step1¶

Purpose:

Compute the dynamic solution to a set of first order differential equations.

Syntax:

[a,da] = step1(K, C, f, a0, bc, ip)
[a,da] = step1(K, C, f, a0, bc, ip, times)
[a,da, ahist, dahist] = step1(K, C, f, a0, bc, ip, times, dofs)

Description:

step1 computes at equal time steps the solution to a set of first order differential equations of the form:

\[\begin{split}\mathbf{C} \dot{\mathbf{a}} + \mathbf{K}\mathbf{a} = \mathbf{f}(x, t), \\ \mathbf{a}(0) = \mathbf{a}_0\end{split}\]

The command solves transient field problems. In the case of heat conduction, K and C represent the \(n \times n\) conductivity and capacity matrices, respectively. a is the temperature and da (i.e., \(\dot{\mathbf{a}}\)) is the time derivative of the temperature.

The matrix f contains the time-dependent load vectors. If no external loads are active, use [] for f. The matrix f is organized as follows:

f = [
time history of the load at dof_1
time history of the load at dof_2
...
time history of the load at dof_n
]

The dimension of f is:

(number of degrees-of-freedom) × (number of timesteps + 1)

The initial conditions are given by the vector a0 containing initial values of a.

The matrix bc contains the time-dependent prescribed values of the field variable a. If no field variables are prescribed, use [] for bc. The matrix bc is organized as follows:

bc = [
dof_1   time history of the field variable
dof_2   time history of the field variable
...
dof_m2  time history of the field variable
]

The dimension of bc is:

(number of dofs with prescribed field values) × (number of timesteps + 2)

The time integration procedure is governed by the parameters given in the vector ip defined as:

ip = [dt, T, alpha]

where dt specifies the length of the time increment, T is the total time, and alpha is the time integration constant. Frequently used values of alpha are:

alpha	Method
0	Forward difference; forward Euler
0.5	Trapezoidal rule; Crank-Nicholson
1	Backward difference; backward Euler

The computed values of a and da are stored in a and da, respectively. The first column contains the initial values, and the following columns contain the values for each time step. The dimension is:

(number of degrees-of-freedom) × (number of time steps + 1)

If the values are to be stored only for specific times, the parameter times specifies at which times the solution will be stored. The values are stored in a and da, one column for each requested time according to times. The dimension is then:

(number of degrees-of-freedom) × (number of requested times + 1)

If the history is to be stored in ahist and dahist for some degrees of freedom, the parameter dofs specifies for which degrees of freedom the history is to be stored. The computed time histories are stored in ahist and dahist, respectively, with one row for each requested degree of freedom. The dimension is:

(number of specified degrees of freedom) × (number of timesteps + 1)

The time history functions can be generated using the command gfunc. If all the values of the time histories of f or bc are kept constant, these values need to be stated only once. In this case, the number of columns in f is one and in bc two.

In most cases, only a few degrees-of-freedom are affected by the exterior load, and hence the matrix contains only a few non-zero entries. In such cases, it is possible to save space by defining f as sparse (a MATLAB built-in function).

Note

Reference: Bathe, K.J.: Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, pp. 511-514, 1982.