Modal Reduced-Order Models

A modal reduced-order model represents a linearization of a full, unreduced model by projecting it onto a set of base vectors in the solution space of the unreduced model. In the simplest case, the basis consists only of eigenmodes to the full unreduced problem. If desired, a set of constraint modes computed as solutions to a set of stationary problems can be added to the basis. In some cases, the model reduction algorithm will also add extra base vectors in order to satisfy the unreduced problem’s initial values and constraints. See The Modal Solver Algorithm for the some theory background.

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 basis itself together with sufficient mesh information 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:

•

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

•

A Frequency Domain, Modal Reduced-Order Model uses a time-harmonic ansatz and represents a linear system of equations in modal degrees of freedom for each frequency for which output is requested. The output for a given frequency depends on the input values for that particular frequency and not on input values for other frequencies.

Both types of modal reduced-order model can be used with either a stateless interface or a stateful interface. In the former case, the modal degrees of freedom are internal to the reduced-order model, which then uses its own internal solver. In the latter case, the modal degrees of freedom are exposed to the solver that uses the reduced-order model, and these degrees of freedom are included in the solution vector of the calling study.

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 (except for time-dependent reduced-order models with a stateful interface, which always use the global time t). 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. The load factor does not multiply the right-hand side of any inhomogeneous constraints.

A modal reduced-order model can provide four 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 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 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 basis as a complete solution object, including the model that generated it.

When the entire basis 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. Use the <rom>.eval() operator to evaluate arbitrary expressions using the reconstructed solution. Reconstruction is always available if there are nonlinear outputs defined. You can also explicitly require that reconstruction capability should be included when creating the reduced-order model.

When a modal reduced-order model has a stateful interface, you can access the reduced state (consisting of mode coefficients) using the <rom>.state(<index>) operator, which takes a mode index as an argument and returns the corresponding state DOF. There are also statet and statett operators for the first and second time derivatives, respectively. The mode indices range from 1 to the total number of modes used by the modal reduced-order model.