Modal Superposition Method: a classic Model Order Reduction

Over the last years, interest has grown in topics related to Model Order Reduction (MOR), due to the great potential in applications where almost instantaneous simulation results are required, as in Digital Twins, used to monitor the efficiency of equipments, for example. With these kinds of techniques, it is possible to reduce the order of the system to an incredibly lower value, making it possible to find real time reliable results.

However, MOR is not such a recent topic, in fact, it has been studied by engineers since 70s. A classic application of a MOR is the well-known Modal Superposition Method, used mainly in dynamic problems such as structural vibration. Consider a dynamic problem described by the differential equation

where M, C and K are the mass, damping and stiffness matrix, u(t) and f(t) are the vectors of displacement and external force. To evaluate (1) in the frequency domain w, Fourier transform may be used. Isolating the displacement vector, it can be obtained

where U(ω) and F(ω) are the vectors of displacement and external force evaluated in the frequency domain. It may be seen that the multiplier matrix of F(ω) describes a relation of input, defined by F(ω) vector, and output, defined by U(ω) vector. Hence, the resultant matrix, function of M, C, K and ω, is called Frequency Response Function (FRF) matrix H(ω).

Now, imagine a matrix generated in a very large FEM model, that would easily reach thousands or even millions of degrees of freedom. The calculation of the FRF matrix would have a high computational cost to be implemented. That is why Modal Superposition Method is a very attractive tool to evaluate dynamic systems. If first it is calculated the natural frequencies and mode shapes of the structure, defined as

where wi and ϕi are the natural frequencies and mode shapes of the structure, thus it can be defined the spectral and modal matrix

where Λ is the spectral matrix, formed by the square of natural frequencies, and Θ is the modal matrix, where the calculated mode shapes form the matrix columns. The r index is the number of eigenvalues and eigenvectors chosen to be evaluated.

If we approximate u(t) using a variable y(t), such that

substituting the equation (7) in equation (1), it can be obtained

multiplying this equation by the transpose of Θ,

If the columns of Θ are the orthonomal eigenvectors, then the equation (9) is reduced to

Where [I] is the identity matrix of order r and Γ is the modal damping matrix, considering the proportional damping theory, defined as

where ζ is the modal damping factor. Using the Fourier transform in equation (10), and isolating Y(ω), it can be obtained

and, finally, multiplying equation (12) by the modal matrix Θ,

where U ̃(ω) is the approximate vector of displacement, in frequency domain. Therefore,

It may be observed that, although the approximate FRF matrix H ̃(ω) has the same order of H(ω), the terms that need to be inverted have order rxr. Therefore, if we consider that the required number of natural frequencies and mode shapes to approximate, with a good agreement, the FRF is such that r <<< n, the calculation of H ̃(ω) is much less computationally expensive then the calculation of H(ω), which also means that it is faster and more efficient.

Basically, the Modal Superposition Method projects the system into a modal subspace, which has an order much smaller than the numerical model, where the matrix we need to deal is also smaller and easier to work with. This is the basic concept of MOR, finding a subspace that is more computationally efficient to simulate a physical problem, but always taking into account the accuracy of the results.