Finite-element flow solvers can utilize high-order meshes to achieve improved accuracy over traditional linear meshes. High order meshes are generally created by elevating linear meshes. For high Reynold’s number viscous flows, the linear mesh is tightly clustered to no-slip surfaces. For curved boundaries the high-order mesh must also curve to match the geometry curvature. An optimization-based node perturbation scheme is described that used a two-component cost function to optimize the high order mesh. The first component uses element Weighted Condition Number (WCN) to enforce element shape. The second component uses a normalized Jacobian to enforce element size and validity. The method is applied to several complex linear meshes with highly curved boundaries and tightly clustered normal spacing.
We define a regularized shape distortion (quality) measure for curved high-order 2D elements on a Riemannian plane. To this end, we measure the deviation of a given 2D element, straight-sided or curved, from the stretching and alignment determined by a target metric. The defined distortion (quality) is suitable to check the validity and the quality of straight-sided and curved elements on Riemannian planes determined by constant and point-wise varying metrics. The examples illustrate that the distortion can be minimized to curve (deform) the elements of a given high-order (linear) mesh and try to match with curved (linear) elements the point-wise alignment and stretching of an analytic target metric tensor.
An improved Lepp based, terminal triangles centroid algorithm for constrained Delaunay quality triangulation is discussed and studied. For each bad quality triangle t, the algorithm uses the longest edge propagating path (Lepp(t)) to find a couple of Delaunay terminal triangles (with largest angles less than or equal to 120∘) sharing a common longest (terminal) edge. Then the centroid of the terminal quadrilateral is Delaunay inserted in the mesh. Bisection of some constrained edges are also performed to assure fast convergence. We prove algorithm termination and that a graded, optimal size, 30∘ triangulation is obtained, for any planar straight line graph (PSLG) geometry with constrained angles greater than or equal to 30∘.
This article introduces an implementation of mesh morphing using radial basis functions (RBF) in a proprietary meshing framework ABIHex. RBF Morphing is a meshless method for mesh morphing that derives its origin from optimisation and machine learning. The method is used to perturb an arbitrarily complex mesh given a sample of displacements at known locations. Using this method, a number of test cases shown in the literature were replicated, and the method was applied to solve some industrial applications for gas turbines. It was found that the RBF mesh morphing method is robust in the face of sparse data, and provides a generic tool for mesh deformation provided the numerical conditions for stability of the method are met in the initial problem setup. A novel application is presented where manufacturing variation data is input to the mesh morphing program to produce a suitable volume mesh from manufactured geometry in order to perform follow on fluid flow simulations.
In this paper we discuss an algorithm to generate anisotropic mesh based on metric tensors. We transform this problem to finding a quasi conformal mapping f defined on an isotropic mesh, such that the image of f is an anisotropic triangulation. According to the metric tensors, the Beltrami coefficients of f can be calculated, then we use discrete Yamabe flow to construct f. The topology of the original triangulation will be updated if necessary, while the number of vertices won't be changed during the process. We also use our method to compute the intersection of functions, and experiments show that the interpolation functions on anisotropic meshes have less errors than the interpolation functions on isotropic mesh.
This paper shows that constraint programming techniques can successfully be used to solve challenging hex-meshing problems. Schneiders' pyramid is a square-based pyramid whose facets are subdivided into three or four quadrangles by adding vertices at edge midpoints and facet centroids. In this paper, we prove that Schneiders' pyramid has no hexahedral meshes with fewer than 18 interior vertices and 17 hexahedra, and introduce a valid mesh with 44 hexahedra. We also construct the smallest known mesh of the octagonal spindle, with 40 hexahedra and 42 interior vertices. These results were obtained through a general purpose algorithm that computes the hexahedral meshes conformal to a given quadrilateral surface boundary. The lower bound for Schneiders' pyramid is obtained by exhaustively listing the hexahedral meshes with up to 17 interior vertices and which have the same boundary as the pyramid. Our 44-element mesh is obtained by modifying a prior solution with 88 hexahedra. The number of elements was reduced using an algorithm which locally simplifies groups of hexahedra. Given the boundary of such a group, our algorithm is used to find a mesh of its interior that has fewer elements than the initial subdivision. The resulting mesh is untangled to obtain a valid hexahedral mesh.
A high-quality block structure of a 3D model can support many important applications; however, the automated generation of a high-quality block structure is still a challenging problem. In this paper, a dual surface-based approach to automated and valid block decomposition of 3D models is proposed. First, dual loops for block decomposition are generated with the help of a computed frame field whose three types of degenerated singularities are corrected. Then, a required dual surfaces set is constructed to suitably separate all the boundary elements of the 3D model and singularities of the frame field with the help of the basis of the non-trivial loops on the boundary surfaces. Finally, a valid block structure is obtained by performing dual operations along the dual surfaces on the hex mesh generated by splitting the tetrahedral mesh of the 3D model. Experimental results showed the effectiveness of the proposed approach.
This paper aims at addressing the following issue. Assume a unit square: Ω = {(x 1, x 2) ∈ [0, 1] × [0, 1]} and a Riemannian metric g ij(x 1, x 2) defined on U. Assume a mesh $$\mathcal T$$ of U that consist in non overlapping valid quadratic triangles that are potentially curved. Is it possible to build a unit quadratic mesh of U i.e. a mesh that has quasi-unit curvilinear edges and quasi-unit curvilinear triangles? This paper aims at providing an embryo of solution to the problem of curvilinear mesh adaptation. The method that is proposed is based on standard differential geometry concepts. At first, the concept of geodesics in Riemannian spaces is quickly presented: the geodesic between two points as well as the unit geodesic starting at a given point with a given direction are the two main tools that allow us to address our issue. Our mesh generation procedure is done in two steps. At first, points are distributed in the unit square U in a frontal fashion, ensuring that two points are never too close to each other in the geodesic sense. Then, a simple isotropic Delaunay triangulation of those points is created. Curvilinear edge swaps as then performed in order to build the unit mesh. Notions of curvilinear mesh quality is defined as well that allow to drive the edge swapping procedure. Examples of curvilinear unit meshes are finally presented.
In this work, we devise surface remesh kernels suitable for massively multithreaded machines. They fulfill the locality constraints induced by these hardware, while preserving accuracy and effectiveness. To achieve that, our kernels rely on: The validity of metric transport is proven. The impact of point projection as well as the accuracy of smoothing kernel are assessed by comparisons with efficient existing schemes, in terms of surface deformation and mesh quality. Kernels compliance are shown by representative examples involving surface approximation or numerical solution field guided adaptations. Finally, their scaling are highlighted by conclusive profiles on recent dual-socket multicore and dual-memory manycore machines.
To achieve the full potential of high-order numerical methods for solving partial differential equations, the generation of a high-order mesh is required. One particular challenge in the generation of high-order meshes is avoiding invalid (tangled) elements that can occur as a result of moving the nodes from the low-order mesh that lie along the boundary to conform to the true curved boundary. In this paper, we propose a heuristic for correcting tangled second- and third-order meshes. For each interior edge, our method minimizes an objective function based on the unsigned angles of the pair of triangles that share the edge. We present several numerical examples in two dimensions with second- and third-order elements that demonstrate the capabilities of our method for untangling invalid meshes.
We optimize a target function de ned by angular properties with a position control term for a basic stencil with a block-structured mesh, to improve element squareness in 2D and 3D. Comparison with the condition number method shows that besides a similar mesh quality regarding orthogonality can be achieved as the former does, the new method converges faster and provides a more uniform global mesh spacing in our numerical tests.
As a feature sensitive meshing investigation, this paper focuses on beads which are tangent continuous, high curvature, raised surfaces meant to stiffen and enhance the durability and specific strength of automotive body panels. An improvised and enhanced medial axis based strategy is proposed for identifying three broad types of bead features. Appropriate boundary discretisation, inclusion of zero medial vertex case for annulus identification, medial axis topology modifications to eliminate undesirable pathologies, T-junction squaring with cubic filtering smoothing highlight some of the improvisations to the medial axis technology employed. Ridge curves representing the crest lines of the bead are extracted and inserted on the face. A combination of multi-blocking, clamping and face node-loop insertion, followed by boundary connection strategies are used to generate high fidelity, feature sensitive, quasi-structured meshes.
We present an algorithm called discrete mesh optimization (DMO), a greedy approach to topology-consistent mesh quality improvement. The method requires a quality metric for all element types that appear in a given mesh. It is easily adaptable to any mesh and metric as it does not rely on differentiable functions. We give examples for triangle, quadrilateral, and tetrahedral meshes and for various metrics. The method improves quality iteratively by finding the optimal position for each vertex on a discretized domain. We show that DMO outperforms other state of the art methods in terms of convergence and runtime.
We present a high-order mesh curving method where the mesh boundary is enforced to match a target virtual geometry. Our method has the unique capability to allow curved elements to span and slide on top of several CAD entities during the mesh curving process. The main advantage is that small angles or small patches of the CAD model do not compromise the topology, quality and size of the boundary elements. We associate each high-order boundary node to a unique group of either curves (virtual wires) or surfaces (virtual shell). Then, we deform the volume elements to accommodate the boundary curvature, while the boundary condition is enforced with a penalty method. At each iteration of the penalty method, the boundary condition is updated by projecting the boundary interpolative nodes of the previous iteration on top of the corresponding virtual entities. The method is suitable to curve meshes featuring non-uniform isotropic and highly stretched elements while matching a given virtual geometry.
This work focuses on the use of evolutionary algorithm to perform automatic blocking of a 2D manifold. The goal of such a blocking process is to completely partition a 2D region into a set of conforming and non-intersecting quadrilaterals to facilitate the generation of an all-quadrilateral, or more preferably an ideal quadrilateral mesh configuration covering the closed 2D region. However, depending on the input shape, the optimal blocking strategy is often unclear and can be very user-dependent. In this work, a novel approach based on evolutionary algorithm is adapted to search for a potential set of such ideal configurations. Based on a selection within a set of candidate vertices from a pre-computed pool, blocking configurations can be derived and ranked based on the collective quality of its blocks. The quality of a block is computed based on objective functions relating to its interior angles and opposite length ratios. Using multi-dimensional ranking criteria, inferior solutions can be slowly filtered away with each successive generation. Based on observations on a range of turbomachinery test cases, it is possible to derive and improve near-optimal blocking configurations by utilizing a large number of generations.
Many mesh optimization applications are based on vertex repositioning algorithms (VrPA). Since the time required for VrPA programs may be large and there is concurrency in processing mesh elements, parallelism has been used to improve performance. In this paper, we propose a performance model for parallel VrPA algorithms that are implemented on memory-distributed computers. This model is validated on two parallel computers and used in a quantitative analysis of performance scalability, load balancing and synchronization and communication overheads. We show that load imbalance and synchronization between boundary partitions are the major causes of the parallel bottlenecks. In order to diminish load imbalance, a new approach to mesh partitioning is proposed. This strategy reduces the imbalance in mesh element evaluations caused by multilevel k-way partitioning algorithms and consequently, improves the performance of parallel VrPA algorithms.
We present a method for simulation-driven optimization of high-order curved meshes. This work builds on the results of Dobrev et al. (The target-matrix optimization paradigm for high-order meshes. ArXiv e-prints, 2018, https://arxiv.org/abs/1807.09807 ), where we described a framework for controlling and improving the quality of high-order finite element meshes based on extensions of the Target-Matrix Optimization Paradigm (TMOP) of Knupp (Eng Comput 28(4):419–429, 2012). In contrast to Dobrev et al. (2018), where all targets were based strictly on geometric information, in this work we blend physical information into the high-order mesh optimization process. The construction of target-matrices is enhanced by using discrete fields of interest, e.g., proximity to a particular region. As these discrete fields are defined only with respect to the initial mesh, their values on the intermediate meshes (produced during the optimization process) must be computed. We present two approaches for obtaining values on the intermediate meshes, namely, interpolation in physical space, and advection remap on the intermediate meshes. Our algorithm allows high-order applications to have precise control over local mesh quality, while still improving the mesh globally. The benefits of the new high-order TMOP methods are illustrated on examples from a high-order arbitrary Lagrangian-Eulerian application (BLAST, High-order curvilinear finite elements for shock hydrodynamics. LLNL code, 2018, http://www.llnl.gov/CASC/blast ).
This paper compares methods for simultaneous mesh untangling and quality improvement that are based on repositioning the vertices. The execution times of these algorithms vary widely, usually with a trade-off between different parameters. Thus, computer performance and workloads are used to make comparisons. A range of algorithms in terms of quality metric, approach and formulation of the objective function, and optimization solver are considered. Among them, two new objective function formulations are proposed. Triangle and tetrahedral meshes and three processors architectures are also used in this study. We found that the execution time of vertex repositioning algorithms is more directly proportional to a new workload measure called mesh element evaluations than other workload measures such as mesh size or objective function evaluations. The comparisons are employed to propose a performance model for sequential algorithms. Using this model, the workload required by each mesh vertex is studied. Finally, the effects of processor architecture on performance are also analyzed.
Curved mesh generation starting from a P 1 mesh relies on mesh deformation and mesh optimization techniques. Mesh optimization techniques consist in locally modifying the mesh in order to improve it with respect to a given quality criterion. This work presents the generalization of two mesh quality-based optimization operators to P 2 meshes. The generalized operators consist in mesh smoothing and generalized swapping. With the use of these operators, P 2 mesh generation starting from a P 1 mesh is more robust and P 2 connectivity-change moving mesh methods for large displacements are now possible.
