October 22 – Théophile Chaumont-Frelet: A posteriori error estimates for Maxwell’s equations based on flux quasi-equilibration

Théophile Chaumont-Frelet: Thursday 22 October at 11:00, A415, Inria Paris. I will present of a novel a posteriori estimator for finite element discretizations of Maxwell’s equations. The construction hinges on a modification of the flux equilibration technique, called quasi-equilibration. The resulting estimator is inexpensive to compute and polynomial-degree-robust, which means that the reliability and efficiency constants are independent of the discretization order. I will first describe the standard flux equilibration technique for the simpler case of Poisson’s problem, and explain why it is hard to directly apply this idea to Maxwell’s equations. Then, I will present in detail the derivation of the proposed estimator through the quasi-equilibration procedure. Numerical examples highlighting the key features of the estimator will be presented, and followed by concluding remarks.

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October 15 – Florent Hédin: A hybrid high-order (HHO) method with non-matching meshes in discrete fracture networks

Florent Hédin: Thursday 15 October at 11:00, Gilles Kahn 1, Inria Paris. We are interested in efficient numerical methods for solving flow in large scale fractured networks. Fractures are ubiquitous in the subsurface. Flow in fractured rocks are of interest for many applications (water resources, geothermal applications, oil/gas extraction, nuclear waste disposal). The networks are modeled as Discrete Fractures Networks (DFN). The main challenges of such flow simulations are the uncertainty regarding the geometry and properties of the subsurface, the observed wide range of fractures length (from centimeters to kilometers) and the number of fractures (from thousands to millions of fractures). In natural rocks, flow is highly channelled, which motivates to mesh finely the fractures that carry most of the flow, and coarsely the remaining fractures. But independent triangular mesh generation from one fracture to another yields non matching triangles at the intersections between fractures. Mortar methods have been developed in the past years to deal with non matching grids. In this presentation, we propose an alternative based on the recent HHO method which naturally handles general meshes (polygons/polyhedral) and face polynomials of order k ≥ 0. Combined with refining/coarsening strategies, we will show how the HHO method allows to save computational time in DFN flow simulations.

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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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October 16 – Nicolas Pignet: Hybrid High-Order method for nonlinear solid mechanics

Nicolas Pignet: Wednesday 16 October at 14:00, A115 Inria Paris. In this thesis, we are interested in the devising of Hybrid High-Order (HHO) methods for nonlinear solid mechanics. HHO methods are formulated in terms of face unknowns on the mesh skeleton. Cell unknowns are also introduced for the stability and approximation properties of the method. HHO methods offer several advantages in solid mechanics: (i) primal formulation; (ii) free of volumetric locking due to incompressibility constraints; (iii) arbitrary approximation order k>=1 ; (iv) support of polyhedral meshes with possibly non-matching interfaces; and (v) attractive computational costs due to the static condensation to eliminate locally cell unknowns while keeping a compact stencil. In this thesis, primal HHO methods are devised to solve the problem of finite hyperelastic deformations and small plastic deformations. An extension to finite elastoplastic deformations is also presented within a logarithmic strain framework. Finally, a combination with Nitsche’s approach allows us to impose weakly the unilateral contact and Tresca friction conditions. Optimal convergence rates of order h^{k+1} are proved in the energy-norm. All these methods have been implemented in both the open-source library DiSk++ and the open-source industrial software code_aster. Various two- and three-dimensional benchmarks are considered to validate these methods and compare them with H¹-conforming and mixed finite element methods.

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September 27 – Ivan Yotov: A nonlinear Stokes-Biot model for the interaction of a non-Newtonian fluid with poroelastic media

Ivan Yotov: Friday 27 September at 15:00, A415 Inria Paris. A nonlinear model is developed for fluid-poroelastic structure interaction with quasi-Newtonian fluids that exhibit a shear-thinning property. The flow in the fluid region is described by the Stokes equations and in the poroelastic medium by the quasi-static Biot model. Equilibrium and kinematic conditions are imposed on the interface. A mixed Darcy formulation is employed, resulting in continuity of flux condition of essential type, which is weakly enforced through a Lagrange multiplier. We establish existence and uniqueness of the solution of the weak formulation using non-Hilbert spaces. A stability and error analysis is performed for the semi-discrete continuous-in-time and the fully discrete formulations. The model is further coupled with an advection-diffusion equation for modeling transport of chemical species within the fluid. Applications to hydraulic fracturing and arterial flows are presented.

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September 5 – Koondi Mitra: A fast and stable linear iterative scheme for nonlinear parabolic problems

Koondi Mitra: Thursday 5 September at 15:00, A415 Inria Paris. We consider nonlinear parabolic problems of the general form ∂tb(u) + ∇ · F(u) = ∇ · [D(u)∇p] + S(u), where the variable p can be expressed as p = f(u,∂u). Such problems find applications in nonlinear diffusion, crowd dynamics and for the main focus of this work, flow through porous media. Due to lack of time regularity of the solutions, backward Euler method is used to discretize such equations in time which gives a sequence of nonlinear elliptic problems. Linear iterative schemes such as the Newton or the Picard scheme are often used to solve the nonlinear elliptic problems. However, their stability can only be ensured under strong constraints on the time step size and provided that the problem does not become degenerate. An alternative is the L-scheme, discussed in [2, 3], which guarantees convergence of the iterations even for degenerate cases with a minor constraint in time step size. However, it is considerably slower compared to the Newton and the Picard scheme [3]. Since for nonlinear parabolic problems, we have a good initial guess to start the iterations in the form of the solution of the previous time step, we propose a modified version of the L-scheme [1] that takes this into account. It is proved that it converges linearly even for degenerate cases with a minor constraint in time step size. The linear convergence rate is propotional to an exponent of the time step size for this scheme, which in practice makes it faster than both the L-scheme and the Picard scheme, and more stable than the Newton and the Picard scheme. This is supported by numerical computations. The scheme is extended to many different problems such as the two phase flow problem and domain decomposition methods.…

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July 11 – Jose Fonseca: Towards scalable parallel adaptive simulations with ParFlow

Jose Fonseca: Thursday 11 July at 11:00, A415 Inria Paris. The accurate simulation of variably saturated flow in a porous media is a valuable component in understanding physical processes occurring in many water resources problems. Such simulations require expensive and extensive computations and efficient usage of the latest high performance parallel computing systems becomes a necessity. The simulation software ParFlow has been shown to have excellent solver scalability for up to 16k processes. In order to scale the code to the full size of current petascale systems, we have reorganized its mesh subsystem to use state of the art mesh refinement and partition algorithms provided by the parallel software library p4est. Evaluating the scalability and performance of our modified version of ParFlow, we demonstrate weak and strong scaling to over 458k processes of the Juqueen supercomputer at the Jülich Supercomputing Centre. In the first part of the talk we will briefly present the algorithmic approach employed to couple both libraries. The enhanced scalability results of ParFlow’s modified version were obtained for uniform meshes. Hence, without explicitly exploiting the adaptive mesh refinement (AMR) capabilities of p4est. We will finish this first part presenting our current efforts to enable the usage of locally refined meshes in ParFlow. In an AMR framework. In such case, the finite difference (FD) method taken by ParFlow will require modifications to correctly deal with different size elements. Mixed finite elements (MFE) are on the other hand better suited for the usage of AMR. It is known that the cell centered FD method used in ParFlow might be reinterpreted as a MFE discretization using Raviart-Thomas elements of lower order. We conclude this talk presenting a block preconditioner for saddle point problems arising from a MFE that retains its robustness in the case of locally refined meshes.

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June 6 – Quanling Deng: High-order generalized-alpha methods and splitting schemes

Quanling Deng: Thursday 6 June at 11:00, A415 Inria Paris. The well-known generalized-alpha method is an unconditionally stable and second-order accurate time-integrator which has a feature of user-control on numerical dissipation. The method encompasses a wide range of time-integrators, such as the Newmark method, the HHT-alpha method by Hilber, Hughes, and Taylor, and the WBZ-alpha method by Wood, Bossak, and Zienkiewicz. The talk starts with the simplest time-integrator, forward/backward Euler schemes, then introduces Newmark’s idea followed by the ideas of Chung and Hulbert on the generalized-alpha method. For parabolic equations, we show that the generalized-alpha method also includes the BDF-2 and the second-order dG time-integration scheme. The focus of the talk is to introduce two ideas to generalize the method further to higher orders while maintaining the features of unconditional stability and dissipation control. We will show third-order (for parabolic equations) and 2n-order (for hyperbolic equations) accurate schemes with numerical validations. The talk closes with the introduction of a variational-splitting framework for these time-integrators. As a consequence, the splitting schemes reduce the computational costs significantly (to linear cost) for multi-dimensional problems.

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April 12 – Menel Rahrah: Mathematical modelling of fast, high volume infiltration in poroelastic media using finite elements

Menel Rahrah: Friday 12 April at 14:30, A315 Inria Paris. Fast, High Volume Infiltration (FHVI) is a new method to quickly infiltrate large amounts of fresh water into the shallow subsurface. This infiltration method would have a great value for the effective storage of rainwater in the underground, during periods of extreme precipitation. To describe FHVI, a model for aquifers is considered in which water is injected. Water injection induces changes in the pore pressure and deformations in the soil. Furthermore, the interaction between the mechanical deformations and the flow of water gives rise to a change in porosity and permeability, which results in nonlinearity of the mathematical problem. Assuming that the deformations are very small, Biot’s theory of linear poroelasticity is used to determine the local displacement of the skeleton of a porous medium, as well as the fluid flow through the pores. The resulting problem needs a considerate numerical methodology in terms of possible nonphysical oscillations. Therefore, a stabilised Galerkin finite element method based on Taylor-Hood elements is developed. Subsequently, the impact of mechanical oscillations and pressure pulses on the amount of water that can be injected into the aquifer is investigated. In addition, a parameter uncertainty quantification is applied using Monte Carlo techniques and statistical analysis, to quantify the impact of variation in the parameters (such as the unknown oscillatory modes and the soil characteristics) on the model output. Since the assumption that the deformations are very small can be violated by imposing large mechanical oscillations, the difference between the linear and the nonlinear poroelasticity equations is analysed in a moving finite element framework using Picard iterations.

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