Real Engineering Structures Research Articles

A fatigue crack prediction method based on inductive semi-supervised learning and Lamb-wave monitoring for orthotropic steel bridge deck

Orthotropic steel bridge deck suffers from fatigue cracking due to its orthotropic configuration and cyclic vehicular load. Hence, fatigue crack monitoring and prediction are crucial for addressing this issue. The Lamb-wave technique has potential in monitoring the growth of fatigue cracks, whereas most of the existing signal processing methods show poor performance in the crack prediction. To improve the accuracy and reliability of fatigue crack prediction, an inductive semi-supervised learning (ISL) approach is proposed in conjunction with Lamb-wave monitoring. The presented method is first verified by conducting a fatigue test of orthotropic steel bridge deck loading by three load conditions which would lead to different types of cracks, where lightweight Lamb-wave monitoring devices were adopted. The approach was further applied in field monitoring to validate its effectiveness in real engineering structures. The results demonstrate that the proposed method can recognize the initiation of fatigue cracks by giving a predicted time interval in which the real initiation moment locates. Similar increasing trends are found between the predicted and real crack lengths during the growth of cracks, while a relative fluctuate is existing in the predicted results, and the prediction accuracy of crack lengths can be regarded as 2 mm. For bridge field monitoring, the real fatigue crack length was also well predicted by the proposed method, exhibiting an average error of 0.82 mm in comparison with the real values. In general, the research findings emphasize the adaptability of the proposed method for fatigue crack prediction in practical engineering.

A piecewise inverse finite element method for shape sensing of the morphing wing fishbone

The shape sensing technology plays a significant role in enhancing the structural performance and flight efficiency of the morphing aircraft by providing feedback to actuation and control systems in real time. The inverse finite element method (iFEM) is a fast, accurate, and robust shape-sensing method based on in-situ surface strain measurements. However, the conventional iFEM becomes less effective when applied to real engineering structures. Thus, this paper proposes a piecewise iFEM (P-iFEM) based on a two-node inverse Hermite beam element (iHB2) for the load-bearing structure of the morphing wing. The P-iFEM method discretizes the structure into a combination of rigid and elastic parts, based on the geometry of the structure when assembling the inverse elements. The Hermite polynomial shape function in iHB2 is adopted to increase the reconstruction efficiency, as only one degree of freedom of deflection is required to achieve the C1-continuity to reconstruct the displacement fields. The high accuracy and efficiency of the proposed method are validated with a numerical fishbone model. Finally, the proposed method is applied to a fishbone structure of a real adaptive deformed wing with high accuracy.

Data-Driven Structural Health Monitoring: Leveraging Amplitude-Aware Permutation Entropy of Time Series Model Residuals for Nonlinear Damage Diagnosis.

In structural health monitoring (SHM), most current methods and techniques are based on the assumption of linear models and linear damage. However, the damage in real engineering structures is more characterized by nonlinear behavior, including the appearance of cracks and the loosening of bolts. To solve the structural nonlinear damage diagnosis problem more effectively, this study combines the autoregressive (AR) model and amplitude-aware permutation entropy (AAPE) to propose a data-driven damage detection method. First, an AR model is built for the acceleration data from each structure sensor in the baseline state, including determining the model order using a modified iterative method based on the Bayesian information criterion (BIC) and calculating the model coefficients. Subsequently, in the testing phase, the residuals of the AR model are extracted as damage-sensitive features (DSFs), and the AAPE is calculated as a damage classifier to diagnose the nonlinear damage. Numerical simulation of a six-story building model and experimental data from a three-story frame structure at the Los Alamos Laboratory are utilized to illustrate the effectiveness of the proposed methodology. In addition, to demonstrate the advantages of the present method, we analyzed AAPE in comparison with other advanced univariate damage classifiers. The numerical and experimental results demonstrate the proposed method's advantages in detecting and localizing minor damage. Moreover, this method is applicable to distributed sensor monitoring systems.

Bisection Constraint Method for Multiple-Loading Conditions in Structural Topology Optimization

Topology optimization (TO) is currently a focal point for researchers in the field of structural optimization, with most studies concentrating on single-loading conditions. However, real engineering structures often have to work under various loading conditions. Approaches addressing multiple-loading conditions often necessitate subjective input in order to determine the importance of each loading condition, aiming for a compromise between them. This paper proposes a so-called bisection constraint method (BCM), offering a unique, user-preference-independent solution for TO problems amidst multiple-loading conditions. It is well-known that minimizing the system’s compliance is commonly used in TO as the objective. Generally, compliance is not as sufficient as stress to be used as a response to evaluate the performance of structures. However, formulations focusing on minimizing stress levels usually pose significant difficulties and instabilities. On the other hand, the compliance approach is generally simpler and more capable of providing relatively sturdy designs. Hence, the formulation of min–max compliance is used as the target problem formulation of the proposed method. This method attempts to minimize compliance under only one loading condition while compliances under the remaining loading conditions are constrained. During the optimization process, the optimization problem is automatically reformulated with a new objective function and a new set of constraint functions. The role of compliance under different loading conditions, i.e., whether it is to be treated as an objective or constraint function, might be changed throughout the optimization process until convergence. Several examples based on the solid isotropic material with penalization (SIMP) approach were conducted to illustrate the validity of the proposed method. Furthermore, the general effectiveness of the compliance approach in terms of stress levels is also discussed. The calculation results demonstrated that while the compliance approach is effective in several cases, it proves ineffective in certain scenarios.

Predicting fracture behavior of quasi-Brittle materials using quantitatively varied geometric specimens: A design method

Fracture toughness (KIC) and tensile strength (ft) are two important parameters for describing the fracture performance of quasi-brittle materials. The accurate determination of both parameters is crucial for the safety design and stability evaluation of real engineering structures. A model and method are developed in this paper for simultaneously determining the KIC and ft for quasi-brittle materials without size effects. And using the geometric structural parameter ae as a key design parameter. This method can get KIC and ft to be obtained by fitting curves to any two sets of quantitative specimens with different geometric structures with ae ratio equal to 3. Additionally, fracture full curves and peak load prediction lines with 95% reliability are established for different fracture modes using quantitative design specimens with an ae ratio of 3. These curves and lines can accurately predict fracture behavior and peak loads of large specimens or engineering structures. The proposed design method is verified by combining experimental studies conducted by other scholars on quasi-brittle materials and the simulation results of ABAQUS software. This paper achieves the objective of accurately predicting the fracture properties of quasi-brittle material structures in the field, using quantitative laboratory specimens of various geometries.

Seismic margin assessment of the real dynamic response of nuclear engineering structures based on the response envelope method

The accuracy of response calculations for nuclear structures is directly related to the reliability of margin assessment results. The seismic response envelope (SRE) method, which is more effective than the traditional mode decomposition response spectrum method, has received increasing attention in studies on structural dynamic responses in nuclear engineering. It is able to consider the characteristics of multiple seismic responses occurring simultaneously, but applying it to the seismic margin assessment of structures beyond design-basis earthquakes becomes a challenge and some practical issues remain unaddressed. Focusing on the characteristics of the seismic response spectrum and regularization of the coordinate system, this study derived an SRE concentric circle equation in the regular coordinate system and defined the seismic safety margin level. A seismic margin assessment strategy for nuclear shear wall structures was developed based on the relationship between the bearing capacity surface controlled by the reinforcement ratio and the concentric circle of the SRE. A comparison between the SRE method and other seismic analysis methods for multiple seismic response calculations verified the effectiveness of the proposed method. Moreover, in particular, the emphasis was on the variation characteristics of the seismic safety margin level considering discrete points under the new and old strategies, and the solution accuracy and efficiency of the proposed strategy were discussed. Finally, the SRE method was used to evaluate the seismic margin of a real nuclear engineering structure, providing reliable data support for the solution strategy in practice.

Synergy of random balance design method and intelligent optimization technique for model updating of the 4 MW offshore wind turbine benchmark

The technical challenges for model updating of real marine engineering structures include the extraction of modal parameters associated with incomplete measurement information and the correction of a large number of structural parameters. As it is difficult to verify the effectiveness of model updating methods for engineering applications, the representative models have to be adopted in finite element (FE) analysis or lab-scale tests, leading to the incomplete reflection of actual structural performances. Based on these factors, a practical model updating method is developed in this study to make full use of field measurement data from a 4 MW offshore wind turbine for the accurate estimation of structural parameters using the random balance designs-Fourier amplitude sensitivity test (RBD-FAST) strategy and an improved particle swarm optimization (IPSO). Leveraging the measured acceleration signals from offshore wind turbines under the conditions including operation, shutdown, collision and typhoon scenarios, the complex exponential decomposition method is applied to accurately extract the time-varying acceleration components (TVAC) for the construction of the frequency response function (FRF). Following that, RBD-FAST is implemented into IPSO with adaptive inertia weights and asynchronously varying learning factors to enable efficient selection of numerous updated physical parameters, thus improving time cost and computational accuracy. The correctness of the proposed method is verified by a numerical jacket platform model. Furthermore, the 4 MW offshore wind turbine benchmark is developed to assess the feasibility of the proposed model updating method using field measured data from different scenarios. Results show that the synergy of RBD-FAST and IPSO in the proposed method can accurately update parameters and minimize a maximum discrepancy of 0.8970% for the first-three orders of natural frequencies between the benchmark and the updated models in the collision scenario. Summarily, the present work provides a potential technique and practical engineering references for model updating of offshore wind turbines subject to harsh marine environments.

Target-free 3D tiny structural vibration measurement based on deep learning and motion magnification

• A novel target-free tiny vibration displacement measurement approach is proposed. • It is based on deep learning and motion magnification techniques. • Binocular vision system is used for measuring 3D vibration measurement. • Deep learning techniques are used for key-points detection, matching and tracking. • Experimental and in-field studies are conducted to validate the accuracy. In the field of structural health monitoring (SHM), computer vision based methods have been usually developed for 2-dimensional (2D) vibration displacement measurement and crack identification of civil engineering structures. However, the accurate measurement of tiny 3-dimensional (3D) vibrations for real civil engineering structures remains a very difficult task. To overcome this challenge, this paper proposes a target-free full-field 3D tiny vibration measurement approach for civil engineering structures by using a binocular vision system. The proposed approach is based on deep learning and motion magnification. A phase-based video motion magnification algorithm is employed to achieve a high measurement accuracy of tiny vibrations at the submillimeter level. The advanced key point detection, matching and tracking algorithms via deep learning techniques are employed to achieve target-free tiny vibration displacement measurement. The accuracy and performance of the proposed approach are evaluated through experimental tests on a steel cantilever beam in the laboratory. In-field experimental tests are conducted on a pedestrian bridge on a university campus to investigate the accuracy of the proposed approach in practical applications. The measured 3D tiny vibration displacement from the proposed approach is compared with those measured by laser displacement sensors, and the derived acceleration responses are compared with those measured from the installed accelerometers on the testing structures. The results demonstrate that the 3D tiny vibration measurements are obtained accurately by the proposed approach.

Vibration based fatigue analysis of a structure integrated on an air vehicle by using experimental and theoretical methods

In this article, fatigue life analyses and tests of a real engineering structure integrated into an air vehicle are performed using frequency domain fatigue approaches and accelerated fatigue life testing concept. The finite element (FE) model of the structure is constructed and verified by experiments. The stress history is obtained using the verified FE model and synthesized operational vibration data. Numerical fatigue calculations are carried out with the developed code and commercial fatigue software. Lastly, the experimental fatigue life of the structure is obtained under accelerated vibration data. The results show considerably good agreement between the numerical analyses and experiments.

Use of dictionary learning for damage localization in complex structures

In this paper, to increase the performance of the sparse reconstruction method in real complex engineering structures an adaptive dictionary learning framework is proposed which updates the dictionary matrix, to allow improved compatibility with the complex structure. This proposed framework was developed by combining analytical modeling with training data sets and learning methods. An experimental evaluation of the proposed dictionary learning framework was performed on an anisotropic composite plate with a stiffener. In this experimental evaluation, a moving magnet was used as the artificial damage to capture the training data set, and both artificial damage in several locations and real impact damage was used for detection and location of the target damage. The obtained results confirmed the concept of the proposed dictionary learning framework for the improved health monitoring of complex structures.

Structural damage detection via phase space based manifold learning under changing environmental and operational conditions

The feasibility and performance of existing vibration-based damage detection methods to real world civil engineering structures are inevitably affected by the varying environmental and operational conditions. Reliable damage detection methods with damage features that are sensitive to structural condition change but robust to environmental and loading effects are desirable for practical applications. This paper proposes a novel structural damage detection approach based on manifold learning for the effective condition assessment of real-world structures under environmental and operational conditions. The phase space representation of the vibration characteristics is reconstructed using the identified natural frequencies of structures. Then, the intrinsic nonlinear manifold between the environmental variables and natural frequencies in the high dimensional phase space is projected to a low-dimensional representation via manifold learning. The Gaussian process regression technique is introduced to extract reliable damage index from the learned manifold structure. The effectiveness and superiority of the proposed approach are demonstrated by two real-world engineering structures, that is, the Dowling Hall Footbridge and Z24 bridge. Damage detection results obtained from the proposed approach are compared with those from the current state-of-the-art Kernel PCA method, which is a representative nonlinear dimensionality reduction method to alleviate the environmental effects. The results demonstrate that the proposed approach is sensitive to structural damage but insensitive to changes in environmental and operational conditions. More importantly, the nonlinear environmental effects can be efficiently characterized by the proposed approach, using only partial datasets with environmental variations in the training datasets.

Three-dimensional surface fatigue crack growth life assessment model of steel welded joints and structures

To investigate the fatigue crack growth of surface cracks in steel welded joints, a three-dimensional surface fatigue crack growth life assessment model considering both the surface crack growth period and the following through crack growth period is proposed. The fatigue crack growth life of a standard steel butt joint specimen is predicted considering the effect of welding residual stress and it is compared with fatigue test results. An example of a welded beam-to-column connection in a steel high-rise building under wind is selected to demonstrate the improvement of the model to make it applicable to stochastic loading and real engineering structures. The life results are compared with those obtained by other fatigue assessment approaches. The mesh sensitivity analysis is made and the effect of extreme welding imperfections and corrosion environment are both discussed. Guidance to select the fatigue life assessment approaches is also proposed. The results show that the growth shape of the edge crack remains a quarter of a circle while that of the center crack remains a quarter of an ellipse during the growth; When the surface crack grows through the plate thickness and becomes a through crack, the remaining fatigue crack growth life is unneglectable for real engineering components especially for the ones with a large plate width; The effect of welding residual stress on the fatigue crack growth life of real engineering component is complex and it depends on the distribution of the residual stress itself; The proposed three-dimensional surface crack engineering model can acquire accurate life results and is applicable to stochastic loading and engineering structures.

Monitoring of Large Diameter Sewage Collector Strengthened with Glass-Fiber Reinforced Plastic (GRP) Panels by Means of Distributed Fiber Optic Sensors (DFOS).

Diagnostics and assessment of the structural performance of collectors and tunnels require multi-criteria as well as comprehensive analyses for improving the safety based on acquired measurement data. This paper presents the basic goals for a structural health monitoring system designed based on distributed fiber optic sensors (DFOS). The issue of selecting appropriate sensors enabling correct strain transfer is discussed hereafter, indicating both limitations of layered cables and advantages of sensors with monolithic cross-section design in terms of reliable measurements. The sensor’s design determines the operation of the entire monitoring system and the usefulness of the acquired data for the engineering interpretation. The measurements and results obtained due to monolithic DFOS sensors are described hereafter on the example of real engineering structure—the Burakowski concrete collector in Warsaw during its strengthening with glass-fiber reinforced plastic (GRP) panels.

SHM system for anomaly detection of bolted joints in engineering structures

In this article, the elastic wave propagation phenomenon is applied as the basis for a system designed to detect anomalies in the pre-tensioned connections of engineering structures. A series of laboratory tests were carried out on a steel frame. This approach, where the connection is a fragment of the complex structure, and not an isolated part, enabled us to analyse behaviours similar to those in real engineering structures. The problem of the similarity of signals corresponding to different types of anomalies was considered. Thanks to the application of principal component analysis and a procedure employing a combination of artificial neural networks, the extraction of the most significant features of the registered signals was possible. The proposed procedure proved to be very sensitive and accurate for the proper detection of anomalies, as well as for the determination of their location and type, even if the loosening of bolts was very slight (i.e., invisible to the naked eye). The most important element of the presented approach is the fact that the combination of the assessment of the condition of local parts along with the application of artificial intelligence techniques can result in a Structural Health Monitoring (SHM) system that allows for the global assessment of structural integrity.

Thin-walled Beam Bending Quartic Deplanation Analytical Function for Improved Experiment Matching

Ensuring the accuracy of an analytical solution is important in modeling real engineering structures. For determining the stress-deformation condition of a thin-walled beam structure in bending, inadequately, the simple beam formula can only provide uniform average stress-deformation distribution at specific cross-sectional elevations. The recently developed analytical solution for determining stress-deformation conditions with consideration of the shear lag effect of a prismatic thin-walled box beam subjected to transverse load causing bending using classical quadratic deplanation function did not provide an accurate but rather a good estimate when compared with finite element analysis results. In this paper, the objective of the study was to see if the accuracy can be improved. The same Vlasov’s method was used. The method was developed using Calculus of Variations based on a Stress-form stationary condition complementary energy which included the shear lag effect. All calculations were computerized using the software MAPLE 18 which assisted in getting quick results (especially for preliminary design studies) for various geometries, material properties, and loading conditions. Several deplanation functions were introduced. Two variants of the quadratic functions, i.e. x 2 y 3 and x 2 y, and the quartic functions, i.e. x 4 y 3 and x4 y were used to find the best match against empirical data and FEA (finite element analysis) results. The finding was that the quartic variant of the deplanation functions provided improved matching with experimental data as well as FEA results. Noteworthy to point out that the characteristic of the functions with quartic-x or x4 of having almost flat variation in the center part enhances matching with the experimental data. Moreover, the characteristic of the function with cubic-y or y3 of having steeper slopes enhances matching with the FEM results at the edge.

Improved Strategy of Two‐Node Curved Beam Element Based on the Same Beam’s Nodes Information

The curved beam with a great initial curvature is the typical structure and applied widely in real engineering structures. The common practice in the current literature employs two‐node straight beam elements as the elementary members for stress and displacement analysis, which needs a large number of divisions to fit the curved beam shape well and increases computational time greatly. In this paper, we develop an improved accurate two‐node curved beam element (IC2) in 3D problems, combining the curved Timoshenko beam theory and the curvature information calculated from the same beam curve. The strategy of calculating the curvature information from the same bean curve in the IC2 beam element and then transferring the curvature information to the two‐node straight beam element can greatly enhance the accuracy of the mechanical analysis with no extra calculation burden. We then introduce the finite element implementation of the IC2 beam element and verify by the complex curved beam analysis. By comparison with simulation results from the straight two‐node beam element in the MIDAS (S2‐MIDAS) and the three‐node curved beam element adopted in the ANSYS (C3‐ANSYS), the simulation results of the typical quarter arc examples under constant or variable curvature show that the IC2 beam element based on curved beam theory is a combination of efficiency and accuracy. And, it is a good choice for analysis of complex engineering rod structure with large initial curvature.

Acoustic emission source location using Lamb wave propagation simulation and artificial neural network for I-shaped steel girder

Acoustic emission (AE) is often used for structural health monitoring (SHM) in the wide field of engineering structures and one of its most beneficial attributes is the ability to localize the damage/crack based on the AE events. The vast majority of ongoing work on AE monitoring focues on geometrically simple structures or a confined area, but the AE source location strategies are rather complicated for real engineering structures. In this paper, an effective method for source localization in realistic structures is presented based on the application of artificial neural networks (ANN), using finite element (FE) simulation results of Lamb waves as the modelling basis. Pencil lead break experiments and related FE simulations on a steel-concrete composite girder are conducted to evaluate the performance of the method. The identification of different wave modes is carried by comparing alternative onset time detection methods. Numerical results are found to be matching closely with the experimental results. To get a reliable ANN model, the validated FE model is used to create a comprehensive database with five different sensor arrangements. It is found that the proposed method is superior to the classical Time of Arrival (TOA) method with the same input data. The results indicate that using trained neural networks based on numerical data is a viable option for AE source location in the case of the I-shaped girder, increasing the likelihood of design and optimization of the AE technique in monitoring realistic structures.

Scientific schools of the department of structural mechanics and strength of materials

The history of the Department of Structural Mechanics and Strength of Materials is entirely related with the development of Prydniprovska State Academy of Civil Engineering and Architecture, which has grown over 90 years into one of the leading technical universities in Ukraine. During this time, the Department has passed a long and difficult path of its formation and development, which can be divided into several stages. In the first period, after the establishment of Dnipropetrovsk Civil Engineering Institute in 1930, due to a small number of students there were no separate departments. However, already in 1932 the Department of Structural Mechanics was organized to provide the more effective teaching of students to the most important engineering disciplines. During the Second World War, from 1941 till 1944, the Institute was evacuated to Novosybirsk, where the staff of the Department carried out teaching on strength of materials and structural mechanics. The Department underwent great difficulties after returning from evacuation in 1944. In 1960, a new stage in the history of the Department started: a course of strength of materials was separated and transferred to the newly organized Department of Strength of Materials. In the period from 1960 to 2000, the Department of Structural Mechanics performed many studies of real engineering structures. The second branch of the history in the period from 1960 to 2000 is associated with the Department of Strength of Materials, which from the beginning was headed by professor A.P. Prusakov. The third stage of the history started after the merger of the both Departments in 2000. The united department was named the Department of Structural Mechanics and Strength of Materials. Nowadays, the staff of the Department makes a lot of efforts to arm the graduates of the Academy with deep knowledge of modern methods for calculating Civil Engineering structures, which is the basis for their effective design.

Experimental and numerical investigation of mortar and ITZ parameters in meso-scale models of concrete

Meso-structural models are presently in common use for analysis of the mechanical behaviour, damage evolution and failure of concrete. These are constructed either from XCT images, or in silico, using statistical information about concrete’s meso-scale constituents. As a minimum, such models include mortar and aggregates as separate phases, while more detailed versions consider the interfacial transition zones (ITZ) between mortar and aggregates, and voids. Analyses of given meso-structures vary further by different constitutive modelling of constituents, with past works focusing on parameters’ calibration using experiments with one loading condition - either tension or compression. Using a detailed meso-structure representation, this work proposes a novel combination of constitutive relations, involving concrete damage plasticity (CDP) for mortar and cohesive zone behaviour for ITZ. CDP parameters are calibrated using both compression and tension experiments with mortar samples. ITZ parameters are calibrated by comparing simulated stress–strain curves and failure patterns with data from compression and tension experiments with concrete samples. This process leads to constitutive relations, applicable to both loading conditions, which has not been demonstrated previously, but is essential to extending the modelling approach to complex stress states encountered in real engineering structures. After establishing the realism of the approach, parametric studies are conducted to investigate the effects of friction between loading plate and specimen, the mortar dilation angle, the ITZ cohesive stiffness, critical stresses, fracture energy and mix mode ratio. The results show that mortar plasticity is the dominant energy dissipation mechanism in both compression and tension, and its rate governs the localisation of damage. The effect of ITZ parameters on the tensile behaviour is found to be negligible. Their effect on the compressive behaviour is found to be limited, but sufficient to propose a set of parameters working for both conditions. Importantly, under both loadings the ITZ is found to control failure localisation into a macroscopic crack in combination with mortar plasticity and damage. Predicted stress-stain curves, damage evolution and macro-crack propagation are shown to be in very good agreement with the experimental observations. This justifies the use of the proposed experimental-modelling strategy for developing models for analysis of complex loading conditions.

Sizing and prestress optimization of Class-2 tensegrity structures for space boom applications

Design of real engineering structures can benefit from an optimization step in which various parameters can be determined to satisfy a given set of requirements and constraints. Optimization of spatial assemblies such as truss or frame structures involves sizing optimization to minimize the mass of the structure. Optimization of tensegrity structures is more challenging as prestress levels should be optimized as well. In this paper, sizing and prestress optimization of Class-2 tensegrity booms are addressed using a particle swarm optimization approach. Nonlinear finite-element models of tensegrity structures and solution methods provide a starting point. Furthermore, a continuum beam modeling technique for tensegrity structures with repeating units is also useful. The particle swarm optimization algorithm is described and two numerical examples are presented. The first example studies the design and single-objective optimization of a deployable Class-2 tensegrity boom with reinforcing cables to maximize the bending stiffness-to-mass ratio. The results indicate that the optimum structure is capable of competing well with the state-of-the-art deployable booms in terms of stiffness-to-mass ratio. The second example investigates a multi-objective optimization problem of a Class-2 tensegrity boom. The objective functions are selected as minimization of the mass and tip displacement, respectively. The objective functions are at least partially conflicting; therefore, Pareto-optimal solutions are obtained to guide future design decisions. The results show the potential of tensegrity structures for implementation as space structures and the robustness of the particle swarm optimization algorithm, even for multi-objective optimization problems.