Optimal Mapped Mesh on the Circle

Transcription

1 Koferece ANSYS 009 Optimal Mapped Mesh o the Circle doc. Ig. Jaroslav Štigler, Ph.D. Bro Uiversity of Techology, aculty of Mechaical gieerig, ergy Istitut, Abstract: This paper brigs out some ideas ad recommedatios how to build mapped mesh o the circle with high quality mesh cells. Keywords: Mapped mesh, circle, 1. Itroductio Quality of mesh sigificatly iflueces the solutio precisio ad covergece rate i case of every umerical solutio of fluid flow. There are several types of mesh. The best results ca be obtaied with Mapped mesh. The cell s shape i this mesh type is close to the square i case of d mesh or to the cube i case of 3d mesh. Whe we are solvig a fluid flow i pipes with the rouded cross-sectio the basic questios arise i our mid: ow we ca make a mapped mesh o circle? Is it possible to fid a optimal solutio how to make mapped mesh o the circuit? Let us try to aswer these questios i the ext chapters.. Differet types of mapped meshes There are two basic ways of mapped mesh creatig. The easiest way is to divide the border of circuit ito four pieces. This is very easy ad quick way. Bat there is a problem with the mesh quality i this case. Quality of such mesh is so bad that the solutio does ot start. This type of mesh is depicted o the fig. 01 a. a) b) ig. 01 Differet types of mapped mesh o circuit.

2 TechSoft gieerig & SVS M The ext way is more difficult. Area of the circuit has to be divided ito some umber of sub-areas which shape is more sufficiet for the mapped meshig. The five sub-areas will be used i our case. The circle cosists of square area i the middle of the circle ad four areas aroud this square. This type of mapped mesh o circuit is depicted i the fig. 01b. The first type of mapped mesh is useless because of their quality. Therefore our attetio will be focused o the secod type of mapped mesh usig mapped sub-areas. Dividig of circuit ito mapped sub-areas ad their dimesios is depicted i the fig.. a) b) ig. 0 Dimesios of mapped sub-areas o circuit. 3. Optimal mapped mesh o a circle with rectagular ier square. Basic ideas of optimal mesh creatig o the circle with ier rectagular square are outlied i detail i a techical report [1]. If we wat divide the circle ito mapped sub-areas, tha the legth of edge has to be kow. Ifluece of size of the edge is clear from the pictures i fig 03. Whe the is small the mesh is very dese i the middle square of circuit ad very rough at the border of circle. This is depicted i fig 03a. I opposite situatio whe the dge is big the cells ear edge are very deformed. The first task is to fid optimal size of edge. a) b) ig. 03 Ifluece of size of square iside of the circuit o the mesh.

3 Koferece ANSYS 009 The square iside circle is the best area for mapped mesh. The problem lies i the four sub-areas roud the square. I case of square the opposite sides have the same legth. But this is ot true i case of other sub-areas of circle. Now the attetio will be focused o oe of the peripheral sub areas of the circle. This sub-area is depicted o the fig. 0b. Now it will be useful to express the legths of edges,, ad. All these leghts will deped o the radius R or o the legth of edge or o both of them. π = R., = R, = R, I the best mapped mesh of such area will be obtaied whet the ext ratios are equal 1. = 1 ad = 1 It is obvious that these coditios caot be accomplished. Both of these coditios are i cotrary o oe to other. irst coditio will be fulfilled i case =0. The secod coditio will be fulfilled i case =R. π/. But caot be bigger tha R.. If it happes the the ier squared area exceeds the circle. Both these coditios caot be satisfied together. It is possible to receive the compromise solutio that both coditios will be equal. = We ca add previous expressios ito this coditio ad we receive quadratic equatio for..r. π + + R. π = 0 This equatio has two solutios i form. 1, π = R ± 1+. π 1 +. π We ca express this solutio as Where = 1, R. a1, a 1 = 3,56854, = 0,9646 0, 96 a Correct solutio is α, because its value lays i a iterval of acceptable values a 0,. The basic geometry of sub-areas is determied. Now we eed to kow a umber of elemets o edges,, ad to be able to create mapped mesh o circle. Numbers of elemets o edges will be siged,,,. I case of mapped mesh the umbers of opposite edges have to be same. It is assumed that successive ratio is 1 for all edges. Tha the umbers of elemets will be determied

4 TechSoft gieerig & SVS M form coditio that legths of elemets are as equal as it is possible. Legths of edges elemets are siged L, L, L, L. These legths ca be calculated L =, L =, L =, L = It is impossible for all edge elemets to be equal. But we ca foud some limits of their legths. These limits are. L L L or L L L or I case of mapped mesh = ad =. Whe all above relatio are take ito cosideratio we receive. 1 a 1 a 1 a 1 1, or 1 a Now it is possible to write that = β. The value of coefficiet b is o the itersectio of above itervals. a β a Number of elemets lies betwee these limit values mi a 1 a 1 = βmi. = 1., max = βmax. =. The umber of elemets is chose ad umber of elemets ca be calculated from above terms. The best way is to take average value of β. We ca calculate for three differet values of a a example. A β mi β max mi max av 0,83 0,3745 0, , ,3957 0, ,0 0, , Tab. 01 Calculatig of for =0 ad for differet value of a. I the case of ratio a opt = 0, optimal value, this iterval of β is oly oe value. Whe the ratio a is bigger tha a opt the iterval for β is empty. Whe the ratio a is smaller tha a opt tha the itersectio iterval is ot empty. The best value is the average valule of β mi ad β max.

5 Koferece ANSYS 009 valuatio of meshes is listed i table 0. The mai criterio C AS ad C VS is costat for all type of these meshes with its value 0,5. So these meshes have to be sorted by differet criterio for example criterio C AR. Usig this criterio the best meshes are for av or close to it. Best mesh is mesh for a=0,964 ad =7 for =0 i a global view. Mesh valuatio of mesh quality by differet criteria. /R N C AR C DR C R C AS C VS C MAS C S 5 800,580 1,0,800 0,5 0,5 0,56 0, ,88 1,0,000 0,5 0,5 0,59 0,40 0, ,665 1,0 1,760 0,5 0,5 0,60 0, ,594 1,0 1,704 0,5 0,5 0,61 0, ,776 1,0 1,900 0,5 0,5 0,61 0, ,500 1,0,660 0,5 0,5 0,63 0, ,500 1,0,700 0,5 0,5 0,55 0,5 0, ,564 1,0 1,704 0,5 0,5 0,59 0, ,79 1,0 1,950 0,5 0,5 0,60 0, ,3 1,0,500 0,5 0,5 0,55 0,48 1, ,58 1,0 1,665 0,5 0,5 0,58 0, ,704 1,0 1,873 0,5 0,5 0,59 0, ,46 1,0,680 0,5 0,5 0,61 0,49 Tab.0. valuatio of mesh quality for =0 ad differet value of a C AR -Aspect Ratio C VS -quisize Skew C DR -Diagoal Ratio C MAS -MidAgle Skew C R -dge Ratio C S -Stretch C AS -quiagle Skew 4. Optimal mapped mesh o a circle with bet ier square. The quality of mesh ca be icreased by usig ot rectagular ier square. The edges will ot be straight but they will be bet with radius r. This type of sub-areas is depicted i fig 04. The expressio for the legth of edge has to be modified ad the expressio of legth J has to be added. J 4.K + 4.K 1 1 = R K =. b a.k.arctg K. = ( 4.b + 1) b.arctg 0,5 b = 4.b

6 TechSoft gieerig & SVS M a) b) Where ig. 04 Dimesios of mapped sub-areas o circuit with bouded ier sub-area. b =. R Now we will take value of the a opt =0, The we will try to fid the optimal ratio b. This optimum value will be searched o the iterval b <0, 1 >. or the case b=bmi =0 it is obtaied rectagular square ier area. I the case b=b max = ( 1) it is obtaied ier area as circuit. This optimum was foud by evaluatig of set meshes with costat ratio a ad with differet values of b o the iterval <b mi, b max >. Results are listed i table 3. The umber of elemets o edge ( ) is calculated similar way as i the case of straight edge oly the legth of edge J is take ito cosideratio istead of edge. The iterval for coefficiet β is β <β bmi, β bmax >. Where β a = 1 b. a b mi, β b max = π a 1 a.m Where ( 4.b + 1) b M =.arctg..b. π 0,5 b There are depedeces of β bmi ad β bmax o the ratio b i the fig. 05. Results of evaluatig of meshes created uder previous coditios are listed i the table 03.

7 Koferece ANSYS 009 Mesh valuatio of mesh quality by differet criteria. b av N C AR C DR C R C AS C VS C MAS C S 0, ,644,100 1,644 0,50 0,50 0,63 0,9 0, ,644 1,90 1,644 0,44 0,44 0,57 0,6 0, ,637 1,744 1,637 0,39 0,39 0,51 0,9 0, ,564 1,616 1,564 0,34 0,34 0,44 0,6 0, ,546 1,665 1,546 0,33 0,33 0,46 0,6 0, ,540 1,679 1,540 0,33 0,33 0,47 0,5 0, ,534 1,70 1,534 0,34 0,34 0,49 0,5 0, ,516 1,784 1,516 0,36 0,36 0,5 0,4 0, ,498 1,846 1,498 0,38 0,38 0,54 0,4 0, ,485 1,910 1,485 0,40 0,40 0,57 0,3 0, ,480,300 1,480 0,50 0,50 0,68 0,3 0, ,480,860 1,480 0,59 0,59 0,78 0,5 0, ,39 3,700 1,39 0,69 0,69 0,86 0,7 0, ,316 5,100 1,316 0,78 0,78 0,9 0,8 0, ,316 7,930 1,316 0,87 0,87 0,96 0,8 0, ,70 17,000 1,70 0,96 0,96 0,99 0,9 Tab.03. valuatio of mesh quality for =30 ad value of a=0, rom both table 03 ad fig. 05 is clear, that umber is decreasig with ratio b icreasig. This is true oly i case that ratio a is a costat. It is possible to fid two optimal values of ratio b. irst is for value b= 0,0613. I this case it is a optimum for two criterio C DR a C MAS. The secod optimum is for value b=0,0675 ad i this case it is optimum for criterio C AS. What optimum will be take ito cosideratio i practice it depeds what criterio we prefer. The depedeces of these three criterios are depicted o the fir 06. The optimum is a itersectio of two differet curves i all three cases. It is obvious, because oe of the curve represets quality of elemets i the outer sub-area ad the secod oe represets quality of elemets of the ier sub-area. I case of criterio C AS it seems to be a itersectio of two straight lies. Whe we icrease bedig of edges of ier square we are gettig worse the elemets quatily of ier sub-area ad gettig better elemets quality of outer sub-area. ig. 05 Coefficiets for settig i depedece o ratio b

8 TechSoft gieerig & SVS M 5. Refereces ig. 06 Coefficiets for settig i depedece o ratio b Štigler, J., 004. Jak vytvořit mapovaou síť a kruhu. Techická zpráva VUT-U QR-03-05, VUT SI v Brě, Bro, pp Appedix Author tried to fid some commo priciples, how to create a optimal mapped mesh o the circle i this paper. Curretly it was foud that best mesh ca be made for these values of ratio a=0,964619, ratio b=0,675, i case that we prefer criterio C AS (quiagle Skew), ad with value of coefficiet β which is a average of β bmi ad β bmax, which is importat for calculatig. These iformatios ca be useful for first desig of mapped mesh o the circle ad ca be useful for automatic mesh geeratio. 7. Ackowledgemet Author is grateful to rat Agecy of Czech Republic for fudig of this research withi scope of project with umber A 101/09/1539 with title Mathematical ad Numerical Modelig of low i Pipe Juctio ad its Compariso with xperimet.