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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

A Frequency-Domain Approach for Transient Dynamic Analysis using Scaled Boundary Finite Element Method (I): Approach and Validation Z. J. Yang 1, 2*, A. J. Deeks 2, H. Hao 2 1 College of Civil Engineering and Architecture, Zhejiang University, Hangzhou, 310027 China 2 School of Civil and Resource Engineering, the University of Western Australia, WA 6009 Australia Email: yang@civil.uwa.edu.au Abstract The semi-analytical scaled boundary finite element method (SBFEM) has been successfully applied in elastostatics. Its application to elastodynamic problems, however, has lagged behind, primarily due to the lack of an effective solution procedure to the governing equilibrium equation system. The authors recently developed an easy-to-follow Frobenius solution procedure to this equation system [1]. This study further develops a frequency-domain approach for the general transient dynamic analysis, through combining the Frobenius solution procedure and the fast Fourier transform (FFT) technique. The complex frequency-response functions (CFRFs) are first computed using the Frobenius solution procedure. This is followed by a FFT of the transient load and an inverse FFT of the CFRFs to obtain the time history of responses. Two transient dynamic problems, subjected to Heaviside step load and triangular blast load, are modelled with a small number of degrees of freedom using the new approach. The numerical results agree very well with the analytical solutions and those from the FEM. Further applications of this approach to transient dynamic fracture problems are presented in the accompanying paper [2]. Keywords: transient dynamic analysis, scaled boundary finite element method, Frobenius solution procedure, frequency domain, fast Fourier transform INTRODUCTION Transient dynamic analyses are very important for understanding responses and failure mechanisms of engineering structures subjected to dynamic loads such as mechanical vibration, earthquake, impact, wind and blast. The finite element method (FEM) is undoubtedly the dominant method for modelling transient dynamic problems at present, because of its powerful capability of simulating a large variety of problems with complex structural geometries, complicated material properties, and various loading and boundary conditions [3]. Although its success in transient dynamic analyses has been firmly established, modelling large-scale linear and nonlinear structural dynamic systems using the FEM still poses great difficulties for structural analysts [4]. In spite of great efforts, the computational cost for the FEM may be still very high for some important problems, such as the structural-soil dynamic interaction involving infinite domains, solids with stress singularity and concentration (e.g., cracks and corners), and the dynamic problems with responses dominated by intermediate and high frequency modes, such as those subjected to impact or blast loads [5]. One alternative to the FEM is the boundary element method (BEM). The BEM discretises the boundaries only and thus reduces the modelled dimensions by one. The application of the BEM in the transient dynamic analysis is currently very active, with many numerical procedures reported [e.g., 6]. BEM/FEM coupling methods have also been developed for linear elastodynamics [7]. However, the need of fundamental solutions limits the applicability of the BEM considerably. The scaled boundary finite-element method (SBFEM), developed recently by Wolf and Song [9, 10], is a semi-analytical method combining the advantages of the FEM and the BEM, and possessing its own attractions at the same time. Specifically, it discretises the boundaries only and thus reduces the modelled spatial dimensions by one as the BEM, but does not need fundamental solutions. The method thus has much wider applicability than the BEM. In addition, the displacement and stress solutions of the SBFEM are approximate in the circumferential direction in the FEM sense, but analytical in the radial direction. This semi-analytical nature makes the method an excellent tool for modelling unbounded domains because the radiation condition at infinity is automatically satisfied. This has been

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demonstrated by a few recent studies of dynamic soil-structure interaction [11, 12]. What’s more, the analytical stress field explicitly represents stress singularities at crack tips, which allows accurate stress intensity factors to be computed directly from the definition. This advantage has been used recently by Song to calculate transient dynamic stress intensity factors [13]. The SBFEM has also a few disadvantages. For example, its current formulation considers only linear elastic material behaviour. For complex domains that are divided into a few subdomains, an eigenproblem with 2n degrees of freedom (DOFs) must be solved for each subdomain of n DOFs, which may result in high computational cost. To date, all of these limited transient dynamic analyses [11-13] based on the SBFEM used direct time-integration schemes, and hence a time-domain approach. The time-domain approach requires a mass matrix in the equations of motion. The mass matrix derived by Song [13] corresponds to the low-frequency expansion of the dynamic stiffness matrix [9, 10] and thus can only accurately represent the inertial effects at low frequencies. This requires that the size of the subdomains or super-elements be small enough to account for the highest frequency component of interest [13]. This may lead to considerable computational cost in solving a significant number of eigenproblems in each time step. A frequency-domain approach has certain advantages over the time-domain approach when modelling elastodynamics, such as no need for a mass matrix, so that fewer subdomains or coarser meshes may be used, and once a complex frequency-response function (CFRF) is obtained, it can be used by the fast Fourier transform (FFT) and the inverse FFT (IFFT) to calculate transient responses for various forms of dynamic loads. The authors recently developed an easy-to-follow analytical solution in the frequency domain using the Frobenius procedure to the governing equilibrium equations of the SBFEM[1]. This study further devises a new frequency-domain approach for general transient dynamic analysis using the SBFEM. The approach involves two steps. The first step is to employ the Frobenius solution procedure in the frequency domain to compute the CFRFs. The second step is to use FFT and IFFT to calculate the time histories from the CFRFs. The following sections describe the basic concept of SBFEM and the Frobenius solution, followed by modelling of a wave propagation problem and a forced vibration problem using the new approach. FUNDAMENTALS OF THE SBFEM A domain of an arbitrary shape is illustrated in Fig. 1(a). The domain is divided into three subdomains. Any scheme of subdivision, with various numbers, shapes and sizes of subdomains, can be used, as long as a scaling centre for each subdomain can be found from which the subdomain boundary is fully visible. Fig. 1(b) shows the details of Subdomain 1. The subdomain is represented by scaling a defining curve S relative to a scaling centre. The defining curve is usually taken to be the domain boundary, or part of the boundary. A normalised radial coordinate ξ is defined, varying from zero at the scaling centre and unit value on S. A circumferential coordinate s is defined around the defining curve S. A curve similar to S defined by ξ=0.5 is shown in Fig. 1(b). The coordinates ξ and s form the local coordinate system, which is used in all the subdomains.

(a) Subdomaining of a domain (b) Subdomain 1

Figure 1: The concept of the scaled boundary finite-element method

The governing equilibrium equations of the SBFEM have been derived within the context of virtual work principle for elastostatics by Deeks and Wolf [14]. An extension to elastodynamics has also been made by Yang et al [1]. They are not repeated here. In a frequency-domain analysis, the first step is to compute the CFRFs. The governing equilibrium equations of the SBFEM under a harmonic excitation with constant frequency f (Hz), e.g. p=p0 ei2πft are [1]

0 10 , 1( ) ( )T

ξ ξξ ξ == +p E u E u (1)

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0 2 0 1 1 2 2 2 0( ), ( ) ( ), ( ) (2 ) ( ) 0T fξξ ξξ ξ ξ ξ ξ π ξ ξ+ + − − + =E u E E E u E u M u (2)

where p0 are the magnitudes of the equivalent nodal loads. E0, E1, E2 and M0 are coefficient matrices that are dependent on the geometry of the subdomain boundaries and the material properties. u(ξ) represent the magnitudes of the nodal displacements and are analytical functions of the radial coordinate ξ. Eq. (1) and Eq. (2) are first applied to the subdomains. The contributions of subdomains to the coefficient matrices and equivalent nodal loads are then assembled in the same way as the contributions of finite elements in FEM. THE FROBENIUS SOLUTION PROCEDURE Eq. (2) is a second-order nonhomogeneous Euler-Cauchy differential equation system with respect to the radial coordinate ξ. The non-homogeneity complicates its solution considerably. An easy-to-follow analytical solution to Eq. (2) was recently developed by Yang et al [1] using the Frobenius procedure. The solution with (k+1) series is

1 2 1( ) ( ) ( )1 2 1

1 1 1( ) ( )

ki i i

n n nk k

i i i i i ii i i

c c cλ λ λξ ξ ξ++ +

= = =

= + +∑ ∑ ∑u φ g g (3)

where 1 1k k

i i i+ +=g G φ (4a)

1 1 2k k ki i i i

+ +=G B B B (4b)

11 2 1 2 0 1 1 1 2 0(2 ) ( ) ( )( )k k k Ti i ifπ λ λ

−+ + +⎡ ⎤= − + − −⎣ ⎦B E E E E M (4c)

1 1 ( )k k ki i i

+ +=d B d (4d)

1i i ic=d φ (4e)

1 2k ki iλ λ+ = + (4f)

where the subscript i ranges from 1 to n for all the variables and n is the DOFs of the problem. Eq. (3) shows that the solution to elastodynamics consists of (k+1) summations of series. The first summation is the solution to the corresponding homogeneous equations of Eq. (2), i.e., the governing equations for elastostatics. The added k summations account for dynamic effects. More series terms are added, more accurately the solution represents the dynamic effects. 1λi = λi and φi are the positive eigenvalues and corresponding eigenvectors of the standard linear eigenproblem [14] formed from the elastostatic governing equations. ci are constants determined by boundary conditions. Eqs. (4a- 4f) describe an iterative process in which the number of added summation terms k in Eq. (3) is determined by a convergence criterion [1]:

max (|R|)< α (5a)

2( )2 0 1

1(2 ) ( )

ki

nk

ii

f λπ ξ+ +

=

= ∑R M d (5b)

where R can be regarded as the residual vector and α is the convergence tolerance. Setting ξ=1 in Eq. (3) leads to

1b 11

k

ξ

+

== =u u cΨ or 1

1 b−=c uΨ (6)

where ub is the nodal displacements on the boundary, c={c1 c2 … cn}T and the matrix

1 1 1

1 1 1 2 22 2 2

k k k

j j jn n

j j j

+ + +

= = =

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦∑ ∑ ∑g g gΨ ϕ ϕ ϕ (7)

Substituting Eq. (3) into Eq. (1) leads to 0 1

0 2 1T= +p E c E cΨ Ψ (8)

where the matrix

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1 1 11 1 1

2 1 1 1 1 2 2 2 22 2 2

k k k

j j j j j jn n n n

j j jλ λ λ λ λ λ

+ + +

= = =

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦∑ ∑ ∑g g gΨ ϕ ϕ ϕ (9)

Substituting Eq. (6) into Eq. (8) gives

( )0 1 10 2 1 b d b

T−= + =p E E u K uΨ Ψ (10)

Therefore, the dynamic stiffness matrix of the domain with respect to DOFs on the boundary is 0 1 1

d 2 1T−= +K E EΨ Ψ (11)

The nodal displacement vector ub can be calculated by Eq. (10) by applying boundary conditions on ub and loading conditions on p0. The integration constants c are then obtained using Eq. (6). Assuming the (k+1)th solution meets the criterion Eq. (5a), the displacement field is recovered as

⎥⎦

⎤⎢⎣

⎡++= ∑∑∑

=

+

==

+n

ii

ki

n

iiii

n

ii cccss i

kii

1

1)(

1

2)()(

1)()()(),(

121

ggφNu λλλ ξξξξ L (12)

where N(s) is the matrix of shape functions at the circumferential direction, which are constructed as in FEM, typically using polynomial functions. The stress field is then obtained as

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

2 1

2 1

1 1 11 1 2 2 1 1

1 1 1

1 1 12 2 1

1 1 1

ki i i

ki i i

n n nk k

i i i i i i i i ii i i

n n nk

i i i i i ii i i

s s c c c

s c c c

λ λ λ

λ λ λ

ξ λ ξ λ ξ λ ξ

ξ ξ ξ

1 +

1 +

− − −+ +

= = =

− − − +

= = =

⎡ ⎤, = + + +⎢ ⎥⎣ ⎦

⎡ ⎤+ + + +⎢ ⎥

⎣ ⎦

∑ ∑ ∑

∑ ∑ ∑

DB g g

DB g g

σ ϕ

ϕ

(13)

where D is the elasticity matrix and B1(s) and B2(s) are relevant matrices [10]. One can see that the dynamic displacement and stress fields from the SBFEM (Eq. (12) and Eq. (13)) are analytical with respect to the radial coordinate ξ. Therefore, these state variables at any position of the domain can be directly calculated once the nodal displacements on the boundary are obtained by solving Eq. (10). NUMERICAL EXAMPLES, RESULTS AND DISCUSSION Two transient dynamic examples, both subjected to the Heaviside-step load and the triangular blast load, are modelled. For each example, the CFRFs of displacements or stresses at desired positions in the domain are first computed for a wide range of sampling frequencies using the Frobenius solution procedure presented in the previous section. A FFT of the transient load is then carried out to obtain the complex discrete Fourier coefficients, followed by an IFFT of the CFRFs to obtain time histories of displacement or stress at desired positions. The readers are referred to [15] for details of the discrete Fourier transform and the fast Fourier transform techniques. The functions fft( ) and ifft( ) in MATLAB [16] are used to conduct FFT and IFFT respectively. For each example, the same CFRFs are used to calculate time histories for all the forms of transient loadings that this example is subjected to. In the frequency-domain SBFEM, the damping effect is taken into account by modifying the elastic moduli to incorporate an internal damping coefficient β, forming the complex Young’s modulus Ec=E(1+i2β) and the complex shear modulus Gc=G(1+i2β), where E and G are the elastic Young’s modulus and the elastic shear modulus respectively. A transient dynamic finite element analysis using ABAQUS is also carried out for both examples for comparison using the implicit time-integration scheme [17]. No physical damping (i.e., pure elastic material behaviour) is considered in all the finite element analyses. The international standard unit system, namely mass in kilograms, length in metres and time in seconds, is used for both examples and thus no units are indicated with all the physical entities. Two-node linear line elements and α = 0.001 in Eq. (5a) are used for all the analyses using the SBFEM. A plane stress condition is assumed for both examples. 1. Example 1: 1D rectangular rod The first example modelled here is the classical wave propagation problem with analytical solutions, which was modelled by many, e.g., [6] and [7], to validate numerical methods. A rectangular rod is subjected to a uniform traction on the right end with the left end fixed. The geometry, boundary and loading conditions are shown in Fig. 2(a). The time responses at the points A, B and C in Fig. 2(a) are examined. Only one subdomain is used to model this example. Fig. 2(b) shows a typical mesh with only 15 nodes. The scaling centre is

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placed at the left-bottom corner so that the two edges connected to it are not discretised. β=1×10−6 is used for this example.

(a) Dimensions, material properties and loading conditions (b) A mesh with 15 nodes

Figure 2: Example 1, an 1D rectangular rod

Figure 3: Horizontal displacement at the point A under the Heaviside step load for the 1st example

Figure 4: Horizontal displacement at the point B under the Heaviside step load for the 1st example

Fig. 3 and Fig. 4 show the time histories of the horizontal (axial) displacements at the free end of the rod (the point A) and the middle of the rod (the point B) respectively under the Heaviside step load. The results from both the frequency-domain SBFEM and the time-domain FEM are in very good agreement with the analytical solutions [18]. The axial stress histories at the middle and the fixed end of the rod are shown in Fig. 5 and Fig. 6 respectively. Good agreement between the numerical results and the analytical solution can also be observed, except that the FEM results oscillate fiercely around the analytical solutions at moments when the stress jumps. This is caused by the sudden application of the step load and cannot be completely corrected by numerical measures. It also happens in other studies [6, 7]. The oscillations also appear in the results from the new frequency-domain approach, but with lower magnitudes and much less degree. Fig. 7 compares the horizontal displacement histories at the points A and B under the triangular blast load. The results from the two numerical methods are virtually identical. Excellent agreement can also be seen in the axial stress histories at the middle and the fixed end of the rod, which are presented in Fig. 8. No oscillation happens because the triangular blast load function is continuous.

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Figure 5: Horizontal stress at the point B under the Heaviside step load for the 1st example

Figure 6: Horizontal stress at the point C under the Heaviside step load for the 1st example

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

0 2 4 6 8 10 12Time

Hor

izon

tal d

ispl

acem

ents

FEM (A)Present (A)FEM (B)Present (B)

Figure 7: Horizontal displacements at the point A and B under the blast load for the 1st example

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 2 4 6 8 10 12Time

Hor

izon

tal s

tres

s

FEM (B)Present (B)FEM (C)Present (C)

Figure 8: Horizontal stress at the point B and C under the blast load for the 1st example

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2. Example 2: 2D simply-supported beam subjected to uniform bending force The forced vibration of a 2D simply-supported beam is modelled as the 2nd example. The dimensions, boundary and loading conditions, and material properties are illustrated in Fig. 9(a). The beam is subjected to a uniformly-distributed bending traction at its upper face. The time responses at the points A and B in Fig. 9(a) are investigated. Both the Heaviside step load and the blast load (only the latter is shown in Fig. 9(a)) are modelled. The beam is modelled with two subdomains consisting of totally 33 nodes using the SBFEM as shown in Fig. 9(b). The two scaling centres are positioned at the beam lower face so that this whole face is not discretised. A material damping coefficient β=1×10−5 is used.

(a) Dimensions, material properties and loading conditions (b) A mesh with 33 nodes

Fig. 9 Example 2: a simply-supported beam subjected to uniformly distributed bending force

Fig. 10 and Fig. 11 show the vertical displacement histories at the point B and the horizontal stress histories at the point A respectively under the Heaviside step load. Also shown in these figures are the analytical solutions [18] taking into account the effects of rotary inertia and shear deformation. It is evident that the results from the time-domain FEM and the frequency-domain SBFEM almost coincide with each other, and the results from both methods agree very well with the analytical solutions, especially for the displacement histories. Because the stresses are derivatives of displacements and thus less accurate, the discrepancies between the analytical stress solution and the numerical results are understandable. Comparing the results here with those reported in [19] using a time-domain FEM also confirms the effectiveness and accuracy of the developed frequency-domain SBFEM. The same conclusion can also be drawn from the results under the blast load, which are shown in Fig. 12 and Fig. 13.

-0.05

-0.045

-0.04

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0 1 2 3 4 5Time

Vert

ical

dis

plac

emen

t at p

oint

B

Analytical Present FEM

Figure 10: Vertical displacement at the point B under the Heaviside step load for the 2nd example

-10-5

05

10

1520

253035

4045

50

0 1 2 3 4 5Time

Hor

izon

tal s

tres

s at

poi

nt A

Analytical Present FEM

Figure 11: Horizontal stress at the point A under the Heaviside step load for the 2nd example

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-0.015

-0.012

-0.009

-0.006

-0.003

0

0.003

0.006

0.009

0.012

0.015

0 1 2 3 4 5Time

Vert

ical

dis

plac

emen

t at p

oint

B

Present FEM

Figure 12: Vertical displacement at the point B under the blast load for the 2nd example

-14

-10.5

-7

-3.5

0

3.5

7

10.5

14

0 1 2 3 4 5Time

Hor

izon

tal s

tres

s at

poi

nt A

Present FEM

Figure 13: Horizontal stress at the point A under the blast load for the 2nd example

CONCLUSIONS This study develops a frequency-domain approach using the SBFEM for modelling general transient dynamic problems. The CFRFs are first computed by the newly-developed analytical Frobenius solution to the governing equations of SBFEM. The FFT and the IFFT are then carried out to calculate time histories of structural responses. This new approach does not need a mass matrix as needed by the time-domain SBFEM, allowing domains to be modelled by fewer subdomains efficiently. The new approach has been used to successfully model transient dynamic behaviour of two examples with very small number of DOFs due to the analytical nature of the Frobenius solution. ACKNOWLEDGEMENT This research is supported by the Australian Research Council (Discovery Project No. DP0452681) and the National Natural Science Foundation of China (No. 50579081).

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domain. Int. J. Num. Meth. Eng., (under review).

2. Yang ZJ, Deeks AJ, Hao H. A frequency-domain approach for transient dynamic analysis using scaled boundary finite element method (II): application to fracture problems. The 10th Int. Conf. Comp. Meth. in Engng.&Sci (EPMESC X). Aug. 21-23, 2006, Sanya, Hainan, China.

3. Zienkiewicz OC, Taylor RL. The finite element method. 5th edition, Butterworth- Heinemann, 2000.

4. Dokainish MA, Subbaraj K. A survey of direct time-integration methods in computational structural dynamics — I. explicit methods. Computers & Structures, 1989; 32(6): 1371-1386.

5. Zeng LF, Wiberg NE. Spatial mesh adaptation in semidiscrete finite element analysis of linear elastodynamics problems. Computational Mechanics, 1992; 9: 315-332.

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6. Carrer JAM, Mansur WJ. Alternative time-marching schemes for elastodynamic analysis with the domain boundary element method formulation. Computational Mechanics, 2004; 34(5): 387-399.

7. Yu GY, Mansur WJ, Carrer JAM. A more stable scheme for BEM/FEM coupling applied to two-dimensional elastodynamics. Computers & Structures, 2001; 79(8): 811-823.

8. Gu YT, Liu GR. A meshless local Petrov–Galerkin (MLPG) method for free and forced vibration analyses for solids. Computational Mechanics, 2001; 27188-198.

9. Wolf JP, Song CM. Finite-element modelling of unbounded media. John Wiley and Sons, Chichester, 1996.

10. Wolf JP. The scaled boundary finite element method. Chichester: John Wiley and Sons, 2003.

11. Zhang X, Wegner JL, Haddow JB. Three-dimensional dynamic soil-structure interaction analysis in the time domain. Earthquake Engineering & Structural Dynamics, 1999; 28(12): 1501-1524.

12. Yan JY, Zhang CH, Jin F. A coupling procedure of FE and SBFE for soil-structure interaction in the time domain. International Journal for Numerical Methods in Engineering, 2004; 59(11): 1453-1471.

13. Song CM. A super-element for crack analysis in the time domain. International Journal of Numerical Methods in Engineering, 2004; 61(8): 1332-1357.

14. Deeks AJ, Wolf JP. A virtual work derivation of the scaled boundary finite-element method for elastostatics. Computational Mechanics, 2002; 28(6): 489-504.

15. Brigham EO. The fast Fourier transform and its applications. Prentice-Hall, 1988.

16. MATLAB V7.1. The MathWorks, Inc., 2005.

17. Hibbitt, Karlsson and Sorensen Inc. ABAQUS/Standard User Manual V6.5, 2004.

18. Timoshenko S, Young DH, Weaver W. Vibration Problems in Engineering. 4th edition, John Wiley & Sons, 1974.

19. Zienkiewicz OC, Li XK, Nakazawa S. Dynamic transient analysis by a mixed, iterative method. International Journal for Numerical Methods in Engineering, 1986; 23(7): 1343-1353.