# Mesh Adaptation Strategies for CFD Simulations Over a Set of Operating Conditions

Hugo Dornier<sup>*1,2*</sup>, Olivier P. Le Maître<sup>3</sup>, Pietro M. Congedo<sup>2</sup>, Itham Salah el Din<sup>1</sup>, Sébastien Bourasseau<sup>4</sup> and Julien Marty<sup>1</sup>

<sup>1</sup>DAAA, ONERA, Institut Polytechnique de Paris, 92160 Meudon, France

<sup>2</sup>Inria, Centre de Mathématiques Appliquées, Ecole polytechnique, IPP, Route de Saclay, 91128 Palaiseau Cedex, France

<sup>3</sup>CNRS, Inria, Centre de Mathématiques Appliquées, Ecole polytechnique, IPP, Route de Saclay, 91128 Palaiseau Cedex, France

<sup>4</sup>DAAA, ONERA, Institut Polytechnique de Paris, 92322 Châtillon, France

## ABSTRACT

In computational fluid dynamics, evaluating accurately quantities of interest (global or local) requires capturing complex local phenomena and interactions, such as shocks and flow separations, while controlling the global error. The latter depends highly on the discretization of the computational domain, hence the mesh. In general, the location of the flow structures within the domain is sensitive to boundary and flow conditions. Proposing an a priori mesh with a discretization effort that concentrates on the demanding parts of the domain is thus usually impossible. Adaptive Mesh Refinement (AMR, [1,2,3,4]) is a method developed to iteratively adjust the local mesh resolution to the computed flow structures and construct meshes that achieve a prescribed accuracy for limited discretization and computational cost.

This work concerns the problem of mesh adaptation when the operating conditions are variable and follow a prescribed continuous distribution. For instance, variability of the flow conditions appears in uncertainty quantification, operating domain analysis, and robust optimization. These analyses typically require many simulations for different conditions, making the cost-accuracy trade-off even more crucial for these problems. Several mesh adaption methods have been proposed for variable conditions ([5,6,7,8]), but they typically focus primarily on error control without simultaneously optimizing for cost.

In this context, we propose two original methodologies. The first one, called *Mean Mesh adaptation* (MMA, [9,10]), builds a unique adapted mesh to minimize the average error over the continuous operating conditions for a given discretization effort. A key ingredient of MMA is using a small sample set of conditions to estimate the local average error at each iteration of the AMR process. The second method, *Error-based Mesh Selection* (EMS), tackles the optimal element selection within a library of adapted meshes to achieve the smallest possible error for any given flow conditions. The library consists of meshes independently adapted for different conditions in an offline stage, for cost efficiency, the selection uses a priori error estimations requiring no additional simulation.

We used analytical and full CFD supersonic simulations [11,12] to analyze the proposed methods. We show that MMA is robust and accurately approximates the optimal mesh minimizing the average error for a limited construction cost (see figures 1 and 2). Similarly, EMS provides a robust approximation of the optimal selection with limited cost overheads (see figure 3). The EMS method is suitable to be extended for progressive library enrichment and *a-posteriori* correction of the error estimates.
<figure><img src="image_1.png" /></figure>

Figure 1: Convergence of the average error with the meshes size for different methods: uniform refinement (UR), MMA, mesh adapted for nominal condition (NMA) and systematic adaptation (SMA). Scramjet problem with inlet Mach number uniformly distributed in [2:9; 3:3]. MMA performance fits between the UR that needs large meshes and SMA that needs a full adaptation at each condition.

<figure><img src="image_2.png" /></figure>

Figure 2: Average local relative error for the MMA mesh (top) and NMA mesh (bottom). Scramjet configuration with inlet Mach number uniformly distributed in [2.9; 3.3] and null incidence. MMA error level is controlled on the whole domain and consistently one order below NMA.

<figure><img src="image_3.png" /></figure>

Figure 3: Illustration of the EMS method for an analytic oblique shock flow with variable inlet Mach number [2; 8] and heat capacity ratio [1.4; 2]. Shown are: a) evolution of the global error with the conditions when using the mesh adapted to the conditions corresponding to grey point. B) results of EMS showing the selection domains of a library of 16 meshes adapted to conditions given by the grey points; c) resulting EMS error for the 16 selection domains shown in b); d) error associated to optimal selection showing that EMS is marginally suboptimal. In fact, EMS yields an average error 5% greater than the minimal achievable, compared to 104% greater for the unique mesh in a).

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