1. Introduction

Fatigue crack growth is one of the most important failure mechanisms that occurs in engineering materials under cyclic loading. It is a gradual process that involves damage accumulation, crack initiation, stable crack propagation and final fracture at a critical crack size. Hence, the prediction of crack growth will be key in the estimation of residual life, the durability analysis and the assessment of the structural integrity [1].

Several computational methods are used to analyze fatigue crack growth. FEM is widely applied to evaluate stress and strain fields near cracks [1, 2]. One of the advantages of XFEM is its ability to represent cracks in a way that is independent of the mesh and minimizes the need for repeated remeshing to model cracks' propagation [2]. Progressive material degradation can be described by damage variables, continuum damage mechanics [3]. Phase-field methods are also well suited for modeling crack initiation, propagation, branching and interaction, but have been shown to be computationally expensive in some cases [4]. Multiscale models are useful if the fatigue behavior depends on the microstructure and machine-learning methods are gaining increasing adoption for crack detection and fatigue prediction [5].

Although such methods exist, it is hard to choose the most appropriate computational method. The more sophisticated the approach, the less likely it is to be the best solution to a given problem. For instance, the use of a phase-field approach might not be necessary for a regular known crack, while classical FEM might not be enough for complex branching or crack paths that are not known. This paper therefore treats the problem of selecting a method as a multi-criteria decision making problem and proposes a TOPSIS based framework for comparing the computational approaches by evaluating them in terms of accuracy, crack-path capability, damage representation, computational cost, data requirements, simplicity of implementation and physical interpretability [6].

2. Literature Review

Recent studies on fatigue crack growth prediction have developed in several main directions: finite element modeling, damage mechanics, phase-field modeling, multiscale approaches, and data-driven methods. Although there are some newer techniques available, the FEM approach is still widely used in computational fracture mechanics for the ability to assess stress fields, crack-tip parameters, and crack propagation behavior. Alshoaibi and Fageehi [1] and Su et al. [9] summarize modern finite element methods such as FEM, XFEM, cohesive zone modeling, virtual crack closure methods, adaptive remeshing, and probabilistic methods. The XFEM formulation developed by Moës, Dolbow and Belytschko [2] is particularly significant as it enables the modelling of crack growth without the need for continuous remeshing.

Furthermore, there are models for materials under cyclic loading which are important for elastoplastic and damage analysis. Vansovich and Yadrov [7] address the problems of elastoplastic modeling of fatigue cracks while Stratula [8] provides a mathematical model of fatigue fracture under high-frequency bending vibrations in titanium alloys. The continuum damage mechanics model represents a material degradation process by means of damage variables [3]. From this direction, Tumanov and Kosov [10] study parameter identification for the damaged viscoplastic media in the prediction of durability.

Another exciting field of research in computational fatigue fracture is phase-field modeling. Cui, Du and Zhang [4] summarize its use for fatigue fracture modeling, and Yan, Schreiber and Müller [11] provide an overview of efficient implementation of phase-field fatigue crack growth models. Kalina et al. [12] compare different phase-field fatigue models in a common framework. The ability to capture the physics of crack initiation, propagation, branching and interaction is demonstrated in these studies, which are limited by the high computational cost of the phase-field models.

The methods of multiscale approach and data-driven methods have also gained more and more relevance. Naimark et al. [13] address scale-invariant regularities in the stages of fracture and laws of fatigue crack growth. Lucarini, Dunne and Martínez-Pañeda [14] propose a computational framework for micromechanical fatigue cracking by combining a crystal plasticity model and a phase field one. In the field of machine learning-based fatigue analysis, Melching et al. [5] are the ones who applied explainable machine learning to fatigue crack-tip detection; Li et al. [15] used automated machine learning to predict fatigue crack propagation in marine structures; Zhan et al. [16] reviewed machine learning methods for fatigue behavior and fatigue crack growth prediction.

Overall, the literature indicates that there is no single computational solution that is generally superior. There are methods that are applicable to certain crack geometry, material behavior, loading, level of accuracy required, data available, and computational resources available. Thus, there is a need for a comparative selection framework. The TOPSIS method is appropriate for this purpose as it ranks the alternatives based on their distance from the ideal solution and negative-ideal solution [6].

3. Theoretical Basis of Fatigue Crack Growth Prediction

In linear elastic fracture mechanics, fatigue crack growth is commonly related to the stress intensity factor range:

where is the range of stress intensity factor, is the geometry correction factor, is the cyclic stress range and is the crack length.

A common expression for fatigue crack growth rate is:

where is the crack extension per loading cycle, and are material constants, and is the number of cycles.

The residual life can be estimated theoretically as follows:

In the equation above, is the remaining number of cycles, is the initial crack length, and is the critical crack length.

These formulas explain the basic mechanics of fatigue crack growth. However, real materials often involve plasticity, variable loading, residual stresses, complex crack paths, and microstructural effects. For this reason, computational approaches are required to extend classical analytical models.

4. Comparative Analysis of Computational Approaches

Before applying the TOPSIS-based selection procedure, the main computational approaches should be compared in terms of their capabilities, limitations, and typical areas of use. The selected methods are listed in Table 1 based on their modeling principle, advantage, limitation and suitable application. This comparison serves as the basis to define the evaluation criteria used later in this multi-criteria decision-making framework.

Table 1

Main computational approaches for fatigue crack growth prediction

The comparison reveals that the selection of the method is dependent on the purpose and nature of the fatigue problem. When the geometry of the crack is known and simple to be described, FEM can be used for simple and interpretable analysis, and when the crack geometry is unknown or is changing with the time, XFEM is more suitable. The distributed degradation can be described by continuum damage mechanics while complex crack initiation, branching and interaction can be described by phase-field modeling. When microstructural effects matter the modeling approach is multiscale, and when there are reliable datasets for prediction or surrogate modeling, machine learning may be applicable.

5. TOPSIS-Based Methodological Framework

TOPSIS ranks the alternatives by comparing their distance from the positive ideal solution and the negative ideal solution. In this paper, the alternatives are:

The evaluation criteria are shown in Table 2.

Table 2

TOPSIS evaluation criteria

The decision matrix is written as:

where is the score for alternative according to criterion .

The normalized matrix is calculated as:

where is the normalized value and is the number of alternatives.

The weighted normalized matrix is:

where is the criterion weight.

The distance from the positive ideal solution is:

The distance from the negative ideal solution is:

The closeness coefficient is:

The highest value indicates the most suitable alternative under the selected criteria and weights.

6. Illustrative Evaluation

In order to illustrate the proposed TOPSIS framework, an illustrative decision matrix is built based on the above defined criteria. The scores are based on a theoretical scale of 5 points and are only meant to demonstrate the ranking procedure and not be used as a general evaluation.

1 = very low suitability; 2 = low; 3 = moderate; 4 = high; 5 = very high.

Table 3

Illustrative decision matrix

This matrix is used as an input data for TOPSIS calculation and helps to compare and rank the selected computational approaches.

After constructing the decision matrix, the TOPSIS procedure is applied to calculate the distance of each method from the positive and negative ideal solutions. Finally, the closeness coefficient is calculated and used to rank the computational approaches, as illustrated in Table 4.

Table 4

Illustrative TOPSIS ranking

The illustrative ranking shows that, based on the criteria and weights, phase-field modeling is the most appropriate approach. This result is primarily tied to its excellent modelling capabilities for crack initiation, branching and complicated crack evolution. XFEM is second as it has a moderate computational complexity and also can model the crack path. The simplicity, interpretability and engineering applicability of FEM make it important. In this example, machine learning has a lower score because the criteria focus on the physical interpretability and the data demands it.

7. Proposed Algorithm

The proposed procedure starts with problem definition as shown in Fig. 1. The fatigue analysis objective, material behavior, crack condition and loading type are determined at this stage. These inputs lead to the choice of candidate methods such as the finite element method (FEM), the extended finite element method (XFEM), continuum damage mechanics, phase-field modeling, multiscale modeling, and machine learning.

The next step is to establish criteria for evaluation including accuracy, crack path, damage modelling, computational cost, data requirements and interpretation. These are then used to build the decision matrix, and use the TOPSIS procedure. Lastly, the methods are compared by the closeness coefficient and the outcome is discussed bearing the assumptions, limitations and future needs for validation.

Fig. 1. Structural model for computational method selection

8. Discussion

In the proposed TOPSIS based framework, the selection of computational approaches is made more transparent because of the comparison of methods, which is done explicitly by means of the criteria, and not only by novelty. The outcomes demonstrate the area of suitability of each method. The model is suitable for complex crack evolution, unknown crack paths (XFEM), known crack geometries (FEM), distributed degradation (continuum damage mechanics), microstructure-sensitive fatigue (multiscale modeling), and data-rich prediction tasks (machine learning). At the same time, the ranking should be understood as conditional. It depends on the chosen criteria, the weights assigned to these criteria, and the scoring assumptions. For instance, if simplicity and low computational cost are key considerations, then FEM is likely to be ranked higher. With good and sufficient data sets, machine learning can be more competitive. The framework can be improved in the future with the help of the expert scoring, sensitivity analysis, fuzzy TOPSIS, AHP-TOPSIS weighting and benchmark crack growth problems for validation purposes.

9. Conclusion

This paper proposed a TOPSIS-based framework for selecting computational approaches to fatigue crack growth prediction in engineering materials. The framework provides comparisons of methods based on accuracy, crack-path capability, damage representation, data requirement, computational cost, simplicity, and interpretability. The main conclusion is that method selection should depend on the characteristics of the fatigue problem rather than on the novelty of the method. Under the weights selected, the best results in the illustrative evaluation were obtained with phase-field modeling, followed by XFEM and FEM. This finding is not a universal one, but rather a methodological one. The proposed framework presents a clear and flexible computational method selection tool for preliminary computational method selection.

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