Structural Shape Optimization Research Articles

Structural Shape Optimization Based on Multi-Patch Weakly Singular IGABEM and Particle Swarm Optimization Algorithm in Two-Dimensional Elastostatics

In this paper, a multi-patch weakly singular isogeometric boundary element method (WSIGABEM) for two-dimensional elastostatics is proposed. Since the method is based on the weakly singular boundary integral equation, quadrature techniques, dedicated to the weakly singular and regular integrals, are applied in the method. A new formula for the generation of collocation points is suggested to take full advantage of the multi-patch technique. The generated collocation points are essentially inside the patches without any correction. If the boundary conditions are assumed to be continuous in every patch, no collocation point lies on the discontinuous boundaries, thus simplifying the implementation. The multi-patch WSIGABEM is verified by simple examples with analytical solutions. The features of the present multi-patch WSIGABEM are investigated by comparison with the traditional IGABEM. Furthermore, the combination of the present multi-patch WSIGABEM and the particle swarm optimization algorithm results in a shape optimization method in two-dimensional elastostatics. By changing some specific control points and their weights, the shape optimizations of the fillet corner, the spanner, and the arch bridge are verified to be effective.

Modified genetic algorithm for optimal design of truss structures

In this paper, topology and shape optimization of truss or frame structures is discussed. The optimization starts from a structure, into which a finite number of nodesare set; all the nodes are connected together by trusses in all possible variants. Unfit variants of the truss system are rejected. Two alternative ways of the optimization are compared: topology optimization starting from initial structure with a larger number of nodes, and topology optimization starting from initial structure with smaller number of nodes but with additional shape optimization of the obtained topology. The topology optimization is solved with original modified genetic algorithm, giving better results in comparison with classical genetic algorithm. Instead of further development of constraint system, the additional step is introduced into algorithm – purification of genotype, which allows complementary improvement of particular population individuals, and together for the optimization process retains more possibilities than stiffening of constraints. The shape optimization is solved by classical genetic algorithm. Both strategies effectively improve the solution, however the common topology/shape optimization requires less computer resources. All numerical examples are obtained with original software developed by the authors.

Physics-informed shape optimization using coordinate projection

The rapid growth of artificial intelligence is revolutionizing classical engineering society, offering novel approaches to material and structural design and analysis. Among various scientific machine learning techniques, physics-informed neural network (PINN) has been one of the most researched subjects, for its ability to incorporate physics prior knowledge into model training. However, the intrinsic continuity requirement of PINN demands the adoption of domain decomposition when multiple materials with distinct properties exist. This greatly complicates the gradient computation of design features, restricting the application of PINN to structural shape optimization. To address this, we present a novel framework that employs neural network coordinate projection for shape optimization within PINN. This technique allows for direct mapping from a standard shape to its optimal counterpart, optimizing the design objective without the need for traditional transition functions or the definition of intermediate material properties. Our method demonstrates a high degree of adaptability, allowing the incorporation of diverse constraints and objectives directly as training penalties. The proposed approach is tested on magnetostatic problems for iron core shape optimization, a scenario typically plagued by the high permeability contrast between materials. Validation with finite-element analysis confirms the accuracy and efficiency of our approach. The results highlight the framework’s capability as a viable tool for shape optimization in complex material design tasks.

Effects of Limiting the Number of Different Cross-Sections Used in Statically Loaded Truss Sizing and Shape Optimization.

This research aims to show the effects of adding cardinality constraints to limit the number of different cross-sections used in simultaneous sizing and shape optimization of truss structures. The optimal solutions for sizing and shape optimized trusses result in a generally high, and impractical, number of different cross-sections being used. This paper presents the influence of constraining the number of different cross-sections used on the optimal results to bring the scientific results closer to the applicable results. The savings achieved using the cardinality constraint are expected to manifest in more than just the minimization of weight but in all the other aspects of truss construction, such as labor, assembly time, total weld length, surface area to be treated, transport, logistics, and so on. It is expected that the optimal weight of the structures would be greater than when not using this constraint; however, it would still be below conventionally sized structures and have the added benefits derived from the simplicity and elegance of the solution. The results of standard test examples for each different cardinality constraint value are shown and compared to the same examples using only a single cross-section on all bars and the overall optimal solution, which does not have the cardinality constraint. An additional comparison is made with results of just the sizing optimization from previously published research where authors first used the same cardinality constraint.

Shape Optimization of Heat Exchanger Fin Structures Using the Adjoint Method and Their Experimental Validation

The freedom of additive manufacturing allows for the production of heat-transferring structures that are optimized in terms of heat transfer and pressure loss using various optimization methods. One question is whether the structural optimizations made can be reproduced by additive manufacturing and whether the adaptations can also be verified experimentally. In this article, adjoint optimization is used to optimize a reference structure and then examine the optimization results experimentally. For this purpose, optimizations are carried out on a 2D model as well as a 3D model. The material chosen for the 3D optimization is stainless steel. Depending on the weighting pairing of heat transfer and pressure loss, the optimizations in 2D result in an increase in heat transfer of 15% compared to the initial reference structure with an almost constant pressure loss or a reduction in pressure loss of 13% with an almost constant heat transfer. The optimizations in 3D result in improvements in the heat transfer of a maximum of 3.5% at constant pressure loss or 9% lower pressure losses at constant heat transfer compared to the initial reference structure. The subsequent experimental investigation shows that the theoretical improvements in heat transfer can only be demonstrated to a limited extent, as the fine contour changes cannot yet be reproduced by additive manufacturing. However, the improvements in pressure loss can be demonstrated experimentally following a cross-section correction. It can therefore be stated that with increasing accuracy of the manufacturing process, the improvements in heat transfer can also be utilized.

Modern software capabilities for shape optimization of shells

The article is devoted to the shape optimization of shell structures in Comsol Multiphysics using three gradient-based methods: IPOPT (Interior Point OPTimizer), SNOPT (Sparse Nonlinear OPTimizer) and MMA or GCMMA (the Method of Moving Asymptotes). Two types of complex shapes, such as right helicoid and developable helicoid are taken for the computational experiment. The task is to investigate the initial design and optimization process of two helicoids. To obtain a more accurate result and an interesting design solution, the calculation is carried out using three physics-controlled mesh sizes: extra coarse, fine and extra fine with varying values of special optimization settings, such as maximum displacement (dmax) and filter radius (Rmin). The results obtained using the three methods allow to conclude that the mesh element size and studied parameters dmax and Rmin have a significant impact on the final optimization result.

Shape optimization of buckling-based deployable stiff structures

This study considers deployable structures and presents a novel optimization method that aims to improve the stiffness at the fully deployed state. The considered structures consist of curved beams that deploy through buckling when rotated. The optimization aims to maximize the structural tangent stiffness by modifying the shape of the beam elements. To accurately predict the deformation, Finite Element Method (FEM) simulations were conducted. A design with an increase of up to 19.6% in structural tangent stiffness was achieved, while conserving the structural volume. The approach is validated by manufacturing the deployable structures using laser cutting and performing compression tests. Experimental results show consistency within a 5% range with the optimization results. These findings support the implementation of the optimization scheme for achieving optimum structural designs of deployable structures.

Shape and size optimization of truss structure by means of improved artificial rabbits optimization algorithm

ABSTRACT The main aim of this article is to use a new metaheuristic algorithm for the optimum design of truss structures. The artificial rabbits optimization (ARO) algorithm is used, in which rabbits’ natural survival strategies, such as detour foraging and random hiding, are considered in developing the main search loop of the algorithm. An improved version, I-ARO, is also proposed, using the diagonal linear uniform (DLU) initialization process instead of conventional random initialization to enhance the overall performance and convergence behaviour of the ARO algorithm. For numerical investigations, five well-known benchmark truss structures with 10, 37, 52, 72 and 120 bars are determined, considering the natural frequencies of the structures as design constraints. I-ARO is shown to be capable of providing better results than the standard ARO and other approaches in the literature. This demonstrates the capability of the DLU initialization method in enhancing the performance of the I-ARO algorithm. Highlights Optimization of truss structures is investigated Improved Artificial Rabbits Optimization (IARO) algorithm is proposed Diagonal Linear Uniform (DLU) initialization process is implemented in IARO

Shape and sizing optimisation of space truss structures using a new cooperative coevolutionary-based algorithm

Optimising the shape and size of large-scale truss frames is challenging because there is a nonlinear interaction between cross-sectional and nodal coordinate forces of structures. Meanwhile, combining the shape and bar size variables creates a multi-modal search space with dynamic constraints, making an expensive optimisation engineering problem. Besides, most of the real truss problems are large-scale, and optimisation algorithms are faced with the issue of scalability by increasing the size of the problem. This paper proposed a novel Cooperative Coevolutionary marine predators algorithm combined with a greedy search (CCMPA-GS) for truss optimisation on shape and sizing. The proposed algorithm used the divide-and-conquer technique to optimise the shape and size separately. Therefore, in each iteration, the CCMPA-GS focuses on shape optimisation initially and then switches to the size of bars and tries to find the best cooperative combination of the solutions in the current population using a context vector (CV). A greedy search is embedded in the following to fix the remaining violations from the structure's stress and displacement. This novel alternative optimisation strategy (CCMPA-GS) compared with 13 established genetic, evolutionary, swarm, and memetic meta-heuristic optimisation algorithms. The comparison is based on optimising two large-scale truss structures consisting of 260-bar and 314-bar configurations. Experimental results demonstrate that the proposed CCMPA-GS method consistently outperforms the other meta-heuristic methods, delivering optimal designs for the 314-bar and 260-bar truss structures that are superior by 52 % and 63.4 %, respectively. This signifies a substantial enhancement in optimisation performance, highlighting the potential of CCMPA-GS as a powerful alternative in the field of structural optimisation.

Combined shape and topology optimisation of shell structures using FE-based parameterisations

Combined shape and topology optimisation of shell structures using FE-based parameterisations

Experimental evaluation of phase-field-based load-specific shape optimization of nature-inspired porous structures

Triply periodic minimal surface (TPMS) structures excel in various research fields, ranging from bone support structures to heat exchangers. By implementing measures for shape alteration, the mechanical properties of the structure can be improved under certain load conditions. While interface-based methods such as the phase-field method have established themselves as powerful simulation techniques for the analysis of microstructure evolution and morphologically complex dynamic processes, they are not yet very well known and widely used for the application of shape optimization in mechanically loaded complex structures. In this study, an experimental procedure to validate shape-optimized samples is presented and applied to validate three computationally derived optimal candidates for sheet-based TPMS structures (Diamond, Gyroid, and Primitive) proposed by applying a mathematical model for shape optimization formulated in terms of the phase-field approach combined with linear elastic continuum mechanics and subject to the constraints of volume conservation. The present experimental study aims to validate recently obtained theoretical research results predict three different TPMS structures were shape-optimized under mechanical stress, using the phase-field method. In the following, the previous theoretical study is validated experimentally. The validation procedure creates a rare intersection between shape optimization phase-field simulations and experimental samples. The measurements show that the shape-optimized structures have a higher average stiffness, which leads to a shift in the plastic deformation range and thus confirms the computationally determined shape optimization.

A self-powered magnetoelectric tactile sensor for material recognition

Due to the complex working environments, robots need to sense and react to their various interactive surroundings with the help of tactile perception. Material recognition, one of the basic applications of tactile perception, can be achieved by further interpreting and representing the tactile data collected by tactile sensors. Suitable control strategies can be implemented to ensure stable and efficient operation of robots if the contacting materials are accurately recognized. Therefore, in this paper, a soft, deformable, and self-powered tactile sensor based on magnetoelectric materials is developed. Structural design, working principles, and shape optimization of this magnetoelectric tactile sensor are first described. Fabrication and assembly of the magnetoelectric tactile sensor are then introduced. The material recognition experiments including the voltage ratio ranges study, test 1 (velocity change), and test 2 (size and shape change) are performed to validate the effectiveness of the proposed magnetoelectric tactile sensor. The experimental results show that the recognition accuracy is high (around 90% for soft materials and 100% for hard materials) regardless of sizes and shapes of the contacting materials.

Member control efficiency and shape control capability of a loaded cable dome

In this paper, the control efficiency of the members and shape control ability of a loaded adaptive cable dome consisting of 16 cables and 4 struts are investigated. The structure uses specially designed cables and electric telescopic struts integrated with an actuation control system, and all members of the structure are length-adjustable. Initially, an intelligent algorithm is employed to solve the structural shape optimization control model, thereby determining the optimal adjustment amounts. Subsequently, active adjustment and active control tests are conducted to evaluate the control efficiency of the members and the shape control capabilities of the cable dome. The results indicate that the ridge and diagonal cables exhibit the most effective control, followed by the upper and lower hoop cables, while the struts show the lowest control efficiency, primarily influencing the displacement of the joints at their two ends. Moreover, the cables' internal force control efficiency is 3–4 times higher than that of the struts. The ridge and diagonal cables' internal force and displacement control efficiency is about 1.5 times higher than that of the upper and lower hoop cables. The loaded test model exhibits the ability to restore the vertical coordinates of upper joints to their target coordinates, while reducing the internal forces of the components by at least 15 %, and the shape control capability of the structure can be enhanced by incorporating various types of active units. The measured values correlate well with the simulation results, validating the accuracy of the simulations.

Isogeometric double-objective shape optimization of free-form surface structures with Kirchhoff–Love shell theory

Free-form surface structures are extensively utilized in practical engineering due to their rich architectural expression and strong visual impact. However, ensuring good mechanical behavior during the generation process of free-form surface structures has become a critical problem in the scientific community. Isogeometric analysis (IGA) is a widely adopted approach in various fields owing to its numerous advantages, such as geometric accuracy, high-order continuity, high precision, without traditional meshing, and ease of integration with CAD. The presented work adopts the NURBS surfaces to create geometric models, and developes an isogeometric discretization for free-form surfaces by formulating a shell analysis method based on Kirchhoff–Love hypothesis. The structural strain energy and the first natural frequency of the free-form surfaces are selected as the static and dynamic performance evaluation indices. The optimization objectives are to minimize the structural strain energy and maximize the first natural frequency, with the control point coordinates considered as the optimization variable. The double-objective shape optimization of the free-form surface structure is carried out using the particle swarm optimization technique. The effectiveness and performance of the proposed method are verified through five numerical examples, and the mechanical properties of the optimized structure are substantially improved.

Design Optimisation for Cable Dome Structures Based on Progressive Collapse Resistance

This study proposed a framework of optimal design for flexible cable dome structures based on progressive collapse resistance. First, a quantitative evaluation method for nonlinear robustness based on robustness control theory to reflect the structural progressive collapse resistance was proposed. Second, an actual engineering structure was used as a case study to evaluate the effects of design parameters on structural robustness. Finally, a genetic algorithm was used as an optimisation algorithm to further optimise the element cross-section and the structural shape and obtain a combined optimisation rate. The results indicated that increasing the element cross-sectional area, decreasing the structural span, and increasing the rise-to-span ratio effectively improved the structural robustness. The structural robustness was also effectively improved through the optimal design of element cross-sections by increasing element cross-sections sensitive to structural robustness and decreasing those insensitive to structural robustness. In this study, the combined optimisation rate was 38.27%, which was not only greater than the individual optimisation rates of 11.2% for element cross-sectional area optimisation and 22.5% for structural shape optimisation but also the sum of these two rates.

Shape optimization of porous structures by phase-field modeling with strain energy density reduction

Triply periodic minimal surfaces (TPMS) are cell structures defined by mathematical formulas. Due to their special structure and properties, such as a high surface-to-volume ratio and good mechanical properties, these structures are used in a wide range of applications, e.g., for implants or heat exchangers. Given the diverse applicability of these structures, both scientific and industrial interest resides in the tailored shape optimization, where structural integrity is preserved on the one hand, while stiffness is selectively enhanced on the other.Therefore, in the following investigation, three TPMS unit cells are shape-optimized under mechanical loading, in the linear-elastic regime, by minimizing the strain energy density, using the phase-field method. The novelty lies in the shape optimization of TPMS structures with respect to the effective stiffness under mechanical load which is addressed with a phase-field method. It is shown that with the present optimization approach, an increase in the effective Young’s modulus of up to 55% can be achieved for specific TPMS structures.

Modern software features for shape optimization of shells

Shape optimization, as one of the types of structural optimization problems, is an important process in the design of shells, since it contributes to the creation of a structure with fine performance characteristics, expansion of design variations and knowledge base to obtain high-quality results. To solve the problems associated with determining the shape and creating more advanced structures, software packages include a special optimization module, which can be based on one or more mathematical methods, the purpose of which is to provide the best solution in the shortest possible time. The research is focused on the process of shape optimization in three well-known universal software packages: Ansys Mechanical, COMSOL Multiphysics and Simulia Abaqus, as well as in Rhinoceros modeling software with a special visual Grasshopper plugin. The purpose of the study is to analyze the technology of shape optimization in four software packages and to compare them with each other in terms of the problem-solving process, user interface, the fullness of libraries, accessibility for educational purposes and system requirements for a computer. The authors specify and describe the characteristic features of each software package. It was found that all the software packages under consideration are equipped with great opportunities for shape optimization of structures and have a variety of functionality for solving this type of tasks. The development of optimization technology in calculation and modeling software packages will allow obtaining the most effective solutions in the process of designing shells of complex shapes.

Chaotic heterogeneous comprehensive learning PSO method for size and shape optimization of structures

This paper proposes a novel chaotic heterogeneous comprehensive learning particle swarm optimization (CLPSO) method for the simultaneous size and shape design of structures. The heterogeneous CLPSO divides the particles into two explorative and exploitative subpopulations. The exploration performs the global searches for the set of best particles experienced solely within its own subpopulation, whilst the exploitation refines the deep searches learnt from the global best particle over an entire population. In essence, the proposed method maintains a good balance between the global explorative and local exploitative optimization schemes. The global searches within the explorative subpopulation are independent to the exploitative simulations even if the latter scheme prematurely converges to the local swarm position. For both subpopulations, the comprehensive learning approach constructs the particles through a cross-positioning process on the individual variable space and avoids the local optimal pitfall. Moreover, the chaotic logistic map within the exploitative optimization tests the global best particle through the set of diversely generated samples and hence enhances the local search ability. Various enriching techniques, including automatic adaptive (inertial weight and acceleration) parameters with dynamic space reduction, are incorporated to improve the likelihood of finding the optimal solution of practical-scale problems at modest computing efforts. The accuracy and robustness of the proposed method are illustrated through a number of planar and spatial truss design benchmarks subjected to the challenging nonconvex and/or nonsmooth programs.

An integrated design approach for simultaneous shape and topology optimization of shell structures

In this work, a novel design approach is developed to carry out shape and topology optimization of shell structures simultaneously. This approach effectively integrates the IsoGeometric Analysis (IGA) method, the Adaptive Bubble Method (ABM) and the Finite Cell Method (FCM) to make full use of their respective advantages in shell structure optimization. To be specific, the IGA method using Non-Uniform Rational B-Splines (NURBS) elements is adopted to analyze the shell structure and realize shape optimization with merits of exact shell surface representation geometrically and rotation-free shell formulation physically; the ABM is utilized to insert deformable holes adaptively in the parametric domain of the NURBS surface for topology optimization of the shell structure; the FCM is employed to facilitate the optimization process by simplifying significantly the discrimination and the numerical integration processes of trimmed elements involved in fixed-mesh analysis of the holed shell structure. Based on this integrated design approach of IGA/ABM/FCM, shape optimization can be implemented by directly optimizing the control-point coordinates of the NURBS surface in the physical space, while topology optimization could be carried out simultaneously in the parametric domain as easily as for 2D planar structures. Finally, representative examples are investigated to demonstrate the effectiveness and advantages of the proposed design approach.

Level set-based shape optimization of deformable structures for achieving desired sound transmission and reflective responses

In this paper, we propose a level set-based shape optimization method for acoustic wave propagation problems with a deformable structure. First, we propose a mathematical model for acoustic wave propagation with a deformed structure based on coordinate transformation and the Eulerian approach. Next, we formulate the shape optimization problem and perform sensitivity analysis based on the shape derivative concept. We then construct an optimization algorithm based on the framework of a level set-based shape and topology optimization method. Finally, we provide two-dimensional optimization examples that demonstrate the effectiveness of our proposed method in providing optimized designs with desired functionality, while considering the structural deformation.