This study has presented an error analysis for anisotropic goal-oriented meshadaptationfor the compressible Navier-Stokes equations.
The simplified feature-based/Hessian-based approach consisting in minimizing interpolation errors has early proved its ef- ficiency and ease of use. Unfortunately, the application to systems is penalized by the arbitrary choice of adequate sensors (features). Indeed, the mathematical model is not taken into account and the genuine purpose of reducing the approximation error is not satisfactorily met. In order to reduce efficiently approximation errors, it is mandatory to use an error estimate. By construction, an a posteriori estimate does not directly involve information concerning the behavior of local error when the mesh is stretched. For that reason, we proposed to consider an a priori analysis of a goal-oriented error formulation in order to express in a natural way the error model in terms of interpolation errors. As in the Hessian-based method, a main advantage of this approach is that interpolation errors are then transformed to account for the coupled influence of mesh size and stretching. Moreover, an analytic expression of the optimal mesh is directly obtained thanks to the continuous mesh framework.
First, a finite-volume discretization for tetrahedral cut-cells is developed that makes possible robust, anisotropic adaptation on complex bodies. Through grid re- finement studies on inviscid flows, this cut-cell discretization is shown to produce similar accuracy as boundary-conforming meshes with a small increase in the degrees of freedom. The cut-cell discretization is then combined with output-based error es- timation and anisotropic adaptation such that the mesh size and shape are controlled by the output error estimate and the Hessian (i.e. second derivatives) of the Mach number, respectively. Using a parallel implementation, this output-based adaptive method is applied to a series of sonic boom test cases and the automated ability to correctly estimate pressure signatures at several body lengths is demonstrated start- ing with initial meshes of a few thousand control volumes.
leading to a null correction as F h (W ) ≈ F h (R h W h ) = F h (W h ) = 0 when W h is a converged numerical solution.
We finally perform an adaptive simulation where we compare standard anisotropic meshadaptation with norm oriented adaptation.
S UBSONIC BUMP EXAMPLE . We consider the extruded bump geometry a , see Figure 1 with an inflow at Mach 0.3. computed from the initial solution and the corrected flow fields on a sequence of uniform meshes composed of 17 723, 134 381, 1 044 943 and 7 682 230 vertices respectively. As the whole flow field is corrected, all functionals of interest can be corrected simultaneously. We restrict ourselves to lift and drag. For the lift, we observe that the corrected lift converges at a higher rates that the uncorrected prediction, see Figure 2 (top right). For each size of mesh, the corrected lift is 2 to 4 times smaller than the initial solution. Similar conclusions hold for the whole flow field. The implicit error kΠ h W − W h k and kΠ h W − W c k on the density field is depicted in Figure 2 (bottom left). The corrected
The methods in the second category are usually refered as "cartesian grid methods", where the forcing accounting for the presence of boundaries is per- formed at the discrete level. The ghost-cell approach, inspired by the ghost- ﬂuid method developed by Fedkiw (, ) for multiphase ﬂows belongs to this category. The ﬁrst developments (, ) were followed by many other extensions , , . Ghost cells are cells in the solid with at least one neighbour in the ﬂuid. The values on these ghost cells are extrapolated from the values in the ﬂuid in order to impose the appropriate boundary conditions at the interface. The sub-mesh penalty method introduced by Sarthou et al.  is also related to this family. The immersed interface method is another ap- proach, developed by Leveque and Li  and extended to ﬂow problems in . It is based on Taylor expansions of the solution on each side of the interface. The "cut-cell" approach belongs also to this class, and was mainly developed forcompressible ﬂows, see for example , , , , . These methods preserve conservation properties near the boundary.
dimensional flows. Soulaïmani and Ben Haj Ali (Soulaïmani and Ben Haj Ali, 2003) proposed a parallel-distributed computing-based approach for the solution of some multiphysics problems. They validated the method on the Agard 445.3 airfoil. Soulaïmani et al (Soulaïmani et al, 2004) proposed an efficient parallel-distributed methodology for solving multiphysics problems. They validated the results on Agard 445.6 airfoil. Ben Haj Ali and Soulaïmani (Ben Haj Ali and Soulaïmani, 2010) proposed a stabilized FEM for solving the compressible Navier-Stokes equations combined with the Spalart-Allmaras model. They validated the code on the 3D boundary layer over a flat plate and on the ONERA-M6 wing. Rebaine (Rebaine, 1997) proposed a numerical method for two-dimensional (2D) compressible laminar and turbulent flows. Rebaine and Soulaïmani (Rebaine and Soulaïmani, 2001) proposed an FEM for simulation of 2D internal compressible turbulent flows. They validated the method in 2D supersonic and thrust augmenting ejectors. Soulaïmani et al (Soulaïmani et al, 2002b) proposed a conservative finite element formulation for the coupled fluid/mesh interaction problem. Soulaïmani et al (Soulaïmani et al, 1994) proposed an FEM for simulation of 2D internal compressible turbulent flows. Soulaïmani and Fortin (Soulaïmani and Fortin, 1994) proposed a method to solve the Navier-Stokes and Euler equations in a conservative form by using the conservation variables.
Figure 4: Surface mesh and velocity iso-values when controlling the sum of the L 1 norm of the in- terpolation error on the density, velocity and pressure (top), the Mach number (middle) and the norm ||W − W h|| L 2 with the norm-oriented approach (bottom).
but is too focused on a particular output and does not produce convergent solution fields. The norm-oriented method has the advantages of both. For elliptic problems, the Hessian- based approach is nearly optimal as suggested by finite-element estimates. However, the presented comparisons seem to indicate that the novel method carries a good improve- ment. We have also proposed a preliminary application to an inviscid compressible flow. New computations will be shown during the conference.
γ = C p /C v 2.35 1.43
Table 1. Fluids parameters for the one dimensional piston test.
We use here the MT-scheme with different space steps. Figure 1 displays the profiles of z, pressure P and velocity u at instants t = 6.66 ms and t = 20 ms. The solid lines represent the approximate solution with 5000 cells, the × symbol lines represent the solution with 1000 cells and the 5 symbol lines represent the solution with 100 cells. This allows to check the good convergence behaviour of the solution when the space step goes to zero. Let us note that some small pressure oscillations appear for the rough mesh, nevertheless their amplitude decrease while refining the space step.
4. Robustness improvement of DPIR
To illustrate this robustness problem let us consider the same test as above on an unstructured mesh (Fig. 5). In some cases the points M i and M i+1 obtained after the first step of DPIR can be to close to the same node
leading to a spike when the algorithm tries to balance the volume fraction. It has been observed, in some cases, that the algorithm cannot balance the volume since the interface ends up outside the cell.
(a) Density field. (b) Viscosity field.
Figure 13: Transonic flow over a bump: results without refinement.
5. Discussion and conclusion
In this work, we have explained how to construct a DG-compliant curvi- linear grid based on a set of rational B´ ezier elements, from boundaries defined by NURBS curves. The proposed method allows both to preserve the CAD geometry and solve governing equations with a DG method including only a few modifications. As underlined in the results obtained for different test-cases, the resolution scheme exhibits a high-order accuracy with quasi-optimal con- vergence rate, and seems to be robust with respect to mesh distorsions. The benefit of accounting for boundary curvature has been demonstrated. Finally, a classical shock capturing strategy based on artificial viscosity has been adapted to the isogeometric context. The oscillations of the control point lattice in the solution are employed to define the local viscosity. In association with a local refinement strategy, the method has proved to provide a sharp shock capture.
A one-dimensional temporal adaptive algorithm was proposed by Sanders and Osher [ 85 ], leading to a first-order time accurate method applied to a scalar equation. Later on, the exten- sion of the procedure to multi-dimensional problems was performed by Dawson [ 86 ]. Both papers used the same procedure in order to allow time synchronisation between cells time inte- grated with their own time step between t n and t n + 1 . In order to keep space-time conservation, the sum of fluxes on small cells is equal to the one of the larger cells at the interface between cells of different sizes. Nevertheless, this method leads to a first-order accurate reconstruction. Dawson and Kirby [ 87 ] proposed an extension to the second order of accuracy. They extended the procedure by means of Total Variation Diminishing (TVD) property. Constantinescu and Sandu [ 88 ] developed a set of Runge-Kutta integrators called Partitioned Runge-Kutta. The interest of these methods is their ability to be adapted to automatic mesh refinement and re- spect strong mathematical properties (Strong Stability Preserving). Later on, the same authors extended their procedure to the famous explicit Adams scheme [ 89 ].
Supersonic flows involve highly anisotropic physical structures. In the case of a supersonic aircraft, a large number of shock structures are present in the flow that correspond to all Mach cones issued from the aircraft geometry, Figure 3 (bottom). The final adapted mesh is highly anisotropic, cf. Figure 2. Figure 3 shows that refinements along Mach cones have been propagated in the whole computational domain (top left) with a high accuracy (or small size) in shock regions (middle left) despite the large scale factor between the mesh accuracy and the domain size.
Among open theoretical problems raised by our numerical experiments, we mention two. A first one is to understand for what reason the dynamical system made of the Volume of Fluid-Machine Learning flux coupled with a more traditional Finite Volume bi-material solver is able to get a good control of the number of flagged cells near the interface. As shown in Section 6.4, this possibility depends crucially of the dataset. Therefore, to get a theoretical answer to this question, it will be necessary to understand the algorithm globally. A second theoretical question is to prove the 1/2 convergence rate for bounded variation data, as it is observed in Section 8.1.4. So far our work is restricted to Cartesian meshes. An open problem of great practical interest is to go beyond such meshes of simple structure. The main difficulty is that adding the local parameters of the mesh in the vector of inputs might result in a very expensive method.
In this report, we consider the last approach. The main interest is its ability perform nu- merical simulations on very complex objects without requiring as input a conformal mesh (e.g. a water-tight and non self-intersecting mesh) of the geometry. Actually, it would be almost impossible to obtain this mesh without manual intervention. This specific type of simulation is of particular interest for simulating shape-varying geometries like veins or aortic valves [47, 48], really complex geometries like helicopters or Formula One cars , or finally multi components bodies like in fragmentation cases [35, 36, 33, 46]. Another advantage relies in the simplicity of the generated volume mesh. As the constraint of the inner object is no longer there, the meshing of the volume part is not an issue and can be easily done, even in a structured fashion. However, following the law that there is constant balance between the work required in the solver and the mesh generation, it implies that the solver has more constraints to follow. Indeed, the solver has to be able to detect the boundary of the object, this boundary being represented by a set of edges or triangles, a CAD model or a level-set function, and to determine, if applicable, the inner and the outer part of the object. Then, the next step consists in applying the appropriate treatment to the elements of the mesh where it is intersected [53, 32, 22]. This step is the most crucial part of the process to get proper boundary conditions. Note that this approach has been developed for both compressible and incompressible Euler/Navier-Stokes discretized with either the Finite Volume Method [54, 55], the Finite Element Method  or a Discontinuous Galerkin Method .
This test case exemplifies the singularity which is met in the simulation of multi-fluid flows with a large deviation between the densities ρ1 and ρ2 of each phase. In the case where a projection algorithm is applied to solve the Navier-Stokes equations for incom- pressible flow, a Poisson problem with discontinuous coefficients has to be solved. An example can be found in . The present case does not satisfy the smoothness as- sumptions introduced for deriving our method. However, a usual expectation in meshadaptation is that the methods should also apply well on non-smooth contexts. We con- sider the equation of Poisson −div( 1 ρ ∇u) = rhs with a discontinuous coefficient taking two different values 1/ρ 1 and 1/ρ 2 on two sub-domains Ω 1 and Ω 2 separated by an in- terface which is a sufficiently smooth curve for having a normal vector. This PDE is mathematically referred as a transmission problem and the solution is continuous across the interface but of discontinuous normal derivatives since:
for each case. The generation of the anisotropic meshes is done with the local remeshing strategy of . The surface meshes and the velocity iso-lines are depicted in Figure 2. Depending on the adaptation strategy, completely different flow fields are observed. The adaptation on the Mach number reveals strong shear layers at the wing tip that are not present in the norm oriented approach. On the contrary, recirculating flows are observed on the norm oriented approach while not being observed on the Mach number adaptation. For each case, the wakes have different features. Note that the accuracy near the body is not equivalent. For the L 1 norm adaptation error and norm oriented approaches, the
3.1 Higher-order (HO) estimates
Main existing HO schemes satisfy the so-called k−exactness property expressing the fact that if the exact solution is a polynomial of order k then the approximation scheme will give the exact solution as answer. The assembly of these schemes in- volves a step of polynomial reconstruction (e.g., ENO schemes), or of polynomial interpolation (e.g., Continuous/discontinuous Galerkin). The main part of the error can then be expressed in terms of a (k + 1)-th term of a Taylor series where the spa- tial increment is related with local mesh size. We want to stress that this is the key of an easy extension of metric-based adaptation to HO schemes. We illustrate this with the computation of 2D Euler flows. Considering a triangulation of the compu- tational domain and its dual cells C i built with triangle medians, the exact solution
r. t. δa in log-log scale.
6.1 Steady inviscid flow in a nozzle
We consider as first test-case the steady inviscid subsonic flow in a two-dimensional nozzle. The Mach number is set to the value M = 0.15. Note that the transonic case yields specific difficulties due to the presence of the shock waves, that are not treated presently (see  for instance). Fig. (3) shows a description of the test-case and Fig. (4) the grid used (23 489 nodes). Note that the mesh should be fine enough to make the approximation error negligible. The local time-step is chosen so that the local CFL number is 10 000. Computations are carried out using four processors. The iterative convergence is plotted in Fig. (5) and the flow field is illustrated in Fig. (6).
modes, which is a suitable property for steady computations but undesirable for time accurate ones. Time
accurate extensions of central IRS schemes (see Ref. 12 , 13 ) have been applied to the simulation of slow
unsteady ﬂows. IRS uses an implicit Laplacian smoothing operator to ﬁlter out high-frequency modes of the residual, which leads to the solution of tridiagonal systems for each space direction and Runge–Kutta stage. The main asset of IRS is that it provides a signiﬁcant increase of the maximum allowable time step compared to an explicit Runge–Kutta scheme, while keeping a computational cost not much higher, thanks to the eﬃcient inversion of scalar tridiagonal matrices. Nevertheless, care must been taken in the selec- tion of the smoothing coeﬃcient, to avoid introducing additional errors that are inconsequential for steady computations but may be unacceptably high for the highly unsteady LES and DNS ones. In the attempt of improving the accuracy of IRS approaches for scale-resolving time-accurate simulations, a higher-order IRS scheme based on the application of a bilaplacian smoothing operator to the residuals is proposed and investigated. Such a scheme leads to the inversion of a scalar pentadiagonal system in each mesh direction,
5.5. Test 5: Interaction acoustic wave with a 2-D stationary vortex
The last test-case has been insipred from the work of Miczek and co-authors . The computational domain represents a pipe section whose diameter is L = 2 m and length is 5 × L = 10 m. The 2-D cartesian mesh is made of 50 × 250 cells. At time t = 0 s, a stationary vortex similar to the one already introduced in subsection (5.3) is present at (x 0 , y 0 ) = (7, 1) . Simultaneously, a smooth but steep pressure pulse is injected along all the pipe section at x 1 = 5 m. Such a pulse reads: