Mathematical Optimization for Geotechnical Problems

Abstract

Almost every aspect of a civil engineering project is to be "optimized". Even if this demand seems a bit abstract and nebulous at first, principals and clients and we as a society are strongly dependent on this optimization being undertaken. This task lies mainly in the hands of the planner. Due to numerous mutual and non-linear dependencies, the resolution of this task is often non-trivial and time consuming. Mathematical optimization algorithms are a valuable tool in practice to support the planner in precisely this task. This article provides an insight into the "why?" and "when?".

You should optimize...

The technical standards of many countries consider it to be part of the designer's task to "optimize" the planned structure. The Swiss article SIA 103 explicitly envisages this, among other things, for preliminary and construction design. This optimization concerns almost every aspect: Costs, feasibility, construction processes, ecological aspects and many more. This very general requirement seems to make perfect sense and is usually also in line with the demands for sustainability.

In geotechnics and civil engineering, the planned structures are usually quite unique, which is why optimizations are often specifically tailored to the construction in question. No two excavations or foundations are the same. Of course, the planning of such structures is based on proven solution patterns and technologies, however the composition and fine-tuning - and not least the subsoil - is always different.

In my experience, this optimization task is approached in many ways, both for different projects and within a single project. Each planner also has his own requirements and approach. This may be linked to the fact that measuring the quality of an optimization of a project is "difficult" at best. What is the objective benchmark for such a one-off? In addition, optimization usually implies a certain amount of extra work for the planner. I, for my part, am very glad that the self-expectation to create an optimized planning is high in my environment.

What do the project participants get out of it?

First and foremost, the principal and client, of course has a great interest in realizing an "optimized" project. However, the purely monetary costs are not the only criterion, which should always solely be placed above all other criteria. In addition to the already above-mentioned aspects, robustness is a criterion that is often underestimated in a wide variety of ways: If a project lacks geometric or schedule robustness, for example, even minor irregularities could result in massive disruptions in the construction process - and cause the oh-so-cost-optimized project to totter. The client is therefore dependent on the planner to optimize the project holistically.

The contractor also has an interest in an optimized project: not only do coordinated construction sequences and the robust design described earlier clearly benefit him and allow him to produce a high-quality structure. To give an example: Optimizing the tie-back of an excavation pit wall helps to avoid unnecessary anchor layers or to reduce the number of anchors to the necessary amount. Superficially, this saves material costs (fewer anchors). Not to mention, the potential reduction of the number of construction conditions and interdependencies, which often clearly outweighs the material costs saved (fewer anchor layers means not least fewer excavation conditions and possibly fewer conditions for an access ramp and also more flexibility during excavation, etc.). As one can see, the benefits for the contractor are very evident.

The planner, or even the project manager, derive their advantages from an optimization by viewing the project as a whole: fewer components, fewer construction states, fewer dependencies, and also better construction processes enable a seamless project and reduce the risk of unforeseen incidents in the course of execution. However, the optimization performed by them is often not trivial: there is a multitude of manipulated variables which in turn have a plethora of dependencies - and these often cause a non-linear response. But there is an intriguing option for them concerning quite a few of the to be executed optimizations: mathematical optimization. Tools for the optimization?

Many aspects of the previously mentioned optimization must be handled with a lot of experience and classical (manual) optimization work. This is a good thing as it leaves room for a sense of proportion – the experience of the planner in particular is a powerful advisor. In addition, the increased use of numerical software in planning creates more and more opportunities to use the powerful tool of mathematical optimization.

Numerical software such as limit equilibrium or finite element programs allow the later response of the structure to be predicted with sufficient accuracy. These tools often have a large number of manipulated variables that need to be skillfully selected: for some of these manipulated variables, there is only one or a handful of permissible settings (e.g. safety factors or shape of the earth pressure distribution), subject to compliance with standards and regulations. For other control variables the planner has more freedom of choice - but then they do need to be applied with the appropriate skill. For example, an excavation pit can be optimized by cleverly choosing the depth and the inclination of the tieback anchors of an excavation pit wall. It may not immediately be obvious at first glance that the "clever" choice of these parameters is not simple to make: The individual parameters (e.g. depth and inclination of the anchor position) influence each other and can have a deviating effect depending on the value of the other parameters and react non-linearly. In the aforementioned example of excavation support, "twice as deep" does not usually mean "twice as expensive". Mathematical optimization algorithms can be very helpful in the adjustment of such parameters.

The optimization algorithms of interest originate from applied mathematics. They treat the to-be-optimized task as a "black box" whose input parameters are to be adjusted in such a way so that the output is either minimized or maximized. Now the task of these mathematical optimization algorithms is to minimize or respectively maximize the output of the black box by skillfully choosing the values of the input parameters. The optimization algorithms usually solve this task through directed probing: Step by step, such an algorithm will choose new combinations for the input parameters and thus try to approach the sought optimum.

The use of mathematical optimization algorithms for design in geotechnical or civil engineering clearly offers itself. Most of our calculations correspond to the black box scheme described above. So why not use them?

For which projects? And where?

For the use of mathematical optimization to be worthwhile, the effect within the project must exceed the costs of the optimization - in a broader sense. In the case of very small projects (such as a single-family house), an optimization is often not worthwhile. Given that construction costs rapidly increase as the project size grows, however, the threshold for use is often lower than expected. Where exactly this threshold lies, strongly depends on the respective project and its boundary conditions. This is where it pays to talk to an expert.

From today's perspective, optimizing any given project in its entirety using mathematical optimization algorithms is still science fiction. Perhaps it is even questionable whether such a thing would be desirable at all, since, among other things, the valuable experience of a planner would no longer be fully utilized.

However, mathematical optimization is easily applicable - joint with experience values - for many partial aspects of a project. To list them all here isn't worthwhile. To give a sense of applicable cases, perhaps a look at two examples will help: