Hybrid Grid Generation for Viscous Flow Simulations in 1 Complex Geometries

: In this paper, we present a hybrid grid generation approach for viscous flow 10 simulations by marching a surface triangulation on viscous walls along certain directions. 11 Focuses are on the computing strategies used to determine the marching directions and 12 distances since these strategies determine the quality of the resulting elements and the reliability 13 of the meshing procedure to a large extent. With respect to marching direction, three strategies 14 featured with different levels of efficiencies and robustness performance are combined to 15 compute the initial normals at front nodes to balance the trade-off between efficiency and 16 robustness. A novel weighted strategy is used in the normal smoothing scheme, which evidently 17 reduce the possibility of early stop of front generation at complex corners. With respect to 18 marching distances, the distance settings at concave and/or convex corners are locally adjusted 19 to smooth the front shape at first; a further adjustment is then conducted for front nodes in the 20 neighbourhood of gaps between opposite viscous boundaries. These efforts, plus other special 21 treatments such as multi-normal generation and fast detection of local/global intersection, as a 22 whole enable the setup of a hybrid mesher that could generate qualitied viscous grids for 23 geometries with industry-level complexities.

In the neighbourhood of small gaps, reducing marching distances appropriately is an option 142 to avoid global intersection of viscous elements propagated from opposite viscous walls. Here, 143 the main issue is efficient computation of gap distances. Normally, an extra data structure (e.g. 144 quadtree in two-dimensional and octree in three dimensional) is required. In [21], an approach 145 relying on constrained DT is suggested. 146 The local adjustment of marching distances may lead to an abrupt change of marching 147 distances at neighbouring front nodes. If this issue happens, Laplacian-type smoothing 148 strategies are usually suggested to resolve it. 149 The more challenging issue is to adapt the mesh to flow solutions or boundary movements. 150 Since this issue is not involved in this study, the discussion is beyond the scope of this paper. 151 Interested readers are referred to [30,31]. 152 3 Outline of the hybrid meshing method 153 Figure 1 presents the main steps included in our hybrid meshing method. Given a valid CAD 154 model, the proposed method mainly takes the following steps to output a hybrid mesh. 155 Step 1. Apply the approach proposed in [32] to compute a sizing function for surface 156 mesh generation and define boundary conditions on surface patches of the model. direction at each front node can be computed by analyzing the manifold of the node. 164 Once the marching directions and marching distances are determined at all the front 165 nodes, a layer of prismatic elements can then be created by connecting the front nodes 166 and their duals after propagating the front. Repeating this front propagation procedure 167 for at most l n times, we can then create semi-structured prismatic elements in the 168 vicinity of the viscous walls.

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Step 4. If a symmetry plane is defined on the domain boundary, layered quadrilateral 170 elements should be created in the vicinity of the common curves of the symmetry plane 171 and viscous walls after Step 3. Therefore, the surface mesh of the symmetry plane, which 172 is initially composed of triangular elements only, need be updated to accommodate these 173 quadrilateral elements.

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Step 5. We can then collect the surface triangles that depict the remaining unmeshed 175 volume region. These triangles include those located at the boundaries with the non-  Figure 1. Flowchart of hybrid mesh generation.

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The above discussion only sketches the main steps in our method. Nevertheless, to be concise, 186 this discussion does not include a few non-trivial techniques incorporated in our method. These 187 techniques are necessary to improve our method for application to real problems. In Sections 4 188 to 6, we will discuss the important technical details involved in the three steps, respectively, 189 with a particular focus on Step 3. In the following subsections, we will present the algorithmic details of the four main steps. cross section and half-cone angle, which can also be called the visibility angle i  .

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Strategy III. This is an iterative algorithm aimed at finding the 'most normal' normal, i.e., the normal that minimizes the maximal angle with the given set of normals [17].

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Weights are given to each face normal depending on the angle created with the current 229 normal. If the angle is high, more weight is given to the normal. See [17] for a pseudo 230 code of this implementation.
conducted in the order listed above. The quality of the normal at the front node is evaluated by 233 the maximal angle between the normal and normals of manifold faces. A hill-climbing scheme 234 is used to ensure the optimal normal is kept always. Meanwhile, the next level of strategy gets 235 no opportunity in order to save computing time when the quality of the present 'optimal' normal 236 is less than 30 degrees. After computing the initial marching directions, a further smooth is executed to ensure a 240 desirable variation across the front and facilitate the following marching process. Here, the 241 smooth is performed by a weighted Laplacian approach, i.e., and n i N are normals at front node pi after n and n-1 iterations, is the 244 normal at neighbouring front node pj after n-1 iterations, and ij w is the weight defined at pj.

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Note that it is benificial to let normals at convex corners be closer to their neighbours, and 246 vice versa for concave corners. To achieve this, ij n is defined as below, Another geometric factor that impacts the computation of marching distances is the gap   Low-quality elements may be created in this step, in particular in the vicinity of concave 306 corners. In this study, we selected scaled aspect ratio [23] to evaluate the quality of a prism.

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This quality measure in effect combines the measures of triangle shapes and edge orthogonality.

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After creating a layer of elements, we pick those elements whose quality values are below 311 0.1 for removal. Meanwhile, we stop propagating the front faces that carry those elements.   Quality of prismatic elements is our major concern. to evaluate the quality of the generated 420 prismatic elements, the scaled aspect ratio quality measure was first adopted in this study. In 421 this study, inverted elements are not allowed, and we refer to elements with ρ(τ)<0.2 as low-422 quality elements. The distributions of scaled aspect ratios of prismatic elements for the F6, 423 rocket and space shuttle models are presented in Figure 11. The ratio of low-quality elements 424 accounts for 0.4%，0.7%，0.03% of the total numbers of prism elements for the three models, 425 respectively.

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The equiangle skewness is another commonly used quality measure for various types of Pointwise stop its propagation much earlier than its counterpart by our method. presented to verify its effectiveness and efficiency.