Mesh Generation y Marshall Bern Paul Plassmann Intro duction A mesh is a discretization of a geometric domain into small simple shap es such as triangles or quadrilaterals in two dimensions and tetrahedra or hexahedra in three Meshes nd use in many application areas In geography and cartography meshes give compact representa tions of terrain data In computer graphics most ob jects are ultimately reduced to meshes b efore rendering Finally meshes are almost essential in the numerical solution of partial dierential equations arising in physical simulation In this chapter we concentrate on al gorithms for this last application assuming an audience including b oth practitioners such as engineers and theoreticians such as computational geometers and numerical analysts Typ es of Geometric Domains We divide the p ossible inputs rst according to dimensiontwo or three We distinguish four typ es of planar domains as shown in Figure For us a simple polygon includes b oth b oundary and interior A polygon with holes is a simple p olygon minus the interiors of some other simple p olygons its b oundary has more than one connected comp onent A multiple domain is more general still allowing internal b oundaries in fact such a domain may b e any planar straightline graph in which the innite face is b ounded by a simple cycle Multiple domains mo del ob jects made from more than one material Curved domains allow sides that are algebraic curves such as splines As in the rst three cases collectively known as polygonal domains curved domains may or may not include holes and internal b oundaries Threedimensional inputs have analogous typ es A simple polyhedron is top ologically equivalent to a ball A general polyhedron may b e multiply connected meaning that it is top ologically equivalent to a solid torus or some other higher genus solid it may also have cavities meaning that its b oundary may have more than one connected comp onent We do assume however that at every p oint on the b oundary of a general p olyhedron a suciently small sphere encloses one connected piece of the p olyhedrons interior and one connected piece of its exterior Finally there are multiple polyhedral domains general p olyhedra with internal b oundariesand threedimensional curved domains which typically have b oundaries dened by spline patches Construction and mo deling of domain geometry lie outside the scop e of this chapter so we shall simply assume that domains are given in some sort of b oundary representation Xerox Palo Alto Research Center Coyote Hill Rd Palo Alto CA b ernparcxeroxcom y Department of Computer Science and Engineering The Pennsylvania State University University Park PA plassmancsepsuedu Supp orted by the Mathematical Information and Computational Sciences Division subprogram of the Oce of Computational and Technology Research US Department of Energy under Contract WEng Figure Typ es of twodimensional inputs a simple p olygon b p olygon with holes c multiple domain and d curved domain without sp ecifying the exact form of this representation Computational geometers typically assume exact combinatorial data structures such as linked lists for simple p olygons and p olygons with holes doubly connected edge lists or quadedge structures for planar multiple domains and wingededge structures for p olyhedral domains In practice complicated domains are designed on computer aided design CAD systems These systems use surface representations designed for visual rendering and then translate the nal design to another format for input to the mesh generator The Stereolithography Tessellation Language STL le format originally develop ed for the rapid prototyping of solid mo dels sp ecies the b oundary as a list of surface p olygons usually triangles and sur face normals The advantages of the STL input format are that a watertight mo del can b e ensured and mo del tolerance deviation from the CAD mo del can b e sp ecied by the user The Initial Graphics Exchange Sp ecication IGES format enables a variety of sur face representations including higherorder representations such as Bsplines and NURBs Perhaps due to its greater complexity or to sloppy CAD systems or users IGES les often contain incorrect geometry either gaps or extra material at surface intersections An alternative approach to format translation is to directly query the CAD system with say p ointenclosure queries and then construct a new representation based on the answers to those queries This approach is most advantageous when the translation problem is dicult as it may b e in the case of implicit surfaces level sets of complicated functions or constructive solid geometry formulas With either approach translation or reconstruction by queries the CAD mo del must b e top ologically correct and suciently accurate to enable meshing We exp ect to see greater integration b etween solid mo deling and meshing in the future Typ es of Meshes A structured mesh is one in which all interior vertices are top ologically alike In graph theoretic terms a structured mesh is an induced subgraph of an innite p erio dic graph such as a grid An unstructured mesh is one in which vertices may have arbitrarily varying lo cal neighb orho o ds A blockstructured or hybrid mesh is formed by a numb er of small structured meshes combined in an overall unstructured pattern In general structured meshes oer simplicity and easy data access while unstructured meshes oer more convenient mesh adaptivity renementderenement based on an initial solution and a b etter t to complicated domains Highquality hybrid meshes enjoy the advantages of b oth approaches but hybrid meshing is not yet fully automatic We shall discuss unstructured mesh generation at greater length than structured or hybrid mesh Figure Typ es of meshes a structured b unstructured and c blo ckstructured Figure a Triangulating quadrilaterals b Sub dividing triangles to form quadrilaterals generation b oth b ecause the unstructured approach seems to b e gaining ground and b ecause it is more closely connected to computational geometry The division b etween structured and unstructured meshes usually extends to the shap e of the elements twodimensional structured meshes typically use quadrilaterals while un structured meshes use triangles In three dimensions the analogous element shap es are hexahedra meaning top ological cub es and tetrahedra There is however no essential rea son for structured and unstructured meshes to use dierent element shap es In fact it is p ossible to sub divide elements in order to convert b etween triangles and quadrilaterals Figure and b etween tetrahedra and hexahedra Organization Section gives a brief survey of numerical metho ds and their implications for mesh genera tion Section discusses the inuence of element shap e on accuracy and convergence time Sections and cover structured and unstructured twodimensional meshes Section discusses threedimensional hexahedral mesh generation including structured hybrid and unstructured approaches Finally Section describ es threedimensional unstructured tetra hedral mesh generation We shall explain the basic computational geometry results as they arise within a larger context however Section concludes with a separate theoretical discussion b ecause un structured planar mesh generation is esp ecially rich in interesting geometric questions Throughout this article we emphasize practical issues an earlier survey by Bern and Epp stein emphasized theoretical results Although there is inevitably some overlap b etween these two surveys we intend them to b e complementary Mesh generation has a huge literature and we cannot hop e to cover all of it There are excellent references on numerical metho ds structured mesh generation and unstructured mesh generation There are also several nice Web sites with uptodate information on mesh generation Numerical Metho ds Scientic computing seeks accurate discrete mo dels for continuous physical phenomena We can divide the pro cess into three interdep endent steps problem formulation mesh generation and equation solution In this section we discuss discretization and solution metho ds and their impact on mesh generation Discrete Formulation There are a numb er of approaches to the discrete approximation of partial dierential equa tions PDEs mo deling a physical system Here we briey review the standard discretization metho ds nite dierence nite element and nite volume Although these metho ds result in linear systems of similar structure the desired characteristics of meshes for these metho ds dier The nite dierence method replaces a continuous dierential op erator with a dierence approximation Consider the partial dierential equation Lu f where L is some dierential op erator and u is a function of p osition and p ossibly also of time We seek an approximate solution of on some geometric domain A standard nite dierence approach replaces L with a discrete stencil Writing u ux for the k k value of u at mesh vertex p osition x the action of the stencil at x can b e approximated k i by X Lux A u i ik k k adj (x ) i where adj x is the set of p oints adjacent to x in the mesh and A is a set of weights i i ik dep ending only on L and the geometry of the mesh The righthand side of can also b e discretized yielding a system of linear equations n X A u f ik k i k =1 to b e solved for the unknowns u Because the nite dierence stencil gives nonzero weight k only to neighb oring vertices this system will b e quite sparse It is convenient