Damping matrix 1D — The Finite Element Method

Now let us compute the damping matrix for the one-dimensional case. The damping matrix is related to the integral of the square of the trial function. Consider the integral for a single segment with vertices $x_i, x_{i+1}$

\int_{x_i}^{x_{i+1}} \upsilon^2 \,dx.

(6.20)

The trial function on the segment has the form $\upsilon_{(i)(i+1)}(x)$ $= q_i \cdot \phi_i(x)$ $+ q_{i+1} \cdot \phi_{i+1}(x)$ . Take into account (5.7) and (5.8) from the section on hat functions and write down the relations for the hat functions

\displaystyle \phi_i(x)

\displaystyle = a_i

\displaystyle + b_i \cdot x

\displaystyle \phi_{i+1}(x)

\displaystyle = a_{i+1}

\displaystyle + b_{i+1} \cdot x

(6.21)

Substitute the trial function into (6.20)

\displaystyle \int_{x_i}^{x_{i+1}} \upsilon_{(i)(i+1)}^2 \,dx

\displaystyle = \int_{x_i}^{x_{i+1}} \left[ q_i \cdot (a_i + b_i \cdot x) + q_{i+1} \cdot (a_{i+1} + b_{i+1} \cdot x) \right]^2 \,dx.

Taking into account (5.8), we can write

\displaystyle \int_{x_i}^{x_{i+1}} \upsilon_{(i)(i+1)}^2 \,dx

\displaystyle = \frac{\displaystyle 1}{\displaystyle (x_i - x_{i+1})^2} \cdot \int_{x_i}^{x_{i+1}} \left[(-q_i \cdot x_{i+1} + q_{i+1} \cdot x_i) + (q_i - q_{i+1}) \cdot x \right]^2 \,dx.

Expand the square and split the integral into three parts

\displaystyle \int_{x_i}^{x_{i+1}} \upsilon_{(i)(i+1)}^2 \,dx

\displaystyle = \frac{\displaystyle 1}{\displaystyle (x_i - x_{i+1})^2} \cdot \Big[ \int_{x_i}^{x_{i+1}} (-q_i \cdot x_{i+1} + q_{i+1} \cdot x_i)^2 \,dx + 2 \cdot \int_{x_i}^{x_{i+1}} (-q_i \cdot x_{i+1} + q_{i+1} \cdot x_i) \cdot (q_i - q_{i+1}) \cdot x \,dx + \int_{x_i}^{x_{i+1}} (q_i - q_{i+1})^2 \cdot x^2 \,dx \Big]

Let us compute each of the three integrals

\displaystyle \int_{x_i}^{x_{i+1}} \upsilon_{(i)(i+1)}^2 \,dx

\displaystyle = \frac{\displaystyle (-q_i \cdot x_{i+1} + q_{i+1} \cdot x_i)^2 \cdot x}{\displaystyle (x_i - x_{i+1})^2} \bigg|_{x_i}^{x_{i+1}}

\displaystyle + \frac{\displaystyle (-q_i \cdot x_{i+1} + q_{i+1} \cdot x_i) \cdot (q_i - q_{i+1}) \cdot x^2}{\displaystyle (x_i - x_{i+1})^2} \bigg|_{x_i}^{x_{i+1}}

\displaystyle + \frac{\displaystyle (q_i - q_{i+1})^2 \cdot x^3}{\displaystyle 3 \cdot (x_i - x_{i+1})^2} \bigg|_{x_i}^{x_{i+1}}.

After simplification we obtain

\displaystyle \int_{x_i}^{x_{i+1}} \upsilon_{(i)(i+1)}^2 \,dx

\displaystyle = - \frac{\displaystyle (q_i^2 + q_i \cdot q_{i+1} + q_{i+1}^2) \cdot (x_i - x_{i+1})^2}{\displaystyle 3 \cdot (x_i - x_{i+1})}

\displaystyle = \frac{\displaystyle x_{i+1} - x_i}{\displaystyle 3} \cdot (q_i^2 + q_i \cdot q_{i+1} + q_{i+1}^2).

We introduce the notation for the segment length

\displaystyle \int_{x_i}^{x_{i+1}} \upsilon_{(i)(i+1)}^2 \,dx

\displaystyle = \frac{l_{(i)(i+1)}}{3} \cdot \left[ q_i^2 + q_i \cdot q_{i+1} + q_{i+1}^2 \right],

where $l_{(i)(i+1)}$ $= x_{i+1}$ $- x_i$ is the length of the segment.

We introduce the notation for the elements of the local damping matrix of the segment

\begin{split} &c_{(i)(i)} = \frac{\displaystyle l_{(i)(i+1)}}{\displaystyle 3}\\ &c_{(i+1)(i+1)} = \frac{\displaystyle l_{(i)(i+1)}}{\displaystyle 3}\\ &c_{(i)(i+1)} = c_{(i+1)(i)} = \frac{\displaystyle l_{(i)(i+1)}}{\displaystyle 6} \end{split}

(6.22)

Thus, the local damping matrix for a one-dimensional element has the form

\begin{aligned}\mathbf{C} = \begin{bmatrix} c_{(i)(i)} & c_{(i)(i+1)}\\ c_{(i+1)(i)} & c_{(i+1)(i+1)} \end{bmatrix} = \frac{\displaystyle l_{(i)(i+1)}}{\displaystyle 6} \begin{bmatrix} 2 & 1\\ 1 & 2 \end{bmatrix}.\end{aligned}

(6.23)

The global damping matrix $\mathbf{C}$ is obtained by summing the contributions from all segments of the mesh using the assembly method: the elements of the local matrices are added to the corresponding elements of the global matrix according to the global node numbering. The dimension of the global damping matrix is $N \times N$ , where $N$ is the total number of nodes in the mesh.

For example, consider a mesh with nodes $0, 1, \ldots, i, i$ $+1, \ldots, N$ . During assembly each segment $(j)(j+1)$ adds $\frac{\displaystyle l_{(j)(j+1)}}{\displaystyle 3}$ to the diagonal and $\frac{\displaystyle l_{(j)(j+1)}}{\displaystyle 6}$ to the adjacent off-diagonal entries; an interior node $i$ receives a diagonal contribution from the two adjacent segments — $(i-1)(i)$ and $(i)(i+1)$ — so its main-diagonal entry is the sum $\frac{\displaystyle l_{(i-1)(i)}}{\displaystyle 3}$ $+ \frac{\displaystyle l_{(i)(i+1)}}{\displaystyle 3}$ . Since the length $l_{(j)(j+1)}$ depends on the segment index, it cannot be taken out as a common factor — each entry of the global matrix keeps the length of its own segment. As a result the matrix is tridiagonal

\displaystyle \mathbf{C}

\displaystyle = \begin{bmatrix} \frac{\displaystyle l_{(0)(1)}}{\displaystyle 3} & \frac{\displaystyle l_{(0)(1)}}{\displaystyle 6} & 0 & \cdots & & & 0\\ \frac{\displaystyle l_{(0)(1)}}{\displaystyle 6} & \frac{\displaystyle l_{(0)(1)}}{\displaystyle 3} + \frac{\displaystyle l_{(1)(2)}}{\displaystyle 3} & \frac{\displaystyle l_{(1)(2)}}{\displaystyle 6} & & & & \\ 0 & \frac{\displaystyle l_{(1)(2)}}{\displaystyle 6} & \ddots & \ddots & & & \\ \vdots & & \ddots & \frac{\displaystyle l_{(i-1)(i)}}{\displaystyle 3} + \frac{\displaystyle l_{(i)(i+1)}}{\displaystyle 3} & \frac{\displaystyle l_{(i)(i+1)}}{\displaystyle 6} & & \vdots\\ & & & \frac{\displaystyle l_{(i)(i+1)}}{\displaystyle 6} & \frac{\displaystyle l_{(i)(i+1)}}{\displaystyle 3} + \frac{\displaystyle l_{(i+1)(i+2)}}{\displaystyle 3} & \ddots & \\ & & & & \ddots & \ddots & \frac{\displaystyle l_{(N-1)(N)}}{\displaystyle 6}\\ 0 & & & \cdots & & \frac{\displaystyle l_{(N-1)(N)}}{\displaystyle 6} & \frac{\displaystyle l_{(N-1)(N)}}{\displaystyle 3} \end{bmatrix}.