Mass matrix - eesd-epfl/OpenSees GitHub Wiki
Considering the displacement field deriving from the kinematics of the macrolement, it is possible to express its consistent mass matrix, which is not diagonal. With respect to the ordering of dofs described in the macrolement formulation, the consistent mass matrix is (where ρ is the material density, H is the total length of the macroelement, and A its cross-section area):
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The matrix is computed assuming that the mass is concentrated along the element height. If the element is defined as "gable", a consistent mass matrix is derived under the same hypotheses considering a triangular distribution of the distributed mass.
Being the matrix non diagonal, its use is advisable in any case in which a non-diagonal mass matrix does not introduce issues in the solution of the nonlinear problem. Check the applicability when combined with particular solvers, or when explicit algorithms, that could be significantly accelerated if the mass matrix is diagonal, are used. For standard applications, with a GenSparse solver, and with a Newmark implicit scheme that would anyway require a new factorisation at every iteration, the consistent mass matrix is surely applicable and advised.
The lumped masses to apply to all translational degrees of freedom, for prismatic elements, are assigned automatically if the user does not specify a consistent mass matrix. They are obtained as:
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If the element is a gable, nodal masses are automatically assigned, ensuring a correct total mass and overturning moments at the middle and base interfaces. The same set of masses is applied both for consistent mass matrix or standard mass matrix. The resulting lumped mass matrix is:
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NOTE: When node offsets are present, a correct treatment of such mass offset would introduce non-diagonal terms in the mass matrix. In order to keep the matrix diagonal, consistently with the formulation of other beam elements in OpenSees, the effect of such offsets is not accounted for. If this approach introduces relevant spurious effects, the user can either use:
- rigid-beam links to avoid node offsets
- a non-diagonal consistent mass matrix