A multiscale flux basis for mortar mixed discretizations of reduced Darcy-Forchheimer fracture models

In this paper, a multiscale flux basis algorithm is developed to efficiently solve a flow problem in fractured porous media. Here, we take into account a mixed-dimensional setting of the discrete fracture matrix model, where the fracture network is represented as lower-dimensional object. We assume the linear Darcy model in the rock matrix and the non-linear Forchheimer model in the fractures. In our formulation, we are able to reformulate the matrix-fracture problem to only the fracture network problem and, therefore, significantly reduce the computational cost. The resulting problem is then a non-linear interface problem that can be solved using a fixed-point or Newton-Krylov methods, which in each iteration require several solves of Robin problems in the surrounding rock matrices. To achieve this, the flux exchange (a linear Robin-to-Neumann co-dimensional mapping) between the porous medium and the fracture network is done offline by pre-computing a multiscale flux basis that consists of the flux response from each degree of freedom on the fracture network. This delivers a conserve for the basis that handles the solutions in the rock matrices for each degree of freedom in the fractures pressure space. Then, any Robin sub-domain problems are replaced by linear combinations of the multiscale flux basis during the interface iteration. The proposed approach is, thus, agnostic to the physical model in the fracture network. Numerical experiments demonstrate the computational gains of pre-computing the flux exchange between the porous medium and the fracture network against standard non-linear domain decomposition approaches

Adaptive modelling of coupled hydrological processes with application in water management

This paper presents recent results of a network project aiming at the modelling and simulation of coupled surface and subsurface flows. In particular, a discontinuous Galerkin method for the shallow water equations has been developed which includes a special treatment of wetting and drying. A robust solver for saturated-unsaturated groundwater flow in homogeneous soil is at hand, which, by domain decomposition techniques, can be reused as a subdomain solver for flow in heterogeneous soil. Coupling of surface and subsurface processes is implemented based on a heterogeneous nonlinear Dirichlet-Neumann method, using the dune-grid-glue module in the numerics software Dune

An ASM type theory for P.L. Lions algorithm – Optimized Schwarz Methods

International audienceWe endow P.L. Lions domain decomposition method with a Additive Schwarz Method like theory. It is done via the introduction of a new preconditioner, named SORAS (symmetrized Optimized Restricted Additive Schwarz). It yields a scalable parallel solver for Stokes and almost incompressible problems with continuous pressure. Numerical results on large scale 3D problems are given

A massively parallel multi-level approach to a domain decomposition method for the optical flow estimation with varying illumination

We consider a variational method to solve the optical flow problem with varying illumination. We apply an adaptive control of the regularization parameter which allows us to preserve the edges and fine features of the computed flow. To reduce the complexity of the estimation for high resolution images and the time of computations, we implement a multi-level parallel approach based on the domain decomposition with the Schwarz overlapping method. The second level of parallelism uses the massively parallel solver MUMPS. We perform some numerical simulations to show the efficiency of our approach and to validate it on classical and real-world image sequences

A Domain Decomposition Method for the Steady-State Navier-Stokes-Darcy Model with Beavers-Joseph Interface Condition

This paper proposes and analyzes a Robin-type multiphysics domain decomposition method (DDM) for the steady-state Navier-Stokes-Darcy model with three interface conditions. In addition to the two regular interface conditions for the mass conservation and the force balance, the Beavers-Joseph condition is used as the interface condition in the tangential direction. The major mathematical difficulty in adopting the Beavers-Joseph condition is that it creates an indefinite leading order contribution to the total energy budget of the system [Y. Cao et al., Comm. Math. Sci., 8 (2010), pp. 1-25; Y. Cao et al., SIAM J. Numer. Anal., 47 (2010), pp. 4239-4256]. In this paper, the well-posedness of the Navier-Stokes-Darcy model with Beavers-Joseph condition is analyzed by using a branch of nonsingular solutions. By following the idea in [Y. Cao et al., Numer. Math., 117 (2011), pp. 601-629], the three physical interface conditions are utilized together to construct the Robin-type boundary conditions on the interface and decouple the two physics which are described by Navier-Stokes and Darcy equations, respectively. Then the corresponding multiphysics DDM is proposed and analyzed. Three numerical experiments using finite elements are presented to illustrate the features of the proposed method and verify the results of the theoretical analysis

A partitioned Newton method for the interaction of a fluid and a 3D shell structure

We review various fluid-structure algorithms based on domain decomposition techniques and we propose a new one. The standard methods used in fluid-structure interaction problems are generally ``nonlinear on subdomains''. We propose a scheme based on the principle ``linearize first, then decompose''. In other words we extend to fluid-structure problems domain decomposition techniques classically used in nonlinear elasticity

A Scalable Preconditioner for a Primal DPG Method

We show how a scalable preconditioner for the primal discontinuous Petrov-Galerkin (DPG) method can be developed using existing algebraic multigrid (AMG) preconditioning techniques. The stability of the DPG method gives a norm equivalence which allows us to exploit existing AMG algorithms and software. We show how these algebraic preconditioners can be applied directly to a Schur complement system arising from the DPG method. One of our intermediate results shows that a generic stable decomposition implies a stable decomposition for the Schur complement. This justifies the application of algebraic solvers directly to the interface degrees of freedom. Combining such results, we obtain the first massively scalable algebraic preconditioner for the DPG system

Highly parallel multi-physics simulation of muscular activation and EMG

Simulation of skeletal muscle activation can help to interpret electromyographic measurements and infer the behavior of the muscle fibers. Existing models consider simplified geometries or a low number of muscle fibers to reduce the computation time. We demonstrate how to simulate a finely-resolved model of biceps brachii with a typical number of 270.000 fibers. We have used domain decomposition to run simulations on 27.000 cores of the supercomputer HazelHen at HLRS in Stuttgart, Germany. We present details on opendihu, our software framework. Its configurability, efficient data structures and modular software architecture target usability, performance and extensibility for future models. We present good parallel weak scaling of the simulations