Residual Stress Estimation in Structures Composed of One-Dimensional Elements via Total Potential Energy Minimization Using Evolutionary Algorithms

Abstract

This study introduces a novel energy-based inverse method for estimating residual stresses in structures composed of one-dimensional elements undergoing elastic–plastic deformation. The problem is reformulated as a global optimization task governed by the principle of minimum total potential energy. Rather than solving equilibrium equations directly, the internal stress distribution is inferred by minimizing the structure’s total potential energy using a real-coded genetic algorithm. This approach avoids gradient-based solvers, matrix assembly, and incremental loading, making it suitable for nonlinear and history-dependent systems. Plastic deformation is encoded through element-wise stress-free lengths, and a dynamic fitness exponent strategy adaptively controls selection pressure during the evolutionary process. The method is validated on single- and multi-bar truss structures under axial tensile loading, using a bilinear elastoplastic material model. The results are benchmarked against nonlinear finite element simulations and analytical calculations, demonstrating excellent predictive capability with stress errors typically below 1%. In multi-material systems, the technique accurately reconstructs tensile and compressive residual stresses arising from elastic–plastic mismatch using only post-load geometry. These results demonstrate the method’s robustness and accuracy, offering a fully non-incremental, variational alternative to traditional inverse approaches. Its flexibility and computational efficiency make it a promising tool for residual stress estimation in complex structural applications involving plasticity and material heterogeneity.

1. Introduction

Residual stresses are self-equilibrated internal stresses that remain in a material after the removal of mechanical or thermal loads [1]. These stresses, which are commonly introduced during manufacturing processes such as welding [2], forming [3], machining [4,5], or additive manufacturing [6], can significantly influence the structural performance of engineering components by affecting fatigue life, dimensional stability, and fracture resistance. Consequently, accurate determination and control of residual stress fields is of critical importance for ensuring the reliability and safety of engineering systems [7], particularly in applications involving high-performance alloys [8], layered composites [9], and advanced energy storage [10] materials.

Various destructive and non-destructive experimental techniques have been developed for residual stress measurement, including hole-drilling [11], contour method [12], ultrasonic testing [13], neutron diffraction [14], synchrotron X-ray diffraction [15], and focused ion beam [16] digital image correlation methods. Among these, diffraction-based methods have been widely used due to their ability to measure strain non-destructively at different depths. However, they typically offer limited spatial resolution and are constrained by experimental accessibility and material anisotropy. More importantly, experimental techniques alone cannot generate full-field residual stress maps, particularly in components with complex geometries or non-uniform loading histories. To overcome these limitations, computational methods, particularly the finite element method [17], are extensively employed to simulate residual stress fields by solving inverse problems. Finite element method allows detailed modelling of the thermomechanical history of materials and can incorporate complex boundary conditions, material nonlinearity, and history-dependent plastic deformation. Nevertheless, traditional finite element method-based inverse approaches suffer from a number of drawbacks, including sensitivity to input assumptions, dependence on mesh quality, and the difficulty of converging to global solutions in highly nonlinear problems [18]. Additionally, these methods typically require the solution of large systems of equations and may be limited in their efficiency and robustness when applied to high-dimensional design spaces [19].

Residual stress reconstruction techniques aim to determine internal stress distributions resulting from prior mechanical [20] or thermal loading [21]. Among the most widely used experimental methods is the contour method [22], which involves cutting a component and measuring the resulting deformation to infer the residual stress field through elastic back-calculation. Another class of approaches is based on the eigenstrain concept, where inelastic strain distributions, such as plasticity [23,24], creep [25,26], or phase transformation strains, are treated as sources of residual stress and reconstructed via inverse finite element analysis [27,28]. While these methods provide valuable insight, they often require detailed measurements, iterative calibration, or assumptions about stress path history. In contrast, the present study proposes an energy-based inverse method that reconstructs the residual stress state without relying on direct stress or strain measurements, instead minimizing the total potential energy to determine the internal self-equilibrated configuration.

In recent years, optimization-based methods have gained prominence in residual stress estimation by recasting the inverse problem as the minimization of an objective function, such as total potential energy or the error between measured and simulated strain fields [29,30,31,32]. Among these, metaheuristic algorithms, such as genetic algorithms, particle swarm optimization, and simulated annealing, offer robust global search capabilities, particularly suited to nonlinear, non-convex problems. Genetic algorithms have been widely applied in structural mechanics for inverse problems like form finding and material identification, evolving candidate solutions through selection, crossover, and mutation based on a fitness function [33,34,35]. In mechanics, this function often reflects strain energy or total potential energy, the latter aligning with the variational principle that a structure reaches equilibrium by minimizing internal energy minus external work [36]. Unlike traditional approaches, metaheuristic methods do not require derivative information, matrix assembly, or incremental steps, enabling efficient exploration of complex energy landscapes [37,38,39,40,41,42,43,44,45]. Prior work in tensegrity and large-displacement systems has demonstrated the viability of this approach, laying the groundwork for its application to residual stress reconstruction in plastically deformed structures [46,47].

In this study, a novel methodology is developed for the estimation of residual stresses in structures with one-dimensional elements using genetic algorithm-based minimization of the total potential energy. The approach leverages a bilinear elastoplastic material model and incorporates plastic deformation history into the energy functional. Unlike conventional forward simulations, the developed method inverts the structural problem by identifying the stress-free length of elements that minimizes the total potential energy after plastic deformation. This allows the residual stress state to be estimated without requiring complex matrix operations or incremental nonlinear solvers. The methodology is validated using multiple loading and unloading cases, and its predictions are compared against analytical calculations and conventional finite element results obtained using SIMULIA Abaqus 2024. This implementation integrates genetic algorithms with total potential energy minimization for inverse residual stress determination in plastically deformed structures with one-dimensional elements as an approach to evaluate residual stresses without direct matrix inversion, while also accounting for plastic deformation via bilinear stress–strain behavior. The method offers a robust, parallelizable alternative to traditional inverse finite element methods and opens a pathway toward more generalized applications in structural inverse problems involving plasticity and residual stress.

2. Methodology

The estimation of a residual stress field from a final, deformed state is a classic inverse problem. Such problems are inherently ill-posed, meaning a unique solution is not guaranteed, and the solution can be highly sensitive to small variations in input data. The novelty of our approach lies in regularizing this ill-posed problem by grounding it in a fundamental physical law that is the principle of minimum total potential energy. This principle dictates that a system in stable equilibrium will occupy a state corresponding to a global minimum of its total potential energy. While multiple local energy minima may exist, representing computationally plausible but physically unstable states, a unique, physically stable equilibrium state is expected. The proposed methodology is therefore justified as it transforms the problem from an ambiguous inverse task into a well-defined global optimization search for this unique, lowest-energy state. This reformulation directly addresses the issues of non-uniqueness and instability.

According to this variational principle, a deformable structure naturally assumes an equilibrium configuration that minimizes its total potential energy, which is the difference between the material’s internal strain energy and the work performed by external forces. In this framework, the unknowns are not displacements or stresses under a given load; instead, they represent the intrinsic, stress-free configuration that a plastically deformed element would adopt if unconstrained after load removal. To evaluate the total potential energy for each candidate stress-free configuration, the algorithm determines the resulting stresses and displacements in the assembled structure. By iteratively identifying the set of these underlying plastic deformations, expressed as stress-free lengths in a system of one-dimensional elements, that minimizes the calculated total potential energy, the residual stress distribution can be inferred. This approach transforms the problem from solving a system of equilibrium equations into a search for a global energy minimum within a high-dimensional solution space. To navigate this space efficiently, the study employs evolutionary algorithms, which effectively capture complex nonlinearities and material history effects without requiring matrix assembly, sensitivity analysis, or incremental solving schemes.

The equilibrium configuration of a structure corresponds to the state at which the total potential energy, defined as the internal strain energy minus the work performed by external forces, is minimized. The structural model is limited to one-dimensional bar elements exhibiting bilinear elastoplastic behavior with isotropic hardening. Each bar element is defined by its initial, stress-free length $L_i^{\prime }$, and deformed length $L_i$, from which the engineering strain ${\epsilon }_i$ is calculated as given in Equation (1).

$${\epsilon }_i={\displaystyle \frac{L_i-L_i^{\prime }}{L_i^{\prime }}}$$

(1)

The internal energy of each element (element $i$) is computed using Equation (2) by integrating the stress–strain curve $\sigma \left({\epsilon }_i\right)$ over the range of elastic deformation, yielding the strain energy density $U_i$.

$$U_i={\int }_0^{{\epsilon }_i}\sigma \left({\epsilon }_i\right)d\epsilon$$

(2)

For a structure consisting of $n$ number of elements, each with cross-sectional area of $A_i$, the total strain energy $U_T$ is obtained using Equation (3) by summing the product of strain energy density and volume that is defined as $V_i=A_i{L}_i^{\prime }$.

$$U_T=\sum _{i=1}^n{U}_i{A}_i{L}_i^{\prime }$$

(3)

External work $W$ performed by applied forces $F_j$ acting along their respective degrees of freedom $d_j$ is calculated using Equation (4).

$$W=\sum _{j=1}^m{F}_j{d}_j$$

(4)

The total potential energy functional of a structure, $\mathsf{\Pi }$, to be minimized is then defined by Equation (5).

$$\mathsf{\Pi }=U_T-W$$

(5)

The total potential energy minimization is carried out using a real-coded genetic algorithm, which is specifically chosen for its effectiveness in navigating the complex, non-convex, and potentially multi-modal energy landscapes characteristic of ill-posed problems. Each individual in the population represents a possible configuration of the structure, encoded as a vector of design variables. The evolutionary process follows the classical structure of genetic algorithms, incorporating selection, crossover, mutation, and elitism. Parameters controlling the evolution, such as population size, mutation rate, and crossover probability, are selected through trial-and-error, guided by the recommendations in Grefenstette’s early work on control parameter optimization [48]. At each generation, a new population is created by applying genetic operations to the current population. The fitness of each individual is evaluated using the total potential energy functional, and the best-performing members are retained or propagated. If the convergence criterion is not met, typically defined in terms of stagnation in fitness improvement, the newly formed population replaces the previous one, and the algorithm proceeds to the next generation.

$$M_l=\left[N_{k=1}^l,N_{k=2}^l,\dots ,N_{k=g}^l\right]$$

(6)

$$N_k^l=[x_k^l,y_k^l,z_k^l]$$

(7)

In the genetic algorithm framework, each candidate solution, referred to as a member of the population, represents a complete configuration of the structure. For a system of one-dimensional elements, this configuration is defined by a vector of nodal coordinates corresponding to the connections of elements with other elements, rigid bodies and external forces. These values, which encode the deformations retained after loading and unloading, act as the genes of a chromosome in the evolutionary algorithm. The member $l$ in the population, denoted as vector $M_l$, consists of $g$ number of genes, each representing the nodal joints of the structure as given in Equation (6). The vector of nodal joint $k$, given in Equation (7), belonging to the chromosome of member $l$ accommodates information about $x$-, $y$- and $z$-coordinates. This representation enables the algorithm to explore a wide range of physically plausible configurations without relying on force equilibrium equations. In this study, each generation consists of 30 such individuals, forming the evolving population that is iteratively optimized to minimize the total potential energy of the structure.

The selection of individuals for mating is governed by a fitness-proportionate mechanism based on a roulette wheel scheme, which probabilistically favors individuals with lower total potential energy values. To control the selective pressure dynamically throughout the optimization process, a fitness exponent is introduced and applied to the raw fitness values when constructing the probability distribution. By raising each fitness value to a specified exponent, the likelihood of selecting high-performing individuals can be modulated. When the exponent is greater than one, the selection becomes more aggressive, concentrating the probability on the fittest members; conversely, an exponent between zero and one flattens the distribution, promoting diversity and exploration as exemplified in Figure 1. At an exponent value of zero, all individuals have equal probability of selection, regardless of fitness. This adaptive control of the selection process ensures a balance between exploitation of current best solutions and the exploration of new regions in the search space, thereby enhancing the algorithm’s ability to avoid premature convergence.

The probability of selecting a given individual with fitness value, ${\mathsf{\Pi }}_l$, of member $l$ is calculated by normalizing its fitness relative to the rest of the population as formulated in Equation (8). Unlike conventional roulette wheel selection, where selection probabilities are directly proportional to fitness, this method introduces a fitness exponent $e$ to control the influence of fitness on selection. Specifically, the probability is determined by raising each fitness value to the power of $e$ and dividing by the sum of all such powered fitness values across the population of $t$ number of members. This exponentiation modifies the selection pressure by either amplifying or attenuating the differences among fitness values. The exponent $e$ is initialized at a constant value and is adaptively updated after each generation to regulate convergence behavior, allowing the algorithm to maintain a balance between refining current solutions and exploring new ones. This exponent also defines the termination criterion, which is triggered when its value falls below the initial fitness exponent.

$$p_i={\displaystyle \frac{{\mathsf{\Pi }}_l^e}{\sum _l^t{\mathsf{\Pi }}_l^e}}$$

(8)

An adaptive termination criterion is implemented based on the evolution of the fitness exponent used during the selection phase. At each generation, the exponent $e$ is updated according to the relative improvement in the fitness of the best-performing individual. If the fitness of the best member in the current generation, denoted $F_N$, is lower than that of the previous generation $F_B$, the exponent is increased by a multiplicative factor $e_p$ equal to 1.01, thereby intensifying the selection pressure. Conversely, if no improvement is observed, the exponent is decreased by a factor $e_n$ equal to 0.99, reducing the influence of fitness differences and promoting diversity within the population as described in Equation (4). This dynamic adjustment enables the algorithm to self-regulate the balance between exploration and exploitation throughout the search process. The optimization continues until the exponent value falls below a predefined threshold, indicating stagnation in the evolution and triggering termination of the process. The entire evolutionary flow, including initialization, selection, recombination, mutation, evaluation, and convergence monitoring, is summarized in the schematic diagram provided in Figure 2.

$$e=\left\{\begin{array}{c}e\times e_p F_B<F_N\\ \\ e\times e_n F_B=F_N\end{array}\right.$$

(9)

The genetic algorithm parameters, summarized in Table 1, were selected to ensure a robust balance between exploration of the solution space and exploitation of promising solutions. A population size of 30 was chosen as a trade-off, providing sufficient genetic diversity for a problem of this scale without incurring excessive computational cost per generation. The crossover probability of 0.80 and mutation probability of 0.16 are standard values recommended in foundational literature, designed to consistently generate novel candidate solutions while preventing premature convergence. The 10% elitism rate guarantees that the best-performing solution is always preserved in the subsequent generation. The remaining members for mating are selected using a roulette-wheel mechanism based on fitness-proportional probabilities. Each mutation involves perturbing a node’s axial coordinate within a 0.1 mm diameter hypersphere to maintain fine-grained local exploration. The optimization process, driven by a dynamic fitness exponent, terminates when the exponent falls below its initial value of 1.0, indicating convergence. The suitability of this entire parameter set is ultimately demonstrated by the accurate and consistent results achieved in the validation studies. The algorithm was implemented by the authors using custom code written in C++.

Figure 2. Flowchart of the genetic algorithm-based energy minimization process. The process begins with an initial population of candidate solutions ($P_B$). It then enters an evolutionary loop involving selection, crossover, and mutation to generate a new population ($P_N$). The Total Potential Energy (TPE) of each population is calculated to evaluate its fitness. A key feature is the adaptive fitness exponent, which is increased or decreased based on whether the new population shows improvement, dynamically balancing exploration and exploitation.

Figure 2. Flowchart of the genetic algorithm-based energy minimization process. The process begins with an initial population of candidate solutions ($P_B$). It then enters an evolutionary loop involving selection, crossover, and mutation to generate a new population ($P_N$). The Total Potential Energy (TPE) of each population is calculated to evaluate its fitness. A key feature is the adaptive fitness exponent, which is increased or decreased based on whether the new population shows improvement, dynamically balancing exploration and exploitation.

Table 1. Key parameters for the genetic algorithm used in all simulations. These values control the evolutionary search process, including population diversity (Population size), generation of new solutions (Crossover and Mutation probability), and preservation of the best-found solution (Elitism). The initial fitness exponent (IFE) and its convergence limit govern the adaptive selection pressure and the termination criterion for the optimization.

Population size	30 members
Elitism (percent of elite members)	10%
Crossover probability	0.80
Mutation probability	0.16
Initial fitness exponent (IFE)	1.00
Convergence limit	IFE

Table 1. Key parameters for the genetic algorithm used in all simulations. These values control the evolutionary search process, including population diversity (Population size), generation of new solutions (Crossover and Mutation probability), and preservation of the best-found solution (Elitism). The initial fitness exponent (IFE) and its convergence limit govern the adaptive selection pressure and the termination criterion for the optimization.

Population size	30 members
Elitism (percent of elite members)	10%
Crossover probability	0.80
Mutation probability	0.16
Initial fitness exponent (IFE)	1.00
Convergence limit	IFE

3. Nonlinear Response and Plastic Deformation

The developed energy-based inverse formulation was employed to investigate nonlinear material behavior under a range of loading scenarios, with a specific focus on plastic deformation in one-dimensional bar structure. The test specimen was defined as a prismatic bar with a total length of 2000 mm and a uniform rectangular cross-sectional area of 100 mm², as illustrated in Figure 3. To ensure consistency with classical beam theory and eliminate the influence of lateral or shear effects, deformation was restricted strictly to the axial direction. This modelling assumption effectively reduces the three-dimensional problem to a one-dimensional formulation, enabling direct comparison with analytical and inverse methods.

A bilinear elastoplastic material model was adopted, characterized by an initial elastic modulus $E$, a defined yield strength ${\sigma }_y$, and a constant plastic modulus $E_p$ beyond the yield point. The properties used in the simulations represent a high-strength material, chosen to clearly demonstrate the method’s ability to handle characteristic nonlinear behavior. These properties are an elastic modulus of 260.2 GPa, a yield strength of 200 MPa, and a plastic modulus of 1.005 GPa. The material behavior was defined by linear elastic response up to ${\sigma }_y$, followed by linear hardening governed by $E_p$. This constitutive model allows for a clear distinction between recoverable elastic deformation and permanent plastic strain accumulation. The stress–strain relationship used in the simulations is presented in Figure 4 and serves as the constitutive basis for all numerical experiments conducted in this study.

Figure 4. The bilinear elastoplastic stress–strain curve used in simulations for the single-bar model and the central bar (Bar 2) of the multi-bar model. The material exhibits linear elastic behavior up to a yield strength of 200 MPa, followed by a linear hardening response.

Figure 4. The bilinear elastoplastic stress–strain curve used in simulations for the single-bar model and the central bar (Bar 2) of the multi-bar model. The material exhibits linear elastic behavior up to a yield strength of 200 MPa, followed by a linear hardening response.

Figure 5. Displacement results for the single-bar model under a 10 kN load, in the direction of red arrow, which is within the elastic regime. The bar displaces by 0.952 mm under load and fully returns to its original configuration upon unloading, indicating the absence of permanent deformation.

Figure 5. Displacement results for the single-bar model under a 10 kN load, in the direction of red arrow, which is within the elastic regime. The bar displaces by 0.952 mm under load and fully returns to its original configuration upon unloading, indicating the absence of permanent deformation.

The structural element was analyzed under a series of axial loading conditions using both the proposed total potential energy minimization framework and conventional finite element simulations for validation. The structure was subjected to monotonic loading and unloading sequences in discrete force increments ranging from 5 to 30 kN, in 5 kN steps. These loading conditions included both elastic and plastic deformation regimes. One end of the bar was fully constrained in all translational directions, while axial loads were applied along the longitudinal $z$-axis at the free end, as shown in Figure 4. For load magnitudes below the yield threshold, the structure returned to its original configuration upon unloading, while for cases exceeding the yield point, load–unload cycles were introduced to evaluate residual stress development and assess post-yield behavior.

Figure 6. Stress–strain evolution of the single-bar model during loading and unloading at 10 kN. The material follows a linear elastic path up to a peak stress and then unloads along the same path back to zero stress and zero strain, as illustrated with red arrows, confirming fully recoverable deformation.

Figure 6. Stress–strain evolution of the single-bar model during loading and unloading at 10 kN. The material follows a linear elastic path up to a peak stress and then unloads along the same path back to zero stress and zero strain, as illustrated with red arrows, confirming fully recoverable deformation.

The accuracy and convergence behavior of the energy minimization process are inherently influenced by the genetic algorithm parameters, including population size, mutation rate, crossover probability, and seed initialization. Given the stochastic nature of genetic algorithms, individual runs may yield slightly different outcomes. To mitigate this variability and ensure robustness, the reported results represent the averaged outcomes of ten independent optimization runs. While this baseline configuration provides satisfactory accuracy, further improvements can be achieved through hyperparameter tuning or ensemble-based strategies.

Finite element analyses were conducted using the commercial software, employing C3D8R elements, linear eight-node brick elements with reduced integration, suitable for general-purpose nonlinear simulations. Due to the high aspect ratio of the bar, with length greatly exceeding cross-sectional dimensions, the structure was assumed to exhibit effectively one-dimensional behavior. Although this modelling simplification may introduce minor discrepancies due to transverse constraints and numerical integration, the resulting stress and displacement profiles were found to be in close agreement with those predicted by the proposed inverse formulation, thereby validating the reliability of the energy-based method in capturing nonlinear mechanical response and residual stress formation.

Figure 7. Displacement and residual deformation of the single-bar model under plastic loading. The initial 30 kN load, in the direction of red arrow, causes a large displacement (101.818 mm). Upon unloading, a permanent residual deformation of 98.982 mm remains. Subsequent reloading cycles to 30 kN and 45 kN (1.5 times the initial load) demonstrate further plastic deformation.

Figure 7. Displacement and residual deformation of the single-bar model under plastic loading. The initial 30 kN load, in the direction of red arrow, causes a large displacement (101.818 mm). Upon unloading, a permanent residual deformation of 98.982 mm remains. Subsequent reloading cycles to 30 kN and 45 kN (1.5 times the initial load) demonstrate further plastic deformation.

The displacements resulting from the application and removal of a 10 kN axial load are illustrated in Figure 5. The applied force induced a displacement of 0.952 mm along the longitudinal $z$-axis, entirely within the elastic regime. Upon unloading, the structure fully returned to its original configuration, confirming the absence of permanent deformation. The corresponding stress–strain response, shown in Figure 6, reveals that a strain of 0.000476 and a peak stress of 96.75 MPa developed during loading. After unloading, the stress fully relaxed to zero, as expected for linear elastic behavior, and the residual strain was negligible, demonstrating the accuracy of the proposed method in predicting recoverable deformation.

The highest applied load in this study was 30 kN, which exceeded the material’s yield strength and induced significant plastic deformation. This load produced an axial displacement of 101.818 mm, as shown in Figure 7. Upon unloading and reapplication of the same 30 kN load, the bar exhibited a slightly reduced displacement of 98.982 mm, indicative of permanent plastic strain. Subsequent loading cycles at the same force level resulted in a displacement of 101.871 mm, exceeding the value observed during initial loading, demonstrating strain accumulation due to cyclic plasticity. To further explore post-yield behavior, the applied load was increased to 45 kN, corresponding to 1.5 times the original peak load. This additional loading step generated a displacement of 251.776 mm, confirming enhanced plastic flow beyond the initial hardened state.

Figure 8 illustrates the stress–strain evolution during successive load cycles. In the first loading step, the strain increased to 0.050936 under a peak stress of 222.04 MPa. Upon unloading, the stress dropped to zero while a residual plastic strain of 0.049491 remained. During the subsequent load cycle, starting from this plastically deformed state, the stress increased further to 255.11 MPa, surpassing the previous maximum stress and indicating the onset of additional strain hardening. The corresponding strain reached 0.125888, significantly beyond the initial elastic–plastic transition, highlighting the material’s evolving resistance and permanent deformation characteristics under repeated mechanical loading.

The displacements and axial strains for all loading stages, as predicted by the potential energy minimization method, are summarized in

Table 2. Total displacement ($d_z$) and axial strain (${\epsilon }_{zz}$) for the single-bar model under various tensile load cases, as predicted by the proposed genetic algorithm-based method. The results show a purely elastic response up to 20 kN, as indicated by the full recovery upon unloading ($U$). For loads of 25 kN and 30 kN, the model exhibits significant plastic deformation, resulting in non-zero residual displacement and strain. Cycled loading ($CL$) at these higher forces demonstrates further strain accumulation.

	5 kN		10 kN		15 kN		20 kN		25 kN		30 kN
	${\mathit{d}}_{\mathit{z}}$	${\mathit{\epsilon }}_{\mathit{z}\mathit{z}}$	${\mathit{d}}_{\mathit{z}}$	${\mathit{\epsilon }}_{\mathit{z}\mathit{z}}$	${\mathit{d}}_{\mathit{z}}$	${\mathit{\epsilon }}_{\mathit{z}\mathit{z}}$	${\mathit{d}}_{\mathit{z}}$	${\mathit{\epsilon }}_{\mathit{z}\mathit{z}}$	${\mathit{d}}_{\mathit{z}}$	${\mathit{\epsilon }}_{\mathit{z}\mathit{z}}$	${\mathit{d}}_{\mathit{z}}$	${\mathit{\epsilon }}_{\mathit{z}\mathit{z}}$
$\mathit{L}$	0.478	0.00024	0.952	0.00048	0.427	0.00071	1.940	0.00097	51.715	0.02586	101.82	0.05091
$\mathit{U}$	0.000	0.00000	0.000	0.00000	0.000	0.00000	0.000	0.00000	49.345	0.02467	98.982	0.04949
$\mathit{C}\mathit{L} \left(\mathit{L}\mathit{x}\mathbf{1.0}\right)$	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	51.781	0.02589	101.87	0.05094
$\mathit{C}\mathit{L} \left(\mathit{L}\mathit{x}\mathbf{1.5}\right)$	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	176.71	0.08836	251.78	0.12589

and

Table 3. Benchmark results for the single-bar model’s total displacement ($d_z$) and axial strain (${\epsilon }_{zz}$) across all loading conditions, obtained from finite element analysis. This table serves as the primary validation dataset for the genetic algorithm-based predictions shown in Table 2, demonstrating close agreement in both the elastic and plastic deformation regimes.

	5 kN		10 kN		15 kN		20 kN		25 kN		30 kN
	${\mathit{d}}_{\mathit{z}}$	${\mathit{\epsilon }}_{\mathit{z}\mathit{z}}$	${\mathit{d}}_{\mathit{z}}$	${\mathit{\epsilon }}_{\mathit{z}\mathit{z}}$	${\mathit{d}}_{\mathit{z}}$	${\mathit{\epsilon }}_{\mathit{z}\mathit{z}}$	${\mathit{d}}_{\mathit{z}}$	${\mathit{\epsilon }}_{\mathit{z}\mathit{z}}$	${\mathit{d}}_{\mathit{z}}$	${\mathit{\epsilon }}_{\mathit{z}\mathit{z}}$	${\mathit{d}}_{\mathit{z}}$	${\mathit{\epsilon }}_{\mathit{z}\mathit{z}}$
$\mathit{L}$	0.474	0.00024	0.949	0.00047	1.423	0.00071	1.897	0.00095	51.270	0.02564	100.80	0.05040
$\mathit{U}$	0.000	0.00000	0.000	0.00000	0.000	0.00000	0.000	0.00000	48.890	0.02445	97.990	0.04899
$\mathit{C}\mathit{L} \left(\mathit{L}\mathit{x}\mathbf{1.0}\right)$	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	51.270	0.02564	100.80	0.05040
$\mathit{C}\mathit{L} \left(\mathit{L}\mathit{x}\mathbf{1.5}\right)$	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	175.20	0.08760	249.50	0.12475

, respectively. The resulting stress magnitudes after loading and residual stress magnitudes following each unloading stage are presented in

Table 4. Comparison of stress results (in MPa) for the single-bar model between the proposed genetic algorithm (GA) and the finite element method (FEM) benchmark. The table shows the peak stress during initial loading ($L$), the residual stress after unloading ($U$), and the peak stress during cycled loading ($CL$) for various applied forces. The close agreement between the GA and FEM results across all cases validates the accuracy of the proposed inverse method.

	5 kN		10 kN		15 kN		20 kN		25 kN		30 kN
	$\mathbf{G}\mathbf{A}$	$\mathbf{F}\mathbf{E}\mathbf{M}$	$\mathbf{G}\mathbf{A}$	$\mathbf{F}\mathbf{E}\mathbf{M}$	$\mathbf{G}\mathbf{A}$	$\mathbf{F}\mathbf{E}\mathbf{M}$	$\mathbf{G}\mathbf{A}$	$\mathbf{F}\mathbf{E}\mathbf{M}$	$\mathbf{G}\mathbf{A}$	$\mathbf{F}\mathbf{E}\mathbf{M}$	$\mathbf{G}\mathbf{A}$	$\mathbf{F}\mathbf{E}\mathbf{M}$
$\mathit{L}$	50.3	50.0	100.4	100.0	150.4	150.0	204.6	200.0	249.8	250.0	299.9	300.0
$\mathit{U}$	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
$\mathit{C}\mathit{L} \left(\mathit{L}\mathit{x}\mathbf{1.0}\right)$	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	249.9	250.0	299.9	300.0
$\mathit{C}\mathit{L} \left(\mathit{L}\mathit{x}\mathbf{1.5}\right)$	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	374.8	375.0	449.9	450.0

. Across the full range of applied forces, from entirely elastic conditions to highly plastic deformation, the proposed energy-based inverse model demonstrates close agreement with finite element predictions. For instance, the predicted stress values in Table 4 deviate from the finite element method results by less than 1% across most load cases, demonstrating high quantitative accuracy. This consistency validates the robustness of the approach for both linear and nonlinear regimes. Although the model in its current implementation provides accurate residual stress reconstruction, further improvements in prediction accuracy could be achieved by systematic tuning of genetic algorithm parameters such as mutation radius, population diversity, and convergence criteria. Overall, the comparative analysis confirms that the metaheuristic potential energy minimization framework effectively captures both the elastic–plastic transition and residual stress development in structures composed of one-dimensional elements.

4. Nonlinear Deformation and Residual Stress

The developed potential energy minimization framework is extended to evaluate residual stress distributions arising from heterogeneous material behavior in constrained multi-bar systems. This representative three-bar assembly serves as a direct validation of the method’s applicability to such heterogeneous systems. A representative three-bar assembly is considered, composed of axially aligned prismatic elements of equal length but with deliberately mismatched mechanical properties to induce non-uniform plastic deformation under loading. As illustrated in Figure 9, one end of the assembly is fixed in all translational directions, while the free ends are joined at a common interface where a uniaxial displacement is applied along the x-direction to initiate axial deformation.

To ensure a purely axial response and eliminate out-of-plane effects, the material properties were deliberately mismatched. The outer bars (Bar 1 and Bar 3) were assigned identical properties characterized by lower stiffness and strength: an elastic modulus of 66.67 GPa, a yield strength of 100 MPa, and a plastic modulus of 0.335 GPa. In contrast, the central bar (Bar 2) was designed with significantly higher stiffness and strength: an elastic modulus of 260.2 GPa, a yield strength of 200 MPa, and a plastic modulus of 1.005 GPa, as summarized in Figure 10. This configuration enables investigation of internal force redistribution and residual stress development due to elastic–plastic mismatch, a condition frequently encountered in welded joints, dissimilar material interfaces, and functionally graded components.

The residual stress state following load removal is both analyzed using the proposed genetic algorithm-based total potential energy minimization method and validated through nonlinear finite element simulations. This case serves as a stringent benchmark for assessing the framework’s ability to resolve self-equilibrated internal stress distributions in the absence of external forces, relying solely on the principle of minimum potential energy and metaheuristic optimization. The modelling strategy builds upon prior research in eigenstrain-based inverse analysis and inelastic beam theory, while extending it to capture internal mechanical interactions in multi-material structure with one-dimensional elements.

The proposed genetic algorithm-based total potential energy minimization method was successfully employed to analyze the three-bar system under identical loading and boundary conditions. A uniaxial load of 50 kN was applied uniformly to the common free-end surface of the central bar, which has a cross-sectional area of 100 mm². To preserve the geometric consistency of the assembly and isolate axial stress effects, the lengths of all three bars were held constant throughout the simulation. The results obtained from the energy minimization framework, as illustrated in the stress–strain diagrams in Figure 11, show that all three bars were strained up to 0.060782 prior to unloading, resulting in axial stresses of 119.860 MPa in the outer bars and 260.085 MPa in the central bar. Upon unloading, a residual strain of 0.059285 remained due to plastic deformation. In contrast to the single-bar homogeneous cases, which return to a stress-free state upon unloading, the heterogeneous system exhibits non-zero self-equilibrated residual stresses: 20.029 MPa tensile in the outer bars and 39.410 MPa compressive in the central bar. This distribution arises from the internal force balance between elements of differing stiffness and plastic resistance. The central bar develops compressive residual stress due to its higher elastic modulus and yield strength, while the outer bars retain tensile residual stress, an outcome that aligns with theoretical expectations of stress redistribution in heterogeneous assemblies. These findings confirm the ability of the proposed inverse formulation to accurately resolve internal residual stress fields in multi-material systems using only energy-based principles and metaheuristic optimization, without requiring explicit equilibrium constraints or matrix-based solvers.

The results obtained from the energy minimization framework were validated against an independent analytical calculation for the three-bar system. This analytical model is governed by two fundamental principles: compatibility, which requires that all three bars, Bar 1, Bar 2 and Bar 3, experience the same uniform axial strain, ${\epsilon }_1={\epsilon }_2={\epsilon }_3=\epsilon$, and equilibrium, where the sum of the internal forces balances the applied external load, $F=F_1+F_2+F_3$. Expressed in terms of stress, $\sigma$, and cross-sectional area, $A$, this becomes $F=A\left({\sigma }_1+{\sigma }_2+{\sigma }_3\right)$. As it was determined by the genetic algorithm solution, all three bars exceed their respective yield points at a 50 kN load, requiring the use of their bilinear elastoplastic stress–strain relationships, ${\sigma }_{\mathrm{1,3}}={\sigma }_{y,\mathrm{1,3}}+E_{p,\mathrm{1,3}}\left({\epsilon }_{max}-{\epsilon }_{y,\mathrm{1,3}}\right)$ and ${\sigma }_2={\sigma }_{y,2}+E_{p,2}\left({\epsilon }_{max}-{\epsilon }_{y,2}\right)$, where ${\epsilon }_{max}$ is maximum total strain, ${\epsilon }_y$ is yield strain and ${\sigma }_y$ is yield strength. Substituting these expressions into the equilibrium condition yields a maximum strain of 0.0609 corresponding to peak stresses of 119.9 MPa in Bars 1 and 3 and 260.2 MPa for Bar 2. Upon removal of the external load, unloading is assumed to be purely elastic. The strain recovery, $∆\epsilon$, is governed by the combined elastic stiffness of the system, $∆F=∆A∆\epsilon \left(E_1+E_2+E_3\right)$. For a load reduction, $∆F$, −50,000 N, the corresponding strain recovery is −0.0015. The associated elastic stress changes are −100 MPa in Bars 1 and 3 and −300 MPa in Bar 2. The residual stress in each bar is obtained by superimposing the elastic stress change on the peak stress, ${\sigma }_{res}={\sigma }_{max}+∆\sigma$, resulting in tensile residual stresses of +19.9 MPa in Bars 1 and 3, and a compressive residual stress of −39.8 MPa in Bar 2. These analytical results show excellent agreement with the outcomes of the genetic algorithm-based total potential energy minimization method, confirming the physical reliability of the proposed approach.

Table 5. Three-way comparison of results for the multi-bar heterogeneous system, validating the genetic algorithm ($GA$) predictions against analytical calculations ($AC$) and the finite element method ($FEM$). The table presents the axial displacement ($d_z$), axial strain (${\epsilon }_{zz}$), and axial stress (${\sigma }_{zz}$) for each of the three bars. The strong agreement across all three methods for both the loaded ($L$) and unloaded ($U$) states confirms the model’s ability to accurately resolve the self-equilibrated residual stresses that arise from material mismatch.

	Bar 1			Bar 2			Bar 3
	${\mathit{d}}_{\mathit{z}}$	${\mathit{\epsilon }}_{\mathit{z}\mathit{z}}$	${\mathit{\sigma }}_{\mathit{z}\mathit{z}}$	${\mathit{d}}_{\mathit{z}}$	${\mathit{\epsilon }}_{\mathit{z}\mathit{z}}$	${\mathit{\sigma }}_{\mathit{z}\mathit{z}}$	${\mathit{d}}_{\mathit{z}}$	${\mathit{\epsilon }}_{\mathit{z}\mathit{z}}$	${\mathit{\sigma }}_{\mathit{z}\mathit{z}}$
$\mathit{L} \left(\mathit{G}\mathit{A}\right)$	121.57	0.06078	119.86	121.565	0.06078	260.08	121.57	0.06078	119.86
$\mathit{U} \left(\mathit{G}\mathit{A}\right)$	118.57	0.05929	20.03	118.570	0.05929	−39.41	118.57	0.05929	20.03
$\mathit{L} \left(\mathit{A}\mathit{C}\right)$	121.80	0.0609	119.90	121.80	0.0609	260.20	121.80	0.0609	119.90
$\mathit{U} \left(\mathit{A}\mathit{C}\right)$	118.80	0.0594	19.90	118.80	0.0594	39.80	118.80	0.0594	19.90
$\mathit{L} \left(\mathit{F}\mathit{E}\mathit{M}\right)$	121.90	0.06095	122.80	121.900	0.06095	254.50	121.90	0.06095	122.80
$\mathit{U} \left(\mathit{F}\mathit{E}\mathit{M}\right)$	118.90	0.05945	22.77	118.900	0.05945	−45.54	118.90	0.05945	22.77

presents the displacements, strains, and stresses computed analytically for both the loaded and unloaded states. The results of analytical calculations show very good agreement with genetic algorithm-based total potential energy minimization method results.

The displacements, axial strains, and stress values obtained after the loading and unloading stages for the genetic algorithm-based potential energy minimization method, analytical calculation and the finite element method are presented in Table 5. While the results from both approaches are in close agreement, slight discrepancies are observed in the magnitudes from analytical calculation, primarily attributed to the use of C3D8R elements in the finite element model, which may introduce localized numerical artifacts due to reduced integration and mesh discretization. In the finite element method, the sum of residual tensile and compressive stresses satisfies global equilibrium, as expected. A similar self-equilibrated state is observed in the results from the energy minimization method; however, a small deviation from analytical calculation remains. This deviation is not indicative of a computational error but rather reflects the stochastic nature of the genetic algorithm. For the purpose of this study, the genetic algorithm parameters were held constant, as defined in the earlier solutions. Although adjusting these parameters on a case-by-case basis could drive the solution closer to exact equilibrium, the current implementation was intentionally preserved to illustrate the intrinsic behavior and convergence characteristics of the method. This minor residual imbalance highlights a key aspect of evolutionary optimization techniques and underscores the importance of parameter tuning when high precision is required.

5. Discussion

The primary advantage of the proposed method over conventional finite element method lies in its reformulation of the inverse problem as a single, global energy minimization task. Unlike traditional nonlinear finite element method, which requires incremental steps, path-dependent loading histories, and the repeated solution of large systems of equations, the presented approach directly searches for the final equilibrium state that minimizes the total potential energy of the structure. This avoids matrix assembly and significantly simplifies the computational process. The use of a genetic algorithm further enhances robustness by enabling a global search of the solution space, reducing the risk of getting trapped in local minima, a common challenge for gradient-based solvers in non-convex energy landscapes. Moreover, because each candidate solution can be evaluated independently, the algorithm is inherently parallelizable, offering excellent scalability and potential for speed-up on modern computing architectures. This approach thus provides a conceptually streamlined and physically grounded alternative to conventional finite element method for residual stress reconstruction.

From a computational cost perspective, the two methods exhibit fundamentally different scaling behaviors. The time complexity of a traditional nonlinear finite element method analysis is typically dominated by the cost of solving a large system of linear equations at each load increment, a process that scales by polynomials with the number of degrees of freedom. In contrast, the cost of the proposed genetic algorithm-based method is determined by the number of generations required for convergence, the population size, and the cost of a single fitness evaluation. This fitness evaluation, which involves calculating the total potential energy, scales only linearly with the number of elements. The most significant efficiency advantage of the evolutionary approach, however, is its inherent parallelizability. The fitness of every individual in the population can be evaluated simultaneously on a multi-core processor. This provides a fundamental architectural advantage over the largely sequential nature of an incremental finite element method solver, offering a clear path to significant speed-up for complex, nonlinear problems without being constrained by the bottlenecks of traditional matrix solvers.

The proposed method is particularly well-suited to structural problems that can be discretized using one-dimensional elements and that undergo plastic deformation due to mechanical overloading. Practical applications include axially loaded truss structures in civil or mechanical engineering, where plastic deformation can occur due to accidental overloading, seismic events, or temporary construction misalignment. Another relevant scenario is the assessment of post-yield behavior in tension members of deployable or reconfigurable mechanical systems, where internal stresses may arise after retraction or repositioning. The method can also be applied to cable-stayed or prestressed frameworks after relaxation or load redistribution, where stress-free configurations are no longer accessible through direct measurement. In all these examples, the novelty of the method lies in its ability to infer the internal stress state without requiring path-dependent simulations or measurement of strain fields. Instead, residual stresses are reconstructed solely from the post-load geometry by minimizing the total potential energy, making this approach uniquely suited to inverse problems where direct observation of plastic history is impractical.

A critical analysis of the results requires understanding the sources of the minor deviations observed between the proposed method, the finite element model, and the analytical solution. These deviations do not indicate a flaw but rather arise from the distinct nature of each approach. Firstly, the genetic algorithm is inherently stochastic; it excels at finding the global region of the energy minimum but may converge to a near-optimal solution that produces a negligible, non-zero force imbalance, as seen in the results. This reflects the trade-off between computational efficiency and achieving perfect mathematical equilibrium. Secondly, the finite element benchmark has its own sources of numerical artifacts, including the choice of C3D8R elements with reduced integration and the specific mesh discretization, which can influence the final stress calculation. Finally, the analytical model represents an idealized case with perfect boundary conditions and material behavior. The fact that our method’s results align so closely with both of these different benchmarks, with well-understood and minor sources of deviation, provides strong confidence in its physical reliability and accuracy.

A more fundamental challenge inherent to any inverse problem is the potential for non-uniqueness of the solution. It is therefore important to clarify how this issue is addressed within the proposed framework. While a global optimization algorithm like a genetic algorithm is effective at finding a global minimum, it cannot, by itself, resolve ambiguity if multiple, degenerate global minima exist. The novelty and justification of the present approach lie in grounding the inverse problem in a fundamental physical law that inherently guides the search to a unique, physically stable state. A purely mathematical formulation might allow for several different internal stress fields that satisfy equilibrium; however, the Principle of Minimum Total Potential Energy provides an essential physical constraint. This principle dictates that a real mechanical system in stable equilibrium will occupy the state corresponding to the single, global minimum of its total potential energy. The method is therefore designed not just to find a minimum, but to find the unique, physically correct equilibrium state that nature itself would select. By reformulating the ill-posed inverse problem as a search for this specific, lowest-energy state, a physical law is directly leveraged to resolve the issue of non-uniqueness.

The reliability of this implementation, and therefore the validity of the results, is established through two key considerations. First, the code is not a proprietary black box but is a direct implementation of the standard, well-documented principles of genetic algorithms, including classical operators for selection, crossover, and mutation. This ensures the methodology is transparent and could be replicated using other standard toolkits, such as those available in MATLAB or Python libraries. Second, and more critically, the ultimate proof of the implementation’s reliability comes from the rigorous validation presented in this work. The close agreement between the algorithm’s predictions and the results from both independent analytical calculations and the commercial finite element software Abaqus serves as a robust external benchmark. While genetic algorithms are inherently stochastic, which can lead to minor deviations from exact equilibrium as noted in the results, the consistent accuracy achieved across multiple test cases provides high confidence in the correctness and robustness of the computational tool used.

A crucial consideration for any method based on metaheuristics, such as the genetic algorithm employed in this study, is the rigor of the obtained solution. Two primary risks inherent to this approach must be addressed. The first is stochastic convergence, as a genetic algorithm is not guaranteed to find the exact mathematical global optimum in a finite number of steps, instead converging to a near-optimal solution. In a physical context, this can manifest as a negligible but non-zero residual force imbalance in the final predicted state. The second risk is parameter sensitivity, as the algorithm’s performance and convergence can be influenced by the initial choice of parameters. Several remedies are employed to mitigate these risks and ensure a rigorous outcome. To address stochastic convergence, the final force imbalance of the predicted state is verified to be acceptably close to zero. Furthermore, confidence in the consistency and robustness of the solution is established by performing multiple independent optimization runs, as was achieved in this study. The issue of parameter sensitivity is addressed by grounding the initial selection in established literature, while acknowledging that a formal hyperparameter tuning process or the use of ensemble-based approaches represents a key area for future work to ensure optimal performance in more complex, generalized applications. Through the acknowledgment of these characteristics and the application of these mitigation strategies, the rigor of the metaheuristic approach can be confidently established.

The performance and convergence behavior of the presented methodology are inherently linked to the choice of the genetic algorithm parameters. While the parameters used in this study yielded accurate and robust results for the validation cases, it is acknowledged that a formal sensitivity analysis was not performed. The selection was based on established literature and was shown to be effective for the specific problems analyzed. However, for broader applications, particularly on problems with higher dimensionality or more complex energy landscapes, the optimal performance would likely benefit from a systematic hyperparameter tuning process. This represents a key area for future work, which could involve automated tuning strategies or developing ensemble-based approaches to mitigate the effects of parameter sensitivity.

The methodology presented in this study was validated on one-dimensional structures, demonstrating its foundational robustness and viability. However, extending the framework to more complex applications presents both challenges and opportunities for future development. A primary limitation is scalability to higher-dimensional systems, such as two- or three-dimensional solids, where direct optimization over high-dimensional stress-free configurations may lead to computational intractability. To address this, future work will explore parameterizing the solution space using reduced-order representations like basis functions and enhancing search efficiency through hybrid metaheuristic or memetic algorithms. While these algorithmic strategies are auxiliary to the core energy-based principle, they serve to improve optimization performance without altering the physical foundation of total potential energy minimization.

Beyond spatial scalability, the method can also be generalized to incorporate other deformation modes, such as bending, by extending the formulation to include nodal rotations and introducing plasticity variables such as stress-free curvature. Since the variational principle of minimum total potential energy is universal, such generalizations remain within the same theoretical framework. Another important direction is the incorporation of anisotropic material behavior, which would involve adapting the energy functional to more complex constitutive models. Finally, while this study provides robust validation against analytical solutions and finite element benchmarks, the ultimate verification of predictive capability requires experimental validation. Future work should thus include a dedicated experimental campaign to physically measure residual deformation after plastic loading and compare it directly with the inverse method’s predictions.

A further consideration is the prediction of ultimate failure or breaking strength. While the current implementation does not explicitly model fracture, the underlying energy-based framework is capable of incorporating such behavior. The prediction of failure is a function of the constitutive material model, not a limitation of the energy minimization principle itself. The bilinear elastoplastic model used in this study assumes continuous hardening without a defined failure point to provide a clear validation of the inverse method for plasticity. To enable the prediction of breaking strength, the methodology could be extended by incorporating a more advanced material model that includes a failure criterion, such as a maximum principal strain or a damage evolution law. In such a scenario, once an element’s state variable exceeds the critical failure threshold, its contribution to the total strain energy in the potential energy functional would be set to zero. This would effectively model the loss of load-carrying capacity for that element, allowing the algorithm to find the equilibrium configuration of the remaining damaged structure. This extension would transform the framework into a tool not only for residual stress estimation but also for predicting ultimate structural strength and failure progression.

6. Conclusions

This study introduced and validated a novel inverse methodology that reformulates residual stress estimation as a global optimization task, governed by the principle of minimum total potential energy and solved with an evolutionary algorithm. The framework was rigorously validated on single- and multi-bar structures, demonstrating excellent agreement with analytical and finite element method benchmarks in predicting load-induced residual stresses, including those arising from material heterogeneity. The primary engineering value of this non-incremental, matrix-free approach lies in its application to structural integrity assessments where a component’s plastic loading history is unknown, as it can successfully determine the residual stress state from the final, post-load geometry alone.

Looking forward, the methodology holds significant potential for extension to more complex, real-world scenarios. As discussed, a key priority for future work is addressing the scalability challenge for two- and three-dimensional structures, which can be achieved through techniques like solution space parameterization and the use of hybrid algorithms. The underlying variational framework is also inherently flexible, allowing for generalization to other loading modes, such as bending, and the incorporation of more advanced material models that include anisotropic behavior or failure criteria. These future directions position the proposed energy-based inverse method as a promising and versatile alternative to traditional computational techniques for a wide range of challenging inverse problems in structural mechanics.