COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 2-23, 26, Sanya, Hanan, Chna 26 Tsnghua Unversty Press & Sprnger A Frequency-Doman Approach for Transent Dynamc Analyss usng Scaled Boundary Fnte Element Method (I): Approach and Valdaton Z. J. Yang, 2 *, A. J. Deeks 2, H. Hao 2 College of Cvl Engneerng and Archtecture, Zhejang Unversty, Hangzhou, 327 Chna 2 School of Cvl and Resource Engneerng, the Unversty of Western Australa, WA 69 Australa Emal: yang@cvl.uwa.edu.au Abstract The sem-analytcal scaled boundary fnte element method (SBFEM) has been successfully appled n elastostatcs. Its applcaton to elastodynamc problems, however, has lagged behnd, prmarly due to the lack of an effectve soluton procedure to the governng equlbrum equaton system. The authors recently developed an easy-to-follow Frobenus soluton procedure to ths equaton system []. Ths study further develops a frequency-doman approach for the general transent dynamc analyss, through combnng the Frobenus soluton procedure and the fast Fourer transform (FFT) technque. The complex frequency-response functons (CFRFs) are frst computed usng the Frobenus soluton procedure. Ths s followed by a FFT of the transent load and an nverse FFT of the CFRFs to obtan the tme hstory of responses. Two transent dynamc problems, subjected to Heavsde step load and trangular blast load, are modelled wth a small number of degrees of freedom usng the new approach. The numercal results agree very well wth the analytcal solutons and those from the FEM. Further applcatons of ths approach to transent dynamc fracture problems are presented n the accompanyng paper [2]. Keywords: transent dynamc analyss, scaled boundary fnte element method, Frobenus soluton procedure, frequency doman, fast Fourer transform INTRODUCTION Transent dynamc analyses are very mportant for understandng responses and falure mechansms of engneerng structures subjected to dynamc loads such as mechancal vbraton, earthquake, mpact, wnd and blast. The fnte element method (FEM) s undoubtedly the domnant method for modellng transent dynamc problems at present, because of ts powerful capablty of smulatng a large varety of problems wth complex structural geometres, complcated materal propertes, and varous loadng and boundary condtons [3]. Although ts success n transent dynamc analyses has been frmly establshed, modellng large-scale lnear and nonlnear structural dynamc systems usng the FEM stll poses great dffcultes for structural analysts [4]. In spte of great efforts, the computatonal cost for the FEM may be stll very hgh for some mportant problems, such as the structural-sol dynamc nteracton nvolvng nfnte domans, solds wth stress sngularty and concentraton (e.g., cracks and corners), and the dynamc problems wth responses domnated by ntermedate and hgh frequency modes, such as those subjected to mpact or blast loads [5]. One alternatve to the FEM s the boundary element method (BEM). The BEM dscretses the boundares only and thus reduces the modelled dmensons by one. The applcaton of the BEM n the transent dynamc analyss s currently very actve, wth many numercal procedures reported [e.g., 6]. BEM/FEM couplng methods have also been developed for lnear elastodynamcs [7]. However, the need of fundamental solutons lmts the applcablty of the BEM consderably. The scaled boundary fnte-element method (SBFEM), developed recently by Wolf and Song [9, ], s a sem-analytcal method combnng the advantages of the FEM and the BEM, and possessng ts own attractons at the same tme. Specfcally, t dscretses the boundares only and thus reduces the modelled spatal dmensons by one as the BEM, but does not need fundamental solutons. The method thus has much wder applcablty than the BEM. In addton, the dsplacement and stress solutons of the SBFEM are approxmate n the crcumferental drecton n the FEM sense, but analytcal n the radal drecton. Ths sem-analytcal nature makes the method an excellent tool for modellng unbounded domans because the radaton condton at nfnty s automatcally satsfed. Ths has been 756
demonstrated by a few recent studes of dynamc sol-structure nteracton [, 2]. What s more, the analytcal stress feld explctly represents stress sngulartes at crack tps, whch allows accurate stress ntensty factors to be computed drectly from the defnton. Ths advantage has been used recently by Song to calculate transent dynamc stress ntensty factors [3]. The SBFEM has also a few dsadvantages. For example, ts current formulaton consders only lnear elastc materal behavour. For complex domans that are dvded nto a few subdomans, an egenproblem wth 2n degrees of freedom (DOFs) must be solved for each subdoman of n DOFs, whch may result n hgh computatonal cost. To date, all of these lmted transent dynamc analyses [-3] based on the SBFEM used drect tme-ntegraton schemes, and hence a tme-doman approach. The tme-doman approach requres a mass matrx n the equatons of moton. The mass matrx derved by Song [3] corresponds to the low-frequency expanson of the dynamc stffness matrx [9, ] and thus can only accurately represent the nertal effects at low frequences. Ths requres that the sze of the subdomans or super-elements be small enough to account for the hghest frequency component of nterest [3]. Ths may lead to consderable computatonal cost n solvng a sgnfcant number of egenproblems n each tme step. A frequency-doman approach has certan advantages over the tme-doman approach when modellng elastodynamcs, such as no need for a mass matrx, so that fewer subdomans or coarser meshes may be used, and once a complex frequency-response functon (CFRF) s obtaned, t can be used by the fast Fourer transform (FFT) and the nverse FFT (IFFT) to calculate transent responses for varous forms of dynamc loads. The authors recently developed an easy-to-follow analytcal soluton n the frequency doman usng the Frobenus procedure to the governng equlbrum equatons of the SBFEM[]. Ths study further devses a new frequency-doman approach for general transent dynamc analyss usng the SBFEM. The approach nvolves two steps. The frst step s to employ the Frobenus soluton procedure n the frequency doman to compute the CFRFs. The second step s to use FFT and IFFT to calculate the tme hstores from the CFRFs. The followng sectons descrbe the basc concept of SBFEM and the Frobenus soluton, followed by modellng of a wave propagaton problem and a forced vbraton problem usng the new approach. FUNDAMENTALS OF THE SBFEM A doman of an arbtrary shape s llustrated n Fg. (a). The doman s dvded nto three subdomans. Any scheme of subdvson, wth varous numbers, shapes and szes of subdomans, can be used, as long as a scalng centre for each subdoman can be found from whch the subdoman boundary s fully vsble. Fg. (b) shows the detals of Subdoman. The subdoman s represented by scalng a defnng curve S relatve to a scalng centre. The defnng curve s usually taken to be the doman boundary, or part of the boundary. A normalsed radal coordnate ξ s defned, varyng from zero at the scalng centre and unt value on S. A crcumferental coordnate s s defned around the defnng curve S. A curve smlar to S defned by ξ=.5 s shown n Fg. (b). The coordnates ξ and s form the local coordnate system, whch s used n all the subdomans. (a) Subdomanng of a doman (b) Subdoman Fgure : The concept of the scaled boundary fnte-element method The governng equlbrum equatons of the SBFEM have been derved wthn the context of vrtual work prncple for elastostatcs by Deeks and Wolf [4]. An extenson to elastodynamcs has also been made by Yang et al []. They are not repeated here. In a frequency-doman analyss, the frst step s to compute the CFRFs. The governng equlbrum equatons of the SBFEM under a harmonc exctaton wth constant frequency f (Hz), e.g. p=p e 2πft are [] T p = E u( ξ), ξ + E u ( ξ ) ξ= () 757
E ξ u( ξ), + ( E + E E ) ξu( ξ), E u( ξ) + (2 π f ) ξ M u ( ξ) = (2) 2 T 2 2 2 ξξ ξ where p are the magntudes of the equvalent nodal loads. E, E, E 2 and M are coeffcent matrces that are dependent on the geometry of the subdoman boundares and the materal propertes. u(ξ) represent the magntudes of the nodal dsplacements and are analytcal functons of the radal coordnate ξ. Eq. () and Eq. (2) are frst appled to the subdomans. The contrbutons of subdomans to the coeffcent matrces and equvalent nodal loads are then assembled n the same way as the contrbutons of fnte elements n FEM. THE FROBENIUS SOLUTION PROCEDURE Eq. (2) s a second-order nonhomogeneous Euler-Cauchy dfferental equaton system wth respect to the radal coordnate ξ. The non-homogenety complcates ts soluton consderably. An easy-to-follow analytcal soluton to Eq. (2) was recently developed by Yang et al [] usng the Frobenus procedure. The soluton wth (k+) seres s n n n 2 k k + ( λ) ( λ) 2 ( λ) k u = cξ φ + ξ c g + ξ + c + g = = = where k+ k+ (3) g = G φ (4a) G = B B B (4b) k+ k+ k 2 k + 2 2 2 (2 ) ( k + ) ( k + )( T = π f λ + λ ) B E E E E M (4c) d = B ( d ) (4d) k+ k+ k d = cφ (4e) k λ = λ + 2 (4f) k+ where the subscrpt ranges from to n for all the varables and n s the DOFs of the problem. Eq. (3) shows that the soluton to elastodynamcs conssts of (k+) summatons of seres. The frst summaton s the soluton to the correspondng homogeneous equatons of Eq. (2),.e., the governng equatons for elastostatcs. The added k summatons account for dynamc effects. More seres terms are added, more accurately the soluton represents the dynamc effects. λ = λ and φ are the postve egenvalues and correspondng egenvectors of the standard lnear egenproblem [4] formed from the elastostatc governng equatons. c are constants determned by boundary condtons. Eqs. (4a- 4f) descrbe an teratve process n whch the number of added summaton terms k n Eq. (3) s determned by a convergence crteron []: max ( R )< α (5a) 2 n 2 ( k λ ) k (2 π f ) ξ + + R = M ( d ) (5b) = where R can be regarded as the resdual vector and α s the convergence tolerance. Settng ξ= n Eq. (3) leads to k + u = u = Ψ c or = b c= Ψ ub (6) ξ where u b s the nodal dsplacements on the boundary, c={c c 2 c n } T and the matrx k+ k+ k+ j j j Ψ = ϕ+ g ϕ2 + g2 ϕ n + gn (7) j= 2 j= 2 j= 2 Substtutng Eq. (3) nto Eq. () leads to p = E Ψ c+ E Ψ c (8) T 2 where the matrx 758
k+ k+ k+ j j j j j j Ψ2 = λϕ+ g λ λ2ϕ2 + g2 λ2 λnϕ n + gn λn (9) j= 2 j= 2 j= 2 Substtutng Eq. (6) nto Eq. (8) gves T p = E ΨΨ + E u = K u () 2 b d b Therefore, the dynamc stffness matrx of the doman wth respect to DOFs on the boundary s K = E ΨΨ + E () T d 2 The nodal dsplacement vector u b can be calculated by Eq. () by applyng boundary condtons on u b and loadng condtons on p. The ntegraton constants c are then obtaned usng Eq. (6). Assumng the (k+)th soluton meets the crteron Eq. (5a), the dsplacement feld s recovered as n n = + + n 2 k + ( λ ) ( λ ) 2 ( λ ) k+ u( ξ, s) N( s) cξ φ ξ c ( g ) L ξ c ( g ) (2) = = = where N(s) s the matrx of shape functons at the crcumferental drecton, whch are constructed as n FEM, typcally usng polynomal functons. The stress feld s then obtaned as n n ( ) n k σ ξ, s = DB s c λ ξ ϕ + c λ ξ ( g ) + + c ( λ ) ξ g k k k k 2 + λ 2 λ 2 + λ + = = = n n ( ) n + DB ϕ + ( g ) + + g 2 + 2 λ λ 2 λ + s cξ cξ cξ = = = where D s the elastcty matrx and B (s) and B 2 (s) are relevant matrces []. One can see that the dynamc dsplacement and stress felds from the SBFEM (Eq. (2) and Eq. (3)) are analytcal wth respect to the radal coordnate ξ. Therefore, these state varables at any poston of the doman can be drectly calculated once the nodal dsplacements on the boundary are obtaned by solvng Eq. (). (3) NUMERICAL EXAMPLES, RESULTS AND DISCUSSION Two transent dynamc examples, both subjected to the Heavsde-step load and the trangular blast load, are modelled. For each example, the CFRFs of dsplacements or stresses at desred postons n the doman are frst computed for a wde range of samplng frequences usng the Frobenus soluton procedure presented n the prevous secton. A FFT of the transent load s then carred out to obtan the complex dscrete Fourer coeffcents, followed by an IFFT of the CFRFs to obtan tme hstores of dsplacement or stress at desred postons. The readers are referred to [5] for detals of the dscrete Fourer transform and the fast Fourer transform technques. The functons fft and fft n MATLAB [6] are used to conduct FFT and IFFT respectvely. For each example, the same CFRFs are used to calculate tme hstores for all the forms of transent loadngs that ths example s subjected to. In the frequency-doman SBFEM, the dampng effect s taken nto account by modfyng the elastc modul to ncorporate an nternal dampng coeffcent β, formng the complex Young s modulus E c =E(+2β) and the complex shear modulus G c =G(+2β), where E and G are the elastc Young s modulus and the elastc shear modulus respectvely. A transent dynamc fnte element analyss usng ABAQUS s also carred out for both examples for comparson usng the mplct tme-ntegraton scheme [7]. No physcal dampng (.e., pure elastc materal behavour) s consdered n all the fnte element analyses. The nternatonal standard unt system, namely mass n klograms, length n metres and tme n seconds, s used for both examples and thus no unts are ndcated wth all the physcal enttes. Two-node lnear lne elements and α =. n Eq. (5a) are used for all the analyses usng the SBFEM. A plane stress condton s assumed for both examples.. Example : D rectangular rod The frst example modelled here s the classcal wave propagaton problem wth analytcal solutons, whch was modelled by many, e.g., [6] and [7], to valdate numercal methods. A rectangular rod s subjected to a unform tracton on the rght end wth the left end fxed. The geometry, boundary and loadng condtons are shown n Fg. 2(a). The tme responses at the ponts A, B and C n Fg. 2(a) are examned. Only one subdoman s used to model ths example. Fg. 2(b) shows a typcal mesh wth only 5 nodes. The scalng centre s 759
placed at the left-bottom corner so that the two edges connected to t are not dscretsed. β= 6 s used for ths example. (a) Dmensons, materal propertes and loadng condtons Fgure 2: Example, an D rectangular rod (b) A mesh wth 5 nodes Fgure 3: Horzontal dsplacement at the pont A under the Heavsde step load for the st example Fgure 4: Horzontal dsplacement at the pont B under the Heavsde step load for the st example Fg. 3 and Fg. 4 show the tme hstores of the horzontal (axal) dsplacements at the free end of the rod (the pont A) and the mddle of the rod (the pont B) respectvely under the Heavsde step load. The results from both the frequency-doman SBFEM and the tme-doman FEM are n very good agreement wth the analytcal solutons [8]. The axal stress hstores at the mddle and the fxed end of the rod are shown n Fg. 5 and Fg. 6 respectvely. Good agreement between the numercal results and the analytcal soluton can also be observed, except that the FEM results oscllate fercely around the analytcal solutons at moments when the stress jumps. Ths s caused by the sudden applcaton of the step load and cannot be completely corrected by numercal measures. It also happens n other studes [6, 7]. The oscllatons also appear n the results from the new frequency-doman approach, but wth lower magntudes and much less degree. Fg. 7 compares the horzontal dsplacement hstores at the ponts A and B under the trangular blast load. The results from the two numercal methods are vrtually dentcal. Excellent agreement can also be seen n the axal stress hstores at the mddle and the fxed end of the rod, whch are presented n Fg. 8. No oscllaton happens because the trangular blast load functon s contnuous. 76
Fgure 5: Horzontal stress at the pont B under the Heavsde step load for the st example Fgure 6: Horzontal stress at the pont C under the Heavsde step load for the st example.75 Horzontal dsplacements.5.25 -.25 -.5 -.75 - FEM (A) Present (A) FEM (B) Present (B) 2 4 6 8 2 Fgure 7: Horzontal dsplacements at the pont A and B under the blast load for the st example 2.5 Horzontal stress.5 -.5 - -.5 FEM (B) Present (B) FEM (C) Present (C) -2 2 4 6 8 2 Fgure 8: Horzontal stress at the pont B and C under the blast load for the st example 76
2. Example 2: 2D smply-supported beam subjected to unform bendng force The forced vbraton of a 2D smply-supported beam s modelled as the 2 nd example. The dmensons, boundary and loadng condtons, and materal propertes are llustrated n Fg. 9(a). The beam s subjected to a unformly-dstrbuted bendng tracton at ts upper face. The tme responses at the ponts A and B n Fg. 9(a) are nvestgated. Both the Heavsde step load and the blast load (only the latter s shown n Fg. 9(a)) are modelled. The beam s modelled wth two subdomans consstng of totally 33 nodes usng the SBFEM as shown n Fg. 9(b). The two scalng centres are postoned at the beam lower face so that ths whole face s not dscretsed. A materal dampng coeffcent β= 5 s used. (a) Dmensons, materal propertes and loadng condtons (b) A mesh wth 33 nodes Fg. 9 Example 2: a smply-supported beam subjected to unformly dstrbuted bendng force Fg. and Fg. show the vertcal dsplacement hstores at the pont B and the horzontal stress hstores at the pont A respectvely under the Heavsde step load. Also shown n these fgures are the analytcal solutons [8] takng nto account the effects of rotary nerta and shear deformaton. It s evdent that the results from the tme-doman FEM and the frequency-doman SBFEM almost concde wth each other, and the results from both methods agree very well wth the analytcal solutons, especally for the dsplacement hstores. Because the stresses are dervatves of dsplacements and thus less accurate, the dscrepances between the analytcal stress soluton and the numercal results are understandable. Comparng the results here wth those reported n [9] usng a tme-doman FEM also confrms the effectveness and accuracy of the developed frequency-doman SBFEM. The same concluson can also be drawn from the results under the blast load, whch are shown n Fg. 2 and Fg. 3. Vertcal dsplacement at pont B.5 -.5 -. -.5 -.2 -.25 -.3 -.35 -.4 -.45 -.5 Analytcal Present FEM 2 3 4 5 Fgure : Vertcal dsplacement at the pont B under the Heavsde step load for the 2 nd example Horzontal stress at pont A 5 45 Analytcal Present FEM 4 35 3 25 2 5 5-5 - 2 3 4 5 Fgure : Horzontal stress at the pont A under the Heavsde step load for the 2 nd example 762
Vertcal dsplacement at pont B.5 Present FEM.2.9.6.3 -.3 -.6 -.9 -.2 -.5 2 3 4 5 Fgure 2: Vertcal dsplacement at the pont B under the blast load for the 2nd example 4.5 Present FEM Horzontal stress at pont A 7 3.5-3.5-7 -.5-4 2 3 4 5 Fgure 3: Horzontal stress at the pont A under the blast load for the 2nd example CONCLUSIONS Ths study develops a frequency-doman approach usng the SBFEM for modellng general transent dynamc problems. The CFRFs are frst computed by the newly-developed analytcal Frobenus soluton to the governng equatons of SBFEM. The FFT and the IFFT are then carred out to calculate tme hstores of structural responses. Ths new approach does not need a mass matrx as needed by the tme-doman SBFEM, allowng domans to be modelled by fewer subdomans effcently. The new approach has been used to successfully model transent dynamc behavour of two examples wth very small number of DOFs due to the analytcal nature of the Frobenus soluton. ACKNOWLEDGEMENT Ths research s supported by the Australan Research Councl (Dscovery Project No. DP45268) and the Natonal Natural Scence Foundaton of Chna (No. 55798). REFERENCES. Yang ZJ, Deeks AJ, Hao H. A Frobenus soluton to the scaled boundary fnte element equatons n frequency doman. Int. J. Num. Meth. Eng., (under revew). 2. Yang ZJ, Deeks AJ, Hao H. A frequency-doman approach for transent dynamc analyss usng scaled boundary fnte element method (II): applcaton to fracture problems. The th Int. Conf. Comp. Meth. n Engng.&Sc (EPMESC X). Aug. 2-23, 26, Sanya, Hanan, Chna. 3. Zenkewcz OC, Taylor RL. The fnte element method. 5th edton, Butterworth- Henemann, 2. 4. Dokansh MA, Subbaraj K. A survey of drect tme-ntegraton methods n computatonal structural dynamcs I. explct methods. Computers & Structures, 989; 32(6): 37-386. 5. Zeng LF, Wberg NE. Spatal mesh adaptaton n semdscrete fnte element analyss of lnear elastodynamcs problems. Computatonal Mechancs, 992; 9: 35-332. 763
6. Carrer JAM, Mansur WJ. Alternatve tme-marchng schemes for elastodynamc analyss wth the doman boundary element method formulaton. Computatonal Mechancs, 24; 34(5): 387-399. 7. Yu GY, Mansur WJ, Carrer JAM. A more stable scheme for BEM/FEM couplng appled to two-dmensonal elastodynamcs. Computers & Structures, 2; 79(8): 8-823. 8. Gu YT, Lu GR. A meshless local Petrov Galerkn (MLPG) method for free and forced vbraton analyses for solds. Computatonal Mechancs, 2; 2788-98. 9. Wolf JP, Song CM. Fnte-element modellng of unbounded meda. John Wley and Sons, Chchester, 996.. Wolf JP. The scaled boundary fnte element method. Chchester: John Wley and Sons, 23.. Zhang X, Wegner JL, Haddow JB. Three-dmensonal dynamc sol-structure nteracton analyss n the tme doman. Earthquake Engneerng & Structural Dynamcs, 999; 28(2): 5-524. 2. Yan JY, Zhang CH, Jn F. A couplng procedure of FE and SBFE for sol-structure nteracton n the tme doman. Internatonal Journal for Numercal Methods n Engneerng, 24; 59(): 453-47. 3. Song CM. A super-element for crack analyss n the tme doman. Internatonal Journal of Numercal Methods n Engneerng, 24; 6(8): 332-357. 4. Deeks AJ, Wolf JP. A vrtual work dervaton of the scaled boundary fnte-element method for elastostatcs. Computatonal Mechancs, 22; 28(6): 489-54. 5. Brgham EO. The fast Fourer transform and ts applcatons. Prentce-Hall, 988. 6. MATLAB V7.. The MathWorks, Inc., 25. 7. Hbbtt, Karlsson and Sorensen Inc. ABAQUS/Standard User Manual V6.5, 24. 8. Tmoshenko S, Young DH, Weaver W. Vbraton Problems n Engneerng. 4th edton, John Wley & Sons, 974. 9. Zenkewcz OC, L XK, Nakazawa S. Dynamc transent analyss by a mxed, teratve method. Internatonal Journal for Numercal Methods n Engneerng, 986; 23(7): 343-353. 764