Dynamic Impulse - time integration

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Stepping Schemes: Time Integration

Many time stepping algorithms have been developed, each having their advantage over others under certain circumstances. However, three main types of solution dominate, backward difference (Implicit), central difference (Crank-Nicolson) & forward difference (Explicit). The explicit & implicit techniques are often referred to as Euler's rule & the backwards Euler's rule respectively.

Explicit schemes, which are conditionally stable (stability of solution not guaranteed), find the response at the end of the time step in terms of the conditions at the start of the time step. In other words, the calculation of the solution at time (t+dt) is obtained by considering the situation at time t. The advantage of this approach is that the underlying system of equations that comprise the model (stiffness matrix, capacitance matrix, flexibility matrix) does not have to be solved at each time step. Furthermore, the material & time matrices can be diagonalised to become uncoupled, and so the solution can be calculated explicitly. Very fast calculations of individual time steps can be achieved as no matrix factorisation is required. However, the technique is much less stable than the implicit method, so very small time steps must be used to ensure an appropriate solution.

Implicit schemes, which are unconditionally stable, find the response at the end of the time step in terms of the conditions at the end of the time step. In other words, the calculation of the solution at time (t+dt) is found by considering the response at time (t+dt). An important point to note is that the solution at each time step involves matrix factorisation (evaluating the system of equations that comprise the model), which is a computationally intensive process. Despite this disadvantage, implicit schemes are often used, as the solution is inherently reliable & robust. Implicit analyses allow much larger time steps than the others, and so the solution can be obtained with fewer calculation increments. As implicit schemes are always stable, the time step length is governed by considerations of accuracy alone.

The Crank-Nicolson approach evaluates the next step of the solution by using the prediction at the centre of the time step. As with the backward difference scheme, this is an implicit solution which is conditionally stable (results in an oscillatory solution if the critical time step for stability is exceeded). The central difference method is more accurate than both the purely implicit or explicit techniques since neither favours the response at the start or end of the time step.