The purpose of the Modal Solver 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.
(20-7)
,
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. The modal solver algorithm requires that a few eigenvectors have been computed. If you form a matrix
Φ whose columns are
m computed eigenvectors, and all appearing Dirichlet boundary conditions are homogeneous, then an approximation
um of the solution
u can be written as
(20-8)
,
where q is a small vector of unknown coefficients. Replacing
u in
Equation 20-7 by
um and premultiplying by
ΦH yield
(20-9)
,
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 20-9. 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 (using a
Time Dependent, Modal study step) can export matrices and the right-hand side for use in further simulations.
For time-dependent studies, the load L is assumed to be of the form
l(
t)
L0, 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.
so that y = u − ud is zero on the boundary.
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 = 0, 1, or 2 is the number of columns that were appended to
Φ (that is, the damping ratios do not affect the
p last diagonal entries). Furthermore, the
load vector,
ΦHL0; the
stiffness matrix times ud,
ΦHKud; the
projection matrix,
Φ; the
initial value vector,
q(0); and the
initial derivative vector,
, can be exported.
(20-10)
,
where ω is the angular frequency of the forcing function and
l(
ω) is the given load factor.
(20-11)
Use the expression of L from
Equation 20-10 and the expression of
q from
Equation 20-11 in
Equation 20-9. Since the coefficient matrices can depend on the frequency, expand around the first frequency
f0 = ω0/(2
π), and truncate after three terms. You then get
(20-12)
An approximate solution to the original problem is given by um = Φq, but as usual only the time-independent factor
Φc is returned. If damping ratios are provided, the term
is added to the sum inside the square bracket of Equation 20-12. 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.
The only type of parameter-dependent Dirichlet boundary conditions that are supported are those that can be written as a scalar frequency-dependent function times a constant vector (that is, the constraint vector M can be written as
M = l(
ω)
M0). For inhomogeneous Dirichlet boundary conditions, a particular solution is needed. To homogenize the original problem, a particular solution
up = vp eiωt is computed from the nonreduced equation
(20-13)
is then subtracted from the right side of Equation 20-12. Here,
,
, and
are the unreduced coefficient matrices from the above expansion around the first frequency. Once an approximate solution,
uh, of the homogeneous problem has been found using modal analysis, an approximate solution of the inhomogeneous problem is given by
uh + up.
For frequency response studies, the following reduced matrices can be exported: the mass matrix,
; the
damping matrix,
; and the s
tiffness 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. The exported load vector is assembled for the last given frequency
ω. You can also export all load vectors (that is,
,
, …,
). This results in a matrix whose columns are all assembled load vectors. If
is independent of
ω, this matrix only contains one column.
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To export the matrices, in the Modal Solver node’s Settings window, expand the Output section and then select the Solution or Reduced matrices check boxes to display check boxes for the various reduced matrices and vectors. Select the check boxes for the matrices and vectors that you want to export.
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Modal in the COMSOL Multiphysics Programming Reference Manual.
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