Peak Feld Approxmaton of Shock Wave Overpressure Based on Sparse Data Yongl Zhang, Taln Han, Yuqun Chen, Enku Zhang, and Xuan Lu Abstract To obtan the shock wave feld dstrbuton, two knds of calber weapons shock wave overpressure by the electrc measurng method are analyzed, and the attenuaton formula n far feld s corrected, and the approxmaton method of overpressure peak feld wth cubc splne nterpolaton s put forward. The sparse array data of small calber muzzle shock wave feld measured by sensor array s frst analyzed and the curve fttng was carred out by usng the correcton formula n the radal drecton; secondly, cubc splne nterpolaton was used n the crcumferental drecton; and lastly, Delaunay trangulaton method was used to dvde the mesh to approxmate the overpressure peak surface. Compared wth the cubc splne nterpolaton results obtaned by ths method, the approxmate sobars showed obvous advantages. Further, the large-dameter muzzle shock wave usng polar coordnate sensors were tested, and the approxmate method of sparse matrx overpressure data processng feld effect s good. The approxmate method provdes reference for the drawng of the shock wave feld by polar coordnate. Keywords peak value of overpressure; curve fttng; cubc splne functon; surface approxmaton. N I. INTRODUCTION UMERICAL smulaton of the ntal flow feld of muzzle shock wave s used to obtan the peak value of overpressure, etc. Wth the propagaton of shock wave (especally n the far-feld) and the enlargement of the calculaton area, the calculaton amount ncreases rapdly and the calculaton accuracy decreases []-[2]. The decrease of the smulaton precson of the far feld of the shock wave caused the error of overpressure peak to ncrease. The shock wave test of conventonal weapon shock wave s mostly carred out by electrometrc method to ensure the accuracy of the data, and the shock wave data s obtaned through sensor array calbrated dynamcally. The overpressure peak surface and ts contour are nterpolated by mathematcal algorthm [3]-[4]. The reasonable sensor placement method n order to reconstruct exploson overpressure feld at the hghest precson s very necessary, especally Interpolaton algorthm methods [5]-[6]. Because of restrctons on the launchng condtons of weapons and space areas, there s a shortage of standards of Taln Han s wth College of Electronc and Informaton Engneerng, Changchun Unversty of Scence and Technology, Changchun 322, Jln, Chna. (Correspondng author; e-mal: hantl@26.com). test method of the safety and the servceablty of gun for placng one hundred sensors that are placed n the 4.5m range of the X axs and the Y axs by usng Cartesan coordnate on the one sde of the muzzle. Polar coordnate s used to reduce the number of sensors for a naval gun test that the number of sensors s twenty-fve [7]. Based on the advantages of polar coordnate [8], the experment of muzzle flow feld was carred out n hgh alttude envronment to obtan shock wave overpressure data [9]. Based on the above analyss, ths study nterpolates the sparse data obtaned through the polar coordnate to obtan the hgh accuracy of the shockwave feld contour. The overpressure surface s approxmated by curve fttng on the radal drecton of propagaton to modfy mathematcal model and ncreasng the data through the cubc splne functon nterpolaton on the crcumferental drecton. In order to verfy the effectveness of the algorthm, the error was compared wth the cubc splne functon nterpolaton. II. PROPAGATION LAW OF SHOCK WAVE OVERPRESSURE In the exstng lteratures, the shock wave overpressure decays exponentally n the far feld propagaton process for the far-feld propagaton of shock wave [], and the mathematcal model s smply expressed as follows: ( ) P = K( θ ) γ α θ () Where, the coeffcent K represents the overpressure value of the shock wave at the begnnng, γ represents the dstance along the ray, and α represents the attenuaton rate of the overpressure value of the shock wave. In polar coordnate, the radal drecton of the fxed angle shows the change of power functon. Equaton () s logarthmcally processed as follows: log P = log K( θ) α( θ)log γ (2) The numercal change of the correspondng formula (2) after the logarthmc process has a lnear relatonshp wth the ndependent varable. III. DATA SOURCES In ths chapter, two dfferent weapons of a mnor-calber gun (7.62mm) [5] and a large-dameter gun [8] are arranged n polar coordnate accordng to the poston of the test ISSN: 998-4464 23
ponts. Consderng the envronmental nterference factors, the smulaton s carred out under the atmospherc pressure envronment. 7.62mm calber gun s placed n the drecton of degrees and the test pont s at the orgn of the muzzle. The nterval between adjacent ponts s 22.5 degrees and the dstance between adjacent measurng ponts n the radal drecton s.25m, that s to say, 4 ponts are placed on the same radal drecton, and 28 ponts are placed n total, and the test results are shown n Table I. Table I Peak overpressure data of a 7.62-mm gun shockwave (kpa ) r/θ.25 m.5 m.75 m. m 22.5 47.4 8.25.73 6.96 45 4.86 4.69 9. 5.67 67.5 29.9.32 5.8 3.76 9 6.76 7.3 4.55 3.6 2.5.66 4.34 2.88 2.36 35 5.76 3.37 2..56 57.5 4. 2.27.26.87 And the large-danmeter gun s placed n the drecton of degrees and the test pont s at the orgn of the muzzle. The nterval between adjacent ponts s 5 degrees and the dstance between adjacent measurng ponts n the radal drecton s m, that s to say, 5 ponts are placed on the same radal drecton, and 4 ponts are placed n total, and the test results are shown n Table II. Table II Peak overpressure data of a large-danmeter gun shockwave (kpa ) r/θ 4 m 5 m 6 m 7 m 9m 5 2.7 7.3 4.9 9.2 3 22. 7.6 4.2.6 9.4 45 2.2 3.9. 8.8 8.7 6 28. 6.8 2.3. 8.9 75 26 9.7 4.7 8.9 9 33.2 2.9 7.7 4.3 2. 5 3.9 25.9 9 5.4 2.6 2 4.3 36.9 22.3 6.4 2.9 35 4.3 37. 22.6 4.8 5.8 IV. OVERPRESSURE PEAK FIELD APPROXIMATION Overpressure peak feld approxmaton belongs to surface modelng. By fttng some dscrete ponts n a certan range around any pont ( x, y ) and ts correspondng dscrete pont f ( x, y ), a mathematcal model of curved surface s establshed to solve the correspondng values f ( x, y ) of the nterpolaton pont ( x, y ). Accordng to the general procedure of the algorthm, frstly, the ndependent curve approxmaton of the axs x and the axs y s carred out to obtan the sngle drecton more value ponts. Secondly, the mesh s encrypted and fnally the surface s approxmated to obtan the surface and contour plot. In the study, the data of the propagaton drecton (radal drecton) are ftted to ensure the accuracy of the data. The cubc splne functon nterpolaton s used to guarantee the frst order and second order contnuty of the fttng curve. The surface of the pressure peak feld s approached by grd encrypton. A. Radal Drecton for Fttng ) Fttng ndex formula The three ndexes of the curve fttng evaluaton ndex are the sum of squares due to error (SSE), Coeffcent of determnaton(r-square), and Root mean squared error (RMSE) []-[2]. The error statstcs are analyzed and the formula s gven as follows: n ' (3) = SSE = Z Z R square = n = SSE ( Z Z ) n k = n ( ) 2 ' RMSE = Z Z 2 (4) (5) In the formula, Z represents the actual measured values, ' Z represents the predcted values, Z represents the mean values of the frst pont, n represents the number of ponts. The closer the SSE s to, the better the fttng s. The closer R-square s to, the stronger the nterpretaton ablty of the fttng data. RMSE reflects the senstvty and extreme value of fttng values. 2) Date analyss Peak overpressure data of a 7.62-mm gun shockwave was compared n seven radal drectons, as shown n Fg. (a). The consstency of the change of data that The greater the overpressure peak near the orgnal pont, the ampltude attenuaton n the radus.5m range s larger, and the outsde change s relatvely gentle, whch s consstent wth the emprcal formula (). The data n dfferent radal drectons are logarthmcally processed, as shown n Fg. (b). The curve change s not a lnear relatonshp, and the curve fttng error s larger. ISSN: 998-4464 24
hgher than.99, and the maxmum value of RMSE s.26. The fttng results of the curve are shown n Fg. 2. Table III Precson ndex of curve fttng error /θ SSE R-square RMSE 22.5.679.9999.266 45.553.9994.79 67.5.9.38 9.93.9999.962 2.5.2.29 35.84.9982.357 7.5.89.9968.375 Fg. 2 Curve fttng of peak value of shock wave overpressure wth radus B. Interpolaton Algorthm n Crcumferental Drecton For a gven set of known ponts ( x, y ), =, 2,, n, m s the slope of type ( x, y ), and the expresson of cubc j splne nterpolaton functon of j secton s as follows: Fg. Varaton of peak value of shock wave at dfferent angles wth radus The fttng error of the data s larger by usng the formula (), so as to elmnate the error, the correcton term s added, and The modfed formula s gven as follows: P K f α ( θ ) = ( θ ) γ + ( γ ) (6) In the formula, f represents the correcton term s lnear wth the radusγ. The modfed formula (6) s used to curve the data of seven radal drectons, and the fttng error s shown n Table III. The error data of the 6 drectons are deal, but the error n the drecton of 45 degrees s slghtly larger. In the radal drecton, the varaton of the error s consstent, the maxmum value of SSE s.679, the value of r-square s y x = a x + b x + c x + d j = n (7) 3 2 j ( ) j j j j, 2,, The coeffcents of nterpolaton functon can be solved []. A comparatve analyss on the crcumferental drecton, the curve n Fg. 3 (a) shows varaton of the data s not obvous, t s dffcult to use the functon fttng unform, usng three order splne functon nterpolaton to ensure smooth curve, nterpolaton results shown n Fg. 3 (b) shows. The data were compared n the crcumferental drecton, as shown n Fg. 3 (a). It s not obvous that the data characterstc changes, and t s dffcult to ft n the unform functon. In order to ensure the smoothness of the curve, nterpolaton s carred out wth cubc splne functon. And ISSN: 998-4464 25
the results of the nterpolaton are shown n Fg. 3 (b). p / k P a p / k P a 5 45 4 35 3 25 2 5 5 r=.25 r=.5 r=.75 r=. 2 4 6 8 2 4 6 θ/ 5 45 4 35 3 25 2 5 5 r=.25 r=.5 r=.75 r=. 2 4 6 8 2 4 6 θ/ Fg. 3 Varaton and nterpolaton of dfferent peak radus of shock wave wth radan C. Feld Surface Approxmaton After the curve fttng of the data ponts n the radal drecton, the nterval of.m n the range of.25m-.m was calculated, and 72 data ponts were added, and the orgnal data pont (4) was ncreased to a precse 9 tmes. The cubc splne functon s used to nterpolate n the crcumference drecton, and the same nterval s.m. Data extracton s carred out n two drectons of radal and crcumferental drectons respectvely. Data extracton s carred out n two drectons of radal and crcumferental drectons. The data s meshed by Delaunay trangulaton method, and the shock wave overpressure peak surface (Fg. 4) and the overpressure contour are plotted. z / k P a 6 5 4 3 2 2.5 x/d Fg. 4 Peak surface of shock wave overpressure From the peak overpressure of shock wave surface n Fg. 4, t can be clearly seen that the shock wave of the chamber has the movement rules, the ntensty attenuaton and the drectvty. In order to verfy the effectveness of the method, the nterpolaton of the cubc splne functon s compared, and the results are shown n Fg.5. The followng conclusons can be seen from fgure 5. The macroscopc analyss shows that the two methods all reflect the characterstc of peach shape dstrbuton, and the ntensty of the overpressure peak s greater than that of the x axs, that s, the blast of blast chamber s drectonal. But the peak value of radal overpressure has a sgnfcant change. The attenuaton of cubc splne nterpolaton s not consstent n the radal drecton of about 67.5 degrees, whch s not consstent wth the prevous data analyss. In about 2.5 drecton, the curve changes, that s to say cubc splne algorthm ensures the curve smooth, but also cause certan nfluence to the dstrbuton of orgnal data. The valdty of ths method can be obtaned by comparng the overpressure contour lnes of two methods n Fgure 5..5.5 y/d 5 45 4 35 3 25 2 5 5 ISSN: 998-4464 26
The larger the calber, the more complex the shock wave. The nfluence of the bore devce and ground reflecton on the far-feld propagaton of the large bore mouth shock wave s dfferent from that of the small dameter, but the radal attenuaton rule s consstent. As can be seen from Fg. 6 (a), 9 degrees s the dvdng lne, and the radal overpressure peak s the strongest between 5~2 degrees, and the 45 degrees s the weakest. It can be seen from Fg. 6 (b) that the dstrbuton of the large-dameter far feld s more unform. Fg. 5 Isoclne contrast map D. Applcaton The sparse array data of small calber muzzle shock wave feld measured by sensor array s analyzed and the curve fttng was carred out by usng the correcton formula n the radal drecton. On the bass of ths, the effect of soclne contrast map s better than the nterpolaton of cubc splne functons. Then, the method s used to process data of a large-dameter gun shown n Table II. The modfed formula (6) s used to curve the data of 9 radal drectons, and the fttng error s shown n Table IV. The error data of the 9 drectons s deal too. Table IV Precson ndex II of curve fttng error /θ SSE R-square RMSE 5.277.9959.3722 3.8854.985.6653 45.44.9824.455 6.8699.9985.286 75.832.99.6443 9.698.9982.294 5.2653.998.3642 2.85.9997.2434 35.943.998.6867 Fg. 6 The results of the large-dameter gun V. CONCLUSION To obtan the overpressure peak feld approxmaton method, the peak value of shock wave overpressure by polar coordnates s analyzed, and the mathematcal model of attenuaton rule s corrected. In the radal drecton, the modfed mathematcal model s adopted to curve fttng, and the overpressure peak surface s approxmated, and the contour lne s drawn. In order to verfy the effect, the nterpolaton of cubc splne functon s compared. The followng conclusons could be drawn: () The modfed formula curve fttng s sutable for dfferent dameter shock wave felds. (2) The overpressure value predcted by the modfed ISSN: 998-4464 27
propagaton formula s more accurate and provdes a reference for the predcton of overpressure value n mddle and far feld. (3) It s more reasonable for the characterstcs of the overpressure feld surface wth the added pont value data nterpolaton and the contour lne precson s hgh. Thus, the ground reflecton s not consdered, and the predcton formula s to be verfed n the far feld. More nfluencng factors should be consdered n future studes. REFERENCES [] Y.Wang, X.H.Jang, Z.Q.Guo, "Numercal Analyss on Physcal Model of Muzzle Blast Wave,"Journal of Ballstcs.vol. 22(),pp. 57-6,2. [2] J.Y.Du, F.Q.Fan, H.Sun, G.H.Zhao,"Present Stuaton and Development Trend of Domestc Weapon Provng Ground Test Technology," Measurement & Control Technology. vol.33(7),pp.-2,24. [3] D. Davydov F. Zelfelder: Scattered data fttng by drect extenson of local polynomals wth bvarate splnes, Adv. Comp. Math., 2 (24), 223 27. [4] M. Bozzn L. Lenarduzz: Bvarate knot selecton procedure and nterpolaton perturbed n scale and shape, n Curve and Surface Desgn Sant-Malo 22, T.Lyche M. L. Mazure L. Schumaker eds., 23, Nashboro Press. [5] M.M.Ba, Y.L.Guo, L.M.Wang,"Study on Optmal Sensor Placement Method Based on the Reconstructon of Exploson Overpressure Feld," Chnese Journal of Sensors and Actuators, vol.27(7),pp.886-892,24. [6] Z.Yang, Z.J.Zhang, Y.L.Xa,"Rescontructon of Shock Ware Overpressure Fled Based on B-Splne Interpolaton," Scence Techonolgy and Engneerng. vol.6(7),pp.236-24,26. [7] F.W.La,Z.J.Zhang,G.M.Hu,J.Y.Zhang, "A Method to Measure Muzzle Shock WavePressure Feld for A Naval Gun," Chnese Journal of Sensors and Actuators,vol.28(),pp. 77-8,25. [8] Y.L.Zhang, T.L.Han, X.Y.Lu, S.Q.Tan, "Shock wave Sgnal Dstrbuted Acquston and Data Processng Technology," In: Proc. of 27 IEEE 2nd Advanced Informaton Technology, Electronc and Automaton Control Conference, Chongqng, Chna: IEEE, 27: 3-34. [9] Z.Q.Guo, Numercal Investgatons on thedynamcs Mechansm of Muzzle Flow, Nanjng Unversty of Scence & Technology,22. [] H.Z.L,D.W.Ma, "The ProPagatng Laws of Muzzle Blast Far Away from the Gun," Journal of Ballstcs, vol,pp. 25-29,992. [] S.M. Rump., "Error estmaton of floatng-pont summaton and dot product." BIT, vol.52. pp. 2-22,2. [2] Z.M.L,J.Q.Tan," Error Analyss of Ratonal Cubc Splnes and Interpolaton of Space Closed Curves,"Journal of computer-aded desgn & computer graphcs.vol.2,pp. 876-88,28. Yongl Zhang was born on Sept. 6, 986. He s a Ph.D. Student wth Electronc and Informaton Engneerng, Changchun Unversty of Scence and Technology. Hs major research nterests s sgnal processng. Taln Han was born on Jun. 9, 969. He s a professor at Electronc and Informaton Engneerng, Changchun Unversty of Scence and Technology. Hs major research nterests nclude communcaton engneerng and sgnal processng. Yuqun Chen was born on May. 5, 979. She s a lecturer at Electronc and Informaton Engneerng, Changchun Unversty of Scence and Technology. Her major research nterests nclude communcaton engneerng and sgnal processng. Enku Zhang was born on Mar. 9, 986. He s a Ph.D. Student wth Electronc and Informaton Engneerng, Changchun Unversty of Scence and Technology. Hs major research nterests s sgnal processng. Xuan Lu was born on Oct. 9, 99. He s a Ph.D. Student wth Electronc and Informaton Engneerng, Changchun Unversty of Scence and Technology. Hs major research nterests s sgnal processng. ISSN: 998-4464 28