FORM start methods in Hydra-Ring

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.

Method 1

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.

Method 2

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.

Method 3

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.

Method 4

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.

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.

Z function at all the analyzed locations in the u-space

Z function at all the analyzed locations in the u-space

Z function, x-y view (discharge Lobith - discharge Lith)

Z function, x-y view (discharge Lobith - discharge Lith)

Z function, x-z view (discharge Lobith - wind speed Deelen)

Z function, x-z view (discharge Lobith - wind speed Deelen)

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Introduction

What is Hydra-Ring?

In the Netherlands, flood defense is a high priority. With 60% of the country being flood-prone, and without any other natural disasters to worry about, flood protection gets a lot of attention. A legally-mandated recurring assessment of all the primary levees in the country takes place every 5-6 years. The history that led to the assessments is rich and interesting, and will be the topic of another post. Born out of that history is the manner in which the levees are assessed.

The heart of the question is: are our levees good enough (strong enough/tall enough/covered enough) to protect us to an acceptable level? But hidden within that question is the deeper question: what is an acceptable level? The ideal way to answer that question is to determine what an acceptable risk would be, in terms of either economic damage or casualties, and to determine the required strength of the levees to meet that goal. The risk is computed as a combination of the probability of a flood occurring and the consequences that result should that flood occur. And then of course taking into account a range of flood magnitudes. Several decades ago, an effort was made to carry out such a risk assessment, but due to the lack of quantitative tools, a lot of qualitative expert knowledge went into the assessment. On the basis of that assessment, criteria, or standards, were determined for individual sections of the levee. For example, in the heavily populated area around the Hague, Amsterdam, and Rotterdam, each levee section should only succumb to failure once every 10,000 years. Sometimes this standard is loosely translated to mean that the levee should be able to withstand a water level with a return frequency of 10,000 years, but that is not equivalent. This will be a topic for a future post. In other, less populated areas, the standards are less stringent, ranging from 1/1250 years to 1/4000 years.

Safety standards for the dike rings in the Netherlands

Safety standards for the dike rings in the Netherlands

The problem with the current approach of defining standards for each levee section, is that it doesn’t tell decision-makers, regulators, or the general public what that actual risk is to the protected people and property. Currently, any levee not meeting its standard is fully funded by the government to make repairs so that it does meet the standard. With the economic situation becoming tighter, this policy cannot be sustained. The fact is, it isn’t necessary to throw all of that money into the levees. Or at least, the assessment tools we use cannot tell us if it is or is not necessary. This idea spawned a large and comprehensive project to look at what the actual state of risk is in the Netherlands. This project is called VNK, an abbreviation for its Dutch name “Veiligheid Nederland in Kaart”. Through this project, which began in the 90’s, a modeling tool was developed which was able to quantitatively predict the flooding probabilities of low-lying protected areas, often referred to in the Netherlands as polders (and the levees protecting them are often referred to as dike rings). This model is sophisticated, and also complex. It computes the flooding probability of protected areas, which means it is able to compute the probability that failure occurs anywhere within the system (dike ring) and due to any failure mechanism. The model is known as PC-Ring, and has been used for over a decade to compute the risk in the Netherlands. The model grew as the project VNK grew, to include new types of water systems (lakes, rivers, tidal rivers, coastal waters), and to improve upon itself. This growth took place rather organically, by which I mean the overall structure of the model was not defined apriori, but rather it took on a hodge-podge character that made it increasingly difficult for outsiders to understand the model. Only a select few who had been actively involved in the project understood the model, and were aware of a large number of ‘fixes’ or ‘tricks’, often quite clever, but rarely documented.

To move towards a risk-based approach of levee assessments in the Netherlands, we need to use this tool. We need to know what the probability of flooding is. That means we need to bring PC-Ring out of the research world and into the practical world, to make it robust, documented, general, and user-friendly. And we need to merge some of the functionality with the Hydra models. The Hydra models are the set of models which has been developed in parallel with PC-Ring, but for use in the current assessments. They are robust and time-tested, but they do not have the capability of combining probabilities in space or over failure mechanisms, and are therefore not suited to a risk-based approach. The Hydra models will serve as a sort of a benchmark for testing the new model and confirming that at the levee-section level, the results agree, and that where they do not, the reasons are clearly understood.

The next step in levee assessments in the Netherlands is thus the transition to a risk-based approach, and the development of a new model is central to this ambition. The new model is called Hydra-Ring. This site is dedicated to describing interesting facets of Hydra-Ring, from larger scale to often very small detailed scale. This site will attempt to provide a central place where some of the sometimes seemingly unwieldy complexity can be nailed down, and described using a variety of media, and in an informal context.