A Fully Automatic hp-Adaptivity

We present an algorithm, and a 2D implementation for a fully automatic hp-adaptive strategy for elliptic problems. Given a mesh, the next, optimally refined mesh, is determined by maximizing the rate of decrease of the hp-interpolation error for a reference solution. Numerical results confirm optimal, exponential convergence rates predicted by the theory of hp methods.     

hp-adaptive Two-Level Methods for Boundary Integral Equations on Curves

p - and hp-versions of the Galerkin boundary element method for hypersingular and weakly singular integral equations of the first kind on curves. We derive a-posteriori error estimates that are based on stable two-level decompositions of enriched ansatz spaces. The Galerkin errors are estimated by inverting local projection operators that are defined on small subspaces of the second level. A p-adaptive and two hp-adaptive algorithms are defined and numerical experiments confirm their efficiency. Received August 30, 2000; revised April 3, 2001     

Constraint-Free Adaptive FEMs on Quadrilateral Nonconforming Meshes

Finite element methods (FEMs) on nonconforming meshes have been much studied in the literature. In all earlier works on such methods , some constraints must be imposed on the degrees of freedom on the edge/face with hanging nodes in order to maintain continuity, which make the numerical implementation more complicated. In this paper, we present two FEMs on quadrilateral nonconforming meshes which are constraint-free. Furthermore, we establish the corresponding residual-based a posteriori error reliability and efficiency estimation for general quadrilateral meshes. We also present extensive numerical testing results to systematically compare the performance among three adaptive quadrilateral FEMs: the constraint-free adaptive $\mathbb Q _1$ FEM on quadrilateral nonconforming meshes with hanging nodes developed herein, the adaptive $\mathbb Q _1$ FEM based on quadrilateral red-green refinement without any hanging node recently proposed in Zhao et al. (SIAM J Sci Comput 3(4):2099–2120, 2010), and the classical adaptive $\mathbb Q _1$ FEM on quadrilateral nonconforming meshes with constraints on hanging nodes. Some extensions are also included in this paper.     

An automatic coarse and fine surface mesh generation scheme based on medial axis transform: Part II implementation

In this paper, we present implementation aspects of a surface finite element (FE) meshing algorithm described in Part I (this volume) [1]. This meshing scheme is based on the medial axis transform (MAT) [2] to interrogate shape and to subdivide it into topologically simple subdomains. The algorithm can be effectively used to create coarse discretization and fine triangular surface meshes. We describe our techniques and methodology used in the implementation of the meshing and MAT algorithms. We also present some running times of our experimental system. We finally report the results we have obtained from several design and analysis applications which include adaptive surface approximations using triangular facets, and adaptiveh- andp-adaptive finite element analysis (FEA) of plane stress problems. These studies demonstrate the potential applicability of our techniques in computer aided design and analysis.     

A goal-oriented hp-adaptive finite element approach to radar scattering problems

The paper presents application of an hp-adaptive finite element method for scattering of electromagnetic waves. The main objective of the numerical analysis is to determine the characteristics of the scattered waves indicating the power being scattered at a given direction––i.e. the radar cross-section (RCS). This is achieved considering the scattered far-field which defines RCS and which is expressed as a linear functional of the solution. Techniques of error estimation for the far-field are considered and an h-adaptive strategy leading to the fast reduction of the error of the far-field is presented. The simulations are performed with a three-dimensional version of an hp-adaptive finite element method for electromagnetics based on the hexahedral edge elements combined with infinite elements for modeling the unbounded space surrounding the scattering object.     

hp Adaptive finite elements based on derivative recovery and superconvergence

In this paper, we present a new approach to hp-adaptive finite element methods. Our a posteriori error estimates and hp-refinement indicator are inspired by the work on gradient/derivative recovery of Bank and Xu (SIAM J Numer Anal 41:2294?C2312, 2003; SIAM J Numer Anal 41:2313?C2332, 2003). For element ?? of degree p, R(? ^p u _hp ), the (piece-wise linear) recovered function of ? ^p u is used to approximate ${|\varepsilon|_{1,\tau} = |\hat{u}_{p+1} - u_{p}|_{1,\tau}}$ , which serves as our local error indicator. Under sufficient conditions on the smoothness of u, it can be shown that ${\|{\partial^{p}(\hat{u}_{p+1} - u_{p})\|_{0,\Omega}}}$ is a superconvergent approximation of ${\|(I - R){\partial}^p u_{hp}\|_{0,\Omega}}$ . Based on this, we develop a heuristic hp-refinement indicator based on the ratio between the two quantities on each element. Also in this work, we introduce nodal basis functions for special elements where the polynomial degree along edges is allowed to be different from the overall element degree. Several numerical examples are provided to show the effectiveness of our approach.     

A template-based approach for parallel hexahedral two-refinement

We provide a template-based approach for generating locally refined all-hex meshes. We focus specifically on refinement of initially structured grids utilizing a 2-refinement approach where uniformly refined hexes are subdivided into eight child elements. The refinement algorithm consists of identifying marked nodes that are used as the basis for a set of four simple refinement templates. The target application for 2-refinement is a parallel grid-based all-hex meshing tool for high performance computing in a distributed environment. The result is a parallel consistent locally refined mesh requiring minimal communication and where minimum mesh quality is greater than scaled Jacobian 0.3 prior to smoothing.     

Inhomogeneous lossy waveguide mode analysis

This paper discusses electromagnetic numerical mode analysis in waveguides with materially inhomogeneous cross-sections and material dissipation. A full-wave formulation of Maxwell’s homogeneous equations including Gauss electric law, stable at vanishing propagation constant is implemented and verified in terms of the hp-adaptive version of the finite element method. It provides the possibility to use high order polynomial enrichments combined with strongly graded meshes. It is considered most efficient in resolving the loss of solution regularity at material interfaces with large contrast. Numerical examples including materially lossless homogeneous and inhomogeneous cross sections with and without losses are analysed to corroborate the implementation. The efficiency of using higher order polynomial enrichments is shown. The approach is anticipated to have a broad application, from modern on-chip interconnect and antenna technologies to the design of low observable aerial vehicles.     

hp-Discontinuous Galerkin Finite Element Methods with Least-Squares Stabilization

We consider a family of hp-version discontinuous Galerkin finite element methods with least-squares stabilization for symmetric systems of first-order partial differential equations. The family includes the classical discontinuous Galerkin finite element method, with and without streamline-diffusion stabilization, as well as the discontinuous version of the Galerkin least-squares finite element method. An hp-optimal error bound is derived in the associated DG-norm. If the solution of the problem is elementwise analytic, an exponential rate of convergence under p-refinement is proved. We perform numerical experiments both to illustrate the theoretical results and to compare the various methods within the family.     

Adaptive mesh refinement using piecewise-linear shape functions based on the blending function method

Use of quadrilateral elements for finite element mesh refinement can lead either to so-called irregular meshes or the necessity of adjustments between finer and coarser parts of the mesh necessary. In the case of irregular meshes, constraints have to be introduced in order to maintain continuity of the displacements. Introduction of finite elements based on blending function interpolation shape functions using piecewise boundary interpolation avoids these problems. This paper introduces an adaptive refinement procedure for these types of elements. The refinement is anh-method. Error estimation is performed using the Zienkiewicz-Zhu method. The refinement is controlled by a switching function representation. The method is applied to the plane stress problem. Numerical examples are given to show the efficiency of the methodology.     

A note on the design of hp-adaptive finite element methods for elliptic partial differential equations

We introduce an hp-adaptive finite element algorithm based on a combination of reliable and efficient residual error indicators and a new hp-extension control technique which assesses the local regularity of the underlying analytical solution on the basis of its local Legendre series expansion. Numerical experiments confirm the robustness and reliability of the proposed algorithm.     

Arbitrary-level hanging nodes and automatic adaptivity in the hp-FEM

In this paper we present a new automatic adaptivity algorithm for the hp-FEM which is based on arbitrary-level hanging nodes and local element projections. The method is very simple to implement compared to other existing hp-adaptive strategies, while its performance is comparable or superior. This is demonstrated on several numerical examples which include the L-shape domain problem, a problem with internal layer, and the Girkmann problem of linear elasticity. With appropriate simplifications, the proposed technique can be applied to standard lower-order and spectral finite element methods.     

Template-based quadrilateral mesh generation from imaging data

This paper describes a novel template-based meshing approach for generating good quality quadrilateral meshes from 2D digital images. This approach builds upon an existing image-based mesh generation technique called Imeshp, which enables us to create a segmented triangle mesh from an image without the need for an image segmentation step. Our approach generates a quadrilateral mesh using an indirect scheme, which converts the segmented triangle mesh created by the initial steps of the Imesh technique into a quadrilateral one. The triangle-to-quadrilateral conversion makes use of template meshes of triangles. To ensure good element quality, the conversion step is followed by a smoothing step, which is based on a new optimization-based procedure. We show several examples of meshes generated by our approach, and present a thorough experimental evaluation of the quality of the meshes given as examples.     

A methodology for quadrilateral finite element mesh coarsening

High fidelity finite element modeling of continuum mechanics problems often requires using all quadrilateral or all hexahedral meshes. The efficiency of such models is often dependent upon the ability to adapt a mesh to the physics of the phenomena. Adapting a mesh requires the ability to both refine and/or coarsen the mesh. The algorithms available to refine and coarsen triangular and tetrahedral meshes are very robust and efficient. However, the ability to locally and conformally refine or coarsen all quadrilateral and all hexahedral meshes presents many difficulties. Some research has been done on localized conformal refinement of quadrilateral and hexahedral meshes. However, little work has been done on localized conformal coarsening of quadrilateral and hexahedral meshes. A general method which provides both localized conformal coarsening and refinement for quadrilateral meshes is presented in this paper. This method is based on restructuring the mesh with simplex manipulations to the dual of the mesh. In addition, this method appears to be extensible to hexahedral meshes in three dimensions. Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC04-94AL85000.     

<Emphasis Type="Italic">hp</Emphasis>-Version <Emphasis Type="Italic">a priori</Emphasis> Error Analysis of Interior Penalty Discontinuous Galerkin Finite Element Approximations to the Biharmonic Equation

We consider the symmetric formulation of the interior penalty discontinuous Galerkin finite element method for the numerical solution of the biharmonic equation with Dirichlet boundary conditions in a bounded polyhedral domain in . For a shape-regular family of meshes consisting of parallelepipeds, we derive hp-version a priori bounds on the global error measured in the L² norm and in broken Sobolev norms. Using these, we obtain hp-version bounds on the error in linear functionals of the solution. The bounds are optimal with respect to the mesh size h and suboptimal with respect to the degree of the piecewise polynomial approximation p. The theoretical results are confirmed by numerical experiments, and some practical applications in Poisson–Kirchhoff thin plate theory are presented.     

Automatic unstructured all-hexahedral mesh generation from B-Reps for non-manifold CAD assemblies

This paper describes an automatic and robust approach to convert non-manifold CAD assemblies into unstructured all-hexahedral meshes conformal to the given B-Reps (boundary-representations) and with sharp feature preservation. In previous works, we developed an octree-based isocontouring method to construct unstructured hexahedral meshes for arbitrary non-manifold and manifold domains. However, sharp feature preservation still remains a challenge, especially for non-manifold CAD assemblies. In this paper, boundary features such as NURBS (non-uniform rational B-Splines) curves and surface patches are first extracted from the given B-Reps. Features shared by multiple components are identified and distinguished. To preserve these non-manifold features, one given surface patch may need to be split into several small ones. An octree-based algorithm is then carried out to create an unstructured all-hexahedral base mesh, detecting and preserving all the sharp features via a curve and surface parametrization. Two sets of local refinement templates are provided for adaptive mesh generation, along with a novel 2-refinement implementation. Vertices in the base mesh are categorized into four groups based on the given non-manifold topology, and each group is relocated using various methods with all sharp features preserved. After this stage, a novel two-step pillowing technique is developed for such complicated non-manifold domains to eliminate triangle-shaped quadrilateral elements along the curves and “doublets”, handling non-manifold and manifold features in different ways. Finally, a combination of smoothing and optimization is used to further improve the mesh quality. Our algorithm is automatic and robust for non-manifold and manifold domains. We have applied our algorithm to several complicated CAD assemblies.     

A finite volume discontinuous Galerkin scheme¶for nonlinear convection–diffusion problems

A popular method for the discretization of conservation laws is the finite volume (FV) method, used extensively in CFD, based on piecewise constant approximation of the solution sought. However, the FV method has problems with the approximation of diffusion terms. Therefore, in several works [17–19, 1, 12, 16, 2], a combination of the FV and FE methods is used. To this end, it is necessary to construct various combinations of simplicial FE meshes with suitable associated FV grids. This is rather complicated from the point of view of the mesh refinement, particularly in 3D problems [20, 21]. It is desirable to use only one mesh. The combination of FV and FE discretizations on the same triangular grid is proposed in [39]. Another possibility is to use the DG method (see [7] or [9] (and the references there) for a general survey). Here we shall use a compromise between the DG FE method and the FV method using piecewise linear discontinuous finite elements over the grid ? _h and piecewise constant approximation of convective terms on the same grid. Dedicated to Professor Ivo Babuška on the occasion of his 75th birthday Received: May 2001 / Accepted: September 2001     

Sparse direct factorizations through unassembled hyper-matrices

We present a novel strategy for sparse direct factorizations that is geared towards the matrices that arise from hp-adaptive Finite Element Methods. In that context, a sequence of linear systems derived by successive local refinement of the problem domain needs to be solved. Thus, there is an opportunity for a factorization strategy that proceeds by updating (and possibly downdating) the factorization. Our scheme consists of storing the matrix as unassembled element matrices, hierarchically ordered to mirror the refinement history of the domain. The factorization of such an ‘unassembled hyper-matrix’ proceeds in terms of element matrices, only assembling nodes when they need to be eliminated. The main benefits are efficiency from the fact that only updates to the factorization are made, high scalar efficiency since the factorization process uses dense matrices throughout, and a workflow that integrates naturally with the application.     

A Bi–Hyperbolic Finite Volume Method on Quadrilateral Meshes

A non-oscillatory, high resolution reconstruction method on quadrilateral meshes in two dimensions (2D) is presented. It is a two-dimensional extension of Marquina’s hyperbolic method. The generalization to quadrilateral meshes allows the method to simulate realistic flow problems in complex domains. An essential point in the construction of the method is a second order accurate approximation of gradients on an irregular, quadrilateral mesh. The resulting scheme is optimal in the sense that it is third order accurate and the reconstruction requires only nearest neighbour information. Numerical experiments are presented and the computational results are compared to experimental data.