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Dynamics of multibody systems
B. TasicDownload PDF (2.4 MB)
Problem SettingFor an autonomous problem
where x(0) is any point in an interval I(0) say, the solution x is called a flow.
Figure 1: Flow examples in 1D and 2D
For stiff problems one needs implicit method to find the solution without stability constraints. In this project we show how we can exploit the autonomous character of the problem and derive explicit methods having the same favourable stability properties.
MethodsThe so-called flow methods, for obtaining the numerical solution, has several main aspects:
Time discretisation. A set of time levels which spans the integration time [0,t] is needed.
Space discretisation. It should be such that enough information about the shape of I(t) and about the vector field is provided. The set of discretisation points may consist of only points of the solution, but it can also be extended to contain additional space points.
Implicit numerical method. This is of interest because of possible stiffness. Example: Euler Backward (EB)
In general, flow methods can be based on the following methods:
Inverse interpolation. We employ the approximation P(x) of the inverse function of present in all implicit numerical methods (listed above) to obtain the solution at the next time-level .
Figure 2: Inverse interpolation (1D) in the flow methods
ResultsThe accuracy of the floow methods is determined by the accuracy of the original (implicit) methods and the interpolation error.
Example 1 A simple scalar nonlinear problem , solved by the flow methods based on BDF (FBDFi) of various order i. Table 1 shows the values in infinity norm of the global error (at t = 1.0, with Δt = 0.1).
Table 1: Accuracy of the flow methods based on BDF methods
Example 2 The flow methods can also be used for higher dimensional problems. The interpolation is then also higher dimensional. Figure 3 shows the evolution in time of the half-ellipse, under the influence of the velocity field . The flow method used is based on the implicit midpoint rule.
Figure 3: Evolution in time of the half-ellipse
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