Transactions on Modelling and Simulation vol 20, 1998 WIT Press, ISSN X

Transcription

1 A BEM-formulation applied to the solution of nonlinear dynamic problems H.J. Holl, A.K. Belyaev & H. Irschik Division of Technical Mechanics, University Linz, Austria uni-linz. ac. at Abstract A semi-analytical time-integration procedure is presented for the integration of discretized dynamic mechanical systems. This method utilizes the advantages of the boundary element method, well known from quasi-static field problems Motivated by these spatial formulations, the present dynamic method is based on influence functions in time, and gives exact solutions in the linear time-invariant case. Similar to domain-type BEM's for nonlinear field problems, the method is extended for different nonlinear dynamic systems having nonclassical damping The numerical stability and accuracy of the semi-analytical method is discussed in two steps for the nonclassical damping and for the nonlinear restoring forces The Duffing oscillator and a system with nonclassical damping are used as representative model problems. A comparison is given to other conventionally used time integration procedures, which shows the efficiency of the present method. 1 Introduction During the last decades, various powerful computational methods have been developed for nonlinear structural dynamics. A main stage of the computational analysis is the space-wise discretization of the governing field equations leading to a system of ordinary differential equations in time. Especially, a semidiscretization by the Finite Element Method (FEM) leads to a dynamic system of second-order equations of the form '* (1) compare Zienkiewicz & Taylor [1], where X denotes the vector of generalised coordinates, M stands for the mass matrix, K and C are matrices of stiffness and

2 300 Boundary Elements damping, respectively, and F gathers the imposed forcing terms. In many practical situations nonlinearities do occur, the matrices in eqn (1) then depending on the generalised coordinates: M = m + M(X,X), C = c + C(X,X), KX = kx + Q(X,X), (2) where m, c and k denote the symmetric and constant portions of the matrices in eqn (1). The damping matrix c is of the Rayleigh type. It has been shown in Zienkiewicz, Taylor [1] that the numerical integration of eqn (1) by a finite element discretization in time provides a strategy unifying many existing singleand multi-step algorithms and providing a variety of new ones. When discussing stability problems of these schemes, it is useful to revert to modally uncoupled equations, see [1] and Hughes [2]: Xk-f2CkO^Xk + o^xk = pk = fk + rk(x,x,x), (3) where the index (k) refers to the number of the mode, which is supressed in the following. The modal analysis transforming eqn (1) into eqn (3) is performed with respect to the constant symmetric matrices m, c and k of eqn (2), while the remaining unsymmetric and nonlinear portions are gathered in the k-th modal force r. The modal force due to F is denoted by f, co is for the natural frequency, and C denotes slight modal damping. With respect to eqn (3), it is noted that our group has been successful to solve various structural elasto-viscoplastic problems by utilizing an analogy between the inelastic strains and a distribution of eigenstrains in a linear-elastic background-structure and by taking the continuous modes of that linear structure for a modal reduction to equations of the type of eqn (3). The nonlinear restoring forces q have been calculated from the constitutive equations in a time-stepping procedure, and a semi-analytic timeintegration of eqn (3) has been performed, see Irschik & Ziegler [3], [4] and Fotiu, Irschik & Ziegler [5] for a recent application relating modal analysis to integral equations for elastic -plastic plate vibrations. For the discretized case of nonlinear rotordynamics, an analogous semi-analytic time-integration scheme has been developed by Holl [6], [7] and extended to systems with time-dependent mass in [8]. Seen from the point of the FEM, however, the time-integration of eqns (1) or (3) by means offiniteelements in time of course is a concise strategy. One should remind, nevertheless, that a formulation of the type of eqn (1) can also be obtained by utilising the Boundary Element Method (BEM) for the space-wise semi-discretization, Nardini & Brebbia [9], see Manolis & Beskos [10] for a review. In that connection, it is the scope of the present contribution to discuss the integration of eqn (3) by means of a BEM in time. It is shown that a timewise BEM corresponds to an exact integration of the modal equations in the linear case, while a domain-boundary element formulation (DBEM) is obtained for the case of nonlinear modal restoring forces, compare [11] for DBEMs of quasi-static inelastic problems. The time-domain integrals of the present

3 Boundary Elements 301 formulation coincide with Duhamel's convolution integral, well-known from the theory of structural vibrations, see e.g. Ziegler [12]. In contrast to an approximation with FEM in time, the present BEM does not approximate the generalised coordinates, but only the integrands of the domain integrals occurring in the nonlinear case are approximated. In the present paper the numerical stability is discussed for the nonlinear case, where a representative approximation of the domain-integrals is used. Results are shown for the Duffing oscillator and are compared to the outcomes of the Newmark method. As a further example a system with time-varying mass and nonclassical damping is analysed. 2 Boundary Integral Formulation in Time Following the fundamental concepts of BEM presented by Brebbia & Dominguez [13], we start with a weighted residual formulation for eqn (3) within the time interval: At [X(t) + 2 C co X(t) + 0)2 X(t) - p(t)] W(t) dt = 0; 0<t<At. (4) W(t) denotes a sufficiently continuous weight function. Integrating by parts twice gives - 2 c co W(t) + of w(t)) x(t) - p(t) w(t)] dt + x(t) w(t) * + 2(toX(t)W(t) '-X(t)\V(t) At = 0- (5) In order to relate eqn (5) to BEM, we follow the singular approach indicated by Brebbia & Dominguez [13]. In the sense of this latter approach, the weight function W is identified by the fundamental solution W, of the modal oscillator due to an unit impulse acting at the inner site of the boundary t=0. W, is governed by the differential equation Vv-i-2Cco\V,+co*Wi=S(O, where (6) Dirac's delta function is denoted by 8, and t+ = lim^ <# - e). W, satisfies homogeneous initial conditions at t=0. As a second fundamental solution, the solution Wj of the modal oscillator due to a unit dipole 6 is introduced:

4 302 Boundary Elements, where [cos (# + ^ sin co^t ] H(Q. (8) % Applying both eqns (7) and (8) to eqn (5), considering the properties of 8 and S, r X(t) 5(t) dt = X(t=0) = Xo, (^ X(t) S(t) dt = - X(t=0) = -*o, (9) Jo Jo using some trigonometric transformation rules and solving for the state-vector at the end of the time-interval, the following transition matrix formulation is obtained: It is thus seen that the evaluation of the weighted residual statement of eqn (5) in the sense of BEM leads to the exact integration of the modal equations (3) by means of Duhamel's integral, since A(At) is the exact transition matrix of the linear oscillator, compare Hughes [2], and see Refs. [3]- [6], where a semianalytical time-stepping procedure analogous to eqn (10) has been used. It is certainly not surprising that the use of BEM in time leads to exact formulations, since the boundaries of the time-interval reduce to a two-point problem, with algebraic relations instead of boundary integrals. In the case of nonlinear problems, the portion r of pin eqn (3) is not known in advance. An iterative procedure has to be established in the sense of the DBEM by approximating the time-evolution of q in eqn (10), see the following Section. The solution for the load transfer matrix of the k-th modal equations of motion due to the excitation forces can be computed using eqn (10), when inserting P(T) = tmco- The excitation force in the integral is approximated in the form of a parabolic interpolation function of the force within the m-th time step according i ( i \* to fkmw = 4mO + ^km^ + 4AM^I 1 We can define a corresponding forcing vector with the three parameters of the interpolation function in the form ^km = { 4m A4m ^4m) ^ the following the index k, which refers to the number of the generalized coordinate, is omitted as it is obvious in these equations. The evaluation of the convolution integral of eqn (10) for the above stated interpolation function (10) gives the corresponding solution of the elements of the load transfer matrix B(At) involving dynamic Green's functions. If we write the result for the generalized coordinate at the end of the m-th time step, we get for a linear system

5 Boundary Elements 393 A'»-PBfp. p = l This is the exact solution if the parabolic interpolation of the excitation force is exact. The suitable application of modal reduction gives additional advantages for the computational effort and can be included with some small changes in the present algorithm. 3 Analysis of Systems with Nonlinear Restoring Forces * The transition-matrix formulation of eqn (10) provides exact solutions for linear vibration^ In order to obtain a measure for the quality of various approximations of the modal force r in the nonlinear case, however, free damped linear vibrations are studied at first, where the damping force formally is treated as a nonlinear restoring force: X + co*x = r(x) = -2CX Using the undamped matrix A(%., the solution of eqn (12) is studied for the approximation of the integrand in eqn (10): VY-\ _ Y j. V. XCD-XO + XOT 2 At Inserting eqn (13) into (10), integrating and rearranging terms gives the form /X(At)\ \*w>r*"\*.,. 04) where A is the approximate transfer matrix. This equation is appropriate for a stability discussion. The stability analysis is performed for a linear initial value problem which is extended to the consideration of the above described effects. The analysis steps for the linear case are taken from [2]. So it can be stated, that the absolute value of the maximum eigenvalues has to be less than one for every numerical algorithm if it is said to be unconditionally stable. The eigenvalues of the exact transfer matrix A are ^=e( C»±i«*)t ^absolute value of the eigenvalues is computedtop max Ap j -/Aj - e * «", which is also called spectral radius. The presented semi-analytic procedure has an approximate transfer mairix A and

6 304 Boundary Elements can be analysed with respect to its numerical behaviour. In [6] the equations for the numerical dissipation, the numerical dispersion and the accuracy are given for the approximate transfer matrix. When considering a linear modal oscillator it is confirmed, that the present BEM-type method is exact. The numerical dissipation is the sum of the modal damping of the oscillator and an additional damping due to the assumptions of the algorithm. The numerical dispersion is characterized by the resulting relative period error. The accuracy is computed from the local error for one time step. The advantage of the present method for the linear system results due to the application of the exact transfer matrix, so the numerical dissipation and dispersion and the local error are zero for the linear symmetric system. 1,2 "S <o 0,8 --* 0,6 '- 0,4-0,2 -, 0 0,5..., 1., _ 1,5 2 dunensionsless lime step At/I Fig. 1: Absolute values of the eigenvalue of the approximate transition matrix The absolute values X for the case of eqn (13) are shown in Fig. 1 as a function of At / T, where T denotes the period of the undamped vibration. Also shown are the exact solutions in the form of dashed lines. The errors accumulated within the time-interval while compared to the exact solution become: (15) It is seen that the accuracy is of second order. (Note that these results are purely artificial, since in practice one would treat linear oscillations by means of the exact transition matrix.) Similar estimates can be obtained for the errors in the vibration period and for the phenomenon of numerical damping (see Holl [6] for details): Here for the above algorithm some additional results are presented for the case of the equation of motion

7 Boundary Elements 305 * (17) The exact transfer matrix formulation for this case is knwon. The eqn (10) is calculated using the convolution integral and inserting the approximation _ Y - A + - T. The approximate transfer matrix A results after some algebraic manipulations. Fig. 2 shows the absolute values of the eigenvalues of the resulting algorithm with the parameters a = 0.3 and = 0.1. An analogous analysis was performed for the Newmark method. Based on the characteristic equations for 7= 1 and 0 = 1, see [2], [6], an application to eqn (17) was computed. In Fig. 2 the corresponding result for the Newmark method can be seen. 1 S? 3 a Newrr ark me od J\ / E act valu *s^ Pres ^s^ ent meth od ^ 0.2 k C=o.i 0=0.3 0 _L J L _L 0 0, Dimensionsless time step At/T 1 Fig. 2 Absolute values of the eigenvalues for the system of eqn (17) The result for the accuracy of the above presented algorithm is where XQ and XQ are the initial values at the beginning of the time step For the Newmark method the local error for = 0 results to (18) (19)

8 306 Boundary Elements We see, that y= ^ gives an algorithm, which has a convergence of second order. This result for the convergence does not change for the linear case a = 0, whereas from eqn (18) it is obvious, that for a = 0 the local error disappears, i.e. K = 0. 4 Examples 4.1 Free Vibrations of an Undamped Duffing Oscillator The above BEM-type time-integration scheme is applied to free vibrations of a Duffing oscillator, r = ~px^ with initial conditions 3 X = 4 of and %o = 0. The approximation (13) is used in a time-stepping procedure with CD At = 2 7t / 100. The increments of the state vector were calculated solving a cubic equation. Fig. 3 shows the error in the displacement e(t) = c exact versus the error in velocity a(t) = for the present method and for the Newmark method. It is seen that the present method is more accurate, while the CPU-time of the present method was smaller by a factor of 1/2. Comparisons with other direct time-integration schemes turned out to give an even better benefit for our method. 100 Present Method Newmark Method e(t) Fig. 3 Phase-plane of the non-dimensional error for the Duffing oscillator

9 Boundary Elements Examples of Systems with Time-dependent Mass As a further example a system with two DOF is analysed, where the matrices are taken from [14]: M = 110] -«<o,t c_[l j [3.8 Ol loir ' M ' *= 0 4 ' = lsin(vt) withv/g>j=l Itfe assumed, that this system has a time varying mass according to the function e with a = 1 /10, which reduces all elements of the mass matrix proporrt^j ^ S»ven system the computed results are shown in Fig. 4 for the first DOF and for homogeneous initial conditions in a dimensionless form. X,, is the static displacements. The four computed cases are: A) with full damping matrix; B) decoupling approximation when neglecting off-diagonal elements- C) as A but with time-variable mass; D) as B but withtime-variablemass I -1,5 0,5 1 1,5 2 Dimensionless time t/t Fig.4 Converged solution for the first DOF 2,5 5 Conclusion The presented semi-analytic algorithm makes use of the methods of linear structural dynamics and applies them in a consistent way to the analysis of transient vibrations of nonlinear rotordynamic systems. It is demonstrated that the resulting algorithm is efficient and accurate. The computation time for solving example problems shows an advantage for the present method when

10 308 Boundary Elements compared to the Newmark method. When using the same time step, the error to a converged solution in the computed results showed an advantage as well. References [1] Zienkiewicz, O.C.; Taylor, R.L.: The Finite Element Method, 4th edition, Vol. 2, McGraw Hill [2] Hughes, T.J.R.: Analysis of Transient Algorithms with Particular Reference to Stability Behavior. In: Computational Methods for Transient Analysis, eds.: Belytschko, T.; Hughes, T.J.R.; North-Holland, Elsevier Science Publishers B.V., [3] Irschik, H.; Ziegler, R: Dynamics of Linear Elastic Structures with Selfstress: A Unified Treatment for Linear and Nonlinear Problems. ZAMM 68 (6), [4] Irschik, H.; Ziegler, F.: Dynamic Processes in Structural Thermo- Viscoplasticity. Applied Mechanics Review 48 (6), , [5] Fotiu, P.; Irschik, H.; Ziegler, F.: Modal Analysis of Elastic-plastic Plate Vibrations by Integral Equations. Engineering Analysis with Boundary Elements 14, Elsevier, [6] Holl, H.J.: Ein effizienter Algorithmus fur nichtlineare Probleme der Strukturdynamik mit Anwendung in der Rotordynamik. Dissertation, University Linz, [7] Holl, H.J.: An Time Integration Algorithm For Time-Varying Systems With Nonclassical Damping Based on Modal Methods. Proceedings of the 15th IMAC, Orlando, ,1997. [8] Holl, H.J.: An Effizient Time Domain Formulation For Nonlinear Rotordynamic Systems Using Modal Reduction. Proceedings of the 13th IMAC, Nashville, [9] Nardini, D.; Brebbia, C.A.: Transient Dynamic Analysis by the Boundary Element Method. In: Boundary Elements, eds.: Brebbia, C.A.; Futagami, T.; Tanaka, M.; Springer Verlag , [10] Manolis, G.D.; Beskos, D.E.: Boundary Element Methods in Elastodynamics. Unwin Hyman [11] Mukherjee, S.: Boundary Element Methods in Creep and Fracture. Applied Science Publisher [12] Ziegler, F.: Mechanics of Solids and Fluids. Springer Verlag [13] Brebbia, C.A.; Dominguez, J.: Boundary Elements. 2nd edition, McGraw Hill [14] Hwang, J.H.; Ma, F.: On the Approximate Solution of Nonclassically Damped Linear Systems. Journal of Applied Mechanics 60, , 1993.