The purpose of the Modal Solver and Model Reduction solvers is to speed up certain simulations by performing a model reduction using eigenpairs, making use of the solution to an eigenvalue or eigenfrequency problem to construct a basis using eigenvectors corresponding to the dominant dynamics. That is, the solution of the underlying system of equations is approximated by a linear combination of parametric or time-dependent coefficients and a few dominant eigenvectors. Optionally, this basis can be extended with constraint modes. Each constraint mode is a solution to a stationary problem with a nonhomogeneous boundary condition. These modes make it possible to extend the validity of the reduced model. They also make it possible to use reduced-model inputs in constraints for more general use in a modal reduced-order model produced by the model reduction.

where E is the mass matrix, D is the damping matrix, K is the stiffness matrix, and L is the load vector. Either E or D can be identically zero. N is the constraint Jacobian and M the constraint load vector. The modal solver algorithm requires that a few eigenvectors have been computed to the corresponding homogeneous harmonic problem. The eigenvalue solvers in COMSOL automatically consider the homogeneous problem when a formulation as the one above is used. That is, in its simplest form, it solves the eigenvalue problem

The constraint modes are optional. A constraint mode Uj is a solution to the stationary problem

Several different constraint load vectors Mj can be used and thereby define several constraint modes. If you form a matrix Φ whose columns are the eigenvectors and the constraint modes, then an approximation um of the solution u can be written as

where q is a small vector of m unknown coefficients. Replacing u in Equation 20-10 by um and premultiplying by ΦH yield

where Em = ΦHEΦ, Dm = ΦHDΦ, Km = ΦHKΦ, Nm = NΦ, and NF,m = ΦHNF.

The damping matrix D may be present when performing the eigenvalue analysis. It is, however, possible to add additional damping by providing damping ratios per mode (or one ratio for all modes). If λi denotes the ith eigenvalue and ξi the associated damping ratio, then

is added to the ith diagonal entry of the reduced damping matrix in Equation 20-12. If E and K are real and symmetric positive definite, D = 0, and Em and Km are diagonal, then ξi can be interpreted as the fraction of critical damping in the ith mode.

The modal solver algorithm supports Model Control Inputs in the load vector and the constraint load vector. Denoting these inputs ν gives

where Br = ΦH∂L/∂ν, Brdot = ΦH∂L/∂, Brdotdot = ΦH∂L/∂, and BM = ∂M/∂ν.

be a vector of output variables, where Y0 is the output bias vector, Cr the output matrix, and F the feedback matrix.

The Modal Solver (using a Time Dependent, Modal study step) and the Time Dependent, Modal Reduced-Order Model can export the reduced matrices and vectors for use in further simulations.

For time-dependent studies, the load L1 is assumed to be of the form l(t)ΦHL0, where L0 is constant, and l(t) is the given load factor. Further, the projection matrix Φ is possibly appended with one or two columns such that the initial values u(0) = u0 and lie in the range of Φ. These initial value vectors are appended before the constraint modes are appended Φ.

If inhomogeneous Dirichlet boundary conditions are present, then Equation 20-10 is rewritten as

so that y = u − ud is zero on the boundary (for ν = 0) and

When using the reduced-order model (see Modal Reduced-Order Model), it is possible to use the constraint modes rather freely. The default is that the constraint mode DOFs are constrained to be equal to a corresponding reduced-model input expression

where qj is a constraint mode DOF connected to the model input expression νk. The motivation for this behavior is that if the constraint modes fulfill the conditions

on its corresponding boundary ∂Ωj and

on the boundaries where i ≠ j, then the default boundary condition for the reduced model approximates the constraint

on the boundary ∂Ωj. Notice that this behavior can be modified from the Modal Reduced-Order Model node in the Constraint Modes section. The map between the constraint mode number i (not the DOF number) and the input expression index k is exported as a vector CImap(i) = k from the modal solver algorithm. This map returns −1 if there is no such expression.

Notice that a simple way to generate constraint modes with the properties a) and b) is to introduce a parameter for each constraint and then enable an Auxiliary sweep for a Stationary study and these parameters. By using the Specified combinations sweep option you can solve for the constraint modes by setting the parameters to zero or one according to Equation 20-14 and Equation 20-15. This way of constructing extra modes is the same as for the classical Craig–Bampton method; see Ref. 44. The order in which these modes are solved for, in such a sweep, also dictates the order in which they are appended to Φ in the modal solver and thereby control the order of qj, the reduced constraint mode DOFs.

The following reduced matrices can be exported: the mass matrix Em, the damping matrix Dm, the stiffness matrix Km, and the damping ratio matrix

where p is the number of columns that were appended to Φ (that is, the damping ratios do not affect the p last diagonal entries coming from initial vectors or constraint mode vectors). Furthermore, the load vector, ΦHL0; the input matrix Br; the time derivative input matrix Brdot; the second time derivative input matrix Brdotdot; the output bias vector Y0; the output matrix Cr; the output feedback matrix F; the stiffness matrix times ud, ΦHKud; the projection matrix, Φ; the constraint mode to input map, CImap; the initial value vector, q(0); and the initial derivative vector, , can also be exported.

In the above, Krxe is the eliminated constraint vector, which is available as the eliminated constraint vector Krxe in an output in a System Matrix node, for example. See below for the reduced system in the frequency domain.

For the frequency domain, the COMSOL software also supports linearized formulations as well as rather general formulations in terms of the frequency. The Equation 20-10 is linear and cannot be used to describe this. The starting point for the model reduction of a frequency-domain problem is a FEM residual vector expansion around a frequency f0 = ω0/(2π). Consider the unreduced residual vector expanded in a three term Taylor expansion

where and . Next, consider a linearization point for u = u0, and the problem

where , , , , and where the load Lload is assumed to be of the type

where ω is the angular frequency of the forcing function and the constraint load vector Mload is assumed to be of the type

The solution to Equation 20-18 is on the form , and we can again use the approximation Φq(ω,t), where , but as usual only the time-independent factor is returned by the solver or when reconstructed by the reduced-order model. Using this approximation in Equation 20-18 and multiplying from the left with ΦH, gives the reduced frequency-domain problem

where the ~ notation is dropped for q, Λ, L, and M from here on. We also introduce and drop the ~ notation. Here, , , and are the reduced coefficient matrices from Equation 20-18 and

is added to the sum inside the square bracket of Equation 20-19. Notice that when the coefficient matrices are independent of the frequency, this damping term coincides with what is added in the corresponding Time Dependent, Modal study.

For inhomogeneous Dirichlet boundary conditions, a particular solution is computed by the modal solver. A particular solution vp is computed from the nonreduced equation

where the matrix is assembled for f0. The term

is then subtracted from the right side of Equation 20-19. Reduced-model inputs in constraints are handled very much like the time-dependent case. Constraint modes, computed as above for a stationary problem, are supported. The produced reduced-order model can rather freely constrain these reduced constraint mode DOFs. Also here the default behavior is to constrain these to their corresponding input expression qj = νk, and also for the frequency domain this behavior can be modified from the Modal Reduced-Order Model node’s settings.

From the Modal Solver and the Frequency Domain, Modal Reduced-Order Model, the following reduced matrices can be exported: the mass matrix, ; the damping matrix, ; and the stiffness matrix . The damping ratio matrix, Dratio; the projection matrix, Φ; the mass matrix times the particular solution, ; the damping matrix times the particular solution, ;and the load vector can also be exported. Furthermore, the constraint mode to input map, CImap; the input matrix Br; the time derivative input matrix Brdot; the second time derivative input matrix Brdotdot; the output bias vector Y0; the output matrix Cr; and the output feedback matrix F can be exported. The exported load vector is assembled for the last given frequency ω. You can also export all load vectors (that is, , , …, ). This export results in a matrix whose columns are all assembled load vectors. If is independent of ω, this matrix only contains one column.