Unstructured Quadrilateral Meshes Research Articles

Efficient quadrature rules for subdivision surfaces in isogeometric analysis

We introduce a new approach to numerical quadrature on geometries defined by subdivision surfaces based on quad meshes in the context of isogeometric analysis. Starting with a sparse control mesh, the subdivision process generates a sequence of finer and finer quad meshes that in the limit defines a smooth subdivision surface, which can be of any manifold topology. Traditional approaches to quadrature on such surfaces rely on per-quad integration, which is inefficient and typically also inaccurate near vertices where other than four quads meet. Instead, we explore the space of possible groupings of quads and identify the optimal macro-quads in terms of the number of quadrature points needed. We show that macro-quads consisting of quads from one or several consecutive levels of subdivision considerably reduce the cost of numerical integration. Our rules possess a tensor product structure and the underlying univariate rules are Gaussian, i.e., they require the minimum possible number of integration points in both univariate directions.The optimal quad groupings differ depending on the particular application. For instance, computing surface areas, volumes, or solving the Laplace problem lead to different spline spaces with specific structures in terms of degree and continuity. We show that in most cases the optimal groupings are quad-strips consisting of (1×n) quads, while in some cases a special macro-quad spanning more than one subdivision level offers the most economical integration.Additionally, we extend existing results on exact integration of subdivision splines. This allows us to validate our approach by computing surface areas and volumes with known exact values. We demonstrate on several examples that our quadratures use fewer quadrature points than traditional quadratures. We illustrate our approach to subdivision spline quadrature on the well-known Catmull–Clark scheme based on bicubic splines, but our ideas apply also to subdivision schemes of arbitrary bidegree, including non-uniform and hierarchical variants. Specifically, we address the problems of computing areas and volumes of Catmull–Clark subdivision surfaces, as well as solving the Laplace and Poisson PDEs defined over planar unstructured quadrilateral meshes in the context of isogeometric analysis.

On Orienting Edges of Unstructured Two- and Three-Dimensional Meshes

Finite element codes typically use data structures that represent unstructured meshes as collections of cells, faces, and edges, each of which require associated coordinate systems. One then needs to store how the coordinate system of each edge relates to that of neighboring cells. However, we can simplify data structures and algorithms if we can a priori orient coordinate systems in such a way that the coordinate systems on the edges follow uniquely from those on the cells by rule. Such rules require that every unstructured mesh allow the assignment of directions to edges that satisfy the convention in adjacent cells. We show that the convention chosen for unstructured quadrilateral meshes in the deal.II library always allows to orient meshes. It can therefore be used to make codes simpler, faster, and less bug prone. We present an algorithm that orients meshes in O ( N ) operations. We then show that consistent orientations are not always possible for 3D hexahedral meshes. Thus, cells generally need to store the direction of adjacent edges, but our approach also allows the characterization of cases where this is not necessary. The 3D extension of our algorithm either orients edges consistently, or aborts, both within O ( N ) steps.

Smooth cubic spline spaces on unstructured quadrilateral meshes with particular emphasis on extraordinary points: Geometric design and isogeometric analysis considerations

We present a framework for geometric design and isogeometric analysis on unstructured quadrilateral meshes. Acknowledging the differing requirements posed by design (e.g., the convenience of an intuitive control net) and analysis (e.g., good approximation behavior), we propose the construction of a separate, smooth spline space for each while ensuring isogeometric compatibility – requiring the geometric models to be members of the analysis-suitable spaces. The methodology is simple and is presented for bi-cubic splines; extensions to higher degrees are possible, and are briefly discussed. The presentation has been structured to show compatibility with T-splines – a state-of-the-art CAD technology – but the approach should extend to other locally refinable spline technologies (based on local tensor-product structures). An instantiation of the framework is presented, and several numerical tests focused on geometric design and isogeometric analysis demonstrate the versatility of the developed framework, and show significantly higher convergence rates than attained previously in the considered setting.

High order spectral difference lattice Boltzmann method for incompressible hydrodynamics

This work presents a lattice Boltzmann equation (LBE) based high order spectral difference method for incompressible flows. In the present method, the spectral difference (SD) method is adopted to discretize the convection and collision term of the LBE to obtain high order (≥3) accuracy. Because the SD scheme represents the solution as cell local polynomials and the solution polynomials have good tensor-product property, the present spectral difference lattice Boltzmann method (SD-LBM) can be implemented on arbitrary unstructured quadrilateral meshes for effective and efficient treatment of complex geometries. Thanks to only first oder PDEs involved in the LBE, no special techniques, such as hybridizable discontinuous Galerkin method (HDG), local discontinuous Galerkin method (LDG) and so on, are needed to discrete diffusion term, and thus, it simplifies the algorithm and implementation of the high order spectral difference method for simulating viscous flows. The proposed SD-LBM is validated with four incompressible flow benchmarks in two-dimensions: (a) the Poiseuille flow driven by a constant body force; (b) the lid-driven cavity flow without singularity at the two top corners–Burggraf flow; and (c) the unsteady Taylor–Green vortex flow; (d) the Blasius boundary-layer flow past a flat plate. Computational results are compared with analytical solutions of these cases and convergence studies of these cases are also given. The designed accuracy of the proposed SD-LBM is clearly verified.

2-D magnetotelluric modeling using finite element method incorporating unstructured quadrilateral elements

In this research, the finite element (FE) method incorporating quadrilateral elements for solving 2-D MT modeling was presented. The finite element software was developed, employing a paving algorithm to generate the unstructured quadrilateral mesh. The accuracy, efficiency, reliability, and flexibility of our FE forward modeling are presented, compared and discussed. The numerical results indicate that our FE codes using an unstructured quadrilateral mesh provide good accuracy when the local mesh refinement is applied around sites and in the area of interest, with superior results when compared to other FE methods. The reliability of the developed codes was also confirmed when comparing both analytical solutions and COMMEMI2D model. Furthermore, our developed FE codes incorporating an unstructured quadrilateral mesh showed useful and powerful features such as handling irregular and complex subregions and providing local refinement of the mesh for a 2-D domain as closely as unstructured triangular mesh but it requires less number of elements in a mesh.

An unstructured quadrilateral mesh generation algorithm for aircraft structures

Because commercial aircraft are built with thin-walled structures, their structural performance is well-modeled using shell-element meshes. However, creating these meshes for the full aircraft configuration can be challenging and presents a bottleneck in the design process, especially in a configuration-level design space. This paper presents an algorithm that automatically creates unstructured quadrilateral meshes for the full airframe based on just the description of the desired structural members. The approach consists in representing each node in the mesh as a linear combination of points on the geometry so that the structural mesh morphs as the geometry changes, as it would, for example, in aerostructural optimization. The algorithm divides the aircraft skin into 4-sided domains based on the underlying B-spline representation of the geometry. It meshes each domain independently using an algorithm based on constrained Delaunay triangulation, triangle merging and splitting to obtain a quadrilateral mesh, and elliptical smoothing. Examples of full-configuration structural meshes are provided, and a mesh convergence study is performed to show that element quality is maintained as the structural mesh is refined. The algorithm is available as part of the open-source aircraft geometry tool suite, GeoMACH.

Isogeometric analysis using manifold-based smooth basis functions

We present an isogeometric analysis technique that builds on manifold-based smooth basis functions for geometric modelling and analysis. Manifold-based surface construction techniques are well known in geometric modelling and a number of variants exist. Common to most is the concept of constructing a smooth surface by blending together overlapping patches (or, charts), as in differential geometry description of manifolds. Each patch on the surface has a corresponding planar patch with a smooth one-to-one mapping onto the surface. In our implementation, manifold techniques are combined with conformal parameterisations and the partition-of-unity method for deriving smooth basis functions on unstructured quadrilateral meshes. Each vertex and its adjacent elements on the surface control mesh have a corresponding planar patch of elements. The star-shaped planar patch with congruent wedge-shaped elements is smoothly parameterised with copies of a conformally mapped unit square. The conformal maps can be easily inverted in order to compute the transition functions between the different planar patches that have an overlap on the surface. On the collection of star-shaped planar patches the partition of unity method is used for approximation. The smooth partition of unity, or blending functions, are assembled from tensor-product b-spline segments defined on a unit square. On each patch a polynomial with a prescribed degree is used as a local approximant. In order to obtain a mesh-based approximation scheme the coefficients of the local approximants are expressed in dependence of vertex coefficients. This yields a basis function for each vertex of the mesh which is smooth and non-zero over a vertex and its adjacent elements. Our numerical simulations indicate the optimal convergence of the resulting approximation scheme for Poisson problems and near optimal convergence for thin-plate and thin-shell problems discretised with structured and unstructured quadrilateral meshes.

Implementation of a GPS-RO data processing system for the KIAPS-LETKF data assimilation system

Abstract. The Korea Institute of Atmospheric Prediction Systems (KIAPS) has been developing a new global numerical weather prediction model and an advanced data assimilation system. As part of the KIAPS package for observation processing (KPOP) system for data assimilation, preprocessing, and quality control modules for bending-angle measurements of global positioning system radio occultation (GPS-RO) data have been implemented and examined. The GPS-RO data processing system is composed of several steps for checking observation locations, missing values, physical values for Earth radius of curvature, and geoid undulation. An observation-minus-background check is implemented by use of a one-dimensional observational bending-angle operator, and tangent point drift is also considered in the quality control process. We have tested GPS-RO observations utilized by the Korean Meteorological Administration (KMA) within KPOP, based on both the KMA global model and the National Center for Atmospheric Research Community Atmosphere Model with Spectral Element dynamical core (CAM-SE) as a model background. Background fields from the CAM-SE model are incorporated for the preparation of assimilation experiments with the KIAPS local ensemble transform Kalman filter (LETKF) data assimilation system, which has been successfully implemented to a cubed-sphere model with unstructured quadrilateral meshes. As a result of data processing, the bending-angle departure statistics between observation and background show significant improvement. Also, the first experiment in assimilating GPS-RO bending angle from KPOP within KIAPS-LETKF shows encouraging results.

A cell-local finite difference discretization of the low-order quasidiffusion equations for neutral particle transport on unstructured quadrilateral meshes

We present a quasidiffusion (QD) method for solving neutral particle transport problems in Cartesian XY geometry on unstructured quadrilateral meshes, including local refinement capability. Neutral particle transport problems are central to many applications including nuclear reactor design, radiation safety, astrophysics, medical imaging, radiotherapy, nuclear fuel transport/storage, shielding design, and oil well-logging. The primary development is a new discretization of the low-order QD (LOQD) equations based on cell-local finite differences. The accuracy of the LOQD equations depends on proper calculation of special non-linear QD (Eddington) factors from a transport solution. In order to completely define the new QD method, a proper discretization of the transport problem is also presented. The transport equation is discretized by a conservative method of short characteristics with a novel linear approximation of the scattering source term and monotonic, parabolic representation of the angular flux on incoming faces. Analytic and numerical tests are used to test the accuracy and spatial convergence of the non-linear method. All tests exhibit O(h2) convergence of the scalar flux on orthogonal, random, and multi-level meshes.

Перестроение неструктурированных четырехугольных и смешанных сеток

Перестроение неструктурированных четырехугольных и смешанных сеток

INTERACTION OF SHOCK WAVE WITH MULTI-FLUIDS INTERFACE USING QUADRILATERAL-BASED ADAPTIVE MESH

In this paper, the interaction of shock waves with multi-fluids interfaces is investigated by numerical simulations using unstructured quadrilateral adaptive meshes. In order to obtain a detailed structure of the interface, a solution adaptive method for compressible multi-fluid flows developed by Zheng et al. is employed. Firstly, the method is verified by a planar shock and interface interaction problem, which is compared with the front tracking method for the Richtmyer–Meshkov instability problem. Following the verification, the interaction between a circular shock and a sinusoidally perturbed circular interface in cylinder vessel is firstly investigated in our paper. The results show that the solution adaptive method can be employed to study the compressible multi-fluid cases with relatively complex geometry as well as capturing the fine details of interfacial structures of the interaction.

Numerical study on the evolution of the shock-accelerated SF6 interface: Influence of the interface shape

The early phases of the shock interaction process on two-dimensional interfaces with different shapes are numerically investigated in this study, which are closely related to the shock refraction and reflection, vorticity production and transport. The numerical method employs an adaptive unstructured quadrilateral mesh, which can capture the wave pattern and interface evolution very well. Simulations are carried out under the conditions of an incident shock Mach number of 1.2 and the light/heavy (air/SF6) interface. Five different shapes are considered in the simulations: rectangle, ellipse, diamond and two kinds of triangle. The results show that the interfacial shapes can influence the wave patterns particularly on the shape and evolution of refracted shock waves. The generation and the distribution of vorticity on the interfaces with five different shapes also have dissimilarities. The circulation deposition on five interfaces is quantitatively investigated and compared with theoretical model. A good agreement is found between the numerical results and the predictions by the theoretical model. Some characteristic scales of the interface are tracked. Under the influence of nonlinear-acoustic effect and vorticity effect, the interfaces present different evolution modes.

Converting an unstructured quadrilateral/hexahedral mesh to a rational T-spline

This paper presents a novel method for converting any unstructured quadrilateral or hexahedral mesh to a generalized T-spline surface or solid T-spline, based on the rational T-spline basis functions. Our conversion algorithm consists of two stages: the topology stage and the geometry stage. In the topology stage, the input quadrilateral or hexahedral mesh is taken as the initial T-mesh. To construct a gap-free T-spline, templates are designed for each type of node and applied to elements in the input mesh. In the geometry stage, an efficient surface fitting technique is developed to improve the surface accuracy with sharp feature preservation. The constructed T-spline surface and solid T-spline interpolate every boundary node in the input mesh, with C 2-continuity everywhere except the local region around irregular nodes. Finally, a Bezier extraction technique is developed and linear independence of the constructed T-splines is studied to facilitate T-spline based isogeometric analysis.

CAM-SE: A scalable spectral element dynamical core for the Community Atmosphere Model

The Community Atmosphere Model (CAM) version 5 includes a spectral element dynamical core option from NCAR’s High-Order Method Modeling Environment. It is a continuous Galerkin spectral finite-element method designed for fully unstructured quadrilateral meshes. The current configurations in CAM are based on the cubed-sphere grid. The main motivation for including a spectral element dynamical core is to improve the scalability of CAM by allowing quasi-uniform grids for the sphere that do not require polar filters. In addition, the approach provides other state-of-the-art capabilities such as improved conservation properties. Spectral elements are used for the horizontal discretization, while most other aspects of the dynamical core are a hybrid of well-tested techniques from CAM’s finite volume and global spectral dynamical core options. Here we first give an overview of the spectral element dynamical core as used in CAM. We then give scalability and performance results from CAM running with three different dynamical core options within the Community Earth System Model, using a pre-industrial time-slice configuration. We focus on high-resolution simulations, using 1/4 degree, 1/8 degree, and T341 spectral truncation horizontal grids.

Numerical simulations of interface with different shape accelerated by a planar shock

The development of Richtmyer-Meshkov instability on interfaces with different shapes is numerically investigated. The numerical method employs an adaptive unstructured quadrilateral mesh, which can capture the interface evolution very well. Simulations are carried out under the condition of a Mach number of 1.2 and light/heavy (air/ SF6) interfaces. Three shapes are considered in the simulations, i.e., rectangle, ellipse and triangle. The differences in the wave system, generation and distribute of the vorticity and evolution of interfaces are presented and discussed for these three different shapes. The results show that the initial shape of the interface will affect the development of the instability, which will cause the differences in wave patterns, vorticity generation and distribution, and eventually will lead to the different evolution modes. These findings indicate that it may be different to the dominant mechanism of Richtmyer- Meshkov instability on different interface shapes.

Converting an unstructured quadrilateral mesh to a standard T-spline surface

This paper presents a novel method for converting any unstructured quadrilateral mesh to a standard T-spline surface, which is C 2-continuous except for the local region around each extraordinary node. There are two stages in the algorithm: the topology stage and the geometry stage. In the topology stage, we take the input quadrilateral mesh as the initial T-mesh, design templates for each quadrilateral element type, and then standardize the T-mesh by inserting nodes. One of two sufficient conditions is derived to guarantee the generated T-mesh is gap-free around extraordinary nodes. To obtain a standard T-mesh, a second sufficient condition is provided to decide what T-mesh configuration yields a standard T-spline. These two sufficient conditions serve as a theoretical basis for our template development and T-mesh standardization. In the geometry stage, an efficient surface fitting technique is developed to improve the geometric accuracy. In addition, the surface continuity around extraordinary nodes can be improved by adjusting surrounding control nodes. The algorithm can also preserve sharp features in the input mesh, which are common in CAD (Computer Aided Design) models. Finally, a Bezier extraction technique is used to facilitate T-spline based isogeometric analysis. Several examples are tested to show the robustness of the algorithm.

有限要素解析におけるソース項の保存型離散化(四辺形要素の場合)

The conventional Galerkin finite element solution is mesh-dependent, and its discretization for Poisson's equation can not satisfy the conservation law over a nodal domain when unstructured linear meshes are used. This research tries to solve these problems by introducing a new concept of the virtual nodal domain(Vnd) for a linear quadrilateral element, and distributing the source term to a nodal algebraic equation in proportion to the area of the Vnd. The Vnd is evaluated using a second-order flux existing within a linear element. We proofed that the total Vnd of the four nodes equals to the area of the element, which guarantees that our scheme is also elementally conservative. Numerical simulation of heat conduction with both Dirichlet and Neumann boundary conditions shows that the accuracy has been improved obviously comparing with the conventional Galerkin FEM for unstructured quadrilateral meshes, especially for bad quality elements. Our scheme can be introduced into any commercial FEM code quite easily.

Quadrilateral Cell-Based Anisotropic Adaptive Solution for the Euler Equations

AbstractAn anisotropic solution adaptive method based on unstructured quadrilateral meshes for inviscid compressible flows is proposed. The data structure, the directional refinement and coarsening, including the method for initializing the refined new cells, for the anisotropic adaptive method are described. It provides efficient high resolution of flow features, which are aligned with the original quadrilateral mesh structures. Five different cases are provided to show that it could be used to resolve the anisotropic flow features and be applied to model the complex geometry as well as to keep a relative high order of accuracy on an efficient anisotropic mesh.

Towards high-quality, untangled meshes via a force-directed graph embedding approach

Abstract High quality meshes are crucial for the solution of partial differential equations (PDEs) via the finite element method (or other PDE solvers). The accuracy of the PDE solution, and the stability and conditioning of the stiffness matrix depend upon the mesh quality. In addition, the mesh must be untangled in order for the finite element method to generate physically valid solutions. Tangled meshes, i.e., those with inverted mesh elements, are sometimes generated via large mesh deformations or in the mesh generation process. Traditional techniques for untangling such meshes are based on geometry and/or optimization. Optimization-based mesh untangling techniques first untangle the mesh and then smoothe the resulting untangled mesh in order to obtain high quality meshes; such techniques require the solution of two optimization problems. In this paper, we study how to modify a physical, force-directed method based upon the Fruchterman-Reingold (FR) graph layout algorithm so that it can be used for untangling. The objectives of aesthetic graph layout, such as minimization of edge intersections and near equalization of edge lengths, follow the goals of mesh untangling and generating good quality elements, respectively. Therefore, by using the force-directed method, we can achieve both steps of mesh untangling and mesh smoothing in one operation. We compare the effectiveness of our method with that of the optimization-based mesh untangling method in [1] and implemented in Mesquite by untangling a suite of unstructured triangular, quadrilateral, and tetrahedral finite element volume meshes. The results show that the force-directed method is substantially faster than the Mesquite mesh untangling method without sacrificing much in terms of mesh quality for the majority of the test cases we consider in this paper. The force-directed mesh untangling method demonstrates the most promise on convex geometric domains. Further modifications will be made to the method to improve its ability to untangle meshes on non-convex domains.

Application of the VOF method based on unstructured quadrilateral mesh

To simulate two-dimensional free-surface flows with complex boundaries directly and accurately, a novel VOF (Volume-of-fluid) method based on unstructured quadrilateral mesh is presented. Without introducing any complicated boundary treatment or artificial diffusion, this method treated curved boundaries directly by utilizing the inherent merit of unstructured mesh in fitting curves. The PLIC (Piecewise Linear Interface Calculation) method was adopted to obtain a second-order accurate linearized reconstruction approximation and the MLER (Modified Lagrangian-Eulerian Re-map) method was introduced to advect fluid volumes on unstructured mesh. Moreover, an analytical relation for the interface’s line constant vs. the volume clipped by the interface was developed so as to improve the method’s efficiency. To validate this method, a comprehensive series of large straining advection tests were performed. Numerical results provide convincing evidences for the method’s high volume conservative accuracy and second-order shape error convergence rate. Also, a dramatic improvement on computational accuracy over its unstructured triangular mesh counterpart is checked.