Dynamical energy analysis (DEA) is a method for numericallymodelling structure borne sound and vibration in complex structures.It is applicable in the mid-to-high frequency range and is in thisregime computational more efficient than traditional deterministicapproaches (such as finite element andboundary element methods).In comparison to conventional statistical approachessuch as statistical energy analysis (SEA),DEA provides more structural details and is less problematic with respectto subsystem division.The DEA method predicts the flow of vibrational wave energy acrosscomplex structures in terms of (linear) transport equations.These equations are then discretized and solved on meshes.
Simulations of the vibro-acoustic properties of complex structures(such as cars, ships, airplanes,...) are routinelycarried out in various design stages.For low frequencies, the established method of choice isthe finite element method (FEM).But high frequency analysis using FEM requires veryfine meshes of the body structure to capture the shorter wavelengths andtherefore is computational extremely costly.Furthermore the structural response at high frequencies isvery sensitive to small variations in material properties,geometry and boundary conditions. This makes the output of a singleFEM calculation less reliable and makes ensemble averagesnecessary furthermore enhancing computational cost.Therefore at high frequencies other numerical methodswith better computational efficiency are preferable.
The statistical energy analysis (SEA)[1] has been developed to dealwith high frequency problems and leads to relatively small and simple models.However, SEA is based on a set of often hard to verify assumptions,which effectively require diffuse wave fields and quasi-equilibrium of wave energywithin weakly coupled (and weakly damped) sub-systems.
One alternative to SEA is to instead consider the original vibrationalwave problem in the high frequency limit, leading to a ray tracing modelof the structural vibrations.[2] The tracking of individual rays acrossmultiple reflection is not computational feasible because of theproliferation of trajectories.Instead, a better approach is tracking densities of rayspropagated by a transfer operator.This forms the basis of the Dynamical Energy Analysis (DEA) method introduced in reference.DEA can be seen as an improvement over SEA where one lifts the diffusive fieldand the well separated subsystem assumption.One uses an energy density which depends both on position and momentum.DEA can work with relatively fine meshes where energy can flow freely betweenneighboring mesh cells.This allows far greater flexibility for the models used by DEA incomparison to the restriction imposed by SEA.No remodeling as for SEA is necessaryas DEA can use meshes created for a FE analysis.As a result, finer structural details than SEA can be resolved by DEA.
The implementation of DEA on meshes is called Discrete Flow Mapping (DFM).We will here briefly describe the idea behind DFM, for details see thereferences[3] [4] [5] [6] [7] [8] below.Using DFM it is possible to compute vibro-acoustic energy densities in complex structuresat high frequencies, including multi-modal propagation and curved surfaces.DFM is a mesh based technique where a transfer operator is used to describe the flow ofenergy through boundaries of subsystems of the structure; the energy flow is representedin terms of a density of rays
\rho
s
p
s
p
\rho(s,p)=\rho(Xs)
Xs=(s,p)
where
\phi(Xs)
s
ps
w(Xs)
In a next step, the transfer operator is discretisedusing a set of basis functions of the phase space.Once the matrix
{\bfB}
\rho
\rho0
The initial density
\rho0
\rho
Concerning the terminology, there is some ambiguity concerning the terms "Discrete Flow Mapping(DFM)"and "Dynamical Energy Analysis". To some extent, one can use one term in place of the other.For example, consider a plate. In DFM, one would subdivide the plate into many small trianglesand propagate the flow of energy from triangle to (neighbouring) triangle.In DEA, one would not subdivide the plate, but use some high order basis functions (both in positionand momentum) on the boundary of the plate. But in principle it would be admissible to describe bothprocedures as either DFM or DEA.
As an example application, a simulation[10] [11] of a carfloor panel is shown here.A point excitation at 2500 Hz with 0.04 hysteretic damping was applied. The results from a frequency averaged FEM simulation are compared with a DEA simulation (for DEA, no frequency averaging is necessary).The results also show a good quantitative agreement. In particular, we see the directional dependence of the energy flow, which is predominantly in the horizontal direction as plotted. This is caused by several horizontally extended out-of-planebulges. It is only in the lower right part of the panel, with negligible energy content, that deviations between the FEM and DFM predictions are visible. The total kinetic energy given by the DFM prediction is within 12% of the FEM prediction.For more details, see the cited works.
As a more applied example, the result of a DEA simulation[12] on a Yanmar tractor model (body in blue: chassis/cabin steel frame and windows) is shown here to the left.In the cited work, the numerical DEA results are compared with experimental measurements at frequencies between 400 Hz and 4000 Hzfor an excitation on the back of the gear casing. Both results agree favorably. The DEA simulation can be extended to predict thesound pressure level at driver's ear.