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 ∈ RN. Using the ansatz q(t) ≈ Vm· q(t), where q ∈ RN, q ∈ Rn, Vm∈ RN ×nand 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 Rm(t),
Rm(t) = Me· Vm· ¨q(t) + De· Vm· ˙q(t) + Ke· Vm· q(t) − Be· u(t), (1) represents the error induced by the reduction of the original system with a reduced system, where Me, De, Ke∈ RN ×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 em(t) = q(t) − Vm· ¯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 ˙x(t) = As· x(t) + Bs· u(t), y(t) = Cs· x(t) is given in [2]. The first order state-space model is reduced with the bi-orthonormal projection matrices Vsand Ws, therefore the differential equation of the error is
˙
es(t) = As· es(t) + Rs(t), (2)
where es = x(t) − Vs· x(t) is the error and Rs = As· Vs· x(t) + Bs· u(t) − Vs· ˙x(t) the residual of the reduced system. The explicit solution for this differential equation can be found in literature such as [3], i.e.
es(t) = Φ(t) · es,0+ Z t
0
Φ(t − τ ) · Rs(t)dτ, (3)
where Φ(t) = eAstis the fundamental matrix of the linear system ˙x = A
s· x and es,0denotes the initial error. In [2] an error bound 4x(t) for the state variable x is derived
kes(t)kGs≤ 4x(t) = C1kes,0kGs+ C1
Z t 0
kRs(τ )kGsdτ, (4)
where es,0 denotes the initial error and the constant C1 is given by C1 ≥ maxtkΦ(t)kGs. The error es(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 ∈ RN ×N. Therefore, k · kGs
is the induced norm with the scaled inner product ha, biG= bT· G · a.
A major advantage of the a-posteriori error estimator is that the estimator is independent of the used reduction tech-nique. 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
k
d
m
q
F
Figure 1: Simple mass spring damper system x y z F 20 7 14 1
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 Me· ¨q(t) + De· ˙q(t) + Ke· q(t) = Be· u(t), y(t) = Ce· q(t), is transformed into a first order system with
˙q(t) ¨ q(t) | {z } ˙ x(t) = 0 I −M−1 e ·Ke −Me−1·De | {z } As ·q(t) ˙ q(t) | {z } x(t) + 0 Be | {z } Bs · u(t), y(t) =Ce 0 | {z } Cs ·q(t) ˙ q(t) | {z } x(t) . (5)
Consequently, the dimensions of the first order and second order systems are related by Ns = 2N . The required bi-orthogonality of Wsand Vscan be ensured for any Wmand Vmby using the projection matrices
WsT =(W T m· Vm)−1· WmT 0 0 M−1e · WT m· Me , Vs= Vm 0 0 Vm . (6)
Furthermore, the relation of the error es(t) and residual Rs(t) to the second order mechanical system is derived es(t) = em(t) ˙ em(t) =q(t) − Vm· q(t) ˙ q(t) − Vm· ˙q(t) , Rs(t) = 0 e Rm(t) = 0 −M−1 e · Rm(t) . (7)
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 over-prediction 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 C1 = maxtkΦ(t)kGs = maxtke
Astk
Gs. However, this is partialy due to the fact that the error estimator determines
an error bound 4x(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 4q(t) will be derived in the next section to improve the output error estimate ignoring the velocities.
If the system ˙x(t) = As· x(t) is asymptotically stable, then all eigenvalues of As have nonpositive real parts and keAstk → 0 with t → ∞. However, this does not necessarily mean that keAstk decreases monotonically as t increases.
If As is not normal, meaning AHs · As 6= As· AHs, then keAstk 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) ¨ q(t) | {z } ˙ x = 0 1 −k m − d m | {z } As ·q(t) ˙ q(t) | {z } x + 01 m | {z } Bs F (t) | {z } u . (9)
If no external force is applied, then F (t) = 0 and the solution of this linear time-invariant differential equation is com-pletely described by the fundamental matrix Φ(t) = eAstand the initial condition x
0and reads x(t) = Φ(t) · x0= Φ11(t) Φ12(t) Φ21(t) Φ22(t) ·q0 ˙ q0 . (10)
As explained in [6] the fundamental matrix of this system can be calculated to be
eAst = T · diag(eλ1t, eλ2t) · T−1 = 1 λ2− λ1 λ2eλ1t− λ1eλ2t eλ2t− eλ1t λ1λ2(eλ1t− eλ2t) λ2eλ2t− λ1eλ1t (11)
where the matrices
T = 1 1 λ1 λ2 , T−1= 1 λ2− λ1 λ2 −1 −λ1 1 (12) follow from the Jordan canonical form of Aswhich is given by JA= T−1· As· T = diag(λ1, λ2) and
λ1=
−d −√d2− 4km
2m , λ2=
−d +√d2− 4km
2m , (13)
are the eigenvalues of As.
Plots of kΦ(t)k exhibit the hump phenomenon for certain parameters of m, d, and k. In Figure 3 the fundamental matrix norm kΦ(t)kGs is depicted and the hump phenomenon can be observed. For the Gs-norm the mass matrix is used
according to (18). The stiffness k = 10 kg/s and damping d = 0.2 kg/s2 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 Φ21(t) in (10) and plots of all entries of Φ(t) confirm that Φ21(t) is responsible for the hump. With the help of (11) and (13) the entry Φ21(t) follows as
Φ21(t) = 1 λ2− λ1 λ1λ2(eλ1t− eλ2t) = 2m 2√d2− 4km 4km 4m2(e λ1t− eλ2t) = ω 2 0 2pδ2− ω2 0 (eλ1t− eλ2t), (14)
where δ = 12md is the damping of the system and ω02=mk the square root of the eigenfrequency of the undamped system. For a stable system the term |(eλ1t− eλ2t)| < 1 is always bounded. Therefore, the term ω2
0/2pδ2− ω02is responsible for the hump phenomena.
This means the larger the eigenfrequency ω0 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 Φ21(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. 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 time [s] m=0.1kg m=0.01kg m=0.001kg
Figure 3: Norm of the fundamental matrix over time
0 0.2 0.4 0.6 0.8 1 40 30 20 10 0 10 m=0.1kg m=0.01kg m=0.001kg time [s]
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) ˙ em(t) =Φ11(t) Φ12(t) Φ21(t) Φ22(t) ·em,0 ˙ em,0 + Z t 0 Φ11(t − τ ) Φ12(t − τ ) Φ21(t − τ ) Φ22(t − τ ) · 0 e Rm(t) dτ, (15)
where the fundamental matrix Φ(t) = eAst ∈ R2N ×2N is decomposed into four submatrices Φ
ij ∈ RN ×N. Since usually only an error bound for em(t) is relevant, (15) is separated into
em(t) = Φ11(t) · em,0+ Φ12(t) · ˙em,0+ Z t
0
Φ12(t − τ ) · eRm(τ )dτ. (16) Be sure to note that only half of the fundamental matrix, i.e. Φ11and Φ12, is relevant for the computation of the error bound of the displacements q. The entry Φ21, which causes the large hump, is no longer required. In the next step, two error bounds e4q(t) and 4q(t) are derived, which fulfill
kem(t)kGm ≤ e4q(t) ≤ 4q(t), (17)
where e4q(t) is more accurate but requires more computation time than 4q(t). In order to maintain consistency to the error estimator for first order systems, the relationship
Gs=
Gm 0
0 Gm
(18)
is compulsory, because this yields the equivalence kRs(t)kGs = k eRm(t)kGm. For the following derivation the bounds
e
C11(t) ≥ kΦ11(t)kGm and eC12(t) ≥ kΦ12(t)kGmare introduced. Applying the triangle inequality and the definition of
the scaled matrix norm yields the error bound
e 4q(t) = eC11(t)kem,0kGm+ eC12(t)k ˙em,0kGm+ Z t 0 e C12(t − τ )k eRm(τ )kGmdτ. (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 C11 ≥ maxtCe11(t) and C12 ≥ maxtCe12(t) can be used to obtain less accurate, but therefore computationally less expensive error bounds from the altered estimate
4q(t) = C11kem,0kGm+ C12k ˙em,0kGm+ C12
Z t 0
k eRm(τ )kGmdτ. (20)
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 e4q(t) and 4q(t) can be used to obtain upper bounds for the output. Due to the relation
y(t) − y(t) = Cm· (q(t) − Vm· q(t)) = Cm· em(t), (21) upper bounds for the output can be computed from
ky(t) − y(t)k2≤ e4y(t) ≤ 4y(t), (22)
where e4y(t) = C24eq(t), 4y(t) = C24q(t) and C2 ≥ kCekGm,2. For the sake of simplicity, only 4y(t) is used in
the following, even though all derivations and conclusions are equally valid for e4y(t). A further improvement can be achieved by splitting the output matrix Ceinto r submatrices for each of the r output dimensions. This results in one error bound per output dimension,
ky(1)(t) − y(1)(t)k2≤ 4y(1)(t) = C2(1)4q(t), .. . ... ... ky(r)(t) − y(r)(t)k2≤ 4y(r)(t) | {z } 4y(t) = C2(r) | {z } C2 4q(t), (23)
which can be summarized into the vector-valued function
4y(t) = C2·4q(t), where C2= kCe(1,:)kGm,2 .. . kCe(r,:)kGm,2 . (24)
This improved output estimate is consistent with (22)) since C2= C2if only one output dimension is considered. Within the error estimation the initial errors em,0 and ˙em,0 are eliminated if all components of the initial condition x0 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 k eRm(t)kGm. 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 20 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 20 in z-direction is chosen as output of the system. The linkage is excited at node 20 by a sinusoidal force F with a frequency of 1 Hz and an amplitude of 100 N in z-direction. The model is reduced to twelve degrees of freedom using Krylov subspaces and the mass matrix is used for the Gm-norm, thus Gm= Me. Furthermore, the integration is performed with an ODE45 integrator from 0 to 2 s in the [m, kg, s] unit system. The output error bounds e4y(t) and 4y(t) are computed for the output with (24) and the results are depicted in Figure 5. The results clearly show the superiority of e4y(t) over 4y(t) for larger times. This is due to the fact that k eRm(τ )kGmand eC12(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 C1= maxtkΦkGs=
4.3492 · 104and C
2 = kCekGm,2= 6.9151 are used. For the new error estimator, however, only the upper half of the
fundamental matrix is considered and the norms C11= maxtkΦ11kGm= 1.0000 and C12= maxtkΦ12kGm= 0.0196 are
utilized. Since C11and C12are significantly smaller than C1, the results are substantially better. Computations confirm that C1= maxtkΦkGs= maxtkΦ21kGm= 4.3492 · 10
4and, therefore, that the submatrix Φ
21causes 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 com-putationally 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 bVm, cWm denote projection matrices and bΦ(t) ≡ eAtb the fundamental matrix of the reduced system with bA =
c
WmT · A · bVm. Considering that the fundamental matrix Φ(t) = eAstentirely describes the system behavior, it can be concluded that bΦ(t) = eAbstcompletely 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 kΦ11(t)kGm and kΦ12(t)kGm are
required. Therefore, we propose an approximation by
kΦ11(t)kGm ≈ k bΦ11(t)kGbm, kΦ12(t)kGm≈ k bΦ12(t)kGbm, (25)
where the reduced matrix bGm= WmT· Gm· Vmis used for the norm, which equals the definition of the reduced mass ma-trix cMein the common case Gm= Me. For (25) to be good approximations, we chose bVmand cWmas modal projection matrices. In Figure 6 we illustrate the resulting values of k bΦ11(t)kGbmusing the most dominant eigenmode (eigenmode 1)
in comparison to kΦ11(t)kGm. 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 2-114 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 eC11(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 obvi-ous that the local maxima yield an exponential envelope which is larger than what the less dominant eigenmodes 2-114 could yield. The same approach holds for eC12(t) as well. This approximation with the exponential envelope slightly worsens the results of the error estimator, depicted in Figure 5 and labeled y ± e4env
y , but significantly saves computation time. The computation of eC11(t), eC12(t), C11and C12with a standard computer (Intel Core 2 Duo CPU P8400, 2.27 Ghz, 4 GB RAM) took 56.251 s with full model and only 0.077 s with the approximation. Consequently, the approximation with eigenmode 1 is computed more than 700 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 Ashas more structure, e.g. as explained in [8] if Asis 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 C11and C12. Unfortunatelly, the class of systems with weak
column diagonally dominant matrices Aswith 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 As.
0 0.5 1 1.5 2 -0.1 -0.05 0 0.05 0.1 time [s] Output y [m]
Figure 5: Displacement of node 20 in z-direction with different error bounds 100.5 100.3 100.1 0 0.1 0.2 0.3 0.4 0.5 time [s] full system eigenmode 1 approximation exponential envelope
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 kΦ(t)k, which yields extremely high values for small but nonzero t. This problem was traced back to the norm of submatrix Φ21(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 Φ11(t) and Φ12(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 time-consuming, 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 310/1) at the University of Stuttgart.
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