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A fuzzy adaptive metaheuristic algorithm for identifying sustainable, economical, lightweight, and earthquake-resistant reinforced concrete cantilever retaining walls

This study proposes a novel adaptive fuzzy-based metaheuristic algorithm to optimize reinforced concrete cantilever retaining walls for seismic resilience, achieving sustainable, economical, and lightweight designs by solving a complex twelve-dimensional constrained optimization problem under nine seismic conditions.

Original authors: Farshid Keivanian, Raymond Chiong, Ali R. Kashani, Amir H. Gandomi

Published 2026-06-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Farshid Keivanian, Raymond Chiong, Ali R. Kashani, Amir H. Gandomi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect tasked with building a massive, heavy-duty wall to hold back a hillside of dirt. This isn't just any wall; it's a Retaining Wall made of reinforced concrete, designed to stand tall even when the ground starts shaking during an earthquake.

The challenge? You need to find the "Goldilocks" design:

Strong enough to not collapse.
Light enough to save on materials.
Cheap enough to fit the budget.
Green enough to produce minimal carbon emissions.

The problem is that there are millions of ways to build this wall (changing the thickness of the base, the height of the stem, the amount of steel inside, etc.). Trying every single combination by hand would take forever. This is where the authors of this paper come in with a new, smart computer program.

Here is a simple breakdown of what they did:

1. The Problem: A Maze of Millions of Paths

Think of designing this wall like trying to find the lowest point in a giant, foggy mountain range. You want to find the absolute bottom (the best design), but there are thousands of valleys (local solutions) that look like the bottom but aren't. Plus, the terrain is full of "No Entry" signs (safety rules). If your design breaks a safety rule, the wall collapses, and you get a huge penalty.

The wall has 12 different dials you can turn (like adjusting the width of the base or the amount of steel). Turning these dials changes the cost, weight, and safety. The goal is to turn them to the perfect setting.

2. The Solution: A Smart, Fuzzy "Swarm" of Explorers

The authors created a new computer algorithm called FAGLSUD. To understand how it works, imagine sending out a team of 100 explorers into that foggy mountain range to find the lowest point.

Instead of just wandering randomly, these explorers are organized into Empires (groups), led by the smartest explorers (called "Imperialists"). The rest are "Colonies" (followers).

The algorithm uses three special tricks to help the team find the best spot faster:

Trick 1: The "Follow the Leader" Move (Global Learning)
If an explorer sees that the leader of their group or the best explorer in the whole mountain is in a better spot, they move toward them. This helps the team zoom in on good solutions quickly.
Trick 2: The "Wander Off" Move (Universal Diversity)
Sometimes, everyone gets stuck in the same small valley. To fix this, the algorithm tells some explorers to look at other groups' best spots and jump in a different direction. This helps the team explore new areas so they don't miss a better valley nearby.
Trick 3: The "Mutate" Move (Local Search)
This is like taking a small step sideways to see if a tiny change makes the spot even better. It's a fine-tuning mechanism to polish the solution.

3. The "Fuzzy" Brain: Making Decisions Without a Rulebook

Here is the clever part. Usually, computer programs need humans to set strict rules (e.g., "If you are 50% done, move 10% faster"). But the real world is messy.

The authors gave their explorers a "Fuzzy Brain." Instead of strict "Yes/No" rules, the brain uses concepts like "Low," "Medium," and "High."

Example: If the team is "stuck" (not finding better spots) and they are "early" in the search, the brain says, "Hey, let's be more adventurous and explore more!"
Example: If the team is "close to the end" and "doing well," the brain says, "Okay, let's stop wandering and focus on polishing the best spot we found."

This "Fuzzy" system allows the computer to learn on the fly and adjust its strategy automatically, without a human needing to tweak the settings.

4. The Earthquake Factor

The wall isn't just sitting there; it has to survive an earthquake. The computer simulates the ground shaking using "pseudo-static" coefficients (basically, pretending the shaking is a heavy weight pushing on the wall). They tested nine different earthquake scenarios (from a light tremor to a heavy shake) to make sure the design holds up in all of them.

5. The Results: The Winner Takes All

The authors tested their new "Fuzzy Swarm" algorithm against other famous computer programs (like Genetic Algorithms, Particle Swarm Optimization, and others).

They ran the simulation 101 times for two different wall sizes to be sure. The results showed that their new algorithm was the champion. It consistently found designs that were:

Cheaper (saved money).
Lighter (used less concrete and steel).
Greener (produced less CO2 pollution).

The Bottom Line

The paper claims that by using this new, self-adjusting "Fuzzy" algorithm, engineers can design earthquake-resistant walls that are safer, cheaper, and better for the environment than what was previously possible with older computer methods. It's like giving a team of explorers a smart, adaptive map that learns as they go, ensuring they find the absolute best path through the fog.

Technical Summary: A Fuzzy Adaptive Metaheuristic for Sustainable RCC Retaining Wall Design

Problem Formulation
This study addresses the complex optimization of Reinforced Concrete Cantilever (RCC) retaining walls, specifically within earthquake-prone zones. The design challenge is characterized as a twelve-dimensional constrained optimization problem involving eight continuous geometric variables (e.g., base width, stem thickness) and four discrete structural variables (steel reinforcement areas). The objective is to identify designs that are simultaneously sustainable (low CO2 emissions), economical (low cost), and lightweight, while maintaining structural integrity under seismic loads.

The analysis employs the pseudo-static method, utilizing the Mononobe-Okabe (M-O) theory to simulate seismic effects through nine distinct combinations of horizontal ( $k_h$ ) and vertical ( $k_v$ ) acceleration coefficients ranging from 0 to 0.3. The optimization is subject to 26 rigorous constraints derived from ACI 318-05 (2005) for structural strength (shear and moment capacity) and DAS (1994) for geotechnical stability (overturning, sliding, and bearing capacity). The solution space is further complicated by the non-linear nature of these constraints and the mixed continuous-discrete variable types.

Methodology: The FAGLSUD Algorithm
To navigate this high-dimensional, constrained search space, the authors propose a novel Fuzzy Adaptive Global Learning Local Search Universal Diversity (FAGLSUD) algorithm. This metaheuristic integrates concepts from Swarm Intelligence (specifically Imperialist Competitive Algorithm and Particle Swarm Optimization) and Evolutionary Algorithms (Differential Evolution), enhanced by Fuzzy Logic Systems (FLS) for adaptive control.

The core components of FAGLSUD include:

Sub-population Structure: The search space is divided into "empires" (sub-populations) containing "imperialists" (best solutions) and "colonies" (other solutions) to facilitate information sharing.
Three Novel Search Operators:
- Global Learning-based Velocity Adaptation (GLVA): Focuses on exploitation by guiding colonies toward their imperialist and personal bests, and imperialists toward the global best.
- Universal Diversity-based Velocity Divergence (UDVD): Focuses on exploration by learning from diverse solutions across the population to prevent premature convergence.
- Enhanced Differential Evolution-based Local Search (EDELS): Mimics gradient descent to enhance information sharing and local refinement through mutation and crossover phases.
Fuzzy Adaptive Mechanisms: Unlike traditional metaheuristics requiring manual parameter tuning, FAGLSUD employs Fuzzy Inference Systems (FIS) to dynamically adjust control parameters (e.g., learning coefficients, scaling factors) and select the most appropriate search operator.
- Input Variables: The FIS utilizes "Normalized Relative Powers" (measuring the distance between current solutions, local bests, and global bests) and the "Normalized Iteration" count.
- Output Actions: The system adapts parameters to balance exploration and exploitation based on the current convergence status (e.g., increasing diversification if the algorithm stagnates).
- Operator Selection: A fuzzy rule-based system evaluates "stagnation" over time windows to probabilistically select which search operator (GLVA, UDVD, or EDELS) should be applied in the next iteration.

Key Contributions

Novel Algorithm: The introduction of FAGLSUD, which uniquely combines sub-population dynamics with three distinct search strategies, all governed by a fuzzy adaptive framework that eliminates the need for manual parameter tuning.
Multi-Objective Seismic Design: The study is the first to apply such an automated optimization technique to RCC walls considering nine pseudo-static seismic loading cases simultaneously, targeting three distinct objective functions: minimum cost, minimum weight, and minimum CO2 emissions.
Sustainability Integration: The explicit calculation of embodied CO2 emissions based on material volumes and standard emission factors (BEDEC database), linking structural optimization directly to environmental impact reduction.
Robustness: The algorithm is designed to handle the high non-linearity and dimensionality of the problem without the computational overhead of extensive parameter sensitivity analysis.

Results and Performance
The efficacy of FAGLSUD was validated using two numerical design examples (Example 1 and Example 2) with varying boundary constraints. The algorithm was tested against a suite of classical and state-of-the-art metaheuristics, including Particle Swarm Optimization (PSO), Grey Wolf Optimization (GWO), Flower Pollination Algorithm (FPA), Biogeography-Based Optimization (BBO), and Simulated Annealing (SA).

Experimental Setup: Each algorithm was run 101 independent times with a population size of 50 and a maximum of 1000 iterations.
Performance Metrics: The study compared the mean objective values (Cost, Weight, CO2) and Standard Deviation (SD) across the nine seismic scenarios.
Findings: FAGLSUD consistently outperformed the comparative algorithms, achieving lower mean costs, weights, and CO2 emissions. The statistical significance of these results was verified using the Wilcoxon rank-sum test. The fuzzy adaptive mechanisms effectively balanced the search process, allowing the algorithm to escape local optima and converge to superior solutions more reliably than the fixed-parameter counterparts.

Significance and Claims
The paper claims that FAGLSUD offers a robust, automated solution for the sustainable design of earthquake-resistant infrastructure. By integrating fuzzy logic into the search mechanism, the method addresses the "parameter tuning" bottleneck common in metaheuristics, making it more accessible and efficient for complex civil engineering problems.

The authors emphasize that their approach fills a research gap by simultaneously addressing algorithmic innovation and the environmental implications of RCC design. The study demonstrates that structural optimization can significantly reduce the embodied carbon of retaining walls without compromising seismic safety or economic viability. The proposed method is presented as a step toward more sustainable construction practices in the face of increasing climate change and seismic risks, providing a tool that maximizes material efficiency while minimizing environmental impact.