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Publication# Coupling non-conforming discretizations of PDEs by spectral approximation of the Lagrange multiplier space

Résumé

This work focuses on the development of a non-conforming method for the coupling of PDEs based on weakly imposed transmission conditions: the continuity of the global solution is enforced by a finite number of Lagrange multipliers defined over the interfaces of adjacent subdomains. The method falls into the class of primal hybrid methods, which include also the well-known mortar method. Differently from the mortar method, we discretize the space of basis functions on the interface by spectral approximation independently of the discretization of the two adjacent domains. In particular, our approach can be regarded as a specialization of the three-field method in which the spaces used to enforce the continuity of the solution and its conormal derivative across the interface are taken equal. One of the possible choices to approximate the interface variational space – which we consider here – is by Fourier basis functions. As we show in the numerical simulations, the method is well-suited for the coupling of problems defined on globally non-conforming meshes or discretized with basis functions of different polynomial degree in each subdomain. We also investigate the possibility of coupling solutions obtained with incompatible numerical methods, namely the finite element method and isogeometric analysis.

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This thesis focuses on the development and validation of a reduced order technique for cardiovascular simulations. The method is based on the combined use of the Reduced Basis method and a Domain Decomposition approach and can be seen as a particular implementation of the Reduced Basis Element method. Our contributions include the application to the unsteady three-dimensional Navier--Stokes equations, the introduction of a reduced coupling between subdomains, and the reconstruction of arteries with deformed elementary building blocks. The technique is divided into two main stages: the offline and the online phases. In the offline phase, we define a library of reference building blocks (e.g., tubes and bifurcations) and associate with each of these a set of Reduced Basis functions for velocity and pressure. The set of Reduced Basis functions is obtained by Proper Orthogonal Decomposition of a large number of flow solutions called snapshots; this step is expensive in terms of computational time. In the online phase, the artery of interest is geometrically approximated as a composition of subdomains, which are obtained from the parametrized deformation of the aforementioned building blocks. The local solution in each subdomain is then found as a linear combination of the Reduced Basis functions defined in the corresponding building block. The strategy to couple the local solutions is of utmost importance. In this thesis, we devise a nonconforming method for the coupling of Partial Differential Equations that takes advantage of the definition of a small number of Lagrange multiplier basis functions on the interfaces. We show that this strategy allows us to preserve the h-convergence properties of the discretization method of choice for the primal variable even when a small number of Lagrange multiplier basis functions is employed. Moreover, we test the flexibility of the approach in scenarios in which different discretization algorithms are employed in the subdomains, and we also use it in a fluid-structure interaction benchmark. The introduction of the Lagrange multipliers, however, is associated with stability problems deriving from the saddle-point structure of the global system. In our Reduced Order Model, the stability is recovered by means of supremizers enrichment.
In our numerical simulations, we specifically focus on the effects of the Reduced Basis and geometrical approximations on the quality of the results. We show that the Reduced Order Model performs similarly to the corresponding high-fidelity one in terms of accuracy. Compared to other popular models for cardiovascular simulations (namely 1D models), it also allows us to compute a local reconstruction of the Wall-Shear Stress on the vessel wall. The speedup with respect to the Finite Element method is substantial (at least one order of magnitude), although the current implementation presents bottlenecks that are addressed in depth throughout the thesis.

Mathematical and numerical aspects of free surface flows are investigated. On one hand, the mathematical analysis of some free surface flows is considered. A model problem in one space dimension is first investigated. The Burgers equation with diffusion has to be solved on a space interval with one free extremity. This extremity is unknown and moves in time. An ordinary differential equation for the position of the free extremity of the interval is added in order to close the mathematical problem. Local existence in time and uniqueness results are proved for the problem with given domain, then for the free surface problem. A priori and a posteriori error estimates are obtained for the semi-discretization in space. The stability and the convergence of an Eulerian time splitting scheme are investigated. The same methodology is then used to study free surface flows in two space dimensions. The incompressible unsteady Navier-Stokes equations with Neumann boundary conditions on the whole boundary are considered. The whole boundary is assumed to be the free surface. An additional equation is used to describe the moving domain. Local existence in time and uniqueness results are obtained. On the other hand, a model for free surface flows in two and three space dimensions is investigated. The liquid is assumed to be surrounded by a compressible gas. The incompressible unsteady Navier-Stokes equations are assumed to hold in the liquid region. A volume-of-fluid method is used to describe the motion of the liquid domain. The velocity in the gas is disregarded and the pressure is computed by the ideal gas law in each gas bubble trapped by the liquid. A numbering algorithm is presented to recognize the bubbles of gas. Gas pressure is applied as a normal force on the liquid-gas interface. Surface tension effects are also taken into account for the simulation of bubbles or droplets flows. A method for the computation of the curvature is presented. Convergence and accuracy of the approximation of the curvature are discussed. A time splitting scheme is used to decouple the various physical phenomena. Numerical simulations are made in the frame of mould filling to show that the influence of gas on the free surface cannot be neglected. Curvature-driven flows are also considered.

Vincent Maronnier, Marco Picasso, Jacques Rappaz

A numerical model is presented for the simulation of complex fluid flows with free surfaces. The unknowns are the velocity and pressure fields in the liquid region, together with a function defining the volume fraction of liquid. Although the mathematical formulation of the model is similar to the volume of fluid (VOF) method, the numerical schemes used to solve the problem are different. A splitting method is used for the time discretization. At each time step, two advection problems and a generalized Stokes problem are to be solved. Two different grids are used for the space discretization. The two advection problems are solved on a fixed, structured grid made out of small rectangular cells, using a forward characteristic method. The generalized Stokes problem is solved using a finite element method on a fixed, unstructured mesh. Numerical results are presented for several test cases: the filling of an S-shaped channel, the filling of a disk with core, the broken dam in a confined domain. (C) 1999 Academic Press.

1999