AUTOMATIC TRIANGULAR MESH GENERATION BASED ON LONGESTEDGE ALGORITHMS MariaCecilia Rivara Nancy Hitschfeld and Carlos Hasan Department of Computer Science University of Chile Casilla Santiago Chile mcrivaradccuchilecl nancydccuchilecl chasandccuchilecl Abstract Several aspects of the LEPP algorithms based on the use of the LongestEdge Propagation Path of the triangles for dealing both with the triangulation improvement problem and with the auto matic quality triangulation problem are reviewed and discussed Ap plications producing quality nonobtuse triangulations and quality sim plication for terrain modeling are also discussed Key Words Mesh generation quality triangulatio n Delaunay algorithm longestedge algorithms b oundary nonobtuse triangulatio n nite volume metho ds terrain mo deling Intro duction In the adaptive nite element context several mathematical algorithms for the renement andor derenement of unstructured triangulatio ns based on the bisection of triangles by its longestedge have b een discussed and used 1{4 in the last years These algorithms guarantee the construction of re ned nested and irregular triangulations of analogous quality as the input Submitted to the International journal of numerical metho ds juli triangulation However the use of a new and related mathematical con 5, 6 cept the longestedge propagation path of a triangle has allowed the development of new longestedge algorithms for dealing with more general asp ects of the mesh generation problem triangulation renement prob lem triangulation improvement problem automatic quality triangu lation problem quality nonobtuse triangulation problem and terrain mo deling triangulation problem In this pap er dierent asp ects of these mesh generation problems and some of the algorithms prop osed to deal with them are reviewed discussed and illustrated In particular the following sp ecic applications are discussed quality triangulations as needed for nite element metho ds quality nonobtuse b oundary triangulatio ns as needed for mixed nite element and nite volume metho ds and quality surface terrain simplication Mesh generation related problems The p olygon triangulatio n problem an imp ortant issue for nite element applications can b e formulated as follows Denition Polygon Triangulation Problem given N representative points of a polygonal region join them by non intersecting straight line segments so that every region internal to the polygon is a triangle The resulting triagu lations is a conforming triangulation the intersection of adjacent triangles is either a common vertex or a common side Many criteria have b een prop osed as to what constitutes a go o d tri angulation for numerical purp oses some of which involve maximizing the smallest angle or minimizing the total edge length The Delaunay algorithm which constructs triangulations satisfying the rst criteria has b een of com mon use in engineering applications followed by a p ostpro cess step which assures the b oundary resp ect of the p olygon In the adaptive nite element context the triangulation renement prob lem is also critical To state this problem some requirements and criteria ab out how to dene the set of triangles to b e rened and how to obtain the desired resolution need to b e sp ecied To simplify we shall intro duce a subregion R to dene the renement area and a condition over the diameter longestedge of the triangles given by a resolution parameter to x the desired resolution Denition Triangulation Renement Problem given an acceptable trian gulation of a polygonal region construct a local ly rened triangulation such that the diameters of the triangles that intersect the renement area R are less than and such that the smal lest or the largest angle is bounded In the case we disp ose of a badquality triangulation of the p olygonal geometry having a nonadequate distribution of vertices the triangulatio n improvement problem has to b e considered To state this problem a triangle quality indicator function qt a tolerance parameter and a lo cal triangle improvement criterion need to b e sp ecied Denition Triangulation Improvement Problem given a nonquality tri angulation of a polygonal region having triangles such that its quality 0 indicator qt construct an improved triangulation such that each triangle t satises qt Note that if an initial coarse triangulatio n of the b oundary p olygonal vertices is considered the more general automatic quality p olygon triangu lation problem can b e stated Denition Quality Triangulation Problem Given an initial boundary triangulation of the boundary vertices which dene the polygonal geometry 0 construct a geometryadapted triangulation such that for each triangle t of q t For nite elementnite volume applications the following Nonobtuse Boundary Triangulati on Problem can b e stated Denition Nonobtuse Boundary Triangulation Problem given a quality Delaunay triangulation of a polygonal region construct a triangulation 0 such that the boundary triangles having at least one boundary or interface edge do not have an obtuse angle opposite to any boundary or interface edge At this p oint the following remarks are in order The triangulation problems stated in Denitions to are essentially dierent than the classical triangulation problem in the following sense in stead of having a xed set of p oints to b e triangulated one has the freedom to cho ose the p oints to b e added in order to construct a mesh either with a desired resolution or with a given meshquality The construction of the mesh is dynamically p erformed Furthermore it is p ossible to exploit the existence of the reference triangulation constructed for instance by means of the De launay algorithm in order to reduce the computational cost to construct the output mesh To cop e with the triangulatio n Renement Problem the longestedge renement algorithms guarantee the construction of go o d quality irregular triangulations section This is due in part to their natural renement propagation strategy farther than the renement area of interest R Fur thermore asymptotically the numb er N of p oints inserted in R to obtain triangles of prescrib ed size is optimal and in spite of the unavoidable prop agation outside the renement region R the time cost of the algorithm is 7 linear in N indep endent of the size of the triangulatio n In the remaining of this pap er longestedge based solutions b oth for the improvement and quality triangulation problems of Denition and will b e discussed in the context of automatic quality triangulations section quality nonobtuse b oundary triangulation s section and triangulation s for terrain mo deling section Note that in all these applications the algorithms take advantage of an LEPP p oint insertion technique based on following the longestedge propagation path of the target triangles over De launay triangulations The longestedge propagation path of a tri angle The longestedge propagation path of a triangle concept is dened as follows Denition For any triangle t of any conforming triangulation the 0 LongestEdge Propagation Path of t wil l be the ordered list of al l the triangles 0 t t t t such that t is the neighbor triangle of t by the longest edge 0 1 2 n i i1 of t for i n In addition we shal l denote it as the LEPPt i1 0 Prop osition For any triangle t of any conforming triangulation of any 0 bounded dimensional geometry the fol lowing properties hold a for any t the LEPPt is always nite b The triangles t t t have strictly 0 1 n1 increasing longest edge if n c For the triangle t of the LongestEdge n Propagation Path of any triangle t it holds that either i t has its longest 0 n edge along the boundary and this is greater than the longest edge of t or n1 ii t and t share the same common longestedge n n1 Denition Two adjacent triangles tt wil l be cal led a pair of terminal triangles if they share their respective common longest edge In addition t wil l be a terminal boundary triangle if its longestedge lies along a boundary edge It should b e p ointed out here that the LongestEdge Propagation Path of any triangle t corresp onds to an asso ciated p olygon which in certain sense measures the lo cal quality of the current p oint distribution induced by t To illustrate these ideas see Figure a where the LongestEdge Propagation Path of t corresp onds to the ordered list of triangles t t t t Moreover 0 0 1 2 3 the pair t t is a pair of terminal triangles 2 3 LEPPBisection Algorithm for the rene ment of quality triangulations By using the LEPPt concept an improved LongestEdge renement algo 5, 6 rithm for nonDelaunay triangulation s can b e formulated where the pure longestedge renement of a target triangle t see Figure essentially means 0 the rep etitive longestedge partition of pairs of terminal triangles asso ciated with the current LEPPt until the triangle t itself is partitioned 0 0 The Figure illustrates the renement of the triangle t over the initial 0 triangulation of Figure a with asso ciated LEPPt ft t t t g The 0 0 1 2 3 triangulations b and c illustrate the rst steps of the LEPPBisection pro cedure and their resp ective current LEPPt while that triangulation d 0 is the nal mesh obtained Note that the new vertices have b een enumerated in the order they were created The LEPPBisection pro cedure schematically describ ed in Figure is a nonrecursive algorithm essentially based on rening pairs of terminal trian gles where the concept of the LongestEdge Propagation Path of the triangle t is