Improving convergence of incremental harmonic balance method using homotopy analysis method

Acta Mech Sin (2009) 25:707 712 DOI 10.1007/s10409-009-0256-4 RESEARCH PAPER Improving convergence of incremental harmonic balance method using homotopy analysis method Yanmao Chen Jike Liu Received: 10 April 2008 / Revised: 5 January 2009 / Accepted: 6 February 2009 / Published online: 28 April 2009 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH 2009 Abstract We have deduced an iteration scheme in the incremental harmonic balance (IHB) method using the harmonic balance plus the Newton-Raphson method. Since the convergence of the iteration is dependent upon the initial values in the iteration, the convergent region is greatly restricted for some cases. In this contribution, in order to enlarge the convergent region of the IHB method, we constructed the zeroth-order deformation equation using the homotopy analysis method, in which the IHB method is employed to solve the deformation equation with an embedding parameter as the active increment. Taking the Duffing and the van der Pol equations as examples, we obtained the highly accurate solutions. Importantly, the presented approach renders a convenient way to control and adjust the convergence. Keywords Incremental harmonic balance method Homotopy analysis method Initial value Convergence 1 Introduction The incremental harmonic balance (IHB) method was firstly presented in 1981 by Lau and Cheung [1]. Compared with The project supported by the National Natural Science Foundation of China (10772202), Doctoral Program Foundation of Ministry of Education of China (20050558032), and Guangdong Province Natural Science Foundation (07003680, 05003295). Y. Chen State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, 800 Dongchuan Road, 200240 Shanghai, China e-mail: chenyanmao@126.com J. Liu (B) Department of Mechanics, Sun Yat-sen University, 510275 Guangzhou, China e-mail: jikeliu@hotmail.com other existing techniques, the IHB method has many advantages. For example, it deals easily with strongly nonlinear systems and provides highly accurate results. Moreover, it is a much simpler and more systematic approach and consequently can be more easily implemented on a computer than the perturbation method. Theoretically, the IHB method is exactly equivalent to the harmonic balance (HB) plus the Newton Raphson method [2]. Therefore, it has been successfully applied to the analysis of many periodic and almost periodic vibrations and related problems [3 8]. Usually, a good enough initial value has to be chosen for ensuring the convergence of the iteration in the implementations of the IHB method. However, sometimes it is really difficult to do so. Therefore, researchers proposed some techniques to make it easier to choose good initial guess with the help of the homotopy method in many applications of the HB method [9,10]. However, most of the aforementioned studies about the IHB method did not pay particular attention to this issue. Thus, it is important to develop new techniques to improve the convergence properties of the IHB method. Following the basic idea of the Liao s homotopy analysis method (HAM) [11 17], the HAM is actually based on the so called zeroth-order deformation equation connecting an auxiliary linear equation and a nonlinear one. As the embedding parameter varies from 0 to 1, the solution of the linear equation becomes that of the nonlinear one. Thus, the HAM has been used to address various nonlinear problems [18 26]. Additionally, the HAM is much easier to choose the exact solution of the linear problem than to directly solve the nonlinear one. On the basis of this idea, we try to choose exact solutions of linear equations as the initial guess of the IHB iterative algorithm. Therefore, in this contribution, we firstly construct a class of zeroth-order deformation equations, and then use the IHB method to solve the deformation equation with the embedding parameter as an active increment. As

708 Y.Chen,J.Liu the embedding parameter increases in the range of 0, 1, the exact solution of the auxiliary linear equation becomes an IHB solution of the considered problem. 2 Zeroth-order deformation equation Consider a nonlinear vibration equation as follows f (x, ẋ, ẍ) = 0, (1) where a top dot denotes a differentiation with respect to time t. Note that, in our studies, Eq. (1) is given as either a conservative or self-excited oscillator. Thus, it possesses at least one periodic (or limit cycle) solution. By introducing a transformation τ = ωt, ω is the frequency of the periodic solution, Eq. (1) becomes f (x,ωx,ω 2 x ) = 0, (2) where a prime denotes a differentiation with respect to τ. Since periodic and limit cycle solutions can be described as sums of a set of base functions as {cos(kτ),sin(kτ) k = 0, 1, 2,...}, one can choose such an auxiliary linear operator as [x(τ)] =x (τ) + x(τ), (3) so that [cos τ] = [sin τ] =0. (4) Considering Eq. (2), one defines the nonlinear operator as ℵ[x(τ), ω] = f [x(τ), ωx (τ), ω 2 x (τ)]. (5) Letting p [0, 1] be an embedding parameter and h a nonzero auxiliary parameter, one constructs such a homotopy in general form H[x, x, x,ω,p] =(1 p) [x(τ)]+hpℵ[x(τ), ω], (6) and calls H[x, x, x,ω,p] =0, (7) the zeroth-order deformation equation. When p = 0,thevariational dynamic system (7) is exactly the linear system as [x(τ)] =0, (8) when p = 1, the system (7) becomes (2). 3 IHB method for zeroth-order deformation equation First of all, we assume that the zeroth-order deformation equation possesses at least one periodic (or limit cycle) solution in our case. In this section, we will use the IHB method to solve Eq. (7) with the embedding parameter p as an active increment (called p-incrementation). When the initial value p 0 = 0, the initial guess x 0 can be easily chosen as the exact solution of Eq. (8). When the embedding parameter p increases from 0 to 1, x varies from x 0 to an HB solution of Eq. (1). If x 0 and ω 0 denote a state of vibration of system (7) with p = p 0, the neighboring state can be expressed by adding the corresponding increments to them as follows x = x 0 + x, ω = ω 0 + ω, p = p 0 + p. (9) Substituting Eq. (9) into Eq. (7) and neglecting small terms of higher order, one obtains the following linearized incremental equation as H x x + H x x + H x x + H ω ω + H p p + R(x 0,ω 0, p 0 ) = 0. (10) And the residue R(x 0,ω 0, p 0 ) = H(x 0, x 0, x 0,ω 0, p 0 ) (11) goes to zero when x 0 is the exact solution of Eq. (7). The second step is the Ritz Garlerkin procedure. We define x 0 = a 0,0 + x = a 0 + x = a 0 + [a n,0 cos(nτ)+ b n,0 sin(nτ)], (12) [ a n cos(nτ)+ b n sin(nτ)], (13) [a n cos(nτ)+ b n sin(nτ)]. (14) Substituting Eqs. (12) and (13) into Eq. (10), using the Ritz Garlerkin procedure, one obtains K a a + K ω ω + K p p + K R = 0, (15) where a =[ a 0, a 1, b 1,..., a N, b N ] T, K a is a square matrix of dimension 2N + 1, and both K R and K ω are vectors of dimension 2N +1. Equation (15) thus contains 2N + 1 equations while 2N + 2 unknowns, i.e., a and ω. The number of equations is one more than that of incremental unknowns in many implementations of the IHB method. Usually, one of the increments is fixed to be zero and then the other unknowns can be determined [5 10]. Whereas, in our studies, an additional equation is constructed to determine all the unknowns simultaneously. Well known, when the system (1) is conservative, the corresponding solution is periodic for any given initial conditions. Without loss of generality, the simple initial conditions x(0) = A, ẋ(0) = 0 (16)

Improving convergence of incremental harmonic balance method 709 are adopted, in which A is a given constant. According to Eqs. (12) (14) and (16), one can let b i = 0 and b i = 0, i = 1, 2,...,N, and an equation can be obtained as a 0 + a 1 + +a N = A. (17) It results in an additional equation in the increments as a 0 + a 1 + + a N = 0. (18) Then one can obtain K c c = K R K p p, (19) [ ] K where c=[ a 0, a 1,..., a N, ω] T, K c = a K ω, V c 0 V c =[1, 1,...,1], and K R = [ K R 0 ] T, K p = [ K p 0 ] T. Because b i (i = 1, 2,...,N) are zeroes, K a, K ω and K R are of dimension N + 1. Then, K c is the square matrix of dimension N + 2, c and K R are of dimension N + 2. Accordingly, the incremental vector c can be uniquely determined by Eq. (19). When Eq. (1) corresponds to a self-excited system possessing at least one limit cycle, considering that the limit solution is independent of initial conditions, one can consider such simple initial conditions as x(0) = α, ẋ(0) = 0, (20) where the amplitude α is to be determined. According to Eqs. (14) and (20), one can attain an equation as ẋ(0) = ωx (0) = ω nb n = 0. (21) Thus, one yields n b n = 0. (22) Then one has K s s = K R K p p. (23) where [ s ] =[ a 0, a 1, b 1,..., a N, b N, ω] T, K s = K a K ω, V V s 0 s =[0, 0, 1, 0, 2,...,0, N], and K R and K p are as determined above. At each iteration stage, Eq. (19) or(23) represents a set of linear algebraic equations in the increments c or s, respectively. Therefore, they can be solved iteratively provided the iteration step p is proscribed. As long as the initial guesses x 0 and ω 0 are properly chosen as the initial solution of Eq. (7) with p = p 0, the iterations can converge. As a result, the solution of Eq. (1) can be obtained by incrementing p from 0 to 1 point by point. It is noticed that, when p 0 = 1 and p = 0, iterations (19) and (23) are exactly the same as those deduced by the IHB method. Thus, the IHB method for Eq. (1) is logically included in the presented approach. 4 Numerical examples 4.1 Duffing equation The Duffing equation under transformation τ = ωt can be described as ω 2 x + k 1 x + k 3 x 3 = 0, x(0) = A, x (0) = 0, (24) where A is a given constant. The nonlinear coefficient k 3 can be both positive and negative denoting hard and soft cubic stiffness, respectively. If k 3 < 0, A is limited as A < k1 /k 3 so that there is no equilibrium points other than the original point (i.e., x(0) = x (0) = 0) in the region of x < A. Note that the zeroth-order deformation equation for Eq. (24) is still a conservative system, and hence possesses periodic solution for any given initial conditions. Because the solution of Eq. (24) does not contain even harmonics, it can be written as m x(τ) = a 2i+1 cos[(2i + 1)τ]. (25) i=0 Using the iteration scheme given by Eq. (19), one can attain the HB solution as long as the initial values of a 2i+1,0 (i = 0, 1,...,m) and ω 0 are properly chosen. The solutions containing five dominant harmonics (i.e., m = 4 or the HB9 solution) of Eq. (24) with positive or negative cubic stiffness are shown in Tables 1 and 2, respectively, in which the initial values chosen are a 1,0 = 1, a 3,0 = a 5,0 = a 7,0 = a 9,0 = 0, ω 0 = 1, p 0 = 0, and p = 0.1. It is shown that the HB9 solutions attained by Eq. (19) are in good agreement with the exact solutions of the Duffing equation with either hard or soft cubic stiffness. Now we turn to discuss the convergence of the iteration schemes (19). Figure 1 shows the convergent points of the IHB method (i.e., iteration (19) with p 0 = 1 and p = 0) in the a 1,0 k 3 plane (with meshes 0.1 0.1) with the other initial values of a 1,0 = A, a 3,0 = a 5,0 = a 7,0 = a 9,0 = 0, and ω 0 = 1. The convergent region with respect to k 3 becomes Table 1 Comparison of HB9 frequency results obtained by iteration (19) with the exact solutions of Duffing equation with k 1 = 1 h A k 3 ω (HB9) ω ex 0.5 1 1 1.317776 1.317764 0.1 100 1 84.72749 84.68921 0.00001 10, 000 1 8258.823 7972.213 1 1 10 2.866640 2.866590 0.1 1 1,000 26.81074 26.80734 0.001 1 100,000 267.9143 267.3675

710 Y.Chen,J.Liu Table 2 Comparison of HB9 frequency results obtained by iteration (19) with the exact solutions of Duffing equation with k 3 = 1and h = 0.5 k 1 A ω (HB9) ω ex 1 0.1 0.996242 0.996233 1 0.5 0.591682 0.600381 1 0.95 0.503496 0.503469 100 9 7.07849 7.07814 10,000 60 85.1952 85.1368 Fig. 2 Convergent radius of iteration (19) with respect to A versus h for Duffing equation with k 1 = k 3 = 1. Solid line obtained with p = 0.1, dash line with p = 0.2, dash-dot line with p = 0.5, dot line with p 0 = 1and p = 0 Fig. 1 Convergent points (the dots) of the IHB method in the a 1,0 k 3 plane for Duffing equation with a 3,0 = a 5,0 = a 7,0 = a 9,0 = 0and ω 0 = 1 smaller and smaller as a 1,0 = A > 0 increasing. Meanwhile, the boundary of the convergent region is not continuous. For the iteration (23), a simple way of choosing the initial guess is letting a 1,0 = A, a 2i+1,0 = 0 (i = 1, 2,...,m), and ω 0 = 1. With these initial values in hand, the convergent radius with respect to a 1,0 (or A) can be found when h and p = p 0 are given, as shown in Fig. 2. With h decreasing, the convergent radius with respect to A becomes larger and larger. As one can see, the convergent region of the iteration (19) is much larger than that of the IHB method. Further, the convergent region can even be enlarged if p is chosen smaller. However, this needs more iteration steps to seek the solution of the considered problem. At any rate, the parameters h and p indeed provide us with a quite convenient way to control and adjust the convergence of the IHB method. 4.2 van der Pol equation The transformed van der Pol equation can be described as ω 2 x + x + ωε(x 2 1)x = 0, (26) where ε is a constant. It is well known that the van der Pol equation possesses a stable limit cycle solution as ε>0 while an unstable one as ε < 0. The zeroth-order deformation Fig. 3 Comparison of frequency solutions of the van der Pol equation obtained by iteration (23) with the numerical solutions. Dash line denotes HB3 results, dot line for HB5 results, solid line for numerical solutions equation can be rewritten in the same form as the van der Pol equation, therefore it possesses one limit cycle solution. Because the limit cycle solution does not contain even type harmonics, it can be written as m x(τ) = {a 2i+1 cos[(2i + 1)τ]+b 2i+1 sin[(2i + 1)τ]}. i=0 (27) Using the iteration scheme (23), the matrix K s in iteration is singular if the initial value of p is chosen as p 0 = 0. To eliminate the singularity, the initial value p 0 should be given as a very small positive constant. For simply, the initial value p 0 is let to be the increment of p, i.e., p 0 = p. With a 1,0 = 1, ω 0 = 1, p 0 = p = 0.1, h = 0.5 and the other initial values are zeroes, the HB3 and HB5 results are obtained by Eq. (23), as shown in Fig. 3 compared with the numerical solutions. These results show that, when ε is

Improving convergence of incremental harmonic balance method 711 Fig. 4 Unconvergent points (the dots) of the IHB method in the a 1,0 b 1,0 plane with ω 0 = 1anda 3,0 = a 5,0 = b 3,0 = b 5,0 = 0 relatively small, the HB results are in good agreement with the numerical solutions, but the accuracy decreases with ε increasing. With the initial values of a 3,0 = a 5,0 = b 1,0 = b 3,0 = b 5,0 = 0, ω 0 = 1 and the value of a 1,0 varying, the initial guess x 0 (τ) is the exact solution of the corresponding auxiliary linear equation. Interestingly, as ε = 1 the convergent region of iteration (23) with respect to a 1,0 > 0 is lager than [10 6, 10 10 ] with h = 0.5and p = p 0 = 0.1. While for the IHB method (i.e., the iteration (23) with p 0 = 1 and p = 0), the corresponding convergent region is only [0.835, 2.495] or so. When the initial values are given as a 3,0 = a 5,0 = b 1,0 = b 3,0 = b 5,0 = 0, ω 0 = 1 and h = 0.5, p = p 0 = 0.1, and the value of ε varying, the convergent region of Eq. (23) with respect to ε is (0, 22.95] as a 1,0 = 1 and (0, 30.97] as a 1,0 = 2, respectively. Further, the corresponding convergent regions can even be enlarged as(0, 58.49] and (0, 323.3], respectively, if p and p 0 decrease as 0.01 and 0.001. For the IHB method, the corresponding convergent region is only (0, 1.080] as a 1,0 = 1 and (0, 1.689] as a 1,0 = 2, respectively. Note that if the iteration (23) is convergent for a given ε, it converges for ε too. Thus, the presented method is both effective for attaining stable and for unstable solutions. For the IHB method, letting a 3,0 = a 5,0 = b 3,0 = b 5,0 = 0 and ω 0 = 1, the unconvergent points in the a 1,0 b 1,0 plane (with meshes 0.1 0.1) for the van der Pol equation with ε = 1 are plotted in Fig. 4. Clearly, one can see that the convergent region is very restricted. Figure 5 shows the convergent radius of iteration (23) with respect to ε for the van der Pol equation, with the initial values as a 1,0 = ω 0 = 1 and a 3,0 = a 5,0 = b 1,0 = b 3,0 = b 5,0 = 0. One can see that the convergent radius of Eq. (23) is much larger than that of the IHB method, and the smaller the auxiliary parameter h is the larger the convergent radius becomes. Fig. 5 Convergent radius of iteration (23) with respect to ε versus h for van der Pol equation. Solid line is obtained with p = p 0 = 0.1; dash line with p = p 0 = 0.2; dash-dot line with p = p 0 = 0.5; dot line with p 0 = 1and p = 0 5 Conclusions We have proposed a new approach to improve the convergence properties of the IHB method. Instead of using the IHB method to solve the studied nonlinear oscillating equation directly, we employed it to solve the zeroth-order deformation equation based on the nonlinear equation. In the proposed technique, the convergence can be controlled and adjusted by the auxiliary parameter and the embedding parameter, respectively. Additionally, the IHB method for the zerothorder deformation equation logically contains the IHB method for the studied equation itself. References 1. Lau, S.L., Cheung, Y.K.: Amplitude incremental variational principle for nonlinear vibration of elastic systems. ASME J. Appl. Mech. 48, 959 964 (1981) 2. Ferri, A.A.: On the equivalence of the incremental harmonic balance method and the harmonic balance-newton Raphson method. ASME J. Appl. Mech. 53, 455 457 (1986) 3. Lau, S.L., Cheung, Y.K., Wu, S.Y.: Incremental harmonic balance method with multiple time scales for aperiodic vibration of nonlinear system. ASME J. Appl. Mech. 50, 871 876 (1983) 4. Lau, S.L., Cheung, Y.K., Wu, S.Y.: Nonlinear vibration of thin elastic plates, part 1: Generalized incremental Hamilton s principle and element formulation; part 2: Internal resonance by amplitude-incremental finite element. ASME J. Appl. Mech. 51, 837 851 (1984) 5. Cheung, Y.K., Chen, S.H., Lau, S.L.: Application of the incremental harmonic balance method to cubic non-linearity systems. J. Sound Vib. 140, 273 286 (1990) 6. Lau, S.L., Yuen, S.W.: The Hopf bifurcation and limit cycle by the incremental harmonic balance method. Comput. Methods Appl. Mech. Eng. 91, 1109 1121 (1991) 7. Lau, S.L., Yuen, S.W.: Solution diagram of nonlinear dynamicsystems by IHB method. J. Sound Vib. 167, 303 316 (1993) 8. Raghothama, A., Narayanan, S.: Non-linear dynamics of a twodimensional airfoil by incremental harmonic balance method. J. Sound Vib. 226, 493 517 (1999)

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