A modal derivatives enhanced Rubin substructuring method for geometrically nonlinear multibody systems
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Abstract
This paper presents a novel model order reduction technique for 3D flexible multibody systems featuring nonlinear elastic behavior. We adopt the meanaxis floating frame approach in combination with an enhanced Rubin substructuring technique for the construction of the reduction basis. The standard Rubin reduction basis is augmented with the modal derivatives of both freeinterface vibration modes and attachment modes to consider the bending–stretching coupling effects for each flexible body. The meanaxis frame generally yields relative displacements and rotations of smaller magnitude when compared to the one obtained by the nodalfixed floating frame. This positively impacts the accuracy of the reduction basis. Also, when equipped with modal derivatives, the Rubin method better considers the geometric nonlinearities than the Craig–Bampton method, as it comprises vibration modes and modal derivatives featuring free motion of the interface. The nonlinear coupling between freeinterface modes and attachment modes is also considered. Numerical tests confirm that the proposed method is more accurate than Craig–Bampton’s, a nodal fixed floating frame counterpart originally proposed in Wu and Tiso (Multibody Syst. Dyn. 36(4): 405–425, [2016]), and produces significant speedups. However, the offline cost is increased because the meanaxis formulation produces operators with decreased sparsity patterns.
Keywords
Geometric nonlinearity Floating frame of reference Modal derivatives Rubin substructuring Meanaxis frame1 Introduction
The simulation of flexible multibody systems (FMBS) often relies on finite element (FE) discretization of flexible components, which are then embedded into a floating frame of reference (FFR) formulation [2, 3]. The FFR represents the position of each body as a superposition of two components: (i) the motion of the reference frame which follows the overall rigid body motion of the flexible body; (ii) the relative motion of the flexible body with respect to the reference frame. The floating frame approach is usually preferred to the description of the multibody motion with respect to the inertial frame (see, for instance, [4]) as it naturally distinguishes the elastic deformation from the rigid body motion. The resulting models often comprise a large number of degrees of freedom (DoFs), which render time integration schemes extremely costly. A relevant example of unaffordable computational burden could be found in the simulation of largescale offshore wind turbines. To assess their fatigue life, thousands of load cases need to be simulated, resulting in disproportionally large computation times. At present, this can be achieved only by relying on extremely simplified beam models that reduce the computational cost to a bearable level. Such models do not inherit the complexity of the actual threedimensional model of the blade, and, as a result, the complex dynamic behavior may not be appropriately represented. For this reason, many model order reduction (MOR) strategies for threedimensional FMBS have been proposed in the past. These techniques are based on classic modal truncation [5, 6] or singular value decomposition (SVD) based MOR techniques as in [7, 8, 9]. In [10], a global modal parametrization based MOR method is proposed, where the motion of the FMBS is described in terms of configuration dependent modes. Using this reduction method, the nonlinear holonomic constraints are naturally satisfied without the adoption of Lagrange Multipliers. However, in most of the MOR techniques, the elastic behavior is assumed to be linear. As discussed in [5], the linear MOR with FFR formulation is only suitable for structures featuring large rigid body motions but small relative displacements with respect to the reference frame, as well as slow rotational speeds. For FMBS featuring high rigid body rotation rates, the centrifugal force is of great significance, and therefore, the centrifugal stiffening effect and foreshortening effect have to be considered.
For many FMBS applications involving finite but moderate relative rotations with respect to the reference frame, neglecting geometrical nonlinearities may lead to incorrect and even diverging solutions [11, 12]. In [13], the geometrical nonlinearities are introduced in the equations of motion. As a result, the internal force vector and tangent matrix need to be recomputed for every iteration within each time step, therefore significantly impacting the computational cost. It is then a must to extend the linear MOR methods to the geometrically nonlinear regime for threedimensional FMBS.
When one substructure of the FMBS features geometrically nonlinear behavior, dominant lowfrequency modes are not sufficient for adequately representing the relative motion with respect to the reference frame. Typically, large slender structures exhibit coupling between bending and axial displacements when excited in the nonlinear regime. The corresponding bending–stretching coupling could be in principle provided by adding membranedominant (usually highfrequency) modes to the bendingdominant (typically lowfrequency) modes based reduction basis. For flat structures, where each vibration mode exhibits purely bending or membrane displacement, such membrane modes can be easily identified and added to the reducedorder basis (ROB). The inclusion of these socalled ad hoc modes has been applied in the FFR formulation in [14, 15]. However, for more complex geometries, the extraction of such modes is (i) challenging, as it is not straightforward to identify membranedominated modes, and (ii) expensive, as several modes need to be extracted.
In previous work [1], the linear Craig–Bampton (CB) substructuring basis [16] was enriched with modal derivatives (MDs) [17, 18] corresponding to lowfrequency fixedinterface modes. The augmented ROB was capable of capturing both the rigid body motions and the nonlinear relative displacement of the FMBS effectively. The nonlinear MOR technique was applied for nodalfixed frame reference [19], which is the most straightforward implementation of the FFR formulation. In this case, the reference frame is attached to specified nodes of the moving body. However, for complex structures, e.g., discretized with shell and solid elements, it is difficult to determine the optimal node whereon the reference frame should be attached. This arbitrary definition of the nodalfixed frame results in significantly different relative displacements and rotations with respect to the reference frame [19], and ultimately degrades the accuracy if the relative displacement and rotations are too large.
The use of meanaxis frame [20], which alleviates the need for the reference frame to be attached to a specified node of the structure, aims at minimizing the relative kinetic energy with respect to the reference frame. As a result, the largest relative displacement and rotation observed from a meanaxis frame will be smaller than the largest one observed when standing at the origin of the nodalfixed frame, as underlined in [19]. This is especially relevant when one assumes geometrical nonlinearities based on the von Kármán kinematic assumption, which is suitable for small strains and moderate rotations [21] with respect to the reference frame. Since the MDs are obtained from a truncated Taylor expansion of the nonlinear static equilibrium around the reference position [22, 23] and are not updated during the time integration, the accuracy of using MDs will be determined by how far the structure departs from the equilibrium position. Therefore, the use of MDs further supports the argument of using the meanaxis formulation.
In this paper, the standard Rubin substructuring technique [24] is enhanced with MDs and then implemented on the meanaxis frame formulation for the construction of reducedorder models (ROMs) for the FMBS featuring moderate relative displacements and rotations with respect to the reference frame. Each body is reduced by forming the ROB with attachment modes, freeinterface modes, and corresponding MDs. The Rubin method fits the meanaxis formulation more naturally than the CB method when applied to the geometrically nonlinear problem, for two reasons. First, the Rubin method is based on a truncated set of freeinterface vibration modes, which naturally describe the elastic deformation of the component with respect to the reference frame (i.e., freeinterface deformation with respect to the reference frame as in meanaxis frame formulation). Second, the nonlinear behavior occurring at the interface is better represented by MDs of both freeinterface modes and attachment modes (related to the Rubin method) than by MDs of fixed interface modes coming from the CB method. In [25], the inclusion of only the MDs relative to rigid body modes (i.e., vibration modes of zero frequency) in the ROB significantly increases the accuracy. In our approach, the MDs relative to rigid body modes are avoided since the rigid body motion has already been described by the reference frame motion. Therefore, a ROB of very limited size can be achieved.
This paper is organized as follows. Section 2 describes the FFR description featuring geometric nonlinearities. The nodalfixed and meanaxis frame are applied to the FFR formulation in Sect. 3. The assembled EoMs of all FMBS, as well as the holonomic joint constraints, are presented in Sect. 4. The nonlinear MOR method based on the enhanced Rubin method is proposed in Sect. 5. Section 6 shows numerical examples to assess the accuracy of the present formulation, especially emphasizing the improvements with respect to [1]. Finally, conclusions are given in Sect. 7.
2 Equations of motion in floating frame of reference
In this work, we adopt the von Kármán kinematic assumption for geometric nonlinearities, which is suitable for small strains and moderate rotations [21]. The elastic force \(\mathbf{f}\) can be directly derived from the differentiation of the strain energy and may be written as a thirdorder polynomial function of the relative DoFs \(\mathbf{q}_{f}\).
It should be noticed that the FE discretized components, without imposed constraints, allow relative rigid body motion of the flexible bodies with respect to the body reference frame. In the FFR formulation, however, the rigid body motion has already been described by the translation and rotation of the reference frame. To define a unique displacement field, we need to eliminate redundant DoFs, by imposing a set of reference constraints. This is discussed in the next section.
3 Floating frame definition
We now briefly summarize the nodalfixed definition [19] and the meanaxis definition [20] of the FFR, together with the embedding technique utilized to impose the constraints introduced by the meanaxis frame definition.
3.1 Nodalfixed frame
3.2 Meanaxis frame
3.3 Embedding of meanaxis and interface constraints
While enforcing Eq. (11) for nodalfixed frame is straightforward, the treatment of Eq. (18) requires more attention, since the constraint conditions are expressed as an explicit form of all relative DoFs \(\mathbf{q}_{f}\). By noticing that the meanaxis frame only yields linear constraints, we apply the socalled embedding technique [5] to obtain a minimum number of equations expressed in terms of independent coordinates. As mentioned in [27], the process of imposing all the reference conditions is actually equivalent to static condensation, where the slave variables are eliminated.
4 Flexible multibody equations
5 Enhanced Rubin substructuring method
5.1 Augmented Rubin reduction bases with modal derivatives
In this section, we extend the standard Rubin substructuring method by augmenting the associated reduction basis with MDs to properly consider geometric nonlinear effects. The ROBs are established for each body separately.
The MDs were first proposed in [17, 18] for a single structure not undergoing rigid body motion, by differentiating the eigenvalue problem associated to the free vibration with respect to the modal amplitude. The computation of MDs is discussed in detail in [30]. The methods in [30] require an explicit form of the internal nonlinear forces. Alternatively, Slaats [29] proposed the use of finite difference, which allows the computation of the MDs by means of standard nonlinear FE programs, as this method does not require access to the nonlinear forces and Jacobians. Related to this property, we applied the simplified definition of MDs by neglecting these inertia related terms. This technique is usually addressed as the definition without mass consideration, or more recently, static MDs [30]. A more theoretical grounding of the validity of MDs is given in [31].
In this paper, the von Kármán kinematic model is applied. Since the internal force vector and stiffness matrix can be explicitly expressed as a polynomial function of the DoFs, the MDs can be computed analytically.
5.2 Reduced equation of motion
In this work, we use the implicit Newmark scheme for the time integration of (50) by setting the integration parameters \(\gamma =\frac{1}{2}\) and \(\beta =\frac{1}{4}\). The artificial damping coefficient \(\alpha \) is set to zero for all the presented examples. The constraint equation is treated as discussed in [33], where the Lagrange multipliers have been set as additional DoFs. Substantial computational cost reduction can be achieved, in comparison to full analysis, thanks to the reduction in size and the efficient treatment of the nonlinear terms (49). The computational efficiency of applying the implicit Newmark scheme has been discussed in [1], and will not be repeated here.
6 Numerical examples

MFRHFML/NL: Linear/Nonlinear response of the High Fidelity Model (HFM) obtained from Meanaxis floating Frame of Reference (MFR);

NFRHFML/NL: Linear/Nonlinear response of the HFM obtained by Nodalfixed floating Frame of Reference (NFR);

MFRERubinNL: Nonlinear response of ROMs obtained by projection on the Enhanced Rubin basis (with MDs) for MFR, as discussed in this work;

NFRECBNL: Nonlinear response of the ROMs obtained by projection on the Enhanced Craig–Bampton basis (with MDs) for NFR, as discussed in [1];

MFRRubinNL: Nonlinear response of ROMs obtained by the projection on the standard Rubin basis (without MDs) for MFR;

NFRCBNL: Nonlinear response of ROMs obtained by the projection on the standard CB basis (without MDs) for NFR.
6.1 Model 1: rotating beam
Size of ROB for the rotating beam model
Number of modes in linear ROB  Number of MDs  Total DoFs  

NFRECBNL  10  10  20 
MFRERubinNL  10  10  20 
NFRCBNL  50  0  50 
MFRRubinNL  50  0  50 
6.2 Model 2: 5 MW/61.5 m wind turbine blade
The blade is assumed to rotate around the \(x\)axis with a constant speed \(\varOmega =8~\text{rad}/\text{s}\) and a physical time of 100 s is simulated. For the time integration, we use a fixed time step of 0.02 s, with updating of the tangential operator at each iteration within one time step, with a convergence criterion on the norm of the force residual relative to the norm of the internal force vector (tolerance set to \(10^{6}\)).
Number of DoFs for the 61.5 m blade model of the NREL 5 MW reference wind turbine
Number of modes in linear ROB  Number of enriched MDs  Total DoFs  

NFRECBNL  10  15  25 
MFRERubinNL  10  15  25 
NFRCBNL  50  0  50 
MFRRubinNL  50  0  50 
The computational time is compared between the ROMs enriched with MDs (NFRECBNL and MFRERubinNL) and the HFMs (MFRHFMNL and NFRHFMNL). All simulations are performed in MATLAB^{®}R2015, on a cluster equipped with 8core Intel^{®} Xeon^{®} CPUs (E52630v3) @ 2.4 GHz and 128 GB RAM.
Computational cost for the 61.5 m wind turbine blade (100 s physical time)
Floating frame  HFM  ROM  Number of iterations  Speed up factor  

Offline  Online  
\(t_{\mathrm{full}}\) (s)  \(t_{\mathrm{off}_{1}}\) (s)  \(t_{\mathrm{off}_{2}}\) (s)  \(t_{\mathrm{on}}\) (s)  \(\mathcal{N}\)  \(\mathcal{S}_{1}\)  \(\mathcal{S}_{2}\)  
Meanaxis  155307  2053  230  72  10598  2157  66 
Nodalfixed  49254  13.25  241  125  17758  394  129.87 
It can be observed that \(t_{\mathrm{full}}\) and \(t_{\mathrm{off}_{1}}\) of the meanaxis frame are larger than their counterparts of the nodalfixed frame, since the stiffness and mass matrices \(\mathbf{M}_{mm}\) and \(\mathbf{K}_{mm}\) feature a worse sparsity pattern due to the condensation of the mean axis constraints. Therefore, the eigenvalue analysis in Eq. (34), the calculation of MDs in Eq. (41), as well as tangent operator calculation in Newmark time integration are more expensive than their correspondents in the nodal fixed frame. The offline cost \(t_{\mathrm{off}_{2}}\) is similar for the meanaxis frame and nodalfixed frame ROMs, as \(t_{\mathrm{off}_{2}}\) is mainly determined by the size of ROBs, i.e., the number of modes included in the reduction basis. On the contrary, \(t_{\mathrm{on}}\) in the meanaxis frame is much smaller than its counterpart in the nodalfixed frame. This is due to the fact that the MFRERubinNL requires fewer iterations for a given time step because of smaller relative DoFs \(\mathbf{q}_{f}\), although NFRECBNL and MFRERubinNL contain the same number of modes in the ROB and their corresponding computational time per iteration is similar.
7 Conclusions
This paper presents a modelorder reduction technique for flexible multibody systems featuring geometrically nonlinear elastic behavior. The overall motion of each body is described with the meanaxis floating frame of reference. The relative displacements of each body are then represented by a basis obtained by enhancing the standard Rubin substructuring basis with modal derivatives computed for both free vibration modes and attachment modes. This allows to accurately capture the geometrically nonlinear elastic behavior of the deformable body. When compared with a previous contribution [1], where a modal derivativesenhanced Craig–Bampton substructuring method is applied in the nodalfixed floating frame, the present approach offers a better representation of the nonlinearity at the interface, since the coupling between attachment modes and freeinterface modes is considered.
For the reducedorder model, the modal derivatives essentially represent secondorder terms of the Taylor expansion of the displacements from the undeformed configuration. As such, it is essential to minimize the relative displacements and rotations with respect to the reference frame. The meanaxis formulation indeed provides generally smaller relative displacements and rotations than their counterpart in the nodalfixed frame, thus improving the accuracy of the reducedorder model, as shown in the numerical examples.
The method provides significant computational gains when tested on the simulation of a flexible wind turbine blade featuring about 24000 degrees of freedom. The necessary offline cost for the computation of reduction basis is higher than the one proposed obtained in [1], since the projected matrices feature worse sparsity pattern due to the embedding of the meanaxis constraints. However, the online speedup is better for the chosen test as compared to the one achieved in [1] for a reducedorder model of equal size. This is due to the fewer iterations within a time step required for convergence, as the relative displacements with respect to the reference frame are smaller. This technique is particularly useful when several load conditions need to be simulated so that the offline cost can be amortized.
Footnotes
Notes
Acknowledgements
Prof. Eleni Chatzi would like acknowledge the support of the ERC Starting Grant award (ERC2015StG #679843) on the topic of “Smart Monitoring, Inspection and LifeCycle Assessment of Wind Turbines”.
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