Iterative Methods for Model Reduction by Domain Decomposition

We propose a method to reduce the computational effort to solve a partial differential equation on a given domain. The main idea is to split the domain of interest in two subdomains, and to use different approximation methods in each of the two subdomains. In particular, in one subdomain we discretize the governing equations by a canonical scheme, whereas in the other one we solve a reduced order model of the original problem. Different approaches to couple the low-order model to the usual discretization are presented. The effectiveness of these approaches is tested on numerical examples pertinent to non-linear model problems including the Laplace equation with non-linear boundary conditions and the compressible Euler equations

Accurate Sharp Interface Scheme for Multimaterials

We present a method to capture the evolution of a contact discontinuity separating two different material. A locally non-conservative scheme allows an accurate and stable simulation while the interface is kept sharp. Numerical illustrations include problems involving fluid and elastic problems

Model order reduction by convex displacement interpolation

We present a nonlinear interpolation technique for parametric fields that exploits optimal transportation of coherent structures of the solution to achieve accurate performance. The approach generalizes the nonlinear interpolation procedure introduced in [Iollo, Taddei, J. Comput. Phys., 2022] to multi-dimensional parameter domains and to datasets of several snapshots. Given a library of high-fidelity simulations, we rely on a scalar testing function and on a point set registration method to identify coherent structures of the solution field in the form of sorted point clouds. Given a new parameter value, we exploit a regression method to predict the new point cloud; then, we resort to a boundary-aware registration technique to define bijective mappings that deform the new point cloud into the point clouds of the neighboring elements of the dataset, while preserving the boundary of the domain; finally, we define the estimate as a weighted combination of modes obtained by composing the neighboring snapshots with the previously-built mappings. We present several numerical examples for compressible and incompressible, viscous and inviscid flows to demonstrate the accuracy of the method. Furthermore, we employ the nonlinear interpolation procedure to augment the dataset of simulations for linear-subspace projection-based model reduction: our data augmentation procedure is designed to reduce offline costs -- which are dominated by snapshot generation -- of model reduction techniques for nonlinear advection-dominated problems

ADER scheme for incompressible Navier-Stokes equations on Overset grids with a compact transmission condition

A space-time Finite Volume method is devised to simulate incompressible viscous flows in an evolving domain. Inspired by the ADER method, the Navier-Stokes equations are discretized onto a space-time overset grid which is able to take into account both the shape of a possibly moving object and the evolution of the domain. A compact transmission condition is employed in order to mutually exchange information from one mesh to the other. The resulting method is second order accurate in space and time for both velocity and pressure. The accuracy and efficiency of the method are tested through reference simulations.Une méthode des volumes finis spatio-temporels est conçue pour simuler des écoulements visqueux incompressibles dans un domaine évolutif. Inspirée de la méthode ADER, les équations de Navier-Stokes sont discrétisées sur un maillage spatio-temporel overset qui est capable de prendre en compte à la fois la forme d’un objet éventuellement en mouvement et l’évolution du domaine. Une condition de transmission compacte est employée afin d’échanger mutuellement des informations d’un maillage à l’autre. La méthode résultante est d’une précision de second ordre dans l’espace et dans le temps pour la vitesse et la pression. La précision et l’efficacité de la méthode sont testées sur des cas test pris de la littérature

A zonal Galerkin-free POD model for incompressible flows

International audienceA domain decomposition method which couples a high and a low-fidelity model is proposed to reduce the computational cost of a flow simulation. This approach requires to solve the high-fidelity model in a small portion of the computational domain while the external field is described by a Galerkin-free Proper Orthogonal Decomposition (POD) model. We propose an error indicator to determine the extent of the interior domain and to perform an optimal coupling between the two models. This zonal approach can be used to study multi-body configurations or to perform detailed local analyses in the framework of shape optimisation problems. The efficiency of the method to perform predictive low-cost simulations is investigated for an unsteady flow and for an aerodynamic shape optimisation problem

Accurate Sharp Interface Scheme for Multimaterials

We present a method to capture the evolution of a contact discontinuity separating two different material. A locally non-conservative scheme allows an accurate and stable simulation while the interface is kept sharp. Numerical illustrations include problems involving fluid and elastic problems

Hollow wakes past arbitrarily shaped obstacles

An analytical solution is presented for steady inviscid separated flows modelled by hollow vortices, that is, by closed vortex sheets bounding a region with fluid at rest. Steady flows past arbitrary obstacles protruding from an infinite wall are considered. The solution is similar to that of the vortex patch model; it depends on two free parameters that define the size of the hollow vortex and the location of the separation point. When a sharp edge constrains the separation point (Kutta condition), the solution depends on a single parameter. As with the vortex patch model, families of growing vortices exist, which represent the continuation of desingularized point vortices. Numerical results are presented for the flows past a semicircular bump, a Ringleb snow cornice and a normal flat plate. The differences from the previous results found in the literature are analysed and discussed with the present solutions for the flow past a normal flat plate. Key words

Modeling and optimization of a propeller by means of an inverse method

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