JOURNAL OF COMPUTATIONAL PHYSICS 124, 14±45 (1996) ARTICLE NO. 0042 Tetrahedral hp Finite Elements: Algorithms and Flow Simulations S. J. SHERWIN AND G. E. KARNIADAKIS* Center for Fluid Mechanics, Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912 Received March 1, 1995; revised June 1995 der ®nite elements and ®nite volumes are the prevailing We present a new discretisation for the incompressible Navier± discretisation [2]. However, with the interest shifted to- Stokes equations that extends spectral methods to three-dimen- wards accurate solutions of the viscous ¯ow equations sional complex domains consisting of tetrahedral subdomains. The around aerodynamic con®gurations, recent research ef- algorithm is based on standard concepts of hp ®nite elements as forts have been focused into developing high-order discret- well as tensorial spectral elements. This new formulation employs isation procedures on unstructured meshes [3, 4]. Also, a hierarchical/modal basis constructed from a new apex co-ordinate system which retains a generalised tensor product. These properties new ®elds such as computational electromagnetics for enable the development of computationally ef®cienct algorithms aerospace design involve the solution of time-dependent for use on standard ®nite volume unstructured meshes. A detailed highly oscillatory solutions for which high-order discreti- analysis is presented that documents the stability and exponential sations are more ef®cient [5]. In particular, for long-time convergence of the method and several ¯ow cases are simulated integration of time-dependent solutions it has been docu- and compared with analytical and experimental results. Q 1996 Aca- demic Press, Inc. mented in [6] that high-order numerical methods provide the most cost-effective approach. Spectral and compact ®nite difference methods provide CONTENTS high-order accuracy but they are practically limited to sim- ple geometries [7, 8]. Spectral element methods have ex- 1. Introduction. tended spectral discretisation to more complex geometries 2. Tetrahedral expansion basis. 2.1. Co-ordinate system. 2.2. Expan- [9, 10] but they require non-standard meshes for unstruc- sion basis. 2.3. Multi-domain connectivity. 3. Local Operations. 3.1. Integration. 3.2. Local projection. 3.3. Dif- tured discretisation and their adaptive capability is limited. ferentiation. For a new numerical method to be useful it has to utilise 4. Global matrix operations. the existing technology of mesh generators for three-di- 5. Boundary transformation and curvilinear elements. 5.1. Boundary mensional geometries [11] and provide faster convergence transformation. 5.2. Curvilinear elements. at costs no higher than standard ®nite element discreti- 6. Linear advection equation. 6.1. Discretisation. 6.2. Spectrum of the weak advection operator. sation. 7. Elliptic Helmholtz equation. 7.1. Discretisation. 7.2. Spectrum of In previous work [12], we have developed an hp ®nite the weak Laplacian operator. 7.3. Results. element method for the numerical solution of the two- 8. Incompressible Navier±Stokes. 8.1. Formulation. 8.2. Simulations. dimensional unsteady Navier±Stokes equations. This for- 8.2.1. Three-dimensional fully developed ¯ows. 8.2.2. Internal ¯ow mulation was implemented in the code N«kTa . The dis- in an conduit expansion. 8.2.3. External ¯ow over a stepped r cylinder. cretisation is based on arbitrary triangulisations in domains 9. Summary. containing complex geometries. On each triangle a spectral expansion is employed. The expansion is constructed using 1. INTRODUCTION Jacobi polynomials of mixed weight and has a hierarchical property which allows the order to be variable in each High-order numerical methods, although very popular subdomain. This construction retains the tensor product in direct and large eddy simulation of turbulent ¯ows [1], property (similar to standard spectral expansions), which have had a very limited use in other areas of computational is key in obtaining computational ef®ciency via the sum ¯uid dynamics. Primarily this is due to use of non-canonical factorisation technique. The expansion also accommodates computational domains. In computational aerodynamics, exact numerical quadrature. The two-dimensional basis for example, where unstructured meshes are involved to was extended to three dimensions in [13], where solutions accommodate the geometric complexity involved, low-or- to the second-order elliptic problems were presented. It was demonstrated that exponential convergence is * Address correspondence to Professor G. E. Karniadakis. achieved for smooth solutions in complex three-dimen- 14 0021-9991/96 $18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. ALGORITHMS AND FLOW SIMULATIONS 15 FIG. 1. Solution to the Helmholtz equation of the form u(x, y, z) 5 sin(fx) sin(fy)(5.4 2 z) in a domain containing K 5 152 elements (left). Convergence due to p-type re®nement (right). sional domains. Such an example is shown in Fig. 1, where [16]. In particular, we analyse the eigenspectra of the ad- we plot the convergence rate due to p-type re®nement of vection and the elliptic Helmholtz equations that constitute the mesh shown on the right. the main components of the semi-discrete system. It is It can be appreciated that the new tetrahedral basis can found that the eigenspectra of the ®rst- and second-order be used in conjunction with standard mesh generators such derivatives scale similarly to the one-dimensional spectral as those described in [11]. Unlike the low-order methods expansions [17]. Finally, we present the formulation of whose convergence may degrade signi®cantly on distorted Navier±Stokes equations and validate the method with a tetrahedra the new high-order discretisation seems to be series of comparisons of ¯ow simulations against analytical less sensitive to mesh distortion as suggested by prelimi- as well as experimental results. nary numerical evidence. In addition to using standard The paper is organised as follows: In Section 2 we intro- tetrahedral meshes, the formulation is also based on stan- duce a new set of coordinates for triangles and tetrahedra dard hp ®nite element concepts. For example, the expan- and describe the three-dimensional bases. In Section 3, we sion basis is decomposed into vertex, edge, face, and inte- describe operations such as integration and differentiation rior modes similar to other hexahedral hp bases [14, 15]. which are performed locally on each tetrahedral element. With this decomposition, the C 0 continuity requirement In Section 4, we discuss the global matrix assembly which for second-order elliptic problems is easily implemented is used in the following algorithms. Then in Section 5, we following a direct stiffness assembly procedure. complete the formulation discussion by describing bound- In this paper we formulate an algorithm for the three- ary transformation and the implementation of curvilinear dimensional unsteady Navier±Stokes equations using the elements. In Section 6, we use the previous formulation new tetrahedral basis and a high-order splitting scheme to discretise the linear advection equation and numerically FIG. 2. Rectangle to triangle transformation. 16 SHERWIN AND KARNIADAKIS analyse the eigenspectra of the weak advection operator. Section 7 contains a similar analysis for the elliptic Helm- holtz equation. Finally in Section 8, we formulate an algo- rithm to solve the incompressible Navier±Stokes equations and present numerical simulations demonstrating the geo- metric ¯exibility of the algorithm which also exhibits expo- nential convergence for smooth solutions. 2. TETRAHEDRAL EXPANSION BASIS We wish to formulate a tetrahedral expansion basis for the solution of the Navier±Stokes equations extending the two-dimensional expansions proposed by Dubiner [18]. A detailed description of this formulation in two dimensions has been given in [12] and the three-dimensional basis formulation can be found in [13]. Here we outline the basic formulation by introducing the co-ordinate system and de- scribe various properties of the tetrahedral expansion. We also elaborate on the non-restrictive constraints imposed by the alignment of sub-domains as required by the co- FIG. 3. Transformation from hexahedral co-ordinate system to tetra- ordinate system. hedral co-ordinate system. 2.1. Co-ordinate System To introduce the co-ordinate system we must ®rst con- the line f 1 c 5 0, except at the point (f 521, c 5 1), sider a basic mapping as illustrated in Fig. 2. Here we see where F is multi-valued. We know that F is bounded in the mapping of a rectangular domain in the (F, C) space R2 and, therefore, we expect the same to be true in T 2. to a triangle in the (f, c) space. Although the basis we To show that this is the case at the degenerate point (f 5 will introduce is not associated with any speci®c set of 21, c 5 1), we consider the substitution f 5211«sin nodes, the co-ordinate mappings shown in Fig. 2 enables u, c 5 1 2«cos u. Here u is de®ned in a counterclock- the use of a more convenient set of co-ordinates from the wise manner from the vertical and « is a small perturba- computational viewpoint. tion such that when «50 we have (f 521, c 5 1). We de®ne the standard triangle and rectangle as Substituting these values into the de®nition of F given by Eq. (1) we can ®nd the limiting behaviour of the T 2 ;h(f,c)u21#f,c;f1c#0j singularity: R2;h(F,C)u21#F,C#1j. 2(1 2 1 1«sin u) F21,1 5 lim 2 1 5 2 tan u 2 1. The rectangular domain R2 can be mapped into the triangu- «R0 (1 2 1 1«cos u) lar domain T 2 by the following transformation: Since 0 # u # f/4 we know that 0 # tan u # 1 and so c 5C 21#F21,1 # 1. It might appear strange to use a co-ordinate (1 1F)(1 2C) system which has a singular point but the singularity in f 5 21, 2 the co-ordinates does not imply that the expansion is singu- lar.