The Algorithm of Simplex Integration in Three-Dimension and

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1 The Algorthm of Smple Integraton n Three-Dmenson and Its Characterstc Analss WU Yan-qang CHEN Guang-q JIANG Za-sen ZHANG Long LIU Xao-a ZHAO Jng The Algorthm of Smple Integraton n Three-Dmenson and Its Characterstc Analss WU Yan-qang CHEN Guang-q JIANG Za-sen ZHANG Long LIU Xao-a and ZHAO Jng ) Insttute of Earthquake Scence CEA Bejng 6 ) Kushu Unverst Fukuoka Japan 86 Abstract An algorthm of ntegraton wrtten b C++ language are proposed based on the analss of smple ntegraton over a polnomal epresson n three-dmenson and the calculatonal procedure for smple ntegraton s descrbed wth concrete eamples n ths paper. Frstl we anale the accurac of the smple ntegraton through comparng the theoretcal results wth calculated results n whch parameters nclude the volume and gravt centers and the ntegral feld s regular but wth hollow. Then the nfluence of graphc condtons on the results of smple ntegraton are dscussed through comparng the theoretcal results wth calculated results n dfferent edge length rato( -5-5 ). Secondl the adaptve characterstcs of smple ntegraton algorthm are dscussed through analng ntegral results n rregular ntegral space wth hollow. Fnall the precson of smple ntegraton are dscussed whch the ntegral term s hgh-order polnomal and the ntegral spaces are also rregular wth hollow. The results show that the relatve error between calculated results and theoretcal results s about and the graphcs condtons have mnmal mpact on ntegral results and the mpact has not sstematc characterstcs. In concluson the dfference between the calculated results and theoretcal results s caused b computng error and the results of smple ntegraton have hgh stablt and accurac. Keword: Smple Integraton Process Of Integraton In Arbtrar Shape Gravt Center Issues Comparatve Analss Influence Of Graphc Condtons Error Analss. Introducton Polnomal ntegraton n three-dmenson felds s often nvolved n mechancal smulaton and engneerng applcaton such as the calculaton of gravt centers the stffness matr of Fnte Element Method (FEM) (Norre. and res 97; Zenkewc and Talor 989; Pepper and Henrch 99; Sarac et al ; Zhang et al ;) the nerta matr of Dscontnuous Deformaton Analss (DDA) and Numercal Manfold Method (NMM) (Sh ; Chen ; Kourepns ; Su ; Kourepns ). Because the real ntegraton feld s often comple we usuall dvde the comple polhedron to man regular elements and calculate ntegratons of these elements usng numercal ntegraton methods for eample Gaussan ntegraton method(feng et al ). Therefore t s worth to stud an eas method whch can calculate polnomal ntegraton n comple Internatonal Journal of Advancements n Computng Technolog(IJACT) olume4numberjune do:.456/jact.vol4.ssue.9 46

2 doman. In order to solve the problem Sh (997) proposed the formulae of smple ntegraton whch can solve above queston easl. However the formulae are ver comple. For ths reason we need to compose an actual algorthm to make the applcaton can be used wdel. In ths paper we present the algorthm of smple ntegraton n three-dmenson wrtten b C++ language and the procedure of ntegraton n polhedron of arbtrar shape. Then we anale the accurac and adaptve characterstcs of the algorthm through comparng the dfference between theoretcal and calculated results. Meanwhle the nfluence of graphc condtons on the results of smple ntegraton are dscussed n dfferent edge length rato( -5-5 ). Lastl the precson of smple ntegraton s dscussed whch the ntegral term s hgh-order polnomal.. The formulae and algorthm of smple ntegraton n three dmenson Eq. () s the formula of smple ntegraton n three dmenson whch s presented b Sh (997). ) ( ) ( ) ( Z Y X n m A l J n m l m n l J ddd n l m n m l n m l () Where J s Jacoban determnant and J n whch are coordnates of the No. verte of three-dmensonal smple and w w w w w. Turnng to consderaton of the comple Eq. () some researchers of NMM select appromate ntegraton methods such as Hammer ntegraton usng b Jang() and Luo(5) Gauss ntegraton usng b Zheng(4). In order to solve the problem of smple ntegraton we present an algorthm n Append A whch wrtten b C++ language. To use ths algorthm the functon of Factoral and Determnant of order 4 should be added n Append A. In order to solve ntegraton ssues n arbtrar shape we should understand the detal of calculaton flow whch s descrbed wth the conve polhedron n Fg.. The ntegraton flow ncludes several steps. ertees of each pl ane are organed n counterclockwse arrangement accordng to ther own outward normal vector such as P 5 P 4 P P P P P 6 P 5 P 5 P 6 P 4 P 4 P 6 P P P 6 P and P P 6 P n Fg.. We select P n arbtrar mode. If we select P at the orgn of coordnate the calculaton of J n Eq. () wll be ver smple. Smple ntegraton are operated n the unt of the plane n polhedron based on the algorthm of Append A. In the plane P 5 P 4 P P P for eample the smple ntegraton elements are P P P P P P P P 4 and P P P 4 P 5. The ntegraton procedure of other planes s as same as the step 4 and the result s of ever smple ntegraton are added to get fnal result. The detal algorthm wrtten b pseudo-code s presented n Append B. On the other hand f we want to calculate the ntegraton of hollow we can organe vertees of each plane of hollow n clockwse arrangement accordng to the outward normal vector of ths plane and the detal step s as same as descrpton above. The Algorthm of Smple Integraton n Three-Dmenson and Its Characterstc Analss WU Yan-qang CHEN Guang-q JIANG Za-sen ZHANG Long LIU Xao-a ZHAO Jng 47

3 The Algorthm of Smple Integraton n Three-Dmenson and Its Characterstc Analss WU Yan-qang CHEN Guang-q JIANG Za-sen ZHANG Long LIU Xao-a ZHAO Jng Fgure. Eample of smple ntegraton of conve polhedron. Precson analss of smple ntegraton n regular space wth hollow Based on the algorthm n Append A and B we can calculate the ntegraton of arbtrar polhedron. In order to anale the accurac of ntegraton results we wll compare the dfference between theoretcal and calculated results frst n whch parameters nclude volume and gravt centers. Eq. () s the formulas of above parameters descrbed b smple ntegraton functon. X Y Z ddd calculate_d_ntegraton(polhedron) ddd ddd calculate_d_ntegraton(polhedron) ddd calculate_d_ntegraton(polhedron ) ddd ddd calculate_d_ntegraton(polhedron) calculate_d_ntegraton(polhedron) ddd ddd ddd calculate_d_ntegraton(polhedron) calculate_d_ntegraton(polhedron) ddd () Where s the volume of ntegraton space; X Y and Z are coordnates of gravt centers. Fgure s an eample of ntegraton n regular space wth hollow. In order to calculate the ntegraton results of Fgure we use the ntegraton flow of Fgure n outer cubods and organe the vertees of planes of nner octahedron n clockwse accordng to the outward normal vector. Then we calculate the ntegraton value of outer and nner polhedron and accumulate them to obtan the fnal result. In fgure the length wdth and heght of outer cubods are 5 and 5. The projecton of nner octahedron n XOZ plane s a regular heagon wth the edge= and the edge whch paralleled wth aes Y s. Based on above nformaton we can calculate theoretcal volume and gravt center of paddng space n fgure. The value are

4 The Algorthm of Smple Integraton n Three-Dmenson and Its Characterstc Analss WU Yan-qang CHEN Guang-q JIANG Za-sen ZHANG Long LIU Xao-a ZHAO Jng X 5. Y 5. and 5. n whch the meanngs of X Y and Z are as same as Eq.(). Fgure. Smple ntegraton n regular space wth hollow In order to compare the dfference between calculated and theoretcal result and anale the nfluence of graphc condtons Tab. presents results of paddng space when coordnate Z oom n or oom out n whch the shape s as same as Fg. and the meanngs of X Y and Z are same as Eq.() too. 49

5 The Algorthm of Smple Integraton n Three-Dmenson and Its Characterstc Analss WU Yan-qang CHEN Guang-q JIANG Za-sen ZHANG Long LIU Xao-a ZHAO Jng Table. Influence of graphc condtons on the results of smple ntegraton n three-dmenson Calculated Calculated X Calculated Y Calculated Z (Paddng space) (Paddng space) (Paddng space) (Paddng space) Z Z scale Z 4 Z Z Z Z Z Z Z 4 Z 5 Z The dfference between theoretcal and calculated results presented b the results n Table s too small to dstngush n 5 th sgnfcant dgt. Meanwle the results presented n Tab. show that the nfluence of graphc condtons s ver small. Because we onl oom n or oom out the coordnate Z the coordnates X and Y of gravt centers should be unchanged whle and the coordnates Z of gravt centers should change wth scale. If we pa attenton to 4 th sgnfcant dgt the volume s scale and the gravt centers are X 5. Y 5. and Z 5. scale n whch the relatve error s about -5. Furthermore the gravt centers ( X andy ) do not show sstematc varaton wth Z scale. As a result graphc condton has no sstematc effect on the results of smple ntegraton. 4. Adaptve characterstcs analss of smple ntegraton n rregular ntegral space wth hollow In order to anale adaptve characterstcs of smple ntegraton Fgure. presents a concave polhedron wth hollow. Because there are not theoretcal results whch can compare wth calculated results we use the above analss method lke Table to llustrate the adaptve characterstcs. Usng Eq. () we can calculate the volume and gravt centers of paddng space easl n whch 58. and the gravt centers are X Y.5 and Z (the meanngs of X Y and Z are as same as Eq.()). 5

6 The Algorthm of Smple Integraton n Three-Dmenson and Its Characterstc Analss WU Yan-qang CHEN Guang-q JIANG Za-sen ZHANG Long LIU Xao-a ZHAO Jng Fgure. Smple ntegraton n rregular ntegral space wth hollow Table presents results when coordnate Y s oomed n or oomed out n whch the shape s as same as Fgure.The results presented n Table show that ntegraton results n rregular ntegral space wth hollow have hgh precson lke that n regular ntegral space. Because we onl oom n or oom out the coordnatey the coordnates X and Z of gravt centers should be unchanged whle and the coordnates Y of gravt centers should change wth scale. If we pa attenton to the 4 th sgnfcant dgt the volume s 58. scale and the gravt centers are X Y.5 scale and Z n whch the relatve error s about -5. On the other hand the gravt centers ( X and Z ) show random varaton wthy scale. As a result the results of smple ntegraton have hgh precson and adaptve characterstcs even though the ntegral space s rregular. 5

7 The Algorthm of Smple Integraton n Three-Dmenson and Its Characterstc Analss WU Yan-qang CHEN Guang-q JIANG Za-sen ZHANG Long LIU Xao-a ZHAO Jng Table. Results of smple ntegraton n rregular ntegral space Y scale Calculated Calculated X Calculated Y Calculated Z Y Y Y Y Y Y Y Y Y Y Y Smple ntegraton of hgh order polnomal n rregular ntegral space wth hollow What we dscuss about smple ntegraton above s nvolved wth ntegraton of order and order polnomal. Then we wll dscuss the ntegral characterstcs of hgh order polnomal whch the ntegral space s as same as Fgure and the epresson of ntegraton s presented n Eq.(). 7 8 () ddd calculate_d_ntegraton(polhedron 78) In order to anale the accurac of smple ntegraton of hgh order polnomal we take the analss flow lke Tab. and the results are presented n Tab.. Because of we onl oom n or oom out the coordnatey the ntegraton result of Eq. () should be scale. The calculated results show the same characterstcs lke the above low order ntegraton and the ntegraton results have hgh accurac too n whch the relatve error s about -5. Table. Results of hgh order smple ntegraton n rregular ntegral space wth hollow Y scale Integraton result Y scale Integraton result Y e+45 Y e+5 Y e+46 Y e+5 Y e+47 Y e+5 Y e+48 4 Y e+5 Y e+49 5 Y e+54 5

8 The Algorthm of Smple Integraton n Three-Dmenson and Its Characterstc Analss WU Yan-qang CHEN Guang-q JIANG Za-sen ZHANG Long LIU Xao-a ZHAO Jng 6. Dscusson and Concluson In ths paper we use volume and gravt centers of polhedron as parameters to anale the accurac and adaptve characterstcs of smple ntegraton n three-dmenson whch the ntegraton s nvolved wth and hgh order polnomal. (a)usng the algorthm and flow descrbed n ths paper we can calculate polnomal ntegraton n polhedron wth arbtrar shape whle the also can be used n smulaton of DDA and NMM (Sh ; Chen ). (b)the algorthm of calculaton volume and gravt centers can be used n ver comple polhedron such as conve polhedron concave polhedron and rregular polhedron wth hollow. Furthermore the algorthm has hgh adaptve characterstcs and can reach hgh precson n whch the relatve error s about -5. (c) When analng to the nfluence of graphc condton the results of volume and gravt centers mantan stead and the dstrbuton of error s random whatever the scale ooms n or ooms out. (d) Results of hgh order polnomal show that the relatve error whch does not change wth the ntegraton order s about -5 and the nfluence of graphc condton s ver small. In concluson the result of smple ntegraton s ver stead and the dstrbuton of the relatve error s satsfed wth randomct. Consderng the characterstcs of the relatve errors we hold the opnon that the manl comes from calculaton error of computer. 7. Acknowledgments. Specal thanks are etended to Prof. Sh G. H. Ths work was fnancall supported b Natonal Natural Scence Foundaton of Chna (444) and JSPS research fee of Japan (base plate of (B) Chen G. Q). 8. Reference [] Chen Guangq Ohnsh Yuo and Ito Takahro Development of Hgh Order Manfold Method Internatonal Journal for Numercal Methods n Engneerngvol.4 ssue 4 pp [] Chen Guangq and Ohnsh Yuo Practcal Computng Formulas of Smple Integraton. ICADD- pp [] Darrell W. Pepper Juan C. Henrch The fnte element method: basc concepts and applcatons. Hemsphere Publshng Corporaton USA99. [4] Dmtros Kourepns Chrs Pearce and Nenad Bćanć. Hgher-order dscontnuous modelng of fracturng n concrete usng the numercal manfold method Internatonal Journal of Computatonal Methods vol.7 Issue () pp.8-6. [5] Douglas H. Norre Gerard De res The fnte element method: fundamentals and applcatons Academc Press USA97. [6] Feng Janhu Che Gangmng and Nu Yufeng. Prncple of data analss Bejng: Scence Publsher.(n Chnese) [7] Jang Dongru and Luo Shaomng -Dmensonal Numercal Manfold Method and a Defnte Algorthm for Its Integral Doman Journal of Shantou Unverst (Natural Scence) vol.7 ssue () pp.9-6. [8] Luo Shaomng Zhang Xangwe Theoretcal Stud of Three-Dmensonal Numercal Manfold Method Appled Mathematcs and Mechancs vol.6 ssue(9)pp.7-5. [9] Olgerd Cecl Zenkewc Robert Lee Talor The Fnte Element Method 4-th Edton. McGraw-Hll USA 989. [] Sh Genhua Goodman. Dscontnuous deformaton analss. In Proceedngs of the 5th U. S. Smposum on Rock Mechancs pp. 69~ [] Sh Genhua Dscontnuous deformaton analss A new numercal model for the statstcs and dnamcs of deformable block structures[j]. Engneerng Computatons vol.9 Issue ()pp.57~ [] Sh Genhua Numercal Manfold Method (NMM) and Dscontnuous Deformaton Analss(DDA) Pe Juemn Translaton Tsnghua Unverst Press P.R.C

9 The Algorthm of Smple Integraton n Three-Dmenson and Its Characterstc Analss WU Yan-qang CHEN Guang-q JIANG Za-sen ZHANG Long LIU Xao-a ZHAO Jng [] Sh Genhua Three dmensonal dscontnuous deformaton analss Rock Mechancs n the Natonal Interest Elsworth Tnucc and Heasle[C] pp.4-48 USA. [4] Su Hadong. Stud on numercal manfold method wth fed meshes Acta Mechanca Snca vol.4 Issue () pp [5] Yoo Ohnsh Tomofum Koama Kaua Yag Applcaton of Manfold Method (MM) to the stablt problems for cut slopes along the natonal roads Geomechancs and Geoengneerng vol.7 Issue pp.9-56 [6] Zhang Xaonng Yang Peng "Fnte Element Analss of Hgh Modulus Asphalt Pavement" IJACT: Internatonal Journal of Advancements n Computng Technolog vol. 4 No. pp [7] Zheng Rongmng and Zhang Yonghu Applcaton and Dscusson of Three Dmensonal Numercal Manfold Method Based on Heahedron Element Chnese Journal of Rock Mechancs and Engneerng vol. ssue () pp [8] aslja Sarac Goce Stefanov "Calculaton of Electromagnetc Felds n Electrcal Machnes usng Fnte Elements Method" IJEI: Internatonal Journal of Engneerng and Industres vol. No. pp.-9 54

10 The Algorthm of Smple Integraton n Three-Dmenson and Its Characterstc Analss WU Yan-qang CHEN Guang-q JIANG Za-sen ZHANG Long LIU Xao-a ZHAO Jng Append A: Algorthm of smple ntegraton n three-dmenson double smple_ntegraton_d(double pont_data[4][]nt lnt mnt n) l m n { // ths functon can be used to calculate ddd n three-dmenson smple //pont_data save the coordnates of vertes of three-dmensonal smple. double a[4]a[4]a[4]a4[4]; //defne varables and set them ntal values; assgn pont_data[][] to a[4]a[4]a[4]a4[4] double jacoban_value; nt ; double dddddvalue; a[]=.; a[]==pont_data[][]; a[]==pont_data[][];a[]==pont_data[][]; a[]=.; a[]==pont_data[][]; a[]==pont_data[][];a[]==pont_data[][]; a[]=.; a[]==pont_data[][]; a[]==pont_data[][];a[]==pont_data[][]; a4[]=.; a4[]==pont_data[][]; a4[]==pont_data[][];a4[]==pont_data[][]; jacoban_value=four_determnant(aaaa4); //calculate the value of J n Eq.() and set t to jacoban_value value=; d=jacoban_valuefactoral(l)factoral(m)factoral(n)/factoral(l+m+n+); //calculate the value of J A( l m n) n Eq.() for(=;<=l;++){ //calculate the result of three-dmensonal smple ntegraton for(=;<=l-;++){ for(=;<=l--;++){ //calucate the value of X n Eq.() and set t to d =l---; d=pow((double))pow((double))pow((double))pow((double))/(factoral() factoral()factoral()factoral()); for(=;<=m;++){ for(=;<=m-;++){ for(=;<=m--;++){ //calculate the value of Y ( ) n Eq.() and set to t d =m---; d=pow((double))pow((double))pow((double)) pow((double))/(factoral()factoral()factoral()factoral()); for(=;<=n;++){ for(=;<=n-;++){ for(=;<=n-- ;++){ //calculate the value of Z ( ) n Eq.() and set t to d =n---; d=pow((double))pow((double))pow((double))p ow( (double))/(factoral()factoral()factoral()factoral ()); //calculate the value of n Eq.() and set t to d d=(factoral(++)factoral(++)factoral(++ ) factoral(++)); value=value+ddddd; //calculate result of ths step and add t to value }}}}}}}}} return value; //return the fnal result } 55

11 The Algorthm of Smple Integraton n Three-Dmenson and Its Characterstc Analss WU Yan-qang CHEN Guang-q JIANG Za-sen ZHANG Long LIU Xao-a ZHAO Jng Append B: Pseudocode of smple ntegraton n three-dmenson polhedron wth arbtrar shape double calculate_d_ntegraton(pol polhedron nt l nt m nt n) { //polhedron save the verte sequence organed b plane l m n //ths functon can be used to calculate ddd n three-dmenson polhedron wth arbtrar shape double ntegraton_result=.; //nt the return value to //set the three-dmenson coordnate of pont P to pont_data[][] for(=;<plane_count;++) //plane_count s plane count of polhedron { //set the three-dmenson coordnate of frst pont n current plane to pont_data[][] k=; //mark the current pont number to k //set the pont count of current plane and set t to node_count for(j=;j<node_count;j++) { //set the three-dmenson coordnate of current pont to pont_data[k][] k++; f(k%4==) //f the condton=true one smple ntgraton operaton wll be performed { //call the functon of smple_ntegraton_d() n Append A and accumulate t to ntegraton_result ntegraton_result+=smple_ntegraton_d(pont_datalmn); //set the value of pont_data[][] to pont_data[][] and set the value of k as pont_data[][]=pont_data[][]; pont_data[][]=pont_data[][]; pont_data[][]=pont_data[][]; k=; } } } return ntegraton_result; } 56