkin spatial discretisation with a multigroup discrete-ordinates discretisation in the momentum variables. Standard multigroup discrete ordinates discretisations may be viewed as employing a piecewise constant approximation space, and possess the particularly attractive characteristic of decoupling the fully-coupled problem into a sequence of three-dimensional linear transport problems which may be solved independently and in parallel. In this talk, we will discuss a new generalisation of these multigroup discrete ordinates schemes. These new schemes employ arbitrarily high order polynomials in the discretisation of the momentum variables, providing high order convergence properties, and offer a familiar Galerkin framework for their analysis. Crucially, moreover, they retain the simple algorithmic structure of their classical counterparts.

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29 April – Lorenzo Mascotto: Enriched nonconforming virtual element methods

Lorenzo Mascotto: Thursday 29 April at 11:00 am ABSTRACT: Solutions to elliptic partial differential equations (PDEs) on smooth domains with smooth data are smooth. However, when solving a PDE on a Lipschitz polygon, its solution is singular at the vertices of the domain. The singular behaviour is known a priori: the solution can be split as a sum of a smooth term, plus series of singular terms that belong to the kernel of the differential operator appearing in the PDE. The virtual element method (VEM) is a generalization of the finite element method (FEM) to polygonal/polyhedral meshes and is based on approximation spaces consisting of solutions to local problems mimicking the target PDE and has been recently generalized to the extended VEM (XVEM). Here, the approximation spaces are enriched with suitable singular functions. A partition of unity (PU) is used to patch local spaces. The approach of the XVEM is close to that of the extended FEM. In this talk, we present a new paradigm for enriching virtual element (VE) spaces. Instead of adding special functions to the global space and eventually patch local spaces with a PU, we modify the definition of the local spaces by tuning the boundary conditions of local problems. By doing this, local VE spaces contain the desired singular functions, but there is no need to patch them with a PU. This results in an effective and slight modification of the already existing implementation of VE codes, as well as in a natural extension for existing theoretical results.

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1 April – André Harnist: Improved error estimates for Hybrid High-Order discretizations of Leray–Lions problems

André Harnist: Thursday 1 April at 11:00 am Abstract: We consider Hybrid High-Order (HHO) approximations of Leray-Lions problems set in W^(1,p) with p in (1,2]. For this class of problems, negative powers of the gradient of the solution can appear in the flux. Depending on the expression of the latter, this can lead to a degeneracy of the problem when the gradient of the solution vanishes or becomes large. The goal of this presentation is to derive novel error estimates depending on the degeneracy of the problem inspired by [1,2,3]. Specifically, we show that, for the globally non-degenerate case, the energy-norm of the error has a convergence rate of (k+1), with k denoting the degree of the HHO approximation. In the globally degenerate case, on the other hand, the energy-norm of the error converges with a rate of (k+1)(p-1), coherently with the estimate originally proved in [4]. We additionally introduce, for each mesh element, a dimensionless number that captures the local degeneracy of the model and identifies the contribution of the element to the global error: from the fully degenerate regime, corresponding to a contribution in (k+1)(p-1), to the non-degenerate regime, corresponding to a contribution in (k+1), through all intermediate regimes. These regime-dependent error estimates are illustrated by a complete panel of numerical experiments. [1] M. Botti, D. Castanon Quiroz, D. A. Di Pietro and A. Harnist, A Hybrid High-Order method for creeping flows of non-Newtonian fluids, Submitted. 2020. URL: https://hal.archives-ouvertes.fr/hal-02519233. [2] D. A. Di Pietro and J. Droniou, The Hybrid High-Order Method for Polytopal Meshes. Modelling, Simulation and Application 19. Springer International Publishing, 2020. ISBN: 978-3-030-37202-6 (Hardcover) 978-3-030-37203-3 (eBook). DOI: 10.1007/978-3-030-37203-3. [3] D. A. Di Pietro and J. Droniou, A Hybrid High-Order method for Leray–Lions elliptic equations on general meshes, Math. Comp., volume 86, 2017, number 307, pages 2159–2191,…

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