Built on top of the LMS Virtual.Lab Structures solutions for component, subsystem and full vehicle modeling, analysis, and post-processing, LMS Virtual.Lab Wave Based Substructuring (WBS) brings innovative technology for fast structural and vibro-acoustic modification predictions and vehicle optimization, capable of speeding up simulation times by 90% while keeping results accuracy.
Introduction
Fast assembly predictions become ever more important in the vehicle development process. Design decisions are more and more based on virtual prototypes, as time-to-market and development costs must be reduced. The
additional trend of mass customization forces engineers to design a higher number of variants on a lower number of platforms.
To predict the structural and vibroacoustic vehicle performance, it is current practice to model the full vehicle
using finite element technology. And although Virtual.Lab developments have
decreased the full vehicle build time and even allowed to construct finite element models early in the development process – using morphing technology – it still is unpractical, even impossible, to perform quick modification predictions and hence optimization studies and make design decisions due to the computational costs on these ever growing full vehicle finite element model sizes.
To partially overcome this practical limit of long computation times, extensive work has been performed on substructuring and Component Mode Synthesis (CMS) techniques. The degrees of freedom (DOFs) of each substructure are expressed in terms of a limited number of component modes; the component models are then synthesized. Recently, Automated Multilevel Substructuring (AMLS) or Automated Component Mode Synthesis (ACMS) has been developed. A vehicle body is recursively divided into dozens of levels of in total thousands of substructures, based on the mathematical structure of the FE models, rather than on the physical composition of the system. Each substructure is separately solved and the results are synthesized. When coupling multiple (levels of) reduced systems, the interface problem size becomes increasingly dominant.
To further speed up the synthesis, the interface representation size between components must be reduced. This
can be done by the condensation of the interface displacements as a linear combination of a limited set of interface basis functions ("waves"). As the required number of basis functions is typically much lower than the number of interface DOFs, faster assembly predictions are obtained. In the Wave-Based Substructuring (WBS) approach available in LMS Virtual.Lab, one performs a single computation of the full assembly model to obtain the interface basis functions. This allows to accurately capturing the interface dynamics, and with modification analysis and optimization in mind, a single full computation is not a large burden any more: an optimization may
consist of numerous iterations involving hundreds of FE runs, so that it is very valuable to speed up the iterations
by reducing the size of the interface representation while maintaining the accuracy.
Wave Based Substructuring Assembly Procedure
Consider an FE substructure in an assembled system. The DOFs x can be divided into interior DOFs xi and junction (coupling) DOFs xj. The system matrices (for the undamped case) are then partitioned into submatrices:
In this formulation, all junction nodes have 6 DOFs. The Wave-Based Substructuring approach restricts the junction DOFs
xj, by expressing them as a linear combination of a set of
N interface basis functions V, weighted
with participation factors
p:
By substituting Equation (2) in Equation (1), one obtains
When the number of basis functions
N is less than the number of junction DOFs
xj, this substitution reduces the size of the FE matrix equations compared to classical CMS techniques. One must then translate the equilibrium and continuity conditions in terms of the interface basis functions.
For a rigid connection between two substructures a and b, the following equilibrium and continuity conditions
apply:
and
In a WBS framework, similar conditions apply on the participation factors
p:
and
An elastic connection between substructures
a and
b is applied with the following equilibrium and continuity condition (where the matrices
K denote the coupling stiffness matrices between the junction DOFs of the substructures):
Following Equation (2), for substructures a and b, the junction DOFs are expressed as a linear combination of sets of interface basis functions V
a and V
b, yielding
As the required number of basis functions is typically much lower than the number of interface DOFs, the Wave Based Substructuring results in a reduced-sized assembly definition, which allows faster assembly calculations.
Modal Reduction Procedure
In general, during vehicle modification and optimization, one will reduce the components which remain invariant. For example, in a vehicle optimization case to reduce the acoustic radiation of a floor panel, one will be interested to apply a reduction step on the remainder of the vehicle. This reduced-sized model of the remainder of the vehicle is then re-assembled with the FE representation of the floor panel, which enables much faster modifications and optimization on this floor panel.
Such a reduction step involves reduction of the number of degrees of freedom of the substructure model, by representing the physical degrees of freedom of each substructure into a reduced number of so-called generalized coordinates. All of them make use of the vibration normal modes of the substructures; they differ in the boundary conditions that are applied to the substructure, and in the selection of enrichment vectors to
these normal modes.
B-pillar re-design case: the full FE model for the body in white (BIW) is shown, next to the reduced assembly model created with Wave-Based Substructuring
(WBS). In this small-sized model, the B-pillars have an FE representation, and a reduced modal model has been created for the remainder of the BIW.
The reduced assembly model created with WBS will result in a much shorter analysis turnaround time, decreasing the simulation time with 99% when compared to the full FE model.
In Virtual.Lab, this reduction is a modal reduction, normal modes of the component in free-free conditions are
used, together with residual flexibility modes (which are the static deformation shapes obtained by successively applying a unit force on one of the interface degrees of freedom, with a zero force on the remaining interface DOFs, and repeating this for all interface DOFs). In such reduction methods, the component modes are generally computed up to a frequency higher than the frequency range of interest for
the assembly, and the enrichment vectors are of vital importance to accurately represent the local flexibility at the connection interface, when the reduced modal model of the substructure is re-assembled to other substructures.
It is clear that the number of enrichment vectors that must be computed, strongly (linearly) depends on the number of interface degrees of freedom, and that this becomes a major bottleneck in the classical CMS substructuring approaches, including AMLS or ACMS, when the number of interface degrees of freedom becomes large. That is why a reduction procedure based on WBS will be much more efficient, since the number of interface degrees of freedom ("waves") in the assembly definition is greatly reduced, hence also the number of residual vectors that needs to be computed.
An application example
The application examples one can think of are legio, and go from fast modification prediction over numerical optimization to even variability analysis and robust design. In general, all applications where the modification is in a relatively small area of the model and connection via a large number
of nodes benefit from the Wave Based Approach. If a small number of connection nodes exist (e.g. 5 or 10), it is better to use the CMS approach, which is also available in Virtual.Lab and can be applied as easily.
Typical applications performed so far can be found in structural and vibroacoustic analysis, such as body pillar re-design, floor plate optimization, panel bead optimization, spot weld variability and layout optimization, windscreen glue or sealing optimization and studying frequency dependent glue characteristics.
In the example here, Wave-Based Substructuring is applied to speed up the re-design of the B-pillars in a BIW model (200k elements). Modifications that can be considered are for panel thickness, material properties, number and position of spot welds, but also re-enforcements to strengthen the B-Pillar. In this case, there are 1788 physical connection DOFs, including the spot welds.
The observations are the following:
- Comparison of computation time, with a computation up to 100Hz is in this case 100x faster than the original run. In other words, one can perform 100 modification loops for the time needed to compute the full vehicle analysis.
Time for standard run: 1hour 42
Time for run using AMLS, in % of previous: 52%
Time for run using WBS: 60 sec
The WBS technique can very accurately predict the modification, as can be seen from the picture to the left.
Comparison versus standard CMS methodology. In the standard methodology, we need to compute 1788 residual vectors. This calculation was not feasible at the workstation available for the job (using 55GB scratch). The calculation of (only) half the required number of enrichment vectors was feasible, but required already much more time (more than 7 hours) just to get halfway the set of enrichment vectors! Moreover, this calculation time increases more than linear when the number of enrichment vectors is increased.
Conclusions
It is clear that the Wave Based Substructuring technology really complements and brings tremendous benefits over the classical substructuring technology. This approach is available inside LMS Virtual.Lab, where it complements the concept modeling and assembly definition and trimming functionality as it allows speeding up modification predictions and thus alleviates the computational burden of industrial vehicle optimization.
