A-Posteriori Error Estimation for Second Order Mechanical Systems

Transcription

1 ENOC 214, July 6-11, 214, Vienna, Austria A-Posteriori Error Estimation for Second Order Mechanical Systems Jörg Fehr, Thomas Ruiner, Bernard Haasdonk, Peter Eberhard Institute of Engineering and Computational Mechanics, University of Stuttgart, Stuttgart, Germany Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Stuttgart, Germany Summary. One important issue for the simulation of flexible multibody systems is the reduction of the flexible body s degrees of freedom. As far as safety questions are concerned, knowledge about the error introduced by reduction of the flexible degrees of freedom is crucial. Here, an a-posteriori error estimator for linear first order systems is extended for error estimation of mechanical second order systems. The error estimator can deliver impractical results for second order mechanical systems originating from the hump phenomenon. Due to the special second order structure of mechanical systems, an improvement of the a-posteriori error estimator is achieved. The decisive advantage of the proposed a-posteriori error estimator is its independence of the reduction technique applied. Therefore, it can be used for reduction processes based on Krylov-subspaces, Gramian matrices or even modal techniques. The capability of the proposed technique is demonstrated by the a-posteriori error estimation of a mechanical system. Introduction One essential step for an efficient simulation of an elastic multibody system (EMBS) is the reduction of the elastic degrees of freedom, see e.g. 1. Frequently, a Petrov-Galerkin ansatz is used to reduce the elastic displacements q R N. Using the ansatz q(t) V m q(t), where q R N, q R n, V m R N n and n N, the flexible coordinates q can be reduced from N to n degrees of freedom. However, the reduction process introduces an error. The residual R m (t), R m (t) M e V m q(t) + D e V m q(t) + K e V m q(t) B e u(t), (1) represents the error induced by the reduction of the original system with a reduced system, where M e, D e, K e R N N denote the flexible mass, damping, and stiffness of the elastic system. Regarding safety questions it is important to know details about the error introduced by the reduction of the flexible degrees of freedom. This is especially important in crash simulations or optimization based on damage values. Existing a-priori and a-posteriori error estimators investigate the error in the frequency-domain, see e.g. 1. This analysis focuses on error estimation in the time domain, where the error is defined as e m (t) q(t) V m q(t). Efficient a-posteriori error estimation in the time-domain is used in the reduced basis community, e.g. a formulation for the first order state-space model ẋ(t) A s x(t) + B s u(t), y(t) C s x(t) is given in 2. The first order state-space model is reduced with the bi-orthonormal projection matrices V s and W s, therefore the differential equation of the error is ė s (t) A s e s (t) + R s (t), (2) where e s x(t) V s x(t) is the error and R s A s V s x(t) + B s u(t) V s ẋ(t) the residual of the reduced system. The explicit solution for this differential equation can be found in literature such as 3, i.e. e s (t) Φ(t) e s, + Φ(t τ) R s (t)dτ, (3) where Φ(t) e Ast is the fundamental matrix of the linear system ẋ A s x and e s, denotes the initial error. In 2 an error bound x (t) for the state variable x is derived e s (t) Gs x (t) C 1 e s, Gs + C 1 R s (τ) Gs dτ, (4) where e s, denotes the initial error and the constant C 1 is given by C 1 max t Φ(t) Gs. The error e s (t) needs to be measured in an adequate norm. Standard norms, such as the 2-norm, are not feasible if elements in a vector or matrix have different units, since e.g. the units of displacement and rotation are hardly comparable. Under these circumstances it is necessary to normalize the various entries, which is achieved by using a scaling matrix G R N N. Therefore, Gs is the induced norm with the scaled inner product a, b G b T G a. A major advantage of the a-posteriori error estimator is that the estimator is independent of the used reduction technique. Therefore, the estimator can be used for moment-matching based, Gramian matrices based or modal based model reduction techniques. For the efficient calculation of the error usually an offline/online decomposition is used 2. In the next section the a-posteriori error estimator (4) is applied to mechanical second order systems and tested for a simple second order MIMO system written as a first order system. A simple example is used to explain the problem due to the hump phenomenon. Afterwards the a-posteriori error estimator is extended for error estimation of mechanical second order systems and the capability of the proposed technique is demonstrated by the a-posteriori error estimation of a stabilization linkage of a car front suspension, see Figure 2. One weakness of the error estimators is the need of the fundamental matrix Φ(t), whose computation is computationally extremely expensive. Consequently, the performance of

2 ENOC 214, July 6-11, 214, Vienna, Austria z y 2 F k d m Figure 1: Simple mass spring damper system q F 1 x 14 7 Figure 2: Constrained stabilization linkage of a car the error estimator can be improved significantly by approximating the fundamental matrix norm. This work is completed by some conclusions. Please consider similar and extended results published by the authors in 4. However, here additonal insight regarding the hump phenomena and further information regarding the possibility to derive even better error bounds are given. A-posteriori error estimation for second order mechanical systems Mechanical systems are usually described as second order dynamical systems and it is very important to use second order model reduction techniques to maintain the second order structure of the system 1. In order to utilize the error estimator for first order systems, a second order system M e q(t) + D e q(t) + K e q(t) B e u(t), y(t) C e q(t), is transformed into a first order system with q(t) q(t) ẋ(t) I q(t) Me 1 K e Me 1 D e q(t) A s x(t) + Be u(t), y(t) C e B s C s q(t). (5) q(t) x(t) Consequently, the dimensions of the first order and second order systems are related by N s 2N. The required biorthogonality of W s and V s can be ensured for any W m and V m by using the projection matrices (W Ws T T m V m ) 1 Wm T Vm M 1 e Wm T, V s. (6) M e V m Furthermore, the relation of the error e s (t) and residual R s (t) to the second order mechanical system is derived em (t) q(t) Vm q(t) e s (t), R ė m (t) q(t) V m q(t) s (t). (7) R m (t) R m (t) M 1 e Hump phenomenon for second order systems After transforming the second order system into a first order system with (5), the error estimator (4) from 2 can be applied. The error estimator might deliver impractical results for second order mechanical systems, as a very large overprediction of the error can be observed. This problem originates from the hump phenomenon, explained e.g. in 5, which causes extreme values of the constant C 1 max t Φ(t) Gs max t e Ast Gs. However, this is partialy due to the fact that the error estimator determines an error bound x (t) for the state variable x and, therefore, a single error bound for both state variables, q and q. The large hump is actually related to the velocity states q. Luckily for elastic body simulations only the position states q are relevant for an error estimation. Therefore, an error bound q (t) will be derived in the next section to improve the output error estimate ignoring the velocities. If the system ẋ(t) A s x(t) is asymptotically stable, then all eigenvalues of A s have nonpositive real parts and e Ast with t. However, this does not necessarily mean that e Ast decreases monotonically as t increases. If A s is not normal, meaning A H s A s A s A H s, then e Ast can grow arbitrarily large for small but nonzero t and decrease afterwards. A mathematical example for this phenomenon is provided e.g. in 5. The phenomenon can be illustrated with a simple mass spring damper system with one degree of freedom, depicted in Figure 1. The equation of motion written as a second order linear time-invariant system reads m q(t) + d q(t) + kq(t) F (t), (8) where the force F (t) is the input into the system. Analogously to (5) this system can be rearranged into q(t) 1 q(t) q(t) k m d + 1 F (t). (9) q(t) m m }{{} }{{} ẋ A s x B s u

3 ENOC 214, July 6-11, 214, Vienna, Austria If no external force is applied, then F (t) and the solution of this linear time-invariant differential equation is completely described by the fundamental matrix Φ(t) e Ast and the initial condition x and reads Φ11 (t) Φ x(t) Φ(t) x 12 (t) q. (1) Φ 21 (t) Φ 22 (t) q As explained in 6 the fundamental matrix of this system can be calculated to be e As t T diag(e λ1t, e λ2t ) T 1 1 λ2 e λ1t λ 1 e λ2t e λ2t e λ1t λ 2 λ 1 λ 1 λ 2 (e λ1t e λ2t ) λ 2 e λ2t λ 1 e λ1t where the matrices T 1 1, T 1 λ 1 λ 2 1 λ 2 λ 1 λ2 1 λ 1 1 follow from the Jordan canonical form of A s which is given by J A T 1 A s T diag(λ 1, λ 2 ) and (11) (12) λ 1 d d 2 4km 2m, λ 2 d + d 2 4km, (13) 2m are the eigenvalues of A s. Plots of Φ(t) exhibit the hump phenomenon for certain parameters of m, d, and k. In Figure 3 the fundamental matrix norm Φ(t) Gs is depicted and the hump phenomenon can be observed. For the G s -norm the mass matrix is used according to (18). The stiffness k 1 kg/s and damping d.2 kg/s 2 are constant and the mass is varied. Even with this simple system the hump phenomenon can be explained visually, e.g. by assuming an initial displacement. Once released, the displacement decays due to the damping and the mass eventually reaches its equilibrium position, possibly after a few oscillations. In order to reach the equilibrium state, however, the velocity must increase. Thus, in contrast to the displacement, the velocity increases at first and causes the hump phenomenon. This connection between initial displacement and velocity is represented by the entry Φ 21 (t) in (1) and plots of all entries of Φ(t) confirm that Φ 21 (t) is responsible for the hump. With the help of (11) and (13) the entry Φ 21 (t) follows as Φ 21 (t) d 1 λ 2 λ 1 λ 1 λ 2 (e λ1t e λ2t ) 2m 2 d 2 4km 4km 4m 2 (eλ1t e λ2t ) ω 2 2 (e λ1t e λ2t ), (14) δ 2 ω 2 where δ 1 2 m is the damping of the system and ω2 k m the square root of the eigenfrequency of the undamped system. For a stable system the term (e λ1t e λ2t ) < 1 is always bounded. Therefore, the term ω/2 2 δ 2 ω 2 is responsible for the hump phenomena. This means the larger the eigenfrequency ω the larger the hump. From a physical point of view this means that the stiffer the spring and the lighter the mass the more the mass is accelerated and the faster it oscillates before reaching the equilibrium state. This is exactly what can be observed in the hump phenomena of the fundamental matrix, depicted in Figure 3. The large value of Φ 21 (t) for small time t, compare Figure 4, is responsible for the hump. The hump increases dramatically when a smaller mass is used. Overall, it can be concluded that the hump is particularly pronounced in case of high-frequency oscillations m.1kg m.1kg m.1kg time s Figure 3: Norm of the fundamental matrix over time 3 m.1kg m.1kg m.1kg time s Figure 4: Entry Φ 21(t) of the fundamental matrix

4 ENOC 214, July 6-11, 214, Vienna, Austria Derivation of the error estimator for second order mechanical system The knowledge from the previous section is utilized now to refine the error estimator. The differential equation of the error (2) is the basis of the refined error estimator. For a second order mechanical system, (2) yields em (t) ė m (t) Φ11 (t) Φ 12 (t) Φ 21 (t) Φ 22 (t) em, ė m, + Φ11 (t τ) Φ 12 (t τ) Φ 21 (t τ) Φ 22 (t τ) R m (t) dτ, (15) where the fundamental matrix Φ(t) e Ast R 2N 2N is decomposed into four submatrices Φ ij R N N. Since usually only an error bound for e m (t) is relevant, (15) is separated into e m (t) Φ 11 (t) e m, + Φ 12 (t) ė m, + Φ 12 (t τ) R m (τ)dτ. (16) Be sure to note that only half of the fundamental matrix, i.e. Φ 11 and Φ 12, is relevant for the computation of the error bound of the displacements q. The entry Φ 21, which causes the large hump, is no longer required. In the next step, two error bounds q (t) and q (t) are derived, which fulfill e m (t) Gm q (t) q (t), (17) where q (t) is more accurate but requires more computation time than q (t). In order to maintain consistency to the error estimator for first order systems, the relationship Gm G s (18) G m is compulsory, because this yields the equivalence R s (t) Gs R m (t) Gm. For the following derivation the bounds C 11 (t) Φ 11 (t) Gm and C 12 (t) Φ 12 (t) Gm are introduced. Applying the triangle inequality and the definition of the scaled matrix norm yields the error bound q (t) C 11 (t) e m, Gm + C 12 (t) ė m, Gm + C 12 (t τ) R m (τ) Gm dτ. (19) High computational cost is caused by the integral which must be reevaluated for each time t. However, the evaluation of the integral could also be limited to the final time or other relevant points in time. For all other times the factors C 11 max t C11 (t) and C 12 max t C12 (t) can be used to obtain less accurate, but therefore computationally less expensive error bounds from the altered estimate q (t) C 11 e m, Gm + C 12 ė m, Gm + C 12 R m (τ) Gm dτ. (2) In this estimate there is no need to reevaluate the integral, because after every time step the new information can simply be added to the integral of the previous time. Both error bounds for the state variable q (t) and q (t) can be used to obtain upper bounds for the output. Due to the relation upper bounds for the output can be computed from y(t) y(t) C m (q(t) V m q(t)) C m e m (t), (21) y(t) y(t) 2 y (t) y (t), (22) where y (t) C 2 q (t), y (t) C 2 q (t) and C 2 C e Gm,2. For the sake of simplicity, only y (t) is used in the following, even though all derivations and conclusions are equally valid for y (t). A further improvement can be achieved by splitting the output matrix C e into r submatrices for each of the r output dimensions. This results in one error bound per output dimension, y (1) (t) y (1) (t) 2 y(1) (t) C 2(1) q (t),. y (r) (t) y (r) (t) 2 y(r) (t) C 2(r) q (t), y(t) C 2 which can be summarized into the vector-valued function C e(1,:) Gm,2 y (t) C 2 q (t), where C 2... (24) C e(r,:) Gm,2.. (23)

5 ENOC 214, July 6-11, 214, Vienna, Austria This improved output estimate is consistent with (22)) since C 2 C 2 if only one output dimension is considered. Within the error estimation the initial errors e m, and ė m, are eliminated if all components of the initial condition x are in the reduced space. However, if this condition is not fulfilled, then the initial error must be taken into consideration. A further essential step in the estimation procedure is the calculation of the residual norm R m (t) Gm. For the efficient calculation of the residual norm an offline/online decomposition is advantageous and a further development of the method is proposed in 2. Application example The application example is the model of a stabilization linkage of a car front suspension, which was introduced in 7. The model consists of 19 3D beam elements and 2 nodes. A fixed displacement boundary condition is applied to the first node, as depicted in Figure 2. This leaves a total of 114 degrees of freedom for the constraint model, since every node has 6 degrees of freedom, namely three displacements and three rotations. In this example, the displacement of node 2 in z-direction is chosen as output of the system. The linkage is excited at node 2 by a sinusoidal force F with a frequency of 1 Hz and an amplitude of 1 N in z-direction. The model is reduced to twelve degrees of freedom using Krylov subspaces and the mass matrix is used for the G m -norm, thus G m M e. Furthermore, the integration is performed with an ODE45 integrator from to 2 s in the m, kg, s unit system. The output error bounds y (t) and y (t) are computed for the output with (24) and the results are depicted in Figure 5. The results clearly show the superiority of y (t) over y (t) for larger times. This is due to the fact that R m (τ) Gm and C 12 (t τ) are fairly large in the beginning, but decay over time and consequently, attenuate each other due to the convolution in the error estimator, see (19). This example also demonstrates the advantage of the new estimator for second order mechanical systems over the original error estimator. This becomes obvious from a comparison between the different norms of the fundamental matrix, which are required for both error estimators. For the original error estimator the norm of the full fundamental matrix is required and the constant C 1 max t Φ Gs and C 2 C e Gm, are used. For the new error estimator, however, only the upper half of the fundamental matrix is considered and the norms C 11 max t Φ 11 Gm 1. and C 12 max t Φ 12 Gm.196 are utilized. Since C 11 and C 12 are significantly smaller than C 1, the results are substantially better. Computations confirm that C 1 max t Φ Gs max t Φ 21 Gm and, therefore, that the submatrix Φ 21 causes these extremely high values analogously to the pictorial example from the previous section. Approximation of the norms of the fundamental matrix One weakness of the derived error estimator is the need of the fundamental matrix Φ(t), whose computation is computationally extremely expensive. Consequently, the performance of the error estimator can be improved significantly by approximating the fundamental matrix norm. This is achieved by using a second low-dimensional reduced model. Let Vm, Ŵm denote projection matrices and Φ(t) eât the fundamental matrix of the reduced system with  Ŵ T m A V m. Considering that the fundamental matrix Φ(t) e Ast entirely describes the system behavior, it can be concluded that Φ(t) eâst completely describes the reduced system behavior. For the error estimation of second order systems the full fundamental matrix Φ is not needed explicitly since only the norms Φ 11 (t) Gm and Φ 12 (t) Gm are required. Therefore, we propose an approximation by Φ 11 (t) Gm Φ 11 (t) Ĝm, Φ 12 (t) Gm Φ 12 (t) Ĝm, (25) where the reduced matrix Ĝm Wm T G m V m is used for the norm, which equals the definition of the reduced mass matrix M e in the common case G m M e. For (25) to be good approximations, we chose V m and Ŵm as modal projection matrices. In Figure 6 we illustrate the resulting values of Φ 11 (t) Ĝm using the most dominant eigenmode (eigenmode 1) in comparison to Φ 11 (t) Gm. We see that eigenmode 1 is not sufficient for all times, even though the approximation is very accurate for the local maxima. The discrepancy in between these maxima is due to the eigenmodes and using more eigenmodes leads to an improvement of the approximation. However, it is difficult to find the required number of eigenmodes and, therefore, we choose as C 11 (t) the exponential envelope of the curve of the most dominant eigenmode, illustrated by the dashed red line. As the most dominant eigenmode 1 is also the least damped eigenmode, it seems obvious that the local maxima yield an exponential envelope which is larger than what the less dominant eigenmodes could yield. The same approach holds for C 12 (t) as well. This approximation with the exponential envelope slightly worsens the results of the error estimator, depicted in Figure 5 and labeled y ± env y, but significantly saves computation time. The computation of C 11 (t), C 12 (t), C 11 and C 12 with a standard computer (Intel Core 2 Duo CPU P84, 2.27 Ghz, 4 GB RAM) took s with full model and only.77 s with the approximation. Consequently, the approximation with eigenmode 1 is computed more than 7 times faster in this example. However, by approximating the fundamental matrix Φ(t) with the envelope of the eigenmodes the mathematically proven error bound derived in (22) is not valid any more. Further work is necessary to derive a fast and error bound approximation of the fundamental matrix Φ(t). For systems where the system matrix A s has more structure, e.g. as explained in 8 if A s is a weak column diagonally dominant matrix with negative main diagonal, refined error estimators can be derived. Such error estimators allow a fast calculation of the factors C 11 and C 12. Unfortunatelly, the class of systems with weak

6 ENOC 214, July 6-11, 214, Vienna, Austria column diagonally dominant matrices A s with negative main diagonal is limited and second order mechanical system do not belong to this class of systems due to the zero upper left block in A s. Output y m full system eigenmode 1 approximation exponential envelope time s Figure 5: Displacement of node 2 in z-direction with different error bounds time s Figure 6: Results of the full system and an approximations with a one degree of freedom system using the first dominant eigenmodes Conclusion In this article the error estimator for first order state-space models from 2 has been applied to second order systems. Therefore, the relationship between reduced first and second order systems has been derived, such that projection matrices from standard reduction techniques can be used. It has been found that the original error estimator delivers impractical results for second order systems representing flexible bodies. This originates from a large hump of the fundamental matrix norm Φ(t), which yields extremely high values for small but nonzero t. This problem was traced back to the norm of submatrix Φ 21 (t) and is related to high frequency oscillations with very low amplitudes, where the hump represents the high velocity of those. However, a modified error estimator has been derived for second order systems from flexible bodies, which does not require this submatrix. This refined error estimator yields good error bounds for the model of a stabilization linkage, which was successfully used as an illustrative example to reproduce every step of the error bound computation. The required submatrices Φ 11 (t) and Φ 12 (t) of the fundamental matrix Φ(t) are computationally extremely expensive, but can be computed during the offline phase prior to the simulation. However, the fundamental matrix is not only timeconsuming, but also very difficult to compute, especially for large systems. In this article an approximation technique for these norms has been developed on the basis of modal analysis. If the envelope of the resulting curves is used, then the fundamental matrix can be reduced to one single degree of freedom using the most dominant eigenmode. This solves both problems and the necessary norms can be computed fast and reliably. Acknowledgements: The authors would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 31/1) at the University of Stuttgart. References 1 Fehr, J. (211). Automated and Error-controlled Model Reduction in Elastic Multibody Systems. Dissertation, Schriften aus dem Institut für Technische und Numerische Mechanik der Universität Stuttgart, Band 21. Shaker Verlag. Aachen. 2 Haasdonk B., Ohlberger M. (211). Efficient reduced models and a-posteriori error estimation for parametrized dynamical systems by offline/online decomposition. Mathematical and Computer Modelling of Dynamical Systems 17(2): Müller P., Schiehlen W. (1985). Linear Vibrations. Martinus Nijhoff Publ., Dordrecht. 4 Ruiner T., Fehr J., Haasdonk B., Eberhard P. (212). A-posteriori error estimation for second order mechanical systems. Acta Mechanica Sinica 28(3): Moler C., Van Loan C. (1978). Nineteen dubious ways to compute the exponential of a matrix. SIAM Review 2: Stykel, T., Mehrmann, V. (212). Differential equations and stability. Handbook of Linear Algebra Chapman: , Chapman and Hall/CRC Press, Boca Raton. 7 Wallrapp, O., Wiedemann, S. (23). Comparison of results in flexible multibody dynamics using various approaches. Nonlinear Dynamics 34: Hasenauer, J., Löhning, M., Khammash, M., Allgöwer, F. (212). Dynamical optimization using reduced order models: A method to guarantee performance. Journal of Process Control 22: