High-order CAD-based surface mesh adaptation with the log-simplex method

This session presents ongoing development on mesh generation and other application of spectral/hp element within Nektar++ framework and comprises the following talks 1. Implementation of the Parareal Algorithm in the Nektar++ Spectral/hp Element Framework with Applications to Linear and Non-Linear Problems 2. NekMesh: High-Order Mesh Generation and Adaptation with NekMesh 3. Locomotion of microparticles close to wallImplementation of the Parareal Algorithm in the Nektar++ Spectral/hp Element Framework with Applications to Linear and Non-Linear ProblemsNektar++ is an open-source spectral-hp element framework designed to support the development of solvers for partial differential equations. The software supports various discretization techniques, notably both continuous and discontinuous Galerkin, in combination with both modal and nodal expansions. Recently, time-parallel integration techniques are recognized as a potential solution to further increase concurrency and speed-up beyond the limits of strong scaling obtained from a pure spatial domain decomposition. Amongst the various time-parallel approaches proposed in the literature, the Parareal algorithm has become one of the most popular one. Parareal is a non-intrusive, iterative-based approach, exploiting a fine and coarse solvers to achieve time-parallelism and can be applied to both linear and non-linear problems. The efficient implementation of the Parareal algorithm in the Nektar++ open-source framework is described in this work. Application to multiple linear and non-linear problems is shown.NekMesh: High-Order Mesh Generation and AdaptationFirst, we present the redesigned high-order mesh generation pipeline from third-party straight-sided meshes for very complex industrial geometries. This workflow integrates all NekMesh high-order mesh curving, surface and volume optimizations, boundary-layer splitting and untangling. We showcase the meshed complex geometries but also the recent advances in the workflow such as multiple sources of CAD-truth, the tools for assessing geometrical accuracy, mesh quality and ensuring periodicities. The main focus of this talk is the mesh modification techniques that extend the mesh generation capabilities but also allow a-posteriori mesh optimization and adaptation. We provide a brief overview of the novel h- and the existing r- and p-adaptation techniques in NekMesh and Nektar++ for both 2D and 3D problems. Then, we combine these in r-p, h-p and h-r-p adaptations, showing the advantages of applying all three adaptation techniques simultaneously for compressible flows.

High-order methods are increasingly popular in computational fluid dynamics, but the construction of suitable curvilinear meshes still remains a challenge. This paper presents a strictly local optimization method to construct high-order triangular surface patches of high quality and accuracy. It combines fitting and energy-minimization, in which approximate bending and stretching functionals are minimized by means of an incremental procedure. The method was applied to analytically defined smooth surfaces as well as scattered surface data derived from scanning data. In both cases the optimization yielded considerable improvements in patch quality, while preserving the accuracy of pure least-squares fitting. As intended, the method achieves the greatest benefit with coarse meshes and high polynomial order.

We consider the problem of reconstructing a high-order surface from a given surface mesh. This problem is important for many meshing operations, such as generating high-order finite elements, mesh refinement, mesh smoothing and mesh adaptation. We introduce two methods, called Weighted Averaging of Local Fittings (WALF) and Continuous Moving Frames (CMF). These methods are both based on weighted least squares polynomial fittings and guarantee C 0 continuity. Unlike existing methods for reconstructing surfaces, our methods are applicable to surface meshes composed of triangles and/or quadrilaterals, can achieve third and even higher order accuracy, and have integrated treatments for sharp features. We present the theoretical framework of our methods, experimental comparisons against other methods, and its applications in a number of meshing operations.

Abstract. The face gear tooth surface is a high-order variable curvature surface, and the curvature of the surface is complicated, so it is difficult to describe the characteristics of the face gear tooth surface by a specific equation. In this paper, the discrete curvature relation of the surface is used instead of an analytical equation to describe a spatial meshing surface, and the traditional meshing theory is neutralized to analyze the characteristics of the face gear tooth surface. Firstly, according to the structural characteristics of the face gear, the sampling method of the face gear tooth surface is analyzed, and the mathematical model of fitting the tooth surface contact point is established. Then, the discrete asymptotic surface development analysis method is studied, and the ellipsoid contact analysis method of the face gear pair is established by simplifying the conjugate surface and its region. Finally, the contact analysis method of the discrete tooth surface is studied, and the instantaneous contact ellipse of the face gear tooth surface is calculated, which formed a new numerical meshing method of space curved surface.

We investigate the problem of optimizing, adapting, and untangling a surface triangulation with high-order accuracy, so that the resulting mesh has sufficient accuracy for high-order numerical methods, such as finite element methods with quadratic or cubic elements or generalized finite difference methods. We show that low-order remeshing, which may preserve the “shape” of the surface, can undermine the order of accuracy or even cause non-convergence of numerical computations. In addition, most existing methods are incapable of accurately remeshing surface meshes with inverted elements. We describe a remeshing strategy that can produce high-quality triangular meshes, while untangling mildly folded triangles and preserving the geometry to high-order accuracy. Our approach extends our earlier work on high-order surface reconstruction and mesh optimization. We present the theoretical framework of our methods, experimental comparisons against other methods, and demonstrate its utilization in accurate solutions for geometric partial differential equations on triangulated surfaces.

Fast and accurate simulations of air-cooled structures

Adaptive surface mesh remeshing based on a sphere packing method and a node insertion/deletion method

We present a method to adapt a tetrahedron mesh together with a surface mesh with respect to a size criterion. The originality of our work lies in the fact that both surface and tetrahedron mesh adaptation are carried out simultaneously and that no CAD is required to adapt the surface mesh. The adaptation procedure consists of splitting or removing interior and surface edges which violate a given size criterion. The enrichment process is based on a bisection technique. In order to guarantee mesh conformity during the refinement process, all possible remeshing configurations of tetrahedra have been examined. Once the tetrahedron mesh has been adapted, surface nodes are projected on a geometrical model. The building of a surface model from discrete data has already been presented in this journal. The method is based on a mesh‐free technique called Hermite Diffuse Interpolation. Surface and volume mesh optimization procedures are carried out during the adaptation and at the end of the process to enhance the mesh. Copyright © 2003 John Wiley & Sons, Ltd.

Direct extraction of surface meshes from implicitly represented heterogeneous volumes

Robust conformal adaptive meshing of complex textile composites unit cells

Circuit parasitic extraction problems are typically formulated using discretized integral equations that use basis functions defined over tesselated surface meshes. The fast multipole method (FMM) accelerates the solution process by rapidly evaluating potentials and fields due to these basis functions. Unfortunately, the FMM suffers from the drawback that its efficiency degrades if the surface mesh has disparately-sized elements in close proximity to each other. Closely-spaced non-uniformly sized elements can appear in realistic situations for a variety of reasons: owing to mesh refinement, due to accurate modeling requirements for fine structural features, and because of the presence of thin doubly-walled structures. In this paper, modifications to the standard multilevel FMM are presented that permit efficient potential and field evaluation over specific non-uniform meshes. The efficiency of the new technique is demonstrated through examples involving large surface meshes with nonuniformly sized elements in close proximity.

A new algorithm to generate three-dimensional (3D) mesh for thin-walled structures is proposed. In the proposed algorithm, the mesh generation procedure is divided into two distinct phases. In the first phase, a surface mesh generator is employed to generate a surface mesh for the mid-surface of the thin-walled structure. The surface mesh generator used will control the element size properties of the final mesh along the surface direction. In the second phase, specially designed algorithms are used to convert the surface mesh to a 3D solid mesh by extrusion in the surface normal direction of the surface. The extrusion procedure will control the refinement levels of the final mesh along the surface normal direction. If the input surface mesh is a pure quadrilateral mesh and refinement level in the surface normal direction is uniform along the whole surface, all hex-meshes will be produced. Otherwise, the final 3D meshes generated will eventually consist of four types of solid elements, namely, tetrahedron, prism, pyramid and hexahedron. The presented algorithm is highly flexible in the sense that, in the first phase, any existing surface mesh generator can be employed while in the second phase, the extrusion procedure can accept either a triangular or a quadrilateral or even a mixed mesh as input and there is virtually no constraint on the grading of the input mesh. In addition, the extrusion procedure development is able to handle structural joints formed by the intersections of different surfaces. Numerical experiments indicate that the present algorithm is applicable to most practical situations and well-shaped elements are generated. Copyright © 2004 John Wiley & Sons, Ltd.

Bias–Variance Analysis for Controlling Adaptive Surface Meshes

Adaptive mesh methods are commonly used to improve the accuracy of numerical solution of problems without essential increase in the number of mesh nodes (Lebedev et al., 2002). Within the scope of all adaptive mesh methods, there is an important class of methods in which the mesh is an image under an appropriate mapping of a fixed mesh over a computational domain (Bern & Plassmann, 1999). Most of widely used conventional methods from the above class, such as equidistribution method (Shokina, 2001), Thompson’s method (Thompson et al., 1985), elliptic method (Liseikin, 1999), etc. determine the mapping by solving a complicated system of nonlinear partial differential equations (PDEs). This often leads to significant difficulties. First, the convergence of numerical solution of these PDEs highly depends on an initial mesh, requires fixing boundary mesh nodes beforehand and imposes quite strong limitations on the properties of mesh density function (Khakimzyanov et al., 2001). Second, efficient parallelization of solvers for the PDEs meets overwhelming difficulties. Finally, the PDEs for mesh construction are not universal and need to be proposed for 1D, 2D or 3D spaces specifically. The complexity of numerical solution of these PDEs essentially grows with increasing the dimensionalities (Khakimzyanov et al., 2001). Moreover, there is no methods and techniques in the above mentioned class that can provide a fully automatic adaptive mesh construction in 3D case. This chapter demonstrates the great ability of the Kohonen’s Self Organizing Maps (SOM) (Kohonen, 2001) to perform high quality adaptive mesh construction. Since the SOM model provides a topology preserving mapping of high-dimensional data onto a low-dimensional space with approximation of input data distribution, the proposed mesh construction method uses the same algorithms for different dimensionalities of a physical domain that proves its universality. In our investigation, the classical SOM model has been studied and modified in order to overcome border effect and provide topology preservation. Based on the ideas in (Nechaeva, 2006), the composition of SOM models of different dimensionalities has been proposed which alternates mesh construction on the border and inside a physical domain. It has been shown that the SOM learning algorithm can be used as a mesh smoothing tool. All the algorithms has been implemented using the GeomBox (Bessmeltsev, 2009) and 9

Feature lines are salient surface characteristics. Their definition involves third and fourth order surface derivatives. This often yields to unpleasantly rough and squiggly feature lines since third order derivatives are highly sensitive against unwanted surface noise. The present work proposes two novel concepts for a more stable algorithm producing visually more pleasing feature lines: First, a new computation scheme based on discrete differential geometry is presented, avoiding costly computations of higher order approximating surfaces. Secondly, this scheme is augmented by a filtering method for higher order surface derivatives to improve both the stability of the extraction of feature lines and the smoothness of their appearance.