High-order mesh curving by distortion minimization with boundary nodes free to slide on a 3D CAD representation

Abstract

We propose a 3D mesh curving method that converts a straight-sided mesh to an optimal-quality curved high-order mesh that interpolates a CAD boundary representation. The main application of this method is the generation of discrete approximations of curved domains that are valid for simulation analysis with unstructured high-order methods. We devise the method as follows. First, the boundary of a straight-sided high-order mesh is curved to match the curves and surfaces of a CAD model. Second, the method minimizes the volume mesh distortion with respect to the coordinates of the inner nodes and the parametric coordinates of the curve and surface nodes. The proposed minimization features untangling capabilities and therefore, it repairs the invalid elements that may arise from the initial curving step. Compared with other mesh curving methods, the only goal of the proposed residual system is to minimize the volume mesh distortion. Furthermore, it is less constrained since the boundary nodes are free to slide on the CAD curves and surfaces. Hence, the proposed method is well suited to generate curved high-order meshes of optimal quality from CAD models that contain thin parts or high-curvature entities. To illustrate these capabilities, we generate several curved high-order meshes from CAD models with the implementation detailed in this work. Specifically, we detail a node-by-node non-linear iterative solver that minimizes the proposed objective function in a block Gauss-Seidel manner.Peer ReviewedPostprint (author's final draft

Connectivity-change moving mesh methods for high-order meshes: Toward closed advancing-layer high-order boundary layer mesh generation

“…It would be harder to implement a directly high-order mesh generator. To curve this mesh, the used models are numerous : PDE or variational models, 1,15,22,23 smoothing and/or optimization procedures, 17,24,27 etc ... Our choice here is to use the linear elasticity equation as a model for the motion of the vertices to generate a P k mesh from a P 1 mesh. For this purpose, let us consider Equation (7) with f = 0 and Dirichlet boundary conditions.…”

“…Some are using a PDE or variational approach to curve a P 1 mesh into a P k mesh, 1,15,22,23 others are based on optimization and smoothing operations and start from a P 1 mesh with a constrained P k curved boundary in order to generate a suitable P k mesh. 17,24,27 In all these approaches, the key feature is to find the best deformation to apply to the initial P 1 mesh.…”

Spline parameterization method for 2D and 3D geometries based on T-mesh optimization

“…To asses the distortion of the entire physical element Ω e we can define a global distortion measure (similarly to [32,33,34]) by means of L 1 norm of the pointwise distortion, namely…”

Insertion of triangulated surfaces into a meccano tetrahedral discretization by means of mesh refinement and optimization procedures

“…To overcome this drawback, we use the regularized distortion measure introduced in Escobar et al and Garanzha and Kaporin for linear elements, in which σ is replaced by where δ is a small parameter that depends on the problem and can be calculated according to Escobar et al and Gargallo‐Peiró et al This regularization is also used for optimizing curved high‐order meshes. ()…”

“…Thus, in the fourth step, we optimize the quality of the final mesh by using a robust untangling and smoothing process based on an optimization of a single distortion measure . Specifically, we optimize a regularized version of the shape distortion measure for tetrahedral elements in which we move all the nodes of the mesh, restricting the movement of the edge and surface nodes along the corresponding entity they belong to . This is achieved by expressing the coordinates of these nodes in terms of their parametric coordinates during the optimization process.…”