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Comparison of Discrete Hodge Star Operators for Surfaces Comparison of discrete Hodge star operators for surfaces Item Type Article Authors Mohamed, Mamdouh S.; Hirani, Anil N.; Samtaney, Ravi Citation Comparison of discrete Hodge star operators for surfaces 2016 Computer-Aided Design Eprint version Post-print DOI 10.1016/j.cad.2016.05.002 Publisher Elsevier BV Journal Computer-Aided Design Rights NOTICE: this is the author’s version of a work that was accepted for publication in Computer-Aided Design. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computer-Aided Design, 10 May 2016. DOI: 10.1016/ j.cad.2016.05.002 Download date 09/10/2021 02:07:50 Link to Item http://hdl.handle.net/10754/609008 Accepted Manuscript Comparison of discrete Hodge star operators for surfaces Mamdouh S. Mohamed, Anil N. Hirani, Ravi Samtaney PII: S0010-4485(16)30022-7 DOI: http://dx.doi.org/10.1016/j.cad.2016.05.002 Reference: JCAD 2417 To appear in: Computer-Aided Design Please cite this article as: Mohamed MS, Hirani AN, Samtaney R. Comparison of discrete Hodge star operators for surfaces. Computer-Aided Design (2016), http://dx.doi.org/10.1016/j.cad.2016.05.002 This is a PDF file of an unedited manuscript that has been accepted for publication. Asa service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. *Highlights (for review) Comparison of discrete Hodge star operators for surfaces Mamdouh S. Mohamed, Anil N. Hirani, Ravi Samtaney Various definitions for discrete Hodge star operators are investigated. Darcy and incompressible Navier-Stokes flows are used for numerical experiments. Barycentric Hodge star reproduces the convergence rate of the circumcentric Hodge. The difference in the computational cost is generally minor. *Manuscript Click here to view linked References Comparison of discrete Hodge star operators for surfaces a,1, b a Mamdouh S. Mohamed ∗, Anil N. Hirani , Ravi Samtaney aMechanical Engineering, Physical Science and Engineering Division, King Abdullah University of Science and Technology, Jeddah, KSA bDepartment of Mathematics, University of Illinois at Urbana-Champaign, IL, USA Abstract We investigate the performance of various discrete Hodge star operators for discrete exterior calculus (DEC) using circumcentric and barycentric dual meshes. The performance is evaluated through the DEC solution of Darcy and incompressible Navier-Stokes flows over surfaces. While the circumcentric Hodge operators may be favorable due to their diagonal structure, the barycentric (geometric) and the Galerkin Hodge operators have the advantage of admitting arbitrary simplicial meshes. Numerical experi- ments reveal that the barycentric and the Galerkin Hodge operators retain the numerical convergence order attained through the circumcentric (diagonal) Hodge operators. Furthermore, when the barycentric or the Galerkin Hodge operators are employed, a super-convergence behavior is observed for the incompressible flow solution over unstructured simplicial surface meshes generated by successive subdivision of coarser meshes. Insofar as the computational cost is concerned, the Darcy flow solutions exhibit a moderate increase in the solution time when using the barycentric or the Galerkin Hodge operators due to a modest decrease in the linear system sparsity. On the other hand, for the incompressible flow simulations, both the solution time and the linear system sparsity do not change for either the circumcentric or the barycentric and the Galerkin Hodge operators. Keywords: Discrete exterior calculus (DEC), Hodge star, Circumcentric dual, Barycentric dual 1. Introduction advantageous not only due to the extra flexibility in mesh gen- eration but also due to significantly facilitating any subsequent Discrete exterior calculus (DEC) is a paradigm for the nu- local/adaptive mesh subdivision. This is important consider- merical solution of partial differential equations (PDEs) on sim- ing that successive subdivision of a Delaunay triangulation with plicial meshes [1, 2]. A main advantage of DEC is the mimetic obtuse-angled triangles would result in a non-Delaunay mesh. behavior of its discrete operators where they retain at the dis- crete level many of the identities/rules of their smooth coun- An alternative choice for the dual mesh is the barycentric terparts. Over the past decade, DEC was used to numerically dual. Since the barycenter of a simplex always lies in its in- solve many physical problems including Darcy [3, 4] and in- terior (unlike circumcenters), the dual barycentric cells are al- compressible flows [5, 6, 7]. The mimetic behavior of the ways non-overlapping for arbitrary simplicial meshes. How- DEC discrete operators generally results in superior conserva- ever, the orthogonality between the primal and dual mesh ob- tion properties for DEC discretizations. There also exist other jects that characterized the circumcentric dual is no longer valid numerical methods to discretize vector PDEs on surfaces that for the barycentric dual. This implies that the DEC operators are not based on DEC [8, 9, 10, 11, 12]. involving metrics may need to be redefined. For DEC appli- The definition of most DEC operators requires a dual mesh cations over surface simplicial meshes, it becomes essential to redefine the Hodge star operator and its inverse 1. Two related to the primal simplicial mesh. A common choice for the ∗1 ∗1− dual mesh is the circumcentric dual. The mutual orthogonality previous discrete definitions for Hodge operators on general of the primal simplices and their circumcentric duals results in simplicial meshes are the Galerkin [14] and the geometrical simple expressions for the discrete Hodge star operators. How- barycentric-based [15, 16] definitions. By barycentric-based ever, using the circumcentric dual limits DEC to only Delaunay definitions here we mean the Hodge star definitions that are simplicial meshes. In the case of non-Delaunay meshes, DEC applicable when a barycentric dual mesh is employed. Both implementations using the circumcentric dual along with the di- the two aforementioned definitions result in a sparse (but non- diagonal) matrix representation for the Hodge star operator . agonal Hodge star definition yields incorrect numerical results ∗1 [13]. Enabling DEC to handle arbitrary simplicial meshes is Such a non-diagonal matrix structure further complicates the 1 representation of the inverse operator 1− . However, it is worth pointing out that on surface simplicial∗ meshes, the DEC dis- ∗Corresponding author cretization of common partial differential equations; e.g. Pois- Email addresses: [email protected] (Mamdouh S. son equation and incompressible Navier-Stokes equations, may Mohamed), [email protected] (Anil N. Hirani), not require the inverse operator 1. An alternative approach [email protected] (Ravi Samtaney) ∗1− 1On leave, Department of Mechanical Design and Production, Faculty of for dealing with non-Delaunay meshes, for the specific case of Engineering, Cairo University, Giza, Egypt. a scalar Laplacian, is through the intrinsic Delaunay triangula- Preprint submitted to Computer-Aided Design May 1, 2016 tion [17]. The purpose of this study is to evaluate the performance of the barycentric-based and the Galerkin Hodge operators com- pared with the circumcentric (diagonal) Hodge star operator. The comparison is carried out through the DEC discretiza- tion of Darcy and incompressible Navier-Stokes flows over sur- faces. The main differences between the circumcentric versus the barycentric dual meshes are first addressed in section 2. The definitions of various Hodge star operators are then provided in section 3. This is followed by numerical experiments for Darcy flow and incompressible Navier-Stokes equations in section 4, demonstrating the behavior of the considered Hodge star op- erators. The results demonstrate the numerical convergence as (a) well as the linear system sparsity and the computational cost for the various operators definitions considered during the current study. The paper closes with conclusions emphasizing the main observations and discussing related future insights. 2. Circumcentric versus barycentric dual meshes Consider a domain Ω approximated with the simplicial com- plex K. This paper focuses only on simplicial meshes over flat/curved surfaces, and therefore the domain is considered to have a dimension N = 2. A k-simplex σk K is defined by the nodes v forming it as σk = [v , ..., v ]. The∈ definition of many i 0 k (b) DEC discrete operators requires the dual complex ?K defined over the primal simplicial complex K. The dual to a primal Figure 1: Sketch for a sample non-Delaunay mesh with: (a) the circumcentric simplex σk is the (N k)-cell denoted by ?σk ?K. The top dual, and (b) the barycentric dual. The primal mesh is in black color and the − ∈ dimensional k simplices/cells are consistently oriented; e.g. dual mesh is in red color. ci jk and bi jk are the circumcenter and barycenter counterclockwise,− in all our examples here. of the triangle [vi, v j, vk], respectively. ci j and bi j are the circumcenter and barycenter of the edge [vi, v j], respectively. Defined on the primal and dual mesh complexes are the spaces of the discrete k-forms denoted by Ck(K) and Dk(?K), respectively. For N = 2, the spaces of the discrete forms are primal edges, the dual to the primal edge [v1, v2] is oriented in related via the discrete exterior derivative dk and the discrete the opposite direction. This is due to the circumcenters of the Hodge star k operators as shown in the following diagram ∗ neighboring triangles being in the reversed order. According to d0 d1 the notation in [13], this implies that the dual edge ?[v , v ] has C0(K) C1(K) C2(K) 1 2 a negative volume.
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