Assemble finite element matrices

The mass matrix

`M`

is nonzero when the model is time-dependent. By using this matrix, you can solve a model with Rayleigh damping. For an example, see Dynamics of Damped Cantilever Beam.For a thermal model, the

`m`

and`a`

coefficients are zeros. The thermal conductivity maps to the`c`

coefficient. The product of the mass density and the specific heat maps to the`d`

coefficient. The internal heat source maps to the`f`

coefficient.For a structural model, the

`a`

coefficient is zero. The Young's modulus and Poisson's ratio map to the`c`

coefficient. The mass density maps to the`m`

coefficient. The body loads map to the`f`

coefficient. When you specify the damping model by using the Rayleigh damping parameters`Alpha`

and`Beta`

, the discretized damping matrix`C`

is computed by using the mass matrix`M`

and the stiffness matrix`K`

as`C = Alpha*M+Beta*K`

.

The full finite element matrices and vectors are as follows:

`K`

is the stiffness matrix, the integral of the`c`

coefficient against the basis functions.`M`

is the mass matrix, the integral of the`m`

or`d`

coefficient against the basis functions.`A`

is the integral of the`a`

coefficient against the basis functions.`F`

is the integral of the`f`

coefficient against the basis functions.`Q`

is the integral of the`q`

boundary condition against the basis functions.`G`

is the integral of the`g`

boundary condition against the basis functions.The

`H`

and`R`

matrices come directly from the Dirichlet conditions and the mesh.

Given these matrices, the `'nullspace'`

technique generates the combined
finite element matrices [`Kc`

,`Fc`

,`B`

,`ud`

] as
follows. The combined stiffness matrix is for the reduced linear
system `Kc = K + M + Q`

. The corresponding combined
load vector is `Fc = F + G`

. The `B`

matrix spans the null space of the columns of `H`

(the Dirichlet condition matrix representing *hu* =
*r*). The `R`

vector
represents the Dirichlet conditions in `Hu = R`

. The
`ud`

vector represents the boundary
condition solutions for the Dirichlet conditions.

From the `'nullspace'`

matrices, you can compute
the solution `u`

as

`u = B*(Kc\Fc) + ud`

.

Internally, for time-independent problems, `solvepde`

uses
the `'nullspace'`

technique, and calculates solutions
using `u = B*(Kc\Fc) + ud`

.

The `'stiff-spring'`

technique returns a matrix `Ks`

and a
vector `Fs`

that together represent a different type
of combined finite element matrices. The approximate solution
`u`

is `u = Ks\Fs`

.

Compared to the `'nullspace'`

technique, the
`'stiff-spring'`

technique generates
matrices more quickly, but generally gives less accurate
solutions.

`PDEModel`

| `StructuralModel`

| `ThermalModel`

| `solve`

| `solvepde`