Modal Reduced-Order Models

A modal reduced-order model represents a linearization of a full, unreduced, model by projecting it onto a basis consisting of eigenmodes to the full problem. See The Modal Solver Algorithm for the full theory. The reduced-order model internally stores projected matrices as well as input and output matrices, which together contain sufficient information for computing linear scalar output values when given values of the inputs. Optionally, a modal reduced-order model can also store the eigenvector basis itself together with mesh information enough to reconstruct a full solution, which is used for evaluating any nonlinear output expressions.

The internal modal matrices can be exported from the reduced-order model feature, and the entire set of modal matrices can be saved as a COMSOL Reduced-Order Model file (.mphrom). Note that export is only possible if all defined outputs are linear. When importing a modal reduced-order model, only the modal matrices are imported; no information about the modal basis is stored. Therefore, reconstruction of the full solution is not possible for an imported reduced-order model.

There are two types of modal reduced-order models:

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A Time Dependent, Modal Reduced-Order Model internally solves a linear time-dependent system of equations in modal degrees of freedom. There is one internal degree of freedom for each eigenvector in the modal basis. The output for given time depends on the input values at all previous times.

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A Frequency Domain, Modal Reduced-Order Modeluses a time-harmonic ansatz and internally solves a linear system of equations in the modal degrees of freedom for each frequency where output is requested. The online model does not have any persistent internal state, making the output values effectively functions of the inputs evaluated at the current frequency.

Modal reduced-order models accept the following inputs:

All modal reduced-order models require one fundamental parameter, which is either time or frequency, depending on the type of model. In most cases, the actual value will be set equal to the time or frequency of evaluation in the calling model, but you can change this behavior in the corresponding Reduced-Order Model feature. For example, you can use a Frequency Domain, Modal Reduced-Order Model in a time-dependent calling model by specifying a fixed frequency as input expression.

In addition to the fundamental parameter, a modal reduced-order model accepts an arbitrary number of global reduced-model inputs. These are expected to enter linearly in the right-hand side of the linearized unreduced model. In practice, this means that they multiply different loads or sources, depending on the type of physics. In a structural mechanics context, you might say that they each multiply a load case. When using the modal reduced-order model in a calling model, you can specify expressions individually for each such input, letting you control the magnitude of each load or source but not its spatial distribution.

The load factor effectively multiplies all loads or sources that are not multiplied by a global reduced-model input. In frequency domain modal reduced-order models, the load factor also multiplies the right-hand side of any inhomogeneous constraints.

A modal reduced-order model can provide three types of outputs:

Linear and nonlinear outputs are handled differently. Linear outputs are incorporated in an output matrix, which is internally multiplied with the modal solution. They can therefore be evaluated directly without access to the eigenvector basis and data structures necessary for evaluating a general expression. A modal reduced-order model defining only linear outputs can therefore be stored on disk in a compact form.

For nonlinear outputs, the modal reduced-order model stores the output expression. When the output value is required, the expression is evaluated on a solution that has been reconstructed using the eigenvector basis, which must therefore also be stored. But interpreting a general expression also requires access to a mesh and variable definitions that are compatible with the basis vectors. If there are nonlinear outputs present, the modal reduced-order model therefore in practice stores the entire eigenfrequency solution, including the model that generated it.

When the entire eigenfrequency solution is stored with the modal reduced-order model, it can be used to reconstruct the entire solution for any required input frequency or time. Using the <rom>.eval() operator, you can evaluate arbitrary expressions using the reconstructed solution. Reconstruction is always available if there are nonlinear outputs defined. But you can also explicitly require that reconstruction capability should be included when creating the reduced-order model.