Transient analysis of wave propagation in 3D soil by using the scaled boundary finite element method

Transcription

1 Southern Cross University 23r Australasian Conference on the Mechanics of Structures an Materials 214 Transient analysis of wave propagation in 3D soil by using the scale bounary finite element metho Xiaojun Chen University of New South Wales Carolin Birk University of New South Wales Chongmin Song University of New South Wales Publication etails Chen, X, Birk, C, Song, C 214, 'Transient analysis of wave propagation in 3D soil by using the scale bounary finite element metho', in ST Smith (e.), 23r Australasian Conference on the Mechanics of Structures an Materials (ACMSM23), vol. II, Byron Bay, NSW, 9-12 December, Southern Cross University, Lismore, NSW, pp ISBN: epublications@scu is an electronic repository aministere by Southern Cross University Library. Its goal is to capture an preserve the intellectual output of Southern Cross University authors an researchers, an to increase visibility an impact through open access to researchers aroun the worl. For further information please contact epubs@scu.eu.au.

2 23r Australasian Conference on the Mechanics of Structures an Materials (ACMSM23) Byron Bay, Australia, 9-12 December 214, S.T. Smith (E.) TRANSIENT ANALYSIS OF WAVE PROPAGATION IN 3D SOIL BY USING THE SCALED BOUNDARY FINITE ELEMENT METHOD Xiaojun Chen* School of Civil an Environmental Engineering, University of New South Wales NSW, 252, Australia. (Corresponing Author) Carolin Birk School of Civil an Environmental Engineering, University of New South Wales NSW, 252, Australia. Chongmin Song School of Civil an Environmental Engineering, University of New South Wales NSW, 252, Australia. ABSTRACT An efficient metho for moelling the propagation of elastic waves in unboune omains is evelope. It is applicable to 3D soil-structure interaction problems involving scalar an vector waves. The scale bounary finite element metho is employe to erive an equation for the isplacement unit-impulse response matrix on the near fiel / far fiel interface. The unboune omain is ivie into multiple subomains where the isplacement unit-impulse response matrices for iniviual subomains are calculate separately an in parallel, leaing to a reuction of computational effort. As the isplacement unit-impulse response matrices approach zero, the convolution integral representing the force-isplacement relationship can be truncate. After the truncation the computational effort only increases linearly with time. Thus, a consierable reuction of computational effort is also achieve by this temporal localization. In aition, a recursive algorithm for calculating the convolution integral base on a piece-wise linear variation of the isplacement unit-impulse response matrix is evelope, so that the time step size for transient analysis can be arbitrarily small without significantly affecting the total computational effort. Numerical examples emonstrate the accuracy an high efficiency of the propose metho. KEYWORDS 3D soil-structure interaction, scale bounary finite element metho, isplacement unit-impulse response matrix, convolution, truncation, multiple subomains, recursive algorithm. INTRODUCTION The numerical moelling of wave propagation in unboune omains is require in a number of engineering applications, such as soil-structure interaction analysis or am-reservoir interaction analysis. Here, the well-establishe finite element metho cannot be use straightforwarly, since outgoing waves are reflecte at the artificial bounaries of the finite element mesh, such that special measures have to be taken to prevent these reflections (Givoli, 1999; Tsynkov, 1998). A popular metho for the analysis of ynamic problems in unboune meia is the bounary element metho, because the funamental solution explicitely fulfils the raiation conition (Beskos, 1987, 1997). The iea of extening the finite element mesh towars infinity has riven the evelopment of infinite element techniques (Bettess, 1992). A recent technique, which is particularly suitable for moelling This work is license uner the Creative Commons Attribution 4. International License. To view a copy of this license, visit 855

3 time-epenent problems in infinite meia, is the scale bounary finite element metho (SBFEM) (Wolf an Song, 1997). This semi-analytical technique is base on a combination of a numerical solution in the circumferential irections with an analytical solution in the irection of wave propagation. Thus, raiation amping is moelle accurately. In a soil-structure interaction problem, the structure (near fiel) can be moelle using either the conventional finite element metho or the scale bounary finite element metho. The soil (far fiel) is moelle by the scale bounary finite element metho. As shown in Figure 1, the relationship between the near fiel an the far fiel, in the time omain can be expresse in terms of the isplacement vector * ( )+ an the interaction force vector * ( )+. The interaction force-isplacement relationship is given in (Wolf an Song, 1997) as, * ( )+, -* ( )+, -* ( )+, ( )-* ( )+ (1) where, - an, - are constant ashpot an spring matrices, respectively. The first two terms in Equation (1) represent the instantaneous isplacement response, whereas the convolution term correspons to the lingering response., ( )- is the co-calle isplacement unit-impulse response matrix. An efficient algorithm for calculating, ( )- by using SBFEM has been propose (Chen, et al., 214), in which a piece-wise linear approximation of, ( )- with an extrapolation parameter an a truncation time is introuce to solve wave propagation problems in 2D half-space. The iea of using a piece-wise linear approximation has first been introuce in a conference paper by Ramanovic an Katz (21). (a) (b) Generalize soil-structure interface *u(t)+ *R(t)+ *R(t)+ *u(t)+ Figure 1. Soil-structure system. (a) Structure (near fiel); (b) Soil (far fiel) In this paper, the propose metho (Chen, et al., 214) is further extene to solve soil-structure interaction problems in 3D half-space. Due to the large size of the computational omain in three imensions, localization techniques are introuce. They are presente in the following orer. In Section 2, a recursive formulation is introuce to represent the interaction force-isplacement relationship. The isplacement unit-impulse response matrix is ivie into a finite number of linear segments an truncate at a certain time, so that the interaction force at any time only epens on a finite number of the previous isplacements, achieving localization in time. In Section 3, the whole unboune omain is ivie into several subomains, so that the fully couple isplacement unitimpulse response matrix is ecouple, achieving localization in space. In Section 4, these localization methos are verifie by a numerical example. Conclusions are presente in Section 5. LOCALIZATION IN TIME In Chen, et al., (213), the isplacement unit-impulse response matrix, ( )- is calculate by assuming a piece-wise linear variation, as shown in Figure 2. After a certain time, referre to as the ACMSM

4 Figure 2. Piece-wise linear variation of, ( )- truncation time, the isplacement unit-impulse response is assume to be zero. By using the z- transform, the convolution term in Equation (1), can be written in a recursive form as, * ( )+, ( )-* ( )+ (2) * + * + * +, - * + (3) where * + is * ( )+ at iscrete time Chapter 5, Wolf, (1988).. The etaile erivation of Equation (3) can be foun in LOCALIZATION IN SPACE In the original metho using SBFEM to moel the unboune omain (Wolf an Song, 1997), the whole unboune omain was treate as one subomain. This results in a fully populate isplacement unit-impulse response matrix, whose size is equal to the total number of egrees of freeom on the interface. Alternatively, the unboune omain can be ivie into several subomains, an thus the interaction force-isplacement relationship (1) can be broken into several smaller matrixvector operations. Note that, iviing the unboune omain into multiple subomains is an approximation, i.e. spurious reflections may occur on the interface between unboune subomains. Also note that wave motion tens to become one-imensional with increasing istance from the source, as state in Chapter 3.9 in (Wolf, 1988). Thus, the spurious reflections on the interface can be reuce by placing the artificial bounary further away from the omain of interest. The omain within the artificial bounary is referre to as the near fiel, which is moelle by FEM or SBFEM an the omain outsie the artificial bounary is referre to as the far fiel, which is moelle by SBFEM. By oing so, the computational effort for calculating the interaction force-isplacement relationship is reuce. This is illustrate in the following parameter stuy. NUMERICAL EXAMPLE A cubic founation embee in homogeneous half-space is investigate, as shown in Figure 3. The founation has a size of an we let. The artificial bounary, in shape, encloses the founation. The surface centre of the near fiel coincies with the surface center of the founation, which is enote as O in Figure 3. The concrete founation has a Young's moulus of, Poisson's ratio of an mass ensity of. The ajacent soil is escribe by a Young's moulus of, Poisson's ratio of an mass ensity of. ACMSM

5 (a) (b) e cubic founation 2b cubic founation Artificial bounary Figure 3. Cubic founation embee in 3D homogeneous half space. (a) sie view; (b) top view The near fiel is moelle with 8-noe isoparametric block finite elements with uniform size of. Generally, for moelling ynamic problems using the finite element metho, at least ten noes per shear wave length are require. Therefore, the minimum shear wave length this mesh can moel is b, which is use as a characteristic parameter to measure the istance between the artificial bounary an actual founation-soil interface. On the near fiel / far fiel interface, 4-noe isoparametric surface scale bounary finite elements are use, which coincie with the FE mesh of the near fiel. The scaling centre is locate at point O. First the unboune omain is treate as one subomain. Then, each element is treate as one subomain, an a group of unit-impulse response matrices for every subomain is calculate, marke as '1 element' subgroup. Then, every an elements are groupe as one subomain, marke as '4 elements', an '9 elements' subgroups, respectively. These sizes of the subomain can also be interprete as 1%, 2% an 3% of the characteristic length b, which is the minimum wave length. A vertical uniformly istribute loa ( ) is applie on the surface of the founation, an its time history is shown in Figure 4. The vertical isplacement of the surface centre of the founation, point O, is evaluate. The isplacement unit-impulse response matrix is calculate with a time step size for five steps, resulting in a truncation time of. In the time omain analysis, the time step size is For verification, an extene FE mesh is use, where a omain of size is moelle using the finite element metho as escribe above. p(t) P P/ Figure 4. Force history of the uniformly istribute loa The results are presente in four groups, corresponing to ifferent locations of the artificial bounary, i.e. =b, =2b, =3b an =4b. In each group, the isplacement is calculate using one subomain, marke as couple, an then either of the three ifferent subomain ivision schemes explaine above. The results are shown in Figure 5. ACMSM

6 Vertical isplacement u y *G/P Vertical isplacement u y *G/P Vertical isplacement u y *G/P Vertical isplacement u y *G/P maximum relative error for 9 elements scheme: <5% -.8 (a) (b) (c) maximum relative error for 1 element scheme: <5% Figure 5. Vertical isplacement of point O for (a) =b, (b) =2b, (c) =3b, () =4b Couple 1 element 4 elements 9 elements Reference (extene mesh) As can be seen in Figure 5, for the case =b, i.e. when the artificial bounary coincies with the actual founation-soil interface, using multiple unboune subomains may cause large errors. The error is reuce by placing the artificial bounary further away from the founation. For the case =2b, the 9 elements scheme is alreay acceptable for engineering practice, whose relative error is less than 5%, an for the case =3b, all the three spatial ecoupling schemes lea to only marginal errors (for 1 element scheme, the relative error is alreay less than 5%). The above schemes, together with the 1 element scheme for =4b, provie acceptable accuracy. The CPU time for calculating the interaction force-isplacement relationship, Equation (1), using the ifferent schemes is recore an shown in Figure 6. It can be seen that the spatial ecoupling leas to a significant reuction of computational effort. However, as the artificial bounary is locate further away from the omain of interest, the size of the near fiel is increase. This will lea to aitional egrees of freeom in the near fiel an to an increase of the total computational size. In large scale engineering practice, the original near fiel may be very large an the increase of the aitional near fiel ue to further location of the artificial bounary can only be marginal. () ACMSM

7 CPU Time (in sec) Couple 9 elements, =2b 9 elements =3b 4 elements =3b 1 element =3b 1 element =4b Figure 6. CPU time require to evaluate the interaction force-isplacement relationship CONCLUSION In this paper, the numerical moelling of 3D ynamic soil-structure interaction problems using SBFEM is aresse. Due to the large geometric size in three imensions, two essential localization techniques are introuce: (1) Temporally, the isplacement unit-impulse response matrix for the unboune omain calculate from the scale bounary finite element metho is ivie into several linear segments. By using the z-transform, the time consuming convolution integral representing the force-isplacement relationship on the interface is transforme into a recursive formulation, where only a small number of previous steps are involve. Together with the introuction of the truncation time, the total number of the operations regaring the convolution integral is reuce significantly; (2) Spatially, since the unboune omain is ivie into a number of inepenent subomains, the fully couple isplacement unit-impulse response matrix for the unboune omain can be broken into several small matrices, so that the computational size in space is reuce. As the interface between two subomains will cause iscontinuity, the artificial bounary nees to be place a certain istance away from the omain of interest. REFERENCES Beskos, D. (1987). Bounary element methos in ynamic analysis. Applie Mechanics Reviews, 4, Beskos, D. (1997). Bounary elements methos in ynamic analysis: Part II. Applie Mechanics Reviews, 5, Bettess, P. (1992). Infinite elements. Sunerlan: Penshaw Press. Chen, X., Birk, C., & Song, C. (214). Numerical moelling of wave propagation in anisotropic soil using a isplacement unit-impulse-response-base formulation of the scale bounary finite element metho. Soil Dynamics an Earthquake Engineering, 65, Givoli, D. (1999). Exact representations on artificial interfaces an applications in mechanics. Applie Mechanics Reviews, 52(22), Ramanovic, B., & Katz, K. (21). A high performance scale bounary finite element metho. IOP Conf. Series: Materials Science an Engineering Tsynkov, S. (1998). Numerical solution of problems on unboune omains: a review. Applie Numerical Mathematics, 27, Wolf, J. P. (1988). Soil-structure-interaction analysis in time omain. New Jersey: Prentice-Hall, Inc. Wolf, J. & Song, C. (1997). Finite element moelling of unboune meia. John Wiley & Sons. ACMSM