Section: New Results
High order embedded and immersed boundary methods

Participants: Héloïse Beaugendre, Mirco Ciallella, Benjamin Constant and Mario Ricchiuto

Corresponding member: Héloïse Beaugendre
In the last years the team has invested some effort in developing the high order embedded method known as "shifted boundary method". In this method, geometrical boundaries are not meshed exactly but embedded in the mesh. Boundary conditions are imposed on a surrogate boundary, toughly defined by the collection of mesh faces “closest” to the true boundary. To recover high order of accuracy, the boundary condition imposed on this surrogate boundary is modified to account for the distance from the true boundary.
This year's work has focused on two aspects. First, we proposed an efficient extension of the method to elliptic diffusion equations in mixed form (e.g., Darcy flow, heat diffusion problems with rough coefficients, etc.) [10] Our aim is to obtain an improved formulation that, for linear finite elements, is at least secondorder accurate for both flux and primary variable, when either Dirichlet or Neumann boundary conditions are applied. Following previous work of Nishikawa and Mazaheri in the context of residual distribution methods, we consider the mixed form of the diffusion equation (i.e., with Darcytype operators), and introduce an enrichment of the primary variable. This enrichment is obtained exploiting the relation between the primary variable and the flux variable, which is explicitly available at nodes in the mixed formulation. The proposed enrichment mimics a formally quadratic pressure approximation, although only nodal unknowns are stored, similar to a linear finite element approximation. We consider both continuous and discontinuous finite element approximations and present two approaches: a nonsymmetric enrichment, which, as in the original references, only improves the consistency of the overall method; and a symmetric enrichment, which enables a full error analysis in the classical finite element context. Combined with the shifted boundary method, these two approaches are extended to highorder embedded computations, and enable the approximation of both primary and flux (gradient) variables with secondorder accuracy, independently on the type of boundary conditions applied. We also show that the the primary variable is thirdorder accurate, when pure Dirichlet boundary conditions are embedded.
Second, using the same ideas underlying the shifted boundary method, a novel approach to handle shock waves has been proposed. In this method shocks are seen as embedded boundaries on which appropriately shifted jump conditions are imposed, allowing to connect the upstream and downstream domains. This new technique, named "shifted shockfitting", has been implemented on twodimensional unstructured grids to deal with shocks by treating them as they were immersed boundary. The new algorithm is aimed at coupling a floating shockfitting technique with the shifted boundary method, so far introduced only to simulate flows with embedded boundaries (see [15], full length paper in revision on J.Comput.Phys.).
A new PhD in collaboration with ONERA has started (Benjamin Constant 's thesis) involving the numerical simulation of unsteady flows around complex geometries in aeronautics. In the CFD simulation process, mesh generation is the main bottleneck when one wishes to study realistic configurations, such as an aircraft landing gear. A mesh can represent a month to several months of engineer time for a specialist, which is prohibitive in the predesign phase where several geometries are evaluated in a very short time frame. For this reason, Inria and Onera have been interested for several years in the development of an immersed boundary method, which does not require representing obstacles by mesh conforming to the wall, thus simplifying the generation of mesh. The consideration of the wall is carried out by the introduction of a forcing term at certain points in the vicinity of obstacles. In our approach, this technique is combined with a method of generating adaptive octree cartesian mesh. This allows us to exploit the advantages of Cartesian mesh (generation and rapid adaptation, performance gains of a dedicated Cartesian solver). In order to model the boundary layer, a wall model is used to avoid an extra cost. This method has been implemented for the simulation of steady turbulent flows around geometries studied in compressible aerodynamics (winged fuselage, engine air intake, helicopter fuselage...), providing a very good compromise between the quality of the aerodynamic solution and the time it takes to return the solution from the definition of geometry. However, the quality of the solution obtained by steady simulations is not sufficient to predict the acoustics satisfactorily. Indeed, oscillations appear for certain sizes of interest (such as turbulent viscosity or pressure fluctuations) in the vicinity of the wall. In addition, the passage of grids of different levels causes reflections, which can greatly degrade the prediction of the acoustic solution. The objective of this thesis is to solve these two problems in order to be able to perform unsteady simulations around complex geometries, such as a landing gear. On the one hand, we will study an algorithm to regularize the solution at the points of interest of the IBM method located near the wall. Together, we will also be interested in improving the wall model, in collaboration with modeling specialists from the department. We will do a study by error estimators to analyze the impact of these improvements on the solution. Validations on academic test cases will be carried out. A second step is to improve the transfer of the solution between grids of different levels (for which the mesh size double). We will propose an algorithm to regularize this passage, by geometrically modifying the mesh and by modifying the transfer formula of the solution at the passage of the fitting. A validation will be carried out on an unsteady case of LEISA profile. Finally, a demonstrative application that is both geometrically complex and of acoustic interest will be performed, typically a LAGOON or Gulfstream landing gear.