Optimization of Sheet-Pile and Retaining-Wall Design: A Review of Methods and Metaheuristic Approaches

Keywords: Sheet-pile wall · Retaining wall · Metaheuristic optimization · Cost minimization · Reliability-based design · Multiobjective optimization · Structural engineering

1 Introduction

The design of sheet-pile and retaining-wall systems has evolved from classical analytical procedures toward modern metaheuristic-based optimization frameworks. Early works primarily focused on limit equilibrium and conventional stability analyses, while contemporary approaches employ numerical and artificial intelligence–assisted optimization. The goal has been consistent: minimizing construction cost and material use while maintaining safety against sliding, overturning, and bearing failures. Optimization frameworks now allow engineers to handle nonlinearity, discrete variables, and multiple design objectives simultaneously. Over the last two decades, studies such as those by Kashani and Camp [1], Sharma and Saha [2], and Liu et al. [3] have demonstrated that metaheuristic methods can outperform classical optimization by achieving superior convergence and solution diversity.

2 Deterministic and Analytical Optimization Approaches

The earliest efforts in retaining-wall optimization employed deterministic formulations based on classical soil mechanics. These models used analytical cost functions and stability constraints, solved by calculus-based or linear programming techniques. Bhatti [8] presented one of the foundational studies by formulating retaining wall optimization in Microsoft Excel Solver, targeting minimal total cost while satisfying safety factors against sliding and overturning. Such deterministic methods offered simplicity but could not easily accommodate discontinuities or multiple objectives.

In subsequent developments, Kalemci and Banu [4] examined seismic effects on reinforced concrete cantilever walls, applying analytical optimization to adjust wall geometry for reduced seismic moments. Their findings established that minor increases in base width could significantly enhance wall stability. These deterministic approaches, although limited in flexibility, provided an important baseline for defining the objective and constraint functions later used in metaheuristic optimization.

3 Metaheuristic and Evolutionary Optimization Methods

The introduction of metaheuristics revolutionized the design of retaining and sheet-pile walls. Algorithms such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO), and Differential Evolution (DE) have been applied to handle the nonlinear, discrete, and multi-constrained nature of wall design.

Sharma and Saha [2] applied GA and PSO for optimizing cantilever wall dimensions to minimize both weight and cost. Their comparative study in Engineering with Computers showed that PSO achieved faster convergence and lower design cost, emphasizing its robustness in continuous parameter spaces. Similarly, Gandomi and Kashani [5] optimized reinforced concrete retaining walls using evolutionary algorithms, proving that evolutionary computation could reduce material consumption by 15–20% compared to traditional manual design.

Mohamed and Fathiyah [7] investigated soil-nailed wall systems and implemented Differential Evolution (DE) for optimization. Their findings highlighted DE’s capability to maintain stability while minimizing construction cost under variable soil conditions. Liu et al. [3] proposed a hybrid algorithm combining swarm intelligence and local search to improve convergence accuracy, successfully demonstrating the efficiency of hybrid metaheuristics for soil–structure interaction problems.

These studies confirmed that metaheuristics, particularly PSO and DE, provide high-quality design solutions for complex geotechnical problems where classical gradient-based algorithms fail to converge.

4 Multiobjective and Reliability-Based Optimization

More recent works extend optimization from single-objective cost minimization toward multiobjective and reliability-based formulations. Kashani and Gandomi [9] addressed the simultaneous minimization of cost and environmental impact for reinforced concrete retaining walls using multiobjective evolutionary algorithms. The resulting Pareto front allowed engineers to select the optimal balance between economy and performance.

Ali and Camp [1] developed a hybrid metaheuristic framework for mechanically stabilized earth walls, considering both cost and safety as objectives. Their model successfully captured the interaction between reinforcement spacing and wall height, demonstrating the advantage of multiobjective search strategies in geotechnical design.

Reliability-based optimization (RBO) has also gained prominence. Gandomi and Kashani [6] integrated metaheuristics with response surface methodology to address uncertainty in soil properties and material strengths. Their results showed that reliability-informed optimization yields safer and more robust designs than deterministic methods. These developments represent an important paradigm shift from safety-factor-based to probability-based design philosophy.

5 Emerging Trends and Research Gaps

Current trends emphasize coupling metaheuristic optimization with finite-element (FEM) or finite-difference (FDM) simulations to account for nonlinear soil–structure interaction. For instance, Liu et al. [3] utilized numerical simulations within a hybrid metaheuristic framework to optimize soil-nail wall systems. Such integration of optimization with physics-based modeling allows better prediction of deflection and stress distribution.

Another growing trend is the adoption of surrogate models and machine learning–aided optimization, where computationally expensive analyses are approximated by regression-based models, neural networks, or Kriging surfaces. While Kashani and Camp [1] highlighted the benefits of hybrid metaheuristics, the use of fully data-driven surrogates remains in its infancy for retaining-wall applications.

Research gaps persist in several areas. First, most optimization studies still neglect construction sequencing and time-dependent effects, which can significantly influence real-world performance. Second, few studies benchmark algorithmic efficiency on standardized test cases, making comparisons between algorithms inconsistent. Third, integration into commercial software environments (e.g., PLAXIS, ETABS, or SAFE) remains limited, restricting the transfer of these techniques to professional practice.

6 Methodological Summary of Applied Metaheuristics

Algorithm	Category	Key Feature	Application Reference
Genetic Algorithm (GA)	Evolutionary	Chromosome-based search with crossover and mutation	Sharma and Saha [2]
Particle Swarm Optimization (PSO)	Swarm-based	Velocity-position update; rapid convergence	Sharma and Saha [2]
Differential Evolution (DE)	Evolutionary	Population-based mutation and crossover	Mohamed and Fathiyah [7]
Hybrid PSO–Local Search	Hybrid	Combines global swarm exploration with local refinement	Liu et al. [3]
Hybrid Metaheuristic Framework	Multiobjective	Integrates multiple algorithms for stability–cost trade-offs	Kashani and Camp [1]
Evolutionary + Response Surface	Reliability-based	Surrogate-assisted uncertainty modeling	Gandomi and Kashani [6]

Across the literature, PSO and hybrid evolutionary algorithms have shown consistent success for engineering optimization due to their few parameters and stable convergence behavior. These frameworks have proven capable of reducing wall cost while maintaining required safety margins under complex constraint sets.

7 Conclusion and Future Directions

The literature reveals a steady progression from deterministic optimization toward advanced metaheuristic and multiobjective frameworks in the design of sheet-pile and retaining-wall systems. Deterministic methods provided foundational understanding, while evolutionary algorithms introduced flexibility and global search capability. Multiobjective and reliability-based formulations have added realism by incorporating multiple design goals and uncertainty quantification.

Nevertheless, further development is required to bridge academic optimization with engineering practice. Future studies should establish standardized benchmark problems, incorporate numerical analysis directly within optimization loops, and promote hybrid AI-optimization frameworks capable of learning from previous design iterations. The integration of optimization tools with commercial design platforms will be critical for real-world adoption.

Ultimately, the next generation of optimization frameworks for retaining structures will likely combine metaheuristic intelligence, numerical modeling, and automation to create fully AI-assisted, code-compliant design environments.

References

1. Ali R. Kashani, Charles V. Camp (2022). Multi-objective optimization of mechanically stabilized earth retaining walls using hybrid metaheuristics. International Journal for Numerical and Analytical Methods in Geomechanics.

2. Sushmita Sharma, Apu Kumar Saha (2021). Optimization of weight and cost of cantilever retaining walls using PSO and GA. Engineering with Computers.

3. Lulu Liu, Ruigang Wu, Surya Prakash (2021). Design optimization of the soil nail wall-retaining system using hybrid metaheuristics. Acta Geotechnica.

4. Elif Nur Kalemci, Sabriye Banu (2020). Design of reinforced concrete cantilever retaining walls under seismic loads. Structures.

5. Amir H. Gandomi, Ali R. Kashani (2017). Optimization of retaining wall design using evolutionary algorithms.Structural and Multidisciplinary Optimization.

6. Amir H. Gandomi, Ali R. Kashani (2015). Optimization of retaining wall design using response surface methodology and metaheuristics. Engineering Structures.

7. Mohd Sukry Mohamed, Fathiyah (2019). Optimization of soil-nailed wall design using differential evolution. —

8. M. Asghar Bhatti (2006). Retaining Wall Design Optimization with MS Excel Solver. —

9. Ali R. Kashani, Amir H. Gandomi (2022). Multi-objective optimization of reinforced concrete retaining walls.Structural and Multidisciplinary Optimization.