Barbera's TRiangler 3.0 BTR User's Manual
Contents
1. gt The point that determines if a triangle belongs or not to a subregion is the centre of gravity of the triangle If this point lies outside the region also the triangle does To improve this research may be useful consider refining regions a little larger than they would otherwise be Page 10 Andrea Barbera BTR 3 0 User s manual OUTPUT BTR has a single output variable geom This variable is a structure that keeps information of every feature of the grid GEOM ELEMENTS COORDINATES geom elements coordinates is a nV x 2 matrix that contains the list of coordinates of every vertex where nV is the number of vertexes present in the mesh So the i esim row of the matrix contains XV yV the x coordinate and the y coordinate of i esim vertex Input vertexes may not coincide with first vertexes of geom elements coordinates but they will surely be memorized in random positions GEOM ELEMENTS TRIANGLES geom elements triangles is a nT x 3 matrix that contains the link between every triangle and vertexes that bound it nT is the number of triangles present in the mesh So the i esim row of the matrix contains V1 V2 V3 the three vertexes that belong to i esim triangle Vertexes are memorized in anticlockwise order GEOM ELEMENTS BORDERS geom elements borders is a nB x 4 matrix that contains the link between every border and vertexes that bound it and triangles that surround it nB is the number of borders present in the2. mesh So the i esim row of the matrix contains V1 V2 T1 T2 V1 and V2 are the two vertexes that belong to i esim border T1 and T2 are the two triangles that surround the i esim border If a border is adjacent to a region that mustn t be meshed the reference in geom elements borders is conventionally 1 GEOM ELEMENTS NEIGHBOURHOOD geom elements neighbourhood is a nT x 9 matrix that contains the link between every triangle and three adjacent triangles and borders nT is the number of triangles present in the mesh So the i esim row of the matrix contains T1 T2 T3 B1 B2 B3 Ref1 Ref2 Ref3 T1 T2 and T3 are the three triangles that are adjacent to the i esim triangle B1 is the border between i esim triangle and T1 B2 is the border between i esim triangle and T2 B3 is the border between i esim triangle and T3 Ref may generically assume values from 1 to 3 if j is a triangle adjacent to i the j esim row of TT will contain the reference to i esim triangle in one of the first three columns The number of column first second or third is Ref for the i esim row of TT six columns after the one in which there is there reference to j esim triangle If a border is adjacent to a region that mustn t be meshed the reference in geom elements triangles is conventionally 1 In this case also related reference assumes value 1 GEOM ELEMENTS VERTEXESNEIGHBOURHOOD geom elements vertexesneighbourhood is an array whose length is nV the number of v
3. POLITECNICO DI TORINO Facolt di Ingegneria Corso di Laurea in Ingegneria Aerospaziale Barbera s TRiangler 3 0 BTR User s Manual Andrea Barbera Dr in Aerospace Engineering ab borgorevel amp yahoo it Andrea Barbera BTR 3 0 User s manual ABSTRACT BTR 3 0 is a library created for Matlab 6 5 that can triangulate a plain surface that may contain every kind of bond on the domain as holes concavities or inner segments To insert the domain it is necessary inserting some vertexes that bound it and specifying in which relation are them Then BTR 3 0 calculates which is the best mesh that fits the domain Parameters of the triangulation are the maximum area of every triangle the smallest angle among the ones added by BTR 3 0 and the regions in which these parameters are active Output of this function is the set of data necessary to apply a finite elements method to the triangulation It includes boundary conditions and elements lists Page 3 Andrea Barbera BTR 3 0 User s manual INPUTS BTR has 3 input variables Domain BC and RefiningOptions Among these Domain is the main input but also other two inputs must be considered We remember that Matlab is case sensitive so problems may arise if users don t pay attention to caps characters ELIGIBLE DOMAINS Every domain or bond must be delimitated by segments ad descrived below DOMAIN VARIABLE Domain is a structure that is composed by different field
4. TR to generate the mesh Every parameter is stored in function global parameter and may be modified by users This may cause or solve some problems according to the parameter PARAMETERS THAT WILL MOST PROBABLY BE MODIFIED TOLERANCE tolerance default value 1e 12 is the value used like a tolerance two numbers are equal if their difference is lesser than tolerance SAFEMESHING If SafeMeshing default value true is activated the program becomes slower but can triangulate also domains that otherwise would crash BTR SafeMeshing must be activated if there are boundary borders very close to other ones RECURSIVEENCROACHINGLIMIT RecursiveEncroachingLimit default value 4 is the limit after which BTR tries to stop recursive borders disecroaching The more this value is high the safer and slower BTR get in presence of small input angles REFININGFOLDERNAME RefiningFolderName default value methods is the name of the folder that contains refining methods PARAMETERS THAT WILL LESS PROBABLY BE MODIFIED SPLITTINGBORDERSLIMITS SplittingBordersLimits default value 10 sets the maximum number of borders that may be contained in single Encroachable subregion XBOXENLARGE YBOXENLARGE XBoxEnlarge default value 0 5 and YBoxEnlarge default value 0 5 are parameters used to create the containing box the rectangle that must contain every input vertex They determines the stretching of the containing box with reference to minimum rec
5. ain Boundary Values 3 4 1 21 but not Domain Boundary Values 1 3 4 21 because there are no borders between Vertexes 1 and 3 or between Vertexes 2 and 4 This sorted list must be open and not closed BTR understands that a border must be inserted between first and last element of Domain Boundary Values and forcing last domain border would give an error DOMAIN HOLES Domain Holes is a structure containing Hole as a field Domain Holes contains the references of every hole in the domain Field Hole is an array whose dimensions are equal to the number of holes presents in the domain Every field Hole has values as sub field This is an array that contains references to some vertexes inserted in Domain InputVertex Domain Holes Hole i Values is a sorted list that says that there is a border of the i esim hole between two adjacent elements of Domain Holes Hole i Values and between its first and last element For example we may insert the same square described before with two triangular holes in the domain We compose Domain InputVertex with every vertex useful for the domain in a random order but the shape of the domain and the holes is accurately described in other fields of Domain Domain InputVertex 0 0 w uB FOU U ALENE UO Page 5 Andrea Barbera BTR 3 0 User s manual 4 41 Domain Boundary Values 1 2 3 41 Domain Holes Hole 1 Values 5 6 71 Il Domain Holes Ho
6. assigned and nDi is the number of these vertexes present in the mesh So the i esim row of the matrix contains iDi BCvalue where iDi is a vertex and BCvalue is a reference to a Dirichlet BC as explained in BC Values GEOM PIVOT NE geom pivot Ne is a nNe x 2 matrix that contains the list of coordinates of every border in which a Neumann boundary condition has been assigned and nNe is the number of these borders present in the mesh So the i esim row of the matrix contains iNe BCvalue where iNe is a vertex and BCvalue is a reference to a Neumann BC as explained in BC Values GEOM SUPPORT TINFO geom support TInfo is an array whose length is nT the number of triangles in the triangulation This array contains informations about every triangle The i esim element of geom support TInfo is composed by geom support TInfo i Area the area of i esim triangle geom support TInfo i B a measure of quality of i esim triangle Calling a in rad the smallest 1 2 sin a geom support TInfo i Circumcenter an array composed by xC yC that contains the coordinates of the circumcenter of i esim triangle geom support TInfo i Circumradius the squared of the circumradius of the i esim triangle angle of the i esim triangle the relation to find Bis B GEOM SUPPORT BINFO geom support BInfo is a NB x 3 matrix that contains information about every border nB is the number of borders present in the mesh So the
7. ertexes in the triangulation This array contains the link of a vertex and adjacent borders and vertexes and these informations are stored in fields called B borders and v verteces So the i esim element of the array is composed by geom elements vertecesneighbourhood B B1 B2 BN if the i esim vertex is adjacent to N borders and geom elements vertecesneighbourhood V V1 V2 VN if the i esim vertex is adjacent to N vertexes Page 11 Andrea Barbera BTR 3 0 User s manual The j esim vertex of geom elements vertecesneighbourhood V belongs always to the j esim border of geom elements vertecesneighbourhood B GEOM NELEMENTS NTRIANGLES geom nelements nTriangles contains the number of triangles in the grid GEOM NELEMENTS NBORDERS geom nelements nBorders contains the number of borders in the grid GEOM NELEMENTS NVERTECES geom nelements nVerteces contains the number of verteces in the grid GEOM PIVOT NODELIST geom pivot nodelist is an array whose length is nV the number of vertexes in the triangulation This array contains informations about BC on every vertex if the i esim component of geom pivot nodelist is equal to zero no BC is assigned for the i esim vertex otherwise the i esim component assumes a value that is a reference explained in BC Values GEOM PIVOT DI geom pivot Di is a nDi x 2 matrix that contains the list of coordinates of every vertex in which a Dirichlet boundary condition has been
8. gles it will be sufficient to impose RefiningOptions CheckAngle equal to n or N RefiningOptions AngleValue is inserted in degrees The algorithm used in the triangulation may converge only if RefiningOptions AngleValue is lesser or equal than 30 REFININGOPTIONS SUBREGIONS It is possible refining not the whole domain but only a subset of it Using RefiningOptions Subregions it is possible to control any number of these subregions RefiningOptions Subregions is a structure that has Subregion as only field This is an array whose length is equal to the number of refining subregions RefiningOptions Subregions Subregion i has some fields similar to ones that RefiningOptions has In particular i esim subregion has these fields RefiningOptions Subregions Subregion i CheckArea RefiningOptions Subregions Subregion i AreaValue RefiningOptions Subregions Subregion i CheckAngle RefiningOptions Subregions Subregion i AngleValue RefiningOptions Subregions Subregion i Type RefiningOptions Subregions Subregion i Descriptors CheckAngle CheckArea AngleValue AreaValue have the same usage described before with the difference that the refining is applied only to the i esim subregion The region is identified by fields Type and Descriptors Type is a generic function in subdirectory methods that gives back a boolean value that becomes true if the point belongs to the subregion Usually Type gives only the s
9. hape of the subregion and values are contained in Descriptors but this is not a rule Users may create their own methods to refine particular subregions Page 9 Andrea Barbera BTR 3 0 User s manual BTR 3 0 contains two methods elliptic and rectangular Description of these methods may be found in the test of the function This is elliptic a sample of method function Flag elliptic Descriptors P Descriptors it specify the limits of the ractangle and it s in the form Descriptors xC yC a Bb c xC yC is the center of the ellipse a is the semiaxis in x direction b is the semiaxis in y direction The ellipse equation used is xP xC 2 a 2 yP yC 2 b 2 1 P the generic point P xP yP that must control if belongs or not no the region de AAP oO o o oO DAL AL o o DAG oO cio Flag the boolean that is true if P belongs to the region if P 1 Descriptors 1 Descriptors 3 2 P 2 Descriptors 2 Descriptors 4 2 lt 1 Flag true else Flag end Il false return Usages of RefiningOptions Subregions are reported in appendixes If refining of subregions isn t required RefiningOptions Subregions must however assume following value RefiningOptions Subregions 1 TROUBLESHOOTING gt Refining a subregion may ruin result of a previous refining this may be for example the case of an area refining that follow a quality refining
10. he domain has a BC that is given by reference by BC Boundary Values n Referring to domain described in figure 1 with Domain InputVertex 0 0 5 0 5 5 0 5 and Domain Boundary Values 1 2 3 4 BC Boundary Values 3 3 2 4 BC Values 3e 1 2e4 0 3e4 1 0 1 4e4 the lower and the right borders have a Dirichlet BC equal to 0 the upper border has a Neumann BC equal to 2e4 and the left border has a Neumann BC equal to 0 BC HoLES AND BG SEGMENTS Usage of BC Holes and BC Segments is similar to usage of BC Boundary with the difference that like for Domain structure these structures admit fields called Hole or Segment that are array in a number coherent with ones in Domain structure Sample of usage of these structures are reported in appendixes If in the domain there are no holes or segments Domain Holes Or Domain Segments must however assume following values Domain Holes Hole or Domain Segments Segment BC must be created even in not interested in its effect of triangulation TRICK NOT INTRESTED IN BC If not interested in BC it will be sufficient impose BC Values 0 and create arrays of ones as sub fields of BC structure of lengths coherent with ones of Domain structure This will not slow the generation of the triangulation Page 8 Andrea Barbera BTR 3 0 User s manual REFININGOPTIONS VARIABLE BTR 3 0 creates a Delaunay triangulation that fits the do
11. i esim row of the matrix contains Info1 Info2 Info2 Info1 says if the i esim border is a domain border value 1 an hole border value 2 a segment border value 3 or an inner border value 0 Info2 says if the i esim border is a limit of the triangulation value 1 or not value 0 Info3 says if the i esim border has some BC assigned value reference in BC Values or not value 0 Page 12 Andrea Barbera BTR 3 0 User s manual GEOM SUPPORT BCIRCLE geom support BCircle is an array whose length is nB the number of borders in the triangulation This array contains informations about the bordercircle of every border The bordercircle is the minimum circle that may contain the border that thereafter is a diameter of the bordercircle The i esim element of geom support BCircle is composed by geom support BCircle i Center an array composed by xC yC that contains the coordinates of the bordercircle of i esim border geom support TInfo i r2 the squared of the circumradius of the i esim triangle GEOM SUPPORT EB geom support EB contains a structure necessary to refine the grid GEOM INPUT DOMAIN geom input Domain contains the Domain input structure that has been used to create the mesh GEOM INPUT BC geom input BC contains the BC input structure that has been used to create the mesh Page 13 Andrea Barbera BTR 3 0 User s manual PARAMETERS There are many parameters that are used by B
12. ith index 8 BC INPUTVERTEXVALUES BC InputVertexValues is list of BC values that are set in InputVertex These conditions must be Dirichlet ones because it has no sense imposing a Neumann condition in a vertex Conventionally instead of the values references to BC Values are used If a vertex is free and there is no BC applied on it a conventional 0 is put in correspondence of it An example of BC InputVertexValues usage is following one with reference to previous BC Values Domain InputVertex 0 0 BC InputVertexValues 0 0 1 3 Page 7 Andrea Barbera BTR 3 0 User s manual This means that on vertexes 0 0 and 5 0 there is no BC setted 5 5 has a Dirichled BC 3e 1 and 0 5 has a Dirichled BC 0 BC BOUNDARY BC Boundary is a structure containing values as a field BC Boundary Values is an array containing the references of the BCs presents on domain boundary borders Every number of this array is a reference to an element of BC InputVertexValues BC Boundary Values is a sorted list that identifies which BC applies to a particular domain border In particular if we know that there is a border between Domain Boundary Values i and Domain Boundary Values i 1 the value to set on that border is given as a reference by BC Boundary Values i Of course border identified by Domain Boundary Values n and Domain Boundary Values 1 where n is number or vertexes that shape t
13. le 1 Values 8 9 10 I This brings to a domain like the one described in figure 2 As for the ones described in Domain Boundary these sorted lists are open and not closed If in the domain there is no hole Domain Holes must however assume following value Domain Holes Hole 1 DOMAIN SEGMENTS Domain Segments is a structure containing Segment as a field and its usage is very similar to the one of Domain Holes Domain Segments contains the references of segments that may be forced in the domain Field Segment is an array whose dimensions are equal to the number of holes presents in the domain Every field Hole has values as sub field This is an array that contains references to some vertexes inserted in Domain InputVertex Domain Segments Segment i Values is a sorted list that says that there is a border of i esim segment between two adjacent elements of Domain Segments Segment i Values Domain Segments may contain references to vertexes that are inserted in the domain only for this purpose or may use vertexes that have been inserted to shape domain or holes As for the ones described in Domain Holes if in the domain there is no segment Domain Segments must however assume following value Domain Segments Segment 1 Sample of usage of segments are reported in appendixes Page 6 Andrea Barbera BTR 3 0 User s manual TRICK FORCE VERTEXES IN GRID Every vertex of Inp
14. les Hole Domain Segments Segment BC Values 1 BC Boundary Values ones 1 106 BC Holes Hole BC Segments Segment BC InputVertexValues ones 1 106 RefiningOptions CheckArea N RefiningOptions CheckAngle Y RefiningOptions AreaValue 0 1 RefiningOptions AngleValue 5 RefiningOptions Subregions Subregion RefiningOptions Subregions Subregion RefiningOptions Subregions Subregion RefiningOptions Subregions Subregion RefiningOptions Subregions Subregion RefiningOptions Subregions Subregion Type rectangular Descriptors 2 0 10 0 CheckArea N CheckAngle Y AreaValue AngleValue 30 ER Er 1 1 1 1 1 1 re _ WH RefiningOptions Subregions Subregion RefiningOptions Subregions Subregion RefiningOptions Subregions Subregion RefiningOptions Subregions Subregion RefiningOptions Subregions Subregion RefiningOptions Subregions Subregion Type elliptic Descriptors 6 6 3 2 CheckArea Y CheckAngle N AreaValue 0 01 AngleValue 30 SNNNND geom btr30 Domain BC RefiningOptions draw grid geom 1 and the result is reported in figure 4 Page 18 Andrea Barbera BTR 3 0 User s manual figure 4 Page 19
15. main Holes Hole 2 Values 8 10 Domain Segments Segment 1 Values 4 10 Domain Segments Segment 2 Values 10 6 Domain Segments Segment 3 Values 6 2 BC Values 2 4 6 8 10 12 BC Boundary Values 1 2 1 2 BC Holes Hole 1 Values 3 3 3 BC Holes Hole 2 Values 4 4 4 BC Segments Segment 1 Values 5 BC Segments Segment 2 Values 6 BC Segments Segment 3 Values 7 BC InputVertexValues 1100333000 RefiningOptions CheckArea Y RefiningOptions CheckAngle N RefiningOptions AreaValue 0 1 RefiningOptions AngleValue RefiningOptions Subregions geom btr30 Domain BC RefiningOptions draw_grid geom 1 The result is reported in figure 3 Red borders are borders to which are assigned Dirichled conditions blue borders are borders to which are assigned Neumann conditions Vertexes that are circled in red are Dirichlet vertexes Page 16 Andrea Barbera BTR 3 0 User s manual figure 3 Page 17 SAMPLE 2 Andrea Barbera BTR 3 0 User s manual This is a sample of triangulation of a simplier domain with many input vertexes on a single border and fewer in other borders Two refining subregions are chosen an elliptic one in the center and a rectangular one on the left for i 0 99 Domain InputVertex i 1 i 10 0 end Domain InputVertex 101 106 100 10 10 6 2 10 610 5 8 10 0 10 Domain Boundary Values 1 106 Domain Ho
16. main and then refines the mesh according to parameter chosen by users inserting new vertexes in best positions To set these parameters users must consider RefiningOptions structure whose fields are RefiningOptions AreaValue RefiningOptions RefiningOptions AngleValue RefiningOptions RefiningOptions CheckArea CheckAngle Subregions Every field written above must be included in RefiningOptions for the right running of BTR REFININGOPTIONS CHECKAREA AND REFININGOPTIONS AREAVALUE One of the possible grid refining continues splitting triangles until the larger triangle has an area value lesser than a threshold value If RefiningOptions CheckArea a character variable assumes values y or Y BTR controls the size of every triangle imposing a threshold value equal to RefiningOptions AreaValue If not interested in size of triangles it will be sufficient to impose RefiningOptions CheckArea equalto n or N REFININGOPTIONS CHECKANGLE AND REFININGOPTIONS ANGLEVALUE The other possible grid refining continues splitting triangles until every newly inserted angle in the domain is greater than a threshold value The value of smallest newly inserted angle is also called quality of the grid If RefiningOptions CheckAngle a character variable assumes values y or Y BTR controls the angles of every triangle imposing a threshold value equal to RefiningOptions AngleValue If not interested in quality of trian
17. s In particular there are Domain InputVertex Domain Boundary Domain Holes Domain Segments Every field written above must be included in Domain for the right running of BTR DOMAIN INPUTVERTEX Domain InputVertex is a matrix containing the coordinates of every vertex useful to describe the domain The i esim row represents the i esim input vertex The first component of i esim row represents the x coordinate of i esim vertex the second component represent the y coordinate An example is Domain InputVertex 0 0 In this example the first vertex is 0 0 the second is 5 0 the third is 5 5 and the fourth is 0 5 The order used to memorise vertexes is irrelevant DOMAIN BOUNDARY Domain Boundary is a structure containing Values as a field Domain Boundary Values is an array containing the references of the boundary of the domain Every number of this array is a reference to a row of Domain InputVertex Domain Boundary Values is a sorted list that says that there is a domain border between two adjacent elements of Domain Boundary Values and between its first and last element Referring to Domain InputVertex written before a squared domain may be described in different ways For example considering domain in figure 1 Page 4 Andrea Barbera BTR 3 0 User s manual 4 3 5L 4L St 2L 1L 2 ol 1 o 1 2 3 4 5 6 figure 1 We may impose Domain Boundary Values 1 2 3 4 or Dom
18. tangle containing every input vertex These values must be positive GENERATIONUNTILRESEARCH GenerationUntilResearch default value 10 says how the research of a particular triangle is made from a starting triangle FIRSTEXTIMATETMULTIPLIER FIRSTEXTIMATEBMULTIPLIER FIRSTEXTIMATEVMULTIPLIER FirstExtimateTmultiplier default value 8 FirstExtimateBmultiplier default value 8 and FirstExtimateVmultiplier default value 6 are multipliers used to make a first estimate of the number of elements of the grid Page 14 Andrea Barbera BTR 3 0 User s manual FIRSTMAXREFININGROOF FirstMaxRefiningRoof default value 1 5 is a multiplier used to give a first estimate of the refining roof the maximum number of triangles that BRT tries to refine at every step INCREASINGREFININGROOF IncreasingRefiningRoof default value 3 is a multiplier used to increase the refining roof after that it has been reached in a previous step RECURSIONSTACKLIMIT RecursionStackLimit default value 1000 is the maximum recursion stack limit that is set in Matlab Page 15 Andrea Barbera BTR 3 0 User s manual APPENDIX SAMPLES SAMPLE 1 This is a sample of triangulation of a complex domain with two holes different segments and different BC assigned to different elements This is the script necessary to generate the mesh Domain InputVertex 0 0 43 5 13 75 Domain Boundary Values 1 4 Domain Holes Hole 1 Values 5 7 Do
19. utVertex will however be inserted in the mesh even if it will not be used to shape the domain a hole or a segment In the same way a single vertex may be used to shape different elements as for example the domain and a segment or two different segments TROUBLESHOOTING Inserting twice the same vertex will give an error gt Elements of triangulations must be coherent and may not intersect each other but may however touch themselves in a single point so if a cross composed by segments must be inserted in the domain also the intersection point must be inserted in InputVertex BC VARIABLE BC is a structure that is composed by different fields In particular there are BC Values BC InputVertexValues BC Boundary BC Holes BC Segments Every field written above must be included in Bc for the right running of BTR BC VALUES BC Values is list of BC values that will be set in domain borders Conventionally Values in even position i e 1 co gg are Dirichlet conditions Values in odd position i e 2 4 are Neumann conditions If it is used a different number of Dirichlet and Neumann condition some 1s may be for example put to fill BC Values An example of BC Values usage is following one BC Value 3e 1 2e4 0 3e4 1 0 1 4e4 In this case are used two Dirichlet condition 3e 1 with index 1 and 0 with index 3 and four Neumann conditions 2e4 with index 2 3e4 with index 4 0 with index 6 and 4e4 w
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