8.2 Basic simulation techniques
8.2.1 Finite element-based techniques
Finite element (FE) methods are computational methods to simulate struc- tural performance by use of numerical models. The basic concept of all FE techniques is to represent the geometry of each single component by a set of numerous small-sized elements (fi nite elements) for which well-known structural performance parameters such as stress, strain, temperature, etc., can be described by analytical (not necessarily linear) formulas [1, 2]. These elements can be simple spring elements, ‘Euler–Timoshenko’ beams, shell elements of triangular or quadrangular size, volume elements like tetrahe- drons or hexahedrons, concentrated mass points (lumped masses), etc.
Several so-called ‘meshing’ tools are available on the market to generate the respective FE models based on CAD geometry. Strictly speaking, the structural performance of each of these fi nite elements can be described by
Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com ETH Zuerich (307-97-768) Sunday, August 28, 2011 12:05:48 AM IP Address: 129.132.208.2
the dynamic equation consisting of the element mass matrix, the element stiffness matrix and the ‘response’ vector.
The overall mass and stiffness matrix as well as the overall response vector are created from the respective individual matrices taking into con- sideration connections between individual elements as well as boundary conditions. Contemporary computer systems – from high-power computer clusters down to single desktop PCs – are capable of effectively solving these multi-degree-of-freedom equations and of generating requested structural results.
The full vehicle noise and vibration FE model consists of several com- ponent FE models, mainly of the following:
• Trimmed body FE model
• (Interior) cavity models
• Fluid–structure boundaries
• Chassis models
• Powertrain model
• Bushing and connections.
The trimmed body itself usually consists of the body in prime (i.e. all body sheet metal with primer and all fi xed glazing; this model is created from shell elements and connections representing spot welds, seam welds or bonding), closures, interior trim (e.g. sound package material), instrument panel, seats, and many further parts rigidly mounted to the body structure.
The latter can be represented with differing degrees of accuracy from simple concentrated masses to a detailed FE model, depending on the respective CAE task.
Cavity models represent air inside the passenger cabin, in the trunk (boot) and for dedicated under-hood (bonnet) noise analyses, even air in the engine bay. It is obvious that modelling of the boundary conditions for the engine bay cavity is non-trivial and requires correlation exercises ahead of any noise and vibration prediction.
Fluid–structure coupling describes the interrelationship between ‘sheet metal vibration’ and in-cabin sound pressure. These fl uid–structure interac- tions need to be capable not only of generating sound pressure due to boundary structural vibrations, but also, vice versa, of generating structural vibrations due to sound pressure excitation. State-of-the-art tools are well capable of meeting these requirements with limited modelling and compu- tational efforts.
Chassis noise and vibration models (e.g. suspension cross-members, lon- gitudinal and lateral suspension links, anti-roll bar, twist beam rear axle, knuckle) can be easily derived from the respective CAE models for strength and durability analyses. Even though non-linear FE models are normally used to analyse strain effects and plastic deformation under extreme load-
Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com ETH Zuerich (307-97-768) Sunday, August 28, 2011 12:05:48 AM IP Address: 129.132.208.2
ings for the latter, linear models can well be used for noise and vibration analyses as even extreme noise loads should never cause any plastic defor- mation of any load-carrying structure.
Powertrain FE models for application to vehicle noise and vibration refi nement represent the engine (this model consists predominantly of solids) and transmission as well as drive shafts. Further FE models repre- sent the exhaust, intake, accessory drive, powertrain mounts, etc. It needs to be noted that real-world operational temperatures need to be taken into account, in particular for exhausts, to correctly represent actual structural performance.
All these individual FE models need to be combined into one single FE model. As almost all currently used FE tools are based on unique ‘grid’
numbers for each node of uniquely numbered elements, using uniquely assigned material numbers to defi ne material properties, each number must be used just once per category. Sophisticated internal numbering conven- tions or effi cient renumbering mechanisms need to be applied for assembly of a full vehicle FE model to avoid any grid, element or material number duplication which would jeopardize the CAE model. Unlike in the early stages of FE analyses, grid numbers no longer need to be manually allo- cated to achieve best ‘narrowband’ system matrices for effi cient solution, as up-to-date FE solvers do an internal reordering of system equations anyhow to achieve best computational performance.
The main task for full vehicle analyses is to predict customer perceptions of noise and vibration and to understand overall vehicle behaviour rather than doing detailed subsystem optimizations (e.g. panel thickness, topology optimization, etc.). Hence effi cient full vehicle analyses can be achieved using ‘superelement’ techniques [3] by substructuring the vehicle FE model into individual pieces (e.g. body structure, closures, suspension compo- nents, etc.). Each superelement is processed individually at defi ned bound- ary conditions; the solutions are then combined to solve the entire model.
The fi nal analysis (in which all of the individual superelement solutions are combined) involves much smaller matrices than would be required to solve the entire model in a single solution. This technique has the advantage of reducing computer resource requirements, especially if changes are made to just one component (superelement) of the vehicle; in this case, only the affected superelement needs to be reanalysed and the fi nal analysis repeated.
For standard superelement analysis each piece is represented by a reduced stiffness, mass and damping matrix, including all connection nodes and analysis points, whereas component mode synthesis is a form of super- element dynamic reduction wherein matrices are defi ned in terms of modal coordinates (corresponding to the superelement modes) and physical coor- dinates (corresponding to the grid points on the superelement boundaries).
The advantage of component modal synthesis is the signifi cantly smaller
Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com ETH Zuerich (307-97-768) Sunday, August 28, 2011 12:05:48 AM IP Address: 129.132.208.2
number of modal coordinates compared to the number of physical coordi- nates of the detailed model. It should be noted that each superelement could be derived also from physical testing, thereby increasing the accuracy of the overall analysis.
A display model for a full vehicle noise and vibration analysis and its components is shown in Fig. 8.1.