Beyond the Best Answer: How Byesian Optimisation Designs Better Bridges

Co-authored by Kashvi Jhamnani and Rudra Limbachiya

1. What Does It Mean to Find the “Best” Design?

Imagine designing a bridge. There is rarely one obvious answer because several things have to be balanced at the same time. The bridge should be strong and safe, but it should also use a reasonable amount of material and ideally have a low environmental impact. Improving one of these factors can sometimes make another one worse.

This is where optimisation comes in. In simple terms, optimisation means finding the best possible solution from many alternatives. However, real-world optimisation is rarely unrestricted. A bridge cannot simply be made thinner because that might reduce material use or emissions. It still has to support loads, remain stable, and satisfy engineering requirements.

This creates the idea of constrained optimisation: finding the best possible solution while staying within a set of requirements.The research paper explored this problem through constrained Bayesian optimisation, using the design of a concrete bridge as a case study. Rather than testing every possible bridge design, the researchers developed a method that could identify promising designs while accounting for structural limitations.

The goal is therefore not simply to find a design that looks best mathematically. It is to find a design that is good, feasible, and safe. This idea also appears in aircraft design. Engineers may want to reduce an aircraft’s weight to improve efficiency, but they cannot remove material indefinitely because the wing still has to withstand the forces acting on it.

2. Framing Bridge Design as a Mathematical Problem

All optimisation problems can be sliced into three fundamental chunks: design variables, an objective function, and constraints. Design variables are what you control. The objective function is what you want to optimise. Constraints set strict limits that you cannot cross.

In this paper, researchers set up a continuous three-span, post-tensioned concrete bridge as their primary test case.

They considered four main design variables:

Girder height (H)

Post-tensioning strand area (Aₜ)

Upper steel reinforcement area (Aₛ,ᵤ)

Lower steel reinforcement area (Aₛ,ₗ)

The primary objective is to minimise the bridge girder’s total CO₂-equivalent emissions, calculated using the mass of the different materials and their associated life-cycle emissions factors. But shrinking the materials’ sizes can affect the structure’s integrity.

Structural limits are evaluated using utilisation ratios.

A utilisation ratio divides the maximum observed strain in a material by its allowed strain limit, which is seen by:

uᵢ = εᵢ,max / εᵢ,lim

If the ratio is greater than 1.0, the design violates that constraint. For a design to be feasible, every utilisation ratio must satisfy:

uᵢ ≤ 1

The mathematical setup can therefore be written simply as:

minimize f(x)

subject to gᵢ(x) ≤ 0

Here, x represents the design variables, f(x) is the objective function, and gᵢ(x) represents the constraints.

But the lowest-emission design is not automatically the best choice. For example, a bridge producing 180 tonnes of CO₂ but having a utilisation ratio of 1.12 cannot be selected, while a 200-tonne design with a ratio of 0.85 is feasible.

3. Finding the Best Design Inside the Rules

This idea becomes easier to understand when we picture the possible designs as a design space. Imagine that the x-axis represents one design variable and the y-axis represents another. Every point on the graph represents a possible design.

Some of these designs satisfy every structural requirement. These points form the feasible region. Other designs fall outside the boundary because they violate at least one constraint, making them infeasible.

There may be a design outside the feasible region that would produce an even lower value of the objective function. Mathematically, it might look like the better answer. However, it cannot actually be selected because it does not satisfy the rules.

This gives us an important distinction between the best overall point and the best allowed point. Constrained optimisation is concerned with the second one: the best solution that remains inside the feasible region.

Figure 1: A 2D graph with a marked feasible region and an infeasible region

For the bridge problem, this means that the algorithm has to search for designs with low CO₂-equivalent emissions without crossing structural limits. The challenge is that this feasible design space can become extremely complicated once several continuous variables and structural constraints are involved.

4. Why Ordinary Optimisation Becomes Difficult

A common question is why can't a computer just run through every possible bridge combination? Because checking structural behaviour takes significant computing effort. It might not be practical to run a large number of these analyses when other alternatives are available.

Engineers evaluate structural behaviour using Finite Element Method (FEM) analyses. These analyses calculate the structural response of the bridge under the specified loading conditions. If we change even one design variable, another structural analysis may be required to determine the resulting stresses and strains.

Now, if we multiply that across four continuous variables, the number of possible combinations becomes very large. FEM operates as a black box from the optimiser's perspective. The optimiser passes design variables into the software and receives structural outputs back, but the internal calculations of the commercial software are not directly accessible for the optimisation process.

Running thousands of these high-fidelity analyses can therefore become impractical because of the computational cost and time involved.

The paper solves this with a surrogate model, or approximation model. Instead of running full FEM analyses continuously, the researchers use results from a smaller set of initially tested designs to construct a faster approximation of the expensive functions. This surrogate can then estimate the objective and constraint values at untested design points.

The same problem occurs when testing aircraft shapes with computational fluid dynamics. Running a detailed simulation for every possible wing shape would be extremely expensive, so a surrogate can help identify which designs are worth testing in detail. This leads directly into Gaussian Process Regression (GPR).

5. Gaussian Process Regression: Learning from Limited Data

Gaussian Process Regression (GPR), also commonly called Kriging in engineering design optimisation, sounds heavy, but the idea is quite simple. We test a set of initial bridge designs, collect the results from the structural analyses, and use those observations to construct a statistical model. Then, we use that model to predict how untested designs might behave.

What makes GPR particularly useful is that it outputs two important pieces of information for a design point: the predicted value and the uncertainty of that prediction.

These can be represented simply as:

μ(x) = predicted value

σ²(x) = predictive variance

The standard deviation is:

σ(x) = √σ²(x)

Imagine the model predicts that a specific bridge design will yield low CO₂-equivalent emissions. If no nearby designs were tested before, the uncertainty may be high. The model is essentially making a less certain prediction. Compare that to a region packed with previous simulation data, where the uncertainty can be much lower because the model has more information.

For example, if two designs have the same predicted emissions but one lies in an area with very little previous data, that design may be more interesting because its prediction is more uncertain.

This uncertainty is an important part of Bayesian optimisation.

GPR provides a faster approximation of the objective and constraint functions, which reduces the number of expensive FEM analyses that need to be performed.

6. How the Model Learns the Design Landscape

The model begins with a limited number of expensive FEM evaluations and uses those results to build an approximation.

The paper uses Latin Hypercube Sampling (LHS) for the initial sampling. The purpose is to distribute the initial design points across the design space so that the surrogate model receives information from different regions rather than concentrating all the observations in one area.

At first, the model only knows about the designs that have actually been tested. Between these points, it estimates what the bridge performance might look like.

As more simulations are performed, the model receives new information and updates its predictions. A design that initially appeared promising may turn out to be infeasible after a real FEM evaluation. Another region that seemed uninteresting may reveal better performance than expected.

This creates a much more efficient optimisation process. However, the model still needs a strategy for deciding which new design should be tested. Testing a design that is already well understood may provide little new information, while testing an entirely uncertain design may waste computational resources.

The optimisation problem therefore becomes a decision-making problem: where should the next expensive simulation be performed? That is the role of Bayesian optimisation.

Figure 2: The Bayesian Optimisation process

7. Bayesian Optimisation: Deciding What to Test Next

We can say that Bayesian optimisation sits directly on top of the GPR surrogate model. While GPR provides predictions and uncertainty, Bayesian optimisation uses an acquisition function to navigate that predicted landscape and select the exact input variables for the next high-fidelity simulation.

It constantly balances two competing ideas: exploitation and exploration.

Exploitation means testing a region where current predictions seem promising. We can think of this as using what the model already knows to search for better solutions.

Exploration means testing a region where the uncertainty of the model is high and data are sparse. We can think of this as investigating areas that are less understood and could potentially contain better solutions.

Leaning too hard on either strategy creates major issues.

Pure exploitation can cause the algorithm to focus too heavily on one promising region and miss superior bridge designs elsewhere. Pure exploration can waste expensive computing power investigating regions that may not lead to useful improvements. Bayesian optimisation solves this by balancing both approaches dynamically.

The paper discusses several acquisition functions, including Probability of Improvement (PI), Expected Improvement (EI), and Max-value Entropy Search (MES). However, for the actual bridge case study, the researchers evaluate two constrained Bayesian optimisation methods:

Constrained Expected Improvement (cEI): Extends Expected Improvement by weighting the expected improvement according to the Probability of Feasibility. This makes designs that are unlikely to satisfy the constraints less attractive.

Constrained Max-value Entropy Search (cMES): Extends Max-value Entropy Search by accounting for the constraints while selecting designs expected to provide useful information about the location of the optimum.

The Probability of Feasibility (PF) is important here. In simple terms, PF measures how likely the surrogate model thinks a design is to satisfy the structural constraints. If a design looks promising in terms of emissions but has a high probability of violating a strain limit, cEI reduces its attractiveness.

The goal is to reach an optimal design using far fewer expensive FEM evaluations while still accounting for uncertainty and structural feasibility.

8. The Bridge Results and the Bigger Idea

The paper’s bridge case study shows that constrained Bayesian optimisation can identify strong design solutions without requiring the same number of expensive structural evaluations as conventional approaches. In particular, cEI and cMES were able to improve the design quickly, with most of the improvement occurring within the early iterations.

The results also highlight that the way an optimisation method handles uncertainty can make a meaningful difference. cEI produced slightly better designs on average, while cMES showed advantages in finding feasible designs when it was sufficiently sampled. In comparison, GPRM was less effective because it relied only on predicted values rather than using uncertainty to guide its decisions.

More importantly, the study shows that optimisation is not about finding a perfect answer in isolation. A solution only becomes useful when it works within the conditions of the real system. For engineering problems, this means that mathematical performance has to remain connected to what can actually be built and used.

The bridge example therefore demonstrates a broader principle: the best solution is not necessarily the theoretical optimum, but the best solution that works in reality. The bigger idea is that this approach can be applied whenever simulations are expensive and several conditions must be satisfied at once. Instead of testing everything, the algorithm learns from a small number of tests and becomes increasingly strategic about where to test next.

Reference:

Røstum, Heine, et al. ‘Constrained Bayesian Optimization for Engineering Bridge Design’. Structural and Multidisciplinary Optimization, vol. 68, no. 1, Jan. 2025, https://doi.org/10.1007/s00158-024-03951-3.