FORM (First order reliability method) is a probabilistic technique that linearizes the failure domain in the design point. FORM is an iterative process that searches for the design point using the partial derivative of the failure function with respect to each of the random variables. These derivatives help the FORM procedure feel its way towards nearest Z=0 point, where Z is the failure function (Z < 0 indicates failure, Z > 0 indicates no failure). The starting point of the FORM procedure can have a big impact on the outcome of a FORM computation. This can be due to irregular failure domains in which local minima are present. Essentially, it is important to start the FORM procedure in the neighborhood of the actual design point, to increase the likelihood of avoiding local minima. In Hydra-Ring there are five start methods; these five start methods will be described in this document.
The u-vector, which represents all the random variables (strengths and loads) in the standard normal space, is set to zero. This means that the variables in the real space are all equal to their mean. The problem with this method is that often the hydrodynamic model output databases do not contain values of the random load variables around the mean because these values do not contribute to failure. As a result, the databases are often extrapolated which leads to unreliable values of the local load, and can throw the FORM procedure off course.
Set the u-vector to the value of 1. This sets each of the variables to a value one standard deviation higher than its mean. This method is not very practical and will likely never be chosen. The main reason that it’s impractical is because it sets the strength variables to higher values while it is actually the values lower than the mean that contribute to failure, and furthermore, the value of 1 is not particularly high for the load variables; not high enough to come into the realm of failure typically.
In this method all the load variables are set to a specified value (in the case of Hydra-Ring, this value is u = 3), and the strength variables are set to u = 0. The thought behind this is that the strength variables can begin at their mean values without any risk to the robustness of FORM, and the load variables begin three standard deviations from their mean, so most likely in the failure realm. There is of course no guarantee that u = 3 brings you into the failure realm. For that, the upcoming methods are required.
This method starts with a normalized vector with a direction along the 45-degree angle (i.e. the u value of each random variable is equal). Then, in small steps, the procedure moves outward along this line until failure is observed. In fact, the way this is programmed is that we start with a u-vector where all load variables are equal to 0.1. We compute the z-value and then increase the u-vector so that all variables are equal to 0.2, and compute the z-value again. We do this in incremental steps until the u-values are equal to 6.1. Of all the computed z-values, the absolute smallest (i.e. the closest to zero) is selected, and the u-vector that produced the smallest z is selected as the starting vector for the FORM computation. The weakness of this method has to do with the complexity of the failure space. If the Z=0 contour is irregular, and the most likely combination of random variables that leads to it is not along (or near) the 45-degree angle, we run the risk of converging to a local minimum. To take into account other directions, we need to advance to Method 5.
This method can be thought of as a balloon placed at the origin in the u-space. In all directions, the u-values are increased incrementally. This is done for a discrete number of angles, and for a discrete number of steps. Essentially it mimics Method 4, only instead of only choosing 45 degrees, it chooses a number of directions.
The film below shows how Method 5 woks for an example case. This example has three random variables, and so the method samples a 3-dimensional u-space. The three dimensions (i.e. random variables) in this example are x = discharge at Lobith, y = discharge at Lith, and z = wind speed at Deelen (all locations in the Netherlands). At each point, the Z function is computed. The Z function in this case is a special one for computing the exceedance probability of a water level. So the Z function is Z = x* – x, where x* is a fixed local water level, and x is a computed local water level. The probability we are computing is the exceedance probability of x*. This is a location in a non-tidal river. The wind has no influence, and two rivers influence the water level. To say it more correctly, the discharge at Lith has influence on the local water level, but the discharge at Lobith has influence on the discharge at Lith (that is, they are correlated).
It’s interesting to also look at what the value of the Z function was at all of the computed points. Recall we are interested in the limit state, where Z=0. Note that pink dots were chosen for values right around Z = 0. This is shown in the figure below. The following two figures show the same plot but rotated to get a two-dimensional view. The first shows the x-y axis (discharge Lobith – discharge Lith), and the second shows the x-z axis (discharge Lobith – wind speed Deelen). The discharge has a big impact on the Z function, and the wind has no impact. Thus, for the x-y axis you see a very clear trend – the Z-function decreases radially with increasing discharge. For the x-z axis, because the wind has no influence, increasing values of the wind should not change the Z-value. This is a bit hard to visualize, but if you look within a thin vertical strip in the third figure, you should see that all the dots in the vertical strip should be the same color. Z should decrease with increasing values of the discharge, and be unaffected by increasing values of the wind.