Asynchronous global-local non-invasive coupling for linear elliptic problems

This paper presents the first asynchronous version of the Global/Local non-invasive coupling, capable of dealing efficiently with multiple, possibly adjacent, patches. We give a new interpretation of the coupling in terms of primal domain decomposition method, and we prove the convergence of the relaxed asynchronous iteration. The asynchronous paradigm lifts many bottlenecks of the Global/Local coupling performance. We illustrate the method on several linear elliptic problems as encountered in thermal and elasticity studies

Domain decomposition methods for domain composition purpose: Chimera, overset, gluing and sliding mesh methods

Domain composition methods (DCM) consist in obtaining a solution to a problem, from the formulations of the same problem expressed on various subdomains. These methods have therefore the opposite objective of domain decomposition methods (DDM). Indeed, in contrast to DCM, these last techniques are usually applied to matching meshes as their purpose consists mainly in distributing the work in parallel environments. However, they are sometimes based on the same methodology as after decomposing, DDM have to recompose. As a consequence, in the literature, the term DDM has many times substituted DCM. DCM are powerful techniques that can be used for different purposes: to simplify the meshing of a complex geometry by decomposing it into different meshable pieces; to perform local refinement to adapt to local mesh requirements; to treat subdomains in relative motion (Chimera, sliding mesh); to solve multiphysics or multiscale problems, etc. The term DCM is generic and does not give any clue about how the fragmented solutions on the different subdomains are composed into a global one. In the literature, many methodologies have been proposed: they are mesh-based, equation-based, or algebraic-based. In mesh-based formulations, the coupling is achieved at the mesh level, before the governing equations are assembled into an algebraic system (mesh conforming, Shear-Slip Mesh Update, HERMESH). The equation-based counterpart recomposes the solution from the strong or weak formulation itself, and are implemented during the assembly of the algebraic system on the subdomain meshes. The different coupling techniques can be formulated for the strong formulation at the continuous level, for the weak formulation either at the continuous or at the discrete level (iteration-by-subdomains, mortar element, mesh free interpolation). Although the different methods usually lead to the same solutions at the continuous level, which usually coincide with the solution of the problem on the original domain, they have very different behaviors at the discrete level and can be implemented in many different ways. Eventually, algebraic- based formulations treat the composition of the solutions directly on the matrix and right-hand side of the individual subdomain algebraic systems. The present work introduces mesh-based, equation-based and algebraicbased DCM. It however focusses on algebraic-based domain composition methods, which have many advantages with respect to the others: they are relatively problem independent; their implicit implementation can be hidden in the iterative solver operations, which enables one to avoid intensive code rewriting; they can be implemented in a multi-code environment

Co-simulation domain decomposition algorithm for hybrid EMT-Dynamic Phasor modeling

An iterative coupling algorithm based on restricted additive Schwarz domain decomposition is investigated to co-simulate electrical circuits with hybrid electromagnetic (EMT) and transient stability (TS) modeled using dynamic phasors. This co-simulation algorithm does not introduce any delay between the data exchanged at the co-simulation step. The pure linear convergence property of the iterative method allows it to be accelerated towards the true solution by a non-intrusive Aitken's acceleration of the convergence post-processing, even if the domain decomposition interface conditions make the iterative method divergent. This provides a method less sensitive to the splitting. Numerical tests on a linear RLC circuit combining EMT and TS modeling are provided.Comment: 30 pages, 18 figures, 2 table

Domain decomposition methods for the coupling of surface and groundwater flows

The purpose of this thesis is to investigate, from both the mathematical and numerical viewpoint, the coupling of surface and porous media flows, with particular concern on environmental applications. Domain decomposition methods are applied to set up effective iterative algorithms for the numerical solution of the global problem. To this aim, we reformulate the coupled problem in terms of an interface (Steklov-Poincaré) equation and we investigate the properties of the Steklov-Poincaré operators in order to characterize optimal preconditioners that, at the discrete level, yield convergence in a number of iterations independent of the mesh size h. We consider a new approach to the classical Robin-Robin method and we reinterpret it as an alternating direction iterative algorithm. This allows us to characterize robust preconditioners for the linear Stokes/Darcy problem which improve the behaviour of the classical Dirichlet- Neumann and Neumann-Neumann ones. Several numerical tests are presented to assess the convergence properties of the proposed algorithms. Finally, the nonlinear Navier-Stokes/Darcy coupling is investigated and a general nonlinear domain decomposition strategy is proposed for the solution of the interface problem, extending the usual Newton or fixed-point based algorithms

Non-intrusive Coupling: Recent Advances and Scalable Nonlinear Domain Decomposition

This paper provides a detailed review of the global/local non-intrusive coupling algorithm. Such method allows to alter a global finite element model, without actually modifying its corresponding numerical operator. We also look into improvements of the initial algorithm (Quasi- Newton and dynamic relaxation), and provide comparisons based on several relevant test cases. Innovative examples and advanced applications of the non-intrusive coupling algorithm are provided, granting a handy framework for both researchers and engineers willing to make use of such process. Finally, a novel nonlinear domain decomposition method is derived from the global/local non-intrusive cou- pling strategy, without the need to use a parallel code or software. Such method being intended to large scale analysis, we show its scalability. Jointly, an efficient high level Message Passing Interface coupling framework is also proposed, granting an universal and flexible way for easy software coupling. A sample code is also given

Global Local Analysis with Robin Parameters: Applications to Crack Propagation in 2D and 3D Models

Global local analysis is a part of the structural analysis that allows to study, with an iterative solution, a coarse global linear model with a specific zone. This zone is defined as a local model with fine mesh and a non-linear behaviour such as crack propagation. However, the current trend in Global Local analysis is to impose displacements on the fine model to later obtain the reactions that will be applied to the global model for each iteration (Primal to Dual solution algorithm). Therefore, we propose a mixed analysis in the local and global models through the application of Robin conditions on the interface, allowing a higher grade of flexibility for the case of the patch or fine model with crack propagation behaviour. As a result, the algorithm converges successfully, presenting kinematic compatibility and good results with respect to the Monolithic (non-decomposed) model. Finally, a sensitivity analysis is performed on some variables regarding the crack propagation for 2D models. Finally, the proposed methodology also allows to improve the performance of the method for cracked models or other nonlinearities when compared with the current global local analysis, presented in the state of the art

Fluid-Structure Interaction Problems in Hemodynamics:Parallel Solvers, Preconditioners, and Applications

In this work we aim at the description, study and numerical investigation of the fluid-structure interaction (FSI) problem applied to hemodynamics. The FSI model considered consists of the Navier-Stokes equations on moving domains modeling blood as a viscous incompressible fluid and the elasticity equation modeling the arterial wall. The fluid equations are derived in an arbitrary Lagrangian-Eulerian (ALE) frame of reference. Several existing formulations and discretizations are discussed, providing a state of the art on the subject. The main new contributions and advancements consist of: A description of the Newton method for FSI-ALE, with details on the implementation of the shape derivatives block assembling, considerations about parallel performance, the analytic derivation of the derivative terms for different formulations (conservative or not) and for different types of boundary conditions. The implementation and analysis of a new category of preconditioners for FSI (applicable also to more general coupled problems). The framework set up is general and extensible. The proposed preconditioners allow, in particular, a separate treatment of each field, using a different preconditioning strategy in each case. An estimate for the condition number of the preconditioned system is proposed, showing how preconditioners of this type depend on the coupling, and explaining the good performance they exhibit when increasing the number of processors. The improvement of the free (distributed under LGPL licence) parallel finite elements library LifeV. Most of the methods described have been implemented within this library during the period of this PhD and all the numerical tests reported were run using this framework. The simulation of clinical cases with patient-specific data and geometry, the comparison on simulations of physiological interest between different models (rigid, FSI, 1D), discretizations and methods to solve the nonlinear system. A methodology to obtain patient-specific FSI simulations starting from the raw medical data and using a set of free software tools is described. This pipeline from imaging to simulation can help medical doctors in diagnosis and decision making, and in understanding the implication of indicators such as the wall shear stress in the pathogenesis

Spectral element approximation of the incompressible Navier-Stokes equations in a moving domain and applications

In this thesis we address the numerical approximation of the incompressible Navier-Stokes equations evolving in a moving domain with the spectral element method and high order time integrators. First, we present the spectral element method and the basic tools to perform spectral discretizations of the Galerkin or Galerkin with Numerical Integration (G-NI) type. We cover a large range of possibilities regarding the reference elements, basis functions, interpolation points and quadrature points. In this approach, the integration and differentiation of the polynomial functions is done numerically through the help of suitable point sets. Regarding the differentiation, we present a detailed numerical study of which points should be used to attain better stability (among the choices we present). Second, we introduce the incompressible steady/unsteady Stokes and Navier-Stokes equations and their spectral approximation. In the unsteady case, we introduce a combination of Backward Differentiation Formulas and an extrapolation formula of the same order for the time integration. Once the equations are discretized, a linear system must be solved to obtain the approximate solution. In this context, we consider the solution of the whole system of equations combined with a block type preconditioner. The preconditioner is shown to be optimal in terms of number of iterations used by the GMRES method in the steady case, but not in the unsteady one. Another alternative presented is to use algebraic factorization methods of the Yosida type and decouple the calculation of velocity and pressure. A benchmark is also presented to access the numerical convergence properties of this type of methods in our context. Third, we extend the algorithms developed in the fixed domain case to the Arbitrary Lagrangian Eulerian framework. The issue of defining a high order ALE map is addressed. This allows to construct a computational domain that is described with curved elements. A benchmark using a direct method to solve the linear system or the Yosida-q methods is presented to show the convergence orders of the method proposed. Finally, we apply the developed method with an implicit fully coupled and semi-implicit approach, to solve a fluid-structure interaction problem for a simple 2D hemodynamics example