MIKE ZERO: Creating 2D Bathymetries. Bathymetry Editor & Mesh Generator. Scientific Documentation

Transcription

1 MIKE ZERO: Creatng D Bathymetres Bathymetry Edtor & Mesh Generator Scentfc Documentaton MIKE 7

2 DHI headquarters Agern Allé 5 DK-97 Hørsholm Denmark Telephone Support Telefax mke@dhgroup.com bathymetrygeneraton_scentfcdoc.docx/jh/jfb/ajs/ DHI

3 PLEASE OTE COPYRIGHT Ths document refers to propretary computer software, whch s protected by copyrght. All rghts are reserved. Copyng or other reproducton of ths manual or the related programmes s prohbted wthout pror wrtten consent of DHI. For detals please refer to your DHI Software Lcence Agreement. LIMITED LIABILITY The lablty of DHI s lmted as specfed n Secton III of your DHI Software Lcence Agreement : I O EVET SHALL DHI OR ITS REPRESETATIVES (AGETS AD SUPPLIERS) BE LIABLE FOR AY DAMAGES WHATSOEVER ICLUDIG, WITHOUT LIMITATIO, SPECIAL, IDIRECT, ICIDETAL OR COSEQUETIAL DAMAGES OR DAMAGES FOR LOSS OF BUSIESS PROFITS OR SAVIGS, BUSIESS ITERRUPTIO, LOSS OF BUSIESS IFORMATIO OR OTHER PECUIARY LOSS ARISIG OUT OF THE USE OF OR THE IABILITY TO USE THIS DHI SOFTWARE PRODUCT, EVE IF DHI HAS BEE ADVISED OF THE POSSIBILITY OF SUCH DAMAGES. THIS LIMITATIO SHALL APPLY TO CLAIMS OF PERSOAL IJURY TO THE EXTET PERMITTED BY LAW. SOME COUTRIES OR STATES DO OT ALLOW THE EXCLUSIO OR LIMITATIO OF LIABILITY FOR COSEQUETIAL, SPECIAL, IDIRECT, ICIDETAL DAMAGES AD, ACCORDIGLY, SOME PORTIOS OF THESE LIMITATIOS MAY OT APPLY TO YOU. BY YOUR OPEIG OF THIS SEALED PACKAGE OR ISTALLIG OR USIG THE SOFTWARE, YOU HAVE ACCEPTED THAT THE ABOVE LIMITATIOS OR THE MAXIMUM LEGALLY APPLICABLE SUBSET OF THESE LIMITATIOS APPLY TO YOUR PURCHASE OF THIS SOFTWARE. MIKE 7

4 COTETS MIKE ZERO: Creatng D Bathymetres Bathymetry Edtor & Mesh Generator Scentfc Documentaton Introducton... Grd Generaton... 3 Grd Interpolaton Land Generaton Polygon fll Extensve Box Groupng Gap Fllng Blnear nterpolaton Trangular nterpolaton Inverse dstance weghted nterpolaton Inverse squared dstances weghted nterpolaton Mesh Generaton Trangular Mesh Generaton Quadrangular Mesh Generaton Algebrac box method Transfnte nterpolaton Generatng Combned Mesh Smoothng the Mesh Refnement by Bsecton... 5 Mesh Interpolaton Interpolaton Accommodatng the Stream Drecton Interpolaton usng Prortsaton of Scatter Data Mesh Analyss and Edtng Mesh Analyss Collapsng Elements Shorelne Fles Baselne Coastlne Edge Map Profle References... 39

5 Introducton Introducton The Bathymetry Edtor and the Mesh Generator provdes you an envronment for creatng, edtng and presentng detaled dgtal D bathymetres. The Bathymetry Edtor generates bathymetres n a rectangular grd (dfs) whereas the Mesh Generator generates bathymetres n a flexble mesh format (mesh). The program provdes you the utltes for mportng raw data from varous external source (.e. xyz soundngs, xyz contours, MIKE / MIKE 3 formatted data), or to manually create data by usng the bult-n drawng tools. To ad the process of manually drawng data and for presentaton, graphcal background mages such as maps can be mported and overlad wth the bathymetry data. Fgure. Example of mport of raw data nto a Bathymetry Edtor workspace Varous nterpolaton optons are avalable to you to provde the best possble method for your type of data. When the bathymetry has been prepared, you can use the export utltes to output bathymetry data n varous formats. The purpose of ths document s to provde the user wth the scentfc background for the D bathymetry generaton. The generaton of a bathymetry s generally dvded nto two phases: The generaton of the grd/mesh The nterpolaton of bathymetry values onto the grd/mesh As a consequence ths document s dvded n smlar way: the generaton of a dfs bathymetry n the Bathymetry Edtor s descrbed n the sectons Grd Generaton and Grd Interpolaton whereas the generaton of a mesh bathymetry s descrbed n the sectons Mesh Generator and Mesh Interpolaton. DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator

6 Grd Generaton Grd Generaton The bathymetry area to descrbe the dfs bathymetry s defned by a map projecton, the orgn of the grd,.e. the centre of cell (.), the orentaton of the grd and the extenson of the grd, defned by the number of grd steps and equdstant grd spacng n the x- and y-drecton, respectvely. In the process of creatng the grd the lmtng bathymetry value for a cell to be recognsed as a land cell must be defned. The resultng grd wll be a rectangle resolved wth rectangular elements (often squares). DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator

7 Grd Interpolaton 3 Grd Interpolaton Before the nterpolaton starts an empty bathymetry s created. Ths means that t contans only delete values. The nterpolaton of a bathymetry s done n 3 steps: Fndng the grd cells that have a centre nsde the land polygons dentfes all the land ponts. Ths s known as Land Generaton. The grd cells that aren t land ponts needs to be assgned a depth. The grd s used for sortng the data (loose ponts, contour ponts and polygon vertces). All ponts used for the nterpolaton are dstrbuted nto optonal lookup-tables for each grd cell. Ths enables a much more effcent search nstead of somethng whch s proportonal to number of horzontal grd cells tmes number of vertcal grd cells tmes number of raw data ponts. Ths s known as Box Groupng Only grd cells defnng land have been assgned an elevaton (z-value). Each of the remanng ponts needs to be assgned. The raw data ponts are used for ths nterpolaton. Ths s the Gap Fllng process. 3. Land Generaton Land generaton s based on an nput data set that conssts of any number of land polygons. 3.. Polygon fll The polygon fll algorthm defnes a set of scan lnes algned wth the bottom of the bathymetry. There s a scan lne defned for each k-grd cell. For each scan lne the lst of polygons s searched and ponts nsde these polygons are marked as land ponts accordng to the algorthm descrbed on the two fgures below. Ths s by far the most effcent algorthm for dentfyng cells nsde polygons. Fgure 3. Polygon Sweep-Lne Algorthm DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 3

8 Grd Interpolaton 3.. Extensve The extensve method for computng land ponts loops over the total number of grd ponts n a bathymetry and check versus nsde/outsde boundares of the polygon. Ths s sgnfcantly more tme consumng than the Polygon fll -algorthm. 3. Box Groupng Group by cell ndex All raw data ponts are attached to the grd cell that they are located n. By dong ths t s very easy to search through the raw data ponts and dentfyng the ponts located close to the grd pont. 3.3 Gap Fllng The gap fllng s based on the concept that we have to calculate the depth n the pont (x c, y c ). We defne ths as the functon z c = f (x c, y c ). If we place our self n ths pont, we can dvde the world up nto four quadrants Q - Q4. From here t s a matter of fndng some ponts from the raw data set relatvely close to ths pont. The search radus for all possble technques can be entered - n grd cell dstance. Ponts outsde ths dstance wll never be taken nto account. Fgure 3. Defnton of quadrants 3.3. Blnear nterpolaton Ths technque fnds four ponts from the raw data set - one n each quadrant. The search s done n the followng way. A mask of relatve ndces s created. The cells n ths mask are sorted accordng to the dstance. For the quadrant Q the cells are sorted n the followng way, the grd pont tself beng excluded. DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 4

9 Grd Interpolaton Fgure 3.3 Illustraton of the neghbourng grd cells beng sorted ote that the grd cells wth a crosshatch pattern contan raw data ponts. When the closest raw data pont n each quadrant s found, we have four ponts that form a quadrangle. Ths quadrangle contans the centre pont, where we want to calculate the z- value. Ths s llustrated n Fgure 3.4. Fgure 3.4 Illustraton of the closest raw data ponts n each quadrant ote that each grd cell mght contan more raw data ponts. If ths s the case, the closest of these s chosen. We now have an rregular quadrangle, where the elevaton s defned n each vertex. We need to compute the elevaton n (x c, y c ). If we transform our quadrangle nto a square, we can perform blnear nterpolaton. Ths s llustrated n Fgure 3.5. DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 5

10 Grd Interpolaton DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 6 Fgure 3.5 Illustraton of blnear nterpolaton Frst the nterpolaton requres the transformaton from quadrangle to a normalsed square. Ths s done n the by computng 8 coeffcents n the followng way: y y y y D x x x x D y y C x x C y y B x x B y A x A (3.) Mappng the coordnates (x c, y c ) to the normalsed square (dx, dy) s done by solvng Equaton (3.). c bx ax (3.) Where the coeffcents are A C A C y C x C c B C C B D A A D D y x D b B D D B a c c c c (3.3) Solvng Equaton (3.) gves us dx a ac b b dx 4 (3.4) Where s used to choose the correct root. dy can now be computed n two ways: dx

11 Grd Interpolaton or dy dy x A B dx c (3.5) x C C A D dx B D dx dx c (3.6) Choosng between Equatons (3.5) and (3.6) s done n such a way that dvson by zero s avoded. (x c, y c ) has been mapped to (dx, dy). The task was to compute the elevaton n the pont (x c, y c ) and ths s done n the followng way usng regular blnear nterpolaton: z c dx dyz dx dyz3 dxdy z dxdy z (3.7) If less than four ponts are found (f one or more quadrants are empty), the double lnear nterpolaton s replaced wth nverse dstance weghted nterpolaton. Ths s done accordng to the followng scheme: w x x y y c c (3.8) w s w (3.9) z c w s w z (3.) The method works farly effcently, but t has one drawback. The quadrant search s heavly dependent on the orentaton of the bathymetry. If the bathymetry s rotated 45 degrees 4 completely dfferent ponts mght be used for the nterpolaton. For ths reason there s also a Trangular nterpolaton method, whch can be used, and ths method should be drecton ndependent Trangular nterpolaton As mentoned prevously the Blnear Interpolaton s dependent on the orentaton of the bathymetry. The Trangular Interpolaton s made as an answer to ths problem. Frst the closest pont to (x c, y c ) s found, cf. Fgure 3.6. DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 7

12 Grd Interpolaton Fgure 3.6 Illustraton of trangular nterpolaton In ths example the pont (x, y, z ) s the closest pont. When ths pont s dentfed, two quadrants are dentfed ndcated by the lght grey and the dark grey areas. The closest ponts n these two quadrants are then found. They can be seen on the fgure as (x, y, z ) and (x, y, z ). The nterpolaton s then done n two steps. Frst the coeffcents descrbng the plane defned by the 3 found ponts are computed: A y y z z y y z z x x y y x x y y B x x z z x x z z x x y y x x y y (3.) C z Ax By And secondly, the actual nterpolaton s done: z c Ax By C (3.) c c If less than 3 ponts are found, nverse dstance weghted nterpolaton s used. The trangular nterpolaton s more tme consumng due to the more complex drecton ndependent search, but better end results should be acheved wth ths method Inverse dstance weghted nterpolaton The Inverse Dstance Weghted Interpolaton s based on the scheme descrbed n Equatons (3.8) - (3.). Please note that the quadrant search s heavly dependent on the orentaton of the bathymetry. Thus, f the bathymetry s rotated 45 degrees four completely dfferent ponts may be used for the nterpolaton. DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 8

13 Grd Interpolaton Inverse squared dstances weghted nterpolaton The Inverse Squared Dstance Weghted Interpolaton s based on the Inverse Dstance Weghted Interpolaton but usng the squared dstance ponts n the search crtera. The appled scheme s thus defned by: w (3.3) x xc y yc w s w (3.4) z c w s w z (3.5) Please note that the quadrant search s heavly dependent on the orentaton of the bathymetry. Thus, f the bathymetry s rotated 45 degrees four completely dfferent ponts may be used for the nterpolaton. DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 9

14 Mesh Generaton 4 Mesh Generaton The mesh generator can construct meshes that consst of both trangular and quadrangular elements. The approach beng that the area of nterest s dvded up nto regons descrbed through polygons. Each polygon may have a trangular or a quadrangular mesh generated wthn. The complementary area s by default populated wth a trangular mesh. The steps to generate such a mesh are the followng: Defne polygons to be used for quadrangular mesh Set propertes for each (default values used f local propertes are not suppled) Generate the mesh wthn each polygon Use the trangular mesh approach for the area not contaned wthn any of the polygons. Thus the overall dea s to generate the quadrangular mesh frst and then to use the trangular mesh to patch the mesh together. 4. Trangular Mesh Generaton The generaton of trangles s based on the Trangle code, developed by Jonathan Shewchuk //. Addtonal nformaton regardng the functonalty can be found on the web page 4. Quadrangular Mesh Generaton The quadrangular grd generaton may be carred out usng two dfferent routnes: A smple algebrac method usng a boxng technque An algebrac grd generator usng transfnte nterpolaton The frst of these s robust but may generate skewed grds. The second s effcent but may not always be successful. 4.. Algebrac box method The am s to generate a mesh for a polygon whch algns tself wth the polylnes that make up the polygon. The man dea s to break any closed polygon nto quadrangles. Below s a polygon consstng of two polylnes l and l wth a number of vertces. The number of vertces on each polylne may be unequal n general. The polylnes are joned by two arcs. These end arcs consst each of only one lne segment. The polygon s consdered as an abstracton of a stream tube.e. the flow drecton s n the drecton of DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator

15 Mesh Generaton the two polylnes and the end arcs are transversal to the flow drecton. Ideally the end arcs should be chosen so that these are perpendcular to the flow drecton. l Vertex ode End arc l Fgure 4. The polygon to be meshed. The vertces are dsplayed along wth the end arcs The procedure s as follows: The length of the two polylnes s determned and the poston of the vertces along these polylnes s gven relatve to ths length.e. the poston of the vertces may thus be dentfed by a scalar value n the range,,. Let ths scalar value be gven by s n,m where n =, (polylne ndex) and m s the ndex of the vertex along the polylne. Each of the vertces along the polylne has a unque value of s n,m. The unon set of s n,m values of the vertces along l and l are then consdered. For each of the two polylnes vertces are ntroduced at the values s n,m, f they are not already present along the polylne. Ths wll gve rse to a polygon where the two polylnes have the same number of vertces. Vertces wth the same value of s wll be referred to as a complementary par. By jonng the complementary vertex pars the polygon s dvded n a number of quadrangles. Each of these quadrangles may then be treated ndvdually. Vertex ode End arc Interpolated complementary vertex Fgure 4. The polygon wth the nterpolated complementary vertces dsplayed. The polygon has now been dvded nto a number of quadrangles DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator

16 Mesh Generaton t s Fgure 4.3 A mesh s appled for each of the quadrangles n the polygon. ote that the number of ntervals n the t drecton s fxed for all ndvdual boxes whereas the number n the s drecton may vary sgnfcantly To generate the mesh for each of the quadrangles the overall maxmum mesh sde length should be consdered. It must be possble to control the element sde length transversal to the stream drecton as well as n the stream drecton. Let the two maxmum element sde lengths be denoted by t and s. These values apply for all the quadrangles n the polygon. The number of mesh lnes n the drecton of the stream needs to be the same for all the quadrangles makng up the polygon. Thus to evaluate the number of mesh lnes n the stream drecton requres evaluatng the maxmum dstance between all complementary vertex pars. The number of ntervals n the t drecton s gven by Celng v t max,... M (4.) t Where v s the length of the vertex par. S (x,t,y,t ) (x s,t,y s,t ) (x,,y, ) S (x s,,y s, ) Fgure 4.4 otaton used for the ndvdual quadrangles The notaton for the quadrangles follows from the above fgure. The ndvdual quadrangles should be meshed n the followng way. Frst the number of ntervals n the s-drecton s determned through DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator

17 Mesh Generaton s max( s, s ) Celng s (4.) Where s y ( xs,, ys,) ( x,,, ) s ( xs, t, ys, t ) ( x, t, y, t ) (4.3) Once the number ntervals s known the node locatons may be found through x, j j t s s x, t j t s x j, t x, s t s j t x, s (4.4) y, j j t s s y, t j t s y j, t y, s t s j t y, s Where,,, s and j, t (4.5) Problematc sectons may arse for some polygons. Such a polygon s dsplayed below. Fgure 4.5 A problematc polygon Problematc sectons may be dentfed as sectons wth opposng sdes havng a scalar product whch s non-postve. Usng the notaton from the earler fgure the crtera s expressed as DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 3

18 Mesh Generaton x y s, o s, o x y,, x y s, t s, t x y, t, t (4.6) 4.. Transfnte nterpolaton The second opton uses a so-called transfnte nterpolaton to determne the grd ponts. The formula uses relatve coordnates along the four polylnes makng up the polygon to be meshed. orth East West South Fgure 4.6 The notaton used for the transfnte nterpolaton grd generaton and vary between to along the arcs. The ntersecton of the boundares are dentfed The ntersecton of the South and the West boundary corresponds to ( he ntersectons are dentfed by: Intersectng boundares (ξ, η) South and West (,) West and orth (,) orth and East (,) East and South (,) DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 4

19 Mesh Generaton The grd s defned through the expressons x y, x, x, x,x, x, x, x, x,, y, y, y,y, y, y, y, y, (4.7) The expresson requres that the four polylnes makng up the polygon have been parametersed usng the parameters and respectvely. These parameters are smply the dstances along the arcs relatve to the full length of the polylnes. The dstrbuton of the grd ponts along the polylnes The grd ponts along the polylnes must obey the locaton of the orgnal vertces. Thus the concept of vertex pars for opposng polylnes should be used. Though note that the vertex pars no-longer by default are connected by a straght lne due to the pecewse lnear nature of the end arcs. orth East West South Fgure 4.7 The polygon wth the orgnal vertces along wth the nterpolated vertces. Opposng polylnes have dentcal number of total vertces. ote that the end arcs also have addtonal nterpolated vertces. The number of grd ponts along the arcs s controlled by the maxmum grd element sde length. More specfcally a length s appled for the and drectons. Ths length s nondmensonalsed by the maxmum arc lengths n the respectve drecton. For example, the maxmum sde length n the drecton s denoted by s. Ths value shall be used to defne the locaton of the mesh nodes along the south and north polylnes. DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 5

20 Mesh Generaton The maxmum length of the south and north polylnes s used to defne the nondmensonal mesh element sde length. Let the two lengths be denoted by l south and l north. The non-dmensonal maxmum sde length s defned by max( l south l north s, ) (4.8) Ths value must be satsfed for all lne segments along the south and north polylnes. The approach s to determne the number of ntervals for the lne segment through Celng, (4.9) Where corresponds to the th vertex. The procedure for the west and east arcs follows along the same lnes. 4.3 Generatng Combned Mesh The present secton outlnes the approach that wll be adopted for mergng the quadrangular mesh polygons wth the surroundng trangular mesh. The trangular mesh generator has the flexblty to nsert so-called Stener ponts. Stener ponts are addtonal vertces used to subdvde lne segments nto several lne segments. Stener ponts are necessary to allow the segments to exst n the mesh whle mantanng the requred propertes of the mesh such as constrants on the mnmum angle and maxmum trangle area. Stener ponts Fgure 4.8 A polygon that has been meshed where addtonal Stener ponts have been nserted The above fgure llustrates the concept of Stener ponts. Stener ponts are useful when the whole area s meshed usng trangles, but do create a problem when the mesh s mxed. To overcome ths problem the trangulsaton routne may be used wth the restrcton that Stener ponts are not nserted. If ths s used on the above case the result would be along the lnes as llustrated n Fgure 4.9. DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 6

21 Mesh Generaton Fgure 4.9 A trangulsaton wth the restrcton that Stener ponts are not allowed It s especally mportant that the mesh routne does not nsert Stener ponts along arcs whch form part of a polygon desgnated for a quadrangular mesh. To overcome ths problem the dea s to run the trangulsaton twce. The frst trangulsaton s done to determne the locaton of Stener ponts along arcs wthout constrants (.e. arcs not part of a polygon for quadrangulsaton). Once the Stener ponts have been dentfed along these free arcs the Stener ponts are nternally converted to vertces. The second trngulsaton s then done wth the constrant that Stener ponts are not allowed. The procedure s outlned on the followng pages. Step : Generate the quadrangular mesh. Step : Insert vertces along the mesh ponts at the arcs used for quadrangular mesh. DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 7

22 Mesh Generaton Step 3: Mark all polygons used for a quadrangular mesh nternally as not ncluded n the mesh. Then mesh the remanng area. Step 4: Convert all mesh ponts along free arcs to vertces. otce the Stener pont at the north arc of the quadrangular mesh polygon. Step 5: Remove the trangular mesh. Step 6: Regenerate the mesh, but ths tme do not allow the trangulaton to nsert Stener ponts. DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 8

23 Mesh Generaton Step 7: Rentroduce the mesh for the quadrangular polygons. Resultng n the fnal mesh. ote that the Stener pont prevously ntroduced at north arc of the box s no longer present. 4.4 Smoothng the Mesh After the generaton of the mesh t s prudent to smooth the mesh to obtan a better applcablty n a smulaton. Smoothng a mesh s the effort to poston the nodes n a way such that the angles n each element and the element area are as large as possble. The smoothng only affects trangular mesh elements. Fgure 4. Defnton: Movng node pont due to smoothng The locaton of the Smoothed pont Q' s found through elements Q' elementsq P (4.) 3 elements The smoother may be appled teratvely and wll converge to a mesh where the followng holds for each pont. Q elements ' P (4.) elements I.e. every pont s placed n the centre of mass of the surroundng polygon. Please note that though the system s convergng towards a unque soluton ths may not be a soluton whch satsfes the mesh constrants as set up n the Mesh Generaton menu (plus local polygons f they exst). The dea s to do the mesh smoothng one step at DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 9

24 Mesh Generaton a tme and then evaluate the mesh and reset any nodes whch gve rse to elements whch do not satsfy the constrants. The measure conssts of the element area and the smallest angle n the element. If one of these s smaller than the allowed values then the calculated measure evaluates to false. Two optons can be ncluded when smoothng the mesh: Smoothng constraned by mesh crteron If ths s not selected the measure wll always evaluate to true Leave mesh nodes at arcs untouched If ths s not selected ALL nternal nodes may be relocated. Otherwse, f a node orgnally was part of a defned arc or s a sngle user defned node then the node s not allowed to be moved durng the smoothng. 4.5 Refnement by Bsecton A mesh may be refned by subdvdng the faces of the exstng elements. The bsecton method s desgned to only change the sze of the elements n a gven mesh wthout changng the overall qualty of the mesh.e. the angles of the refned mesh wll not become less than the orgnal mesh. The mesh bsecton method follows some clear prncples: Take all elements faces of the orgnal mesh and place a node at the md ponts (hence bsecton). Consder each element n the orgnal mesh. Dependng on whether the element s trangular or quadrlateral the approach vares slghtly. For a trangular element the approach s as llustrated n Fgure 4.. Fgure 4. Bsecton of trangular element As can be seen from Fgure 4., the operaton wll gve rse to 4 new elements per trangular element n the orgnal mesh and 3 new element faces. For a quadrlateral element the approach s as llustrated n Fgure 4.. DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator

25 Mesh Generaton Fgure 4. Bsecton of quadrangular element As can be seen from Fgure 4. also for the quadrangular elements the process ntroduces 4 new elements per quadrangular element n the orgnal mesh and 4 element faces. ote that the quadrangular elements also generate an addtonal mesh node on top of the ones caused by the face bsecton. The refnement may n prncple be appled an arbtrary number of tmes. The table below summarses the number of topologcal elements (faces, elements, nodes) as a functon of the number of refnements (n). umber of topologcal elements Orgnal mesh st refnement nd refnement n th refnement (n) Faces F F+3T+4Q 4F+8T+4Q n F + (3T + 4Q) n = 4 n Trangular elements T 4T 6T 4 n T Quadrangular elements Q 4Q 6Q 4 n Q odes +F+Q +3F+3T+9Q n n + F + Q 4 = = n j +(3T + 4Q) ( 4 j ) j= = DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator

26 Mesh Interpolaton 5 Mesh Interpolaton The Mesh Generator gves two possbltes wth respect to nterpolaton for trangular elements. The two possble nterpolaton routnes are atural neghbour Lnear nterpolaton These nterpolaton methods are vald ndependent of the type of mesh used. The nterpolaton requres values only at the mesh nodes and wll base the nterpolaton solely on the scatter data. Thus whether the nodes are part of a trangular or quadrangular mesh s of no sgnfcance. These methods may however not be the correct ones to use for rver beds or other water ways n an n land settng. In partcular the nterpolated surface generated for a strongly meanderng rver may gve rse to a hghly erroneous rver bed (see Fgure 5.). A B E C D Fgure 5. A meanderng rver wth a pont (O) at whch the topography s to be nterpolated usng the natural neghbour method. The crosses ndcate locaton at whch data s located. The grey polygon llustrates the weghts of the values at A-E As can be seen from the above fgure, the pont B wll have a sgnfcant weght and thus mpact on the nterpolated value at O. Pont B s located on the opposte bank of the nterpolaton locaton O and thus the bed level may be very dfferent from the true value at O. One would assume that the actual level at O s closer to the values at A and C n the stream drecton. Further the nterpolaton routne also takes the values at D and E nto account even though the ponts are located outsde the man rver. To create sound nterpolated topographes n rver beds an nterpolaton routne that emphasses values n the stream drecton and does not nterpolate across rverbanks has been developed for quadrangular elements. DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator

27 Mesh Interpolaton 5. Interpolaton Accommodatng the Stream Drecton To accommodate hgher weghts on ponts postoned n the stream drecton a modfed verson of the nverse dstance method s appled. The nverse dstance s a smple nterpolaton routne. A neghbourhood about the nterpolated pont s dentfed and a weghted average s taken of the observaton values wthn ths neghbourhood. The weghts are a decreasng functon of dstance r and are typcally gven by w p p r (5.) r for pont where p s greater than. The dstance r s often gven by the Pythagorean dstance but may be any measure. The latter may be utlsed to accommodate the weghtng of ponts n the stream drecton. The coordnate system wll be constructed so that at the mesh nodes the coordnates take on nteger values. At ntermedate locatons the coordnates are found through blnear nterpolaton. The rver segment or polygon when pctured n the mesh based coordnate system s smply a rectangular as llustrated n Fgure 5.. E A B E A B C D C D Fgure 5. The meanderng rver on the left wthout the mesh dsplayed. The rght fgure llustrates the rver overlad wth the mesh k A j B C Fgure 5.3 The polygon dsplayed n the mesh coordnate system (j,k) In the new coordnate system the ponts A, B and C wll have the followng coordnates: A: (j A,k A ) B: (j B,k B ) C: (j C,k C ) where j and k may take on non-nteger values. DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 3

28 Mesh Interpolaton The locaton at whch the nterpolaton s to take place has coordnates (j o,k o ). The z-value at O may then be expressed as z O A, B, C j j p O k k q O A, B, C j j p O z k k q O (5.) where q > p. The latter s to ensure that the effect of the j coordnate decays more slowly than the effect of the k-coordnate. Both p and q should be user defned (advanced opton). Some expermentaton s needed to dentfy proper default values of p and q. The blnear nterpolaton to be used for determnng the (j,k) coordnates for scatter ponts not placed at a mesh node s found through (x,y ) (x,y ) (x,y) (x,y ) (x,y ) Fgure 5.4 A mesh element wth four nodes wth mesh coordnates (j,k), (j+,k),(j,k+) and (j+,k+). The mesh coordnates of (x,y) are sought The coordnates of (x,y) satsfy the followng equaton x x y y x t y x s y x s y x y x y x y x s y x y (5.3) Where s,t les wthn the nterval [;]. Ths equaton may be re-arranged as x x x s y y y x x x st y y y x y x y x t y x y (5.4) whch n turn may be expressed as d d a s a b t b c st c (5.5) DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 4

29 Mesh Interpolaton Ths equaton may be rewrtten as a set of two equatons a c ac s ( ab ab dc dc ) s bd bd d sa t b sc (5.6) of whch the frst s second order algebrac equaton n s. Once the two solutons are found t may be determned from equaton number. The true soluton s the one that gves rse to a set of s and t lyng wthn the nterval to. The mesh coordnates of the pont (x,y) are thus gven by (j+s,k+t). 5. Interpolaton usng Prortsaton of Scatter Data The prortsaton s carred out by applyng weghts to the ndvdual scatter data sets. These weghts may be assgned globally and locally. The methodology used for prortsaton durng nterpolaton s the followng: Assume n sources of data each beng suppled n any of the accepted data formats (xyz, dfs, dfsu, mesh). Let these data sets be denoted by D, D, D n. The nterpolaton s done separately for each data source D. Hereby there wll be a mesh M wth values only dependent on data source D. Each data source has an assocated weght w n the range to. The resultng mesh s obtaned through the sum n M = M w = (5.7) Where the weghts satsfy n w = = (5.8) The concept s also used wth localsed weghts assocated wth user specfed regons. The process s llustrated below usng a set-up wth three dfferent data sources. DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 5

30 Mesh Interpolaton Fgure 5.5 Mesh pont P n setup wth three dfferent data sources. Each scatter data source has assocated weght After nterpolaton, the mesh pont P wll have a z value based on the three nterpolatons from data source to 3 weghted accordng to the weghts: Z(P) = w Z(P, D ) + w Z(P, D ) + w 3 Z(P, D 3 ) (5.9) Where Z(P,D ) s the result of an nterpolaton only ncludng data source D. To apply local weghts prortsaton polygons must be defned. Prortsaton polygons are user defned areas where a specfc set of prortsaton weghts are to be used. Prortsaton polygons are dfferent from the polygons used for specfyng mesh resolutons and mesh type. DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 6

31 Mesh Analyss and Edtng 6 Mesh Analyss and Edtng Durng the process of creatng the mesh t s possble to analyse the mesh for ts applcablty n a smulaton and to edt local mesh elements. 6. Mesh Analyss It s possble to analyse a mesh for ts applcablty n a smulaton by evaluatng the restrctng tme step for each mesh element. The analyses of a mesh conssts of consderng The element area The smallest angle wthn each element The CFL stablty crtera Fgure 6. Defnton of node pont and poston n mesh analyss The area of a trangular element s gven by A x x y y y y x x 3 3 (6.) The area of a quadrlateral s calculated as the sum of areas of the two trangles that matches the quadrlateral. The smallest angle may be evaluated by calculatng the normalsed scalar product: x x x x y y y y x x y y x x y y s (6.) The quantty S vares between and where refers to the smallest angle and the largest. The tme step Δt evaluated on bass of the CFL number s gven by mn t CFL facelength gh (6.3) DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 7

32 Mesh Analyss and Edtng where h s water depth (water level - topography) and g s 9.8 m /s. 6. Collapsng Elements Effectvely the mesh element (trangular or quadrangular) s collapsed by takng all nodes n the mesh and collapsng them to the centre pont of the element (centre of mass). ote that the procedure may remove up to four mesh elements. The collapse of an element may be acheved by subsequent collapses of cells faces and a fnal move of the node to the centre of mass of the orgnal mesh element. The process s llustrated n Fgure 6. for a trangular element. The procedure for the quadrangular element requres four face collapses nstead of three. Fgure 6. Process for collapsng trangular element The z value at the new node s obtaned through nterpolaton based on the user settngs. The code value of the mesh node s set to the maxmum of the nodes of the collapsed element. The data used as scatter data for the nterpolaton are the values at the mesh nodes. DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 8

33 Shorelne Fles 7 Shorelne Fles The shorelne fles are a collecton of dfferent fles especally formatted for use n the Shorelne Morphology model, whch can be appled from wthn ST n MIKE Coupled Model FM. 7. Baselne A baselne s defned by an arc wth arc attrbute value Baselne_code. The start pont of the baselne, and hence the orentaton, s defned by the lowest arc attrbute of the two node ponts n each end of the arc. In case the arc attrbute for the two end ponts are the same the start node s the defned by the western-most pont. When savng the baselne to a xyz asc fle, the method of dervng the arc data s smlar to the method of exportng selected arcs to a boundary fle, - the contents of the xyz fle s just a bt dfferent wth only three columns (x,y,z), correspondng to column, and 4 n a fle exported as boundary. ote that the coordnates of the node ponts and vertces must be saved n order wth respect to orentaton,.e. frst wrte the coordnates for the node pont wth the lowest attrbute number, next the coordnates for the vertces on the baselne (n the gven order ether forwards or backwards, dependng on the baselne orentaton), and then the coordnates for the remanng node pont at the other end of the arc. Fgure 7. Illustraton of baselne coordnates and assumed order DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 9

34 Shorelne Fles Example of output fle (Baselne.xyz): (pont d 7, attrbute - last pont n arc, frst pont on baselne) (pont d 4) (pont d ) (pont d ) (pont d ).5.5 (pont d 3, attrbute - frst pont n arc, last pont on baselne) 7. Coastlne A coastlne s defned by an arc wth arc attrbute value Coastlne_code. The start pont of the coastlne s defned by the lowest arc attrbute of the two node ponts n each end of the arc. In case the arc attrbute for the two end ponts are the same the start node s the defned by the western-most pont. The coastlne arc can ntally be defned by any number of ponts, however n order to match the baselne the number of ponts on the coastlne needs to be + compared to the number of ponts on the baselne, and the poston of the coastlne ponts should be located at the centre of the baselne sectons when projected perpendcular to the baselne. Fgure 7. Coastlne arc before modfyng the poston of vertces to be postoned at centre of baselne sectons DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 3

35 Shorelne Fles Example: contents n mdf-fle before coastlne modfcaton: [POITS] Data = ' ' EndSect // POITS [ARCS] Data = ' ' EndSect // ARCS [ARCS] The method of modfyng the coastlne ponts to match the baselne s descrbed n the followng.. An array holdng the poston of the coastlne ponts (n order) s defned. For each baselne secton the followng s done: a. the centre pont poston (x,y) s located b. the lne that starts n (x,y) and s perpendcular to the baselne secton s defned: DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 3

36 Shorelne Fles c. A new coastlne pont (xc,yc) s found at the poston where the orgnal coastlne arc ntersects wth the perpendcular lne from the baselne: 3. Fnally, a new coastlne s created usng the frst and last node pont n the ntal coastlne fle and the updated coastlne ponts. DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 3

37 Shorelne Fles Example: contents n mdf-fle after coastlne modfcaton: [POITS] Data = ' ' EndSect // POITS [ARCS] Data = ' ' EndSect // ARCS When savng the coastlne to a xyz asc fle, the method of dervng the arc data s smlar to the method of exportng selected arcs to a boundary fle, - the contents of the xyz fle s just a bt dfferent wth only three columns (x,y,z), correspondng to column, and 4 n a fle exported as boundary (z-value can be set to ). ote that the coordnates of the node ponts and vertces are saved n order wth respect to orentaton,.e. frst the coordnates for the node pont wth the lowest attrbute number are wrtten, next the coordnates for the vertces on the modfed coastlne (n the gven order ether forwards or backwards, dependng on the baselne orentaton), and fnally, the coordnates for the remanng node pont at the other end of the arc. Example of output fle (Coastlne.xyz): (new pont d, attrbute - last pont n arc, frst pont on coastlne) (new pont d 9, attrbute - frst pont n arc, last pont on baselne) DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 33

38 Shorelne Fles 7.3 Edge Map The edge map defnes whch coastlne edge/secton each element n the mesh belongs to. The edge map s defned n an area lmted by the baselne and an arc formng a lmtng edge map lne n a varable dstance from the baselne as shown below. The edge map values are saved n a dfsu fle, whch are based on the generated (or loaded) mesh. The creaton of an edge map thus requres a mesh, a baselne and an edge map lne. An edge map lne s defned by an arc wth arc attrbute value Edge_map_code. The edge map arc can be defned by any number of ponts and s used to from an outer lmt of the edge map. Along wth the baselne and leadng lnes, extendng from the baselne ponts and perpendcular to the local baselne orentaton, the edge map lne forms polygons that defnes whch secton the ndvdual elements n the mesh belong to. The mesh elements, whose centre poston s located wthn an edge map polygon, are allocated the nteger value that corresponds to the coastlne secton (.e. baselne secton number + ). DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 34

39 Shorelne Fles Example: contents n mdf-fle contanng mesh, baselne arc and edge map arc: [POITS] Data = ' ' EndSect // POITS [ARCS] Data = ' ' EndSect // ARCS [MESH] ode_count = 9 Max_ode_Count = 5 Element_Count = 87 Max_Element_Count = 5 Segment_Count = Max_Segment_Count = 5 Tr_Pont_Count = 9 Tr_Element_Count = 87 odes_ = ' ' Elements_ = ' ' EndSect // MESH DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 35

40 Shorelne Fles The method of creatng the edge map s descrbed n the followng.. An array holdng the poston of the edge map lne arc s defned. For each baselne pont, : a. the leadng lne that s perpendcular to the averaged baselne orentaton s found b. the coordnate (x,y) for the pont where the leadng lne ntersects wth the edge map lne s found 3. For each baselne secton a polygon s created based on the two ponts on the baselne and the correspondng ponts where the leadng lnes cross the edge map lne. The polygons are saved n an array. 4. An array for hold the edge map values n the mesh s defned. DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 36

41 Shorelne Fles 5. For each mesh element the centre poston s compared wth the calculated polygons. c. If the element s located wthn one of the polygons the element value s set to the nteger value correspondng to the baselne secton for the gven polygon (e.g. for baselne secton located between baselne pont and, the edge map value s set to ). d. If the element s located outsde any polygon the value s set to a delete value. When Exportng the edge map the values are saved to a dfsu fle. DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 37

42 Shorelne Fles 7.4 Profle A profle s defned by the dstance and z-level along a lne, gong through the coastlne pont and perpendcular to the related baselne secton. The profle lne does not necessarly cross the baselne. The profle data s defned by the dstance behnd and n front of the coastlne, the number of ponts and z-levels derved from the underlyng mesh fle or external fles. The method of creatng one profle from the mesh for a gven coastlne pont s descrbed n the followng:. Based on the chosen coastlne pont (CoastID), the coastlne poston (x,y) and the local orentaton of the related baselne secton s found.. The equaton for the lne extendng the gven dstance behnd and n front of the coastlne s found. 3. For each pont on the profle lne, the dstance to the coastlne pont (to be saved n tem of the dfsu fle) s found, and the z-level s lnearly nterpolated from the nearest ponts (to be saved to tem n the dfsu fle). If there s several profles to be saved n a dfs fle, the dstance and z-level for the defned profles s found, and lnear nterpolaton s used to calculate the profle data for all remanng coastlne ponts. In case the profle s to be created from external fles, the dstance and z-level s found from lnear nterpolaton. DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 38

43 References 8 References // Jonathan Rchard Shewchuk, Trangle: Engneerng a D Qualty Mesh Generator and Delaunay Trangulator, n ``Appled Computatonal Geometry: Towards Geometrc Engneerng'' (Mng C. Ln and Dnesh Manocha, edtors), volume 48 of Lecture otes n Computer Scence, pages 3-, Sprnger-Verlag, Berln, May 996. DHI - MIKE ZERO: Creatng D Bathymetres - Bathymetry Edtor & Mesh Generator 39