COMSOL Multiphysics™

assemble
Purpose
assemble
Assemble the stiffness matrix, right-hand side, mass matrix, damping matrix, and
constraints of a PDE problem.
Syntax
Description
[K,L,M,N] = assemble(fem,...)
[K,L,M,N,D] = assemble(fem,...)
[K,L,M,N,D,E] = assemble(fem,...)
[D,M,...] = assemble(fem,'Out',{'D' 'M' ...}, ...)
assemble is a fundamental function in COMSOL Multiphysics. It assembles a PDE
problem using a FEM discretization.
For time-dependent problems, the FEM discretization is the system of ODEs
0 = LUU·
( , , U·· , t) – NUt ( , ) T Λ
0 = MUt ( , )
where L is the residual vector, M is the constraint residual vector, U is the solution
vector, and Λ is the Lagrange multiplier vector. The linearization of this system uses
the stiffness matrix K, the damping matrix D, the mass matrix E, and the constraint
Jacobian matrix N given by
K
=
∂L
– ,
∂U
∂L
D = – ,
∂U·
E
∂U··
∂L
= – ,
N
= –
∂M
∂U
All these matrices can depend on the solution vector U. The matrices K, D, and E
can also depend on the time derivatives U·
and U·· .
For a stationary problem, the discretization is
and the linearized problem is
0 = LU ( )–
NU ( ) T Λ
0 = MU ( )
KU ( – U 0
) = L–
N T Λ
NU = M
where K, L, M, and N are evaluated for some linearization “point” U = U 0 .
For an eigenvalue problem, the discretization reads
U – ( λ – λ 0
)DU + ( λ–
λ 0
) 2 EU = – N T
NU = M
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