Plant Methods

Abstract Node-based shape optimization has been successfully and consistently formulated via shape and sensitivity filtering methods. This work studies the union of the Vertex Morphing method, a shape parameterization technique that uses an explicit shape filtering approach, and shell thickness optimization. On this occasion, thickness variables are explicitly filtered similarly to the shape, but their filter is computed on the initial geometry configuration throughout the optimization. The problem is formulated such that both design variable types are optimized concurrently. Gradient-based algorithms are employed to solve the optimization problem, which have proven well suited for the Vertex Morphing method. Due to the different dimensionalities of the shape and thickness variables, a design variable scaling method between the two types is proposed, improving the convergence behavior without the necessity of second-order information. Academic examples and the application of a structure with industrial significance illustrate the method’s success.

Abstract We propose a topology optimization (TO) formulation and related optimization scheme for designing compliant mechanisms following a user-defined trajectory. To ensure the broad applicability and achieve precisely control of the outputs, geometric nonlinearity with incremental solutions are considered. A challenge in the design optimization of these structures is the development of formulations with satisfactory balance between (i) precise trajectory control and (ii) proper connectivity between the input/output ports and the support. Previously proposed density-based topology optimization formulations typically lack the promotion of the desired load-transferring connections, or usually complicate the design using mixed shape, size, and topology variables to enforce a minimum connectivity. To simplify design progress using exclusive topology variables, i.e., purely density-based TO methods, we propose a relatively straightforward formulation involving commonly used response functions, such as compliance and volume as constraints. For the constraints, the paper provides a scheme for defining corresponding upper limits. Numerical examples of challenging shell and plate design optimization problems demonstrate the effectiveness of the proposed formulation and scheme in the generation of load-transferring connections while limiting the impact on the performance of the path generation functionality.

Abstract Topology optimization methods employing binary (also known as discrete) design variables currently lack mathematical formulations to ensure length scale control in their solutions. This paper proposes and applies a morphology-mimicking filtering scheme to provide a minimum size control (often also referred to as minimum length scale control) in this class of binary designs. The Topology Optimization of Binary Structures (TOBS) method was chosen as the foundational framework for this length scale control study. Thermal and structural compliance scenarios were explored under this approach. Numerical results show that the proposed filter efficiently imposes the desired minimum length scale. The optimized designs were also less dependent on the filtering parameters when compared to designs optimized using standard techniques that employ continuous design variables.

Abstract A method is proposed to automatically increase the threshold projection parameter in three-field density-based topology optimization to achieve near binary designs. The three-field method is composed of an element-wise design density field that is filtered and then passed through a smooth threshold projection function to compute the projected density field, which is then used to compute element properties, e.g., using a power law for stiffness. The sharpness of the threshold projection function is controlled by a parameter $$\beta $$ β . In this paper, a method is introduced to automatically increase this parameter during optimization by linking it to the change in objective function. Furthermore, the gray value indicator is added as a stopping criterion to guarantee the solution is near binary. This results in a method that does not need to be tuned for specific problems, or optimizers, and the same set of user-defined parameters can be used for a wide range of problems. However, a high value of the threshold projection parameter may cause convergence issues for some optimizers, such as the optimally criteria method, and an adaptive move limit strategy is introduced to overcome this problem. It is also shown that some problems require length-scale control to achieve a near binary design. The effectiveness of the method is demonstrated on several benchmark problems, including linear compliance, linear buckling, compliant mechanism, heat conduction, and geometrically nonlinear problems.

Abstract A compliant structure under fluidic pressure can undergo relatively large shape change, but the design of such type of structure is challenging as the pressure distribution depends on detailed structural geometry. In this study, a novel mixed substructure-density (MSD) model is proposed for topology representation and update in the optimal design of nonlinear compliant structures under quasi-static fluidic pressure. An optimization algorithm is developed via implementing the present model by using super-elements in commercial finite element analysis (FEA) software. Numerical examples are presented to validate the present model, algorithm, and designs numerically via full linear and nonlinear FEAs. A planar cellular network with five cells arranged in parallel is then designed for representing a pressurized wing rib structure capable of modulating airfoil thickness variation. The test results of the single-cell and five-cell PCS specimens prototyped using polyurethane material show that the respective cell thickness can be reduced by 11.9 and 6.4% respectively under a cell pressure of 250 kPa.

Abstract Optimization of reinforcing bar (rebar) design represents a preponderant factor in reducing material usage and wastes for reinforced concrete (RC) structures. The assessment of constructability of such rebar designs is crucial to improve their practicality and reduce construction costs, which makes the problem multi-objective (MO). However, when applying optimization methods for the design of rebar in RC structures, little attention has been paid on columns, in comparison to beams and slabs. Meta-heuristic algorithms (MA) have been the ones mostly deployed for these types of elements, which have proven to be of high computational cost. Additionally, an existing gap in the literature as to how to relate the design and construction stage of rebar in RC structures through constructability analysis is evident. In this regard, research has been focused mainly at the building level but not at the element level. This works presents a novel algorithmic framework using Machine Learning (ML)-enhanced meta-heuristics for the optimal design of rebar on rectangular RC columns. To assess the constructability of the resulting rebar layouts a Buildability Score (BS) model at the element level is proposed. The complexity analysis of rebar design under the constructability restrictions, through combinatorial optimization (CO), is used to assess the global time efficiency of the framework. The Non-Sorting Genetic Algorithm II (NSGA-II) was deployed for showcase and five different ML algorithms were used to enhance it, namely the k-NN classifier, SVM regression, ANN, Gauss Process (GP) regression, and Tree Ensembles (TE), where the latter three showed the best performance.

Abstract The optimization results of level-set methods typically suffer from a strong dependence on the initial design. To mitigate this dependence, this work presents a density-based hole nucleation method for level-set topology optimization of flow problems. This is achieved by defining both the level-set and density values as functions of the design variables. To preserve the crispness of the geometry definition afforded by the level-set method, the fluid flow is modeled using the Heaviside enriched eXtended Finite Element Method (XFEM) for the laminar incompressible Navier–Stokes equation, which is augmented by a Brinkman model. The boundary conditions are enforced weakly using Nitsche’s method. A face-oriented ghost stabilization scheme is applied to stabilize the XFEM formulation. Additional terms ensuring stability are added to Nitsche’s method and the ghost stabilization to account for the Brinkman term in the Navier–Stokes equation. The necessity of adding these terms is highlighted by numerical studies. Two- and three-dimensional fluid manifolds are optimized to minimize fluid power dissipation while achieving a predefined mass flow distribution among the outlets. The optimization results show that the proposed method bypasses the need for an initial hole seeding and speeds up the convergence of the optimization process.

This paper introduces a novel efficient topology optimization methodology for beams’ torsion using the warping function formulation. The finite element method is used to discretize the cross-section and an efficient gradient-based optimization problem is formulated to optimize the relevant parameters corresponding to the torsion and warping constants of the beam. As a result, for the first time, one can optimize a beam for problems where the warping behavior is dominant. Density-based optimization is defined where the SIMP approach is utilized to penalize intermediate element densities. A key challenge of the optimization that arises in the warping function framework is the design-dependent nature of the problem. That is, the forcing vector varies during the optimization as it depends on the cross-section boundaries, which are functions of the updating topology. To this end, a differentiable boundary recognition algorithm is proposed. The methodology is applied to design beam cross-sections in which both torsion and warping constants are of interest. While intuitive topologies are obtained in the case of optimized torsion constant, this is not the case for the warping constant. The latter shows unique material distributions and a special dependence on the allowable material density.

Electric shovel (ES) is a large mining equipment crucial for energy security. The traditional design of the structure and control system of ES is carried out in stages, and the influence of the structural uncertainty for the system is not considered, which makes it difficult to obtain the optimal parameters of the system. Facing the demand of intelligent development, ES designed using traditional deterministic methods is difficult to meet the working demand of unmanned mining electric shovel (UMES). To address these challenges, this paper proposes an uncertainty-based multidisciplinary multi-objective optimization (UMMO) framework for UMES. Within this framework, the mechanical structure of the front-end mechanism was analyzed, excavation trajectories were planned based on a polynomial point-to-point motion strategy, and models for the excavation resistance of the dipper and the dynamical model of the front-end working device were constructed. Then, optimization objective functions were constructed with excavation energy consumption, excavation efficiency, and full dipper rate as targets. By analyzing the working characteristics of UMES, essential constraints were introduced for the mechanical system, control system and hardware. The UMMO optimization model was established to enhance the reliability of the UMES production process. Finally, the mechanical structure dimensions and control system parameters are optimized to generate the optimal physical structure and excavation trajectory considering uncertainties. The numerical results show that compared with the deterministic optimization results, the optimized structure of the proposed UMMO strategy is more compact and the mechanical structure is more reliable in the production process.

Complex engineering systems that involve multiple disciplines or scales are often decomposed into multiple subsystems in a nested or hierarchical manner to enhance the analysis efficiency. However, uncertainties inherent in input parameters will propagate with hierarchy, and severely threaten the reliability of engineering systems. Adaptive surrogate modeling technique is a potent tool to alleviate the computational burden of reliability analysis, especially involving time-consuming computer experiments. Conventional black-box adaptive surrogate modeling framework did not incorporate nested characteristic, which is inefficient for the reliability analysis of complex systems with nested or hierarchical characteristics. This paper develops a collaborative adaptive Kriging-based algorithm for the reliability analysis of nested systems. First, we propose a nested U-function to propel the adaptive updating of underlying Kriging models and derive its approximate closed form based on a defined most probable misclassification point. Then, an accuracy enhancement stage is devised to compensate for the inaccuracies of first-order approximation in early iterations. A parallel radius-based importance sampling technique is presented to mitigate the computational effort at multiple candidates. Finally, an index considering the reduction of model uncertainty is exploited to quantify the contribution of individual Kriging model and select the to-be-refined Kriging model in one iteration. Through numerical examples and case studies, the superiority of the proposed methodology is comprehensively illustrated compared with other benchmark methods.

We present a new framework for efficiently solving general topology optimization (TO) problems that find an optimal material distribution within a design space to maximize the performance of a part or structure while satisfying design constraints. These problems can involve convex or non-convex objective functions and may include multiple candidate materials. The framework is designed to greatly enhance computational efficiency, primarily by diminishing optimization iteration counts and thereby reducing the frequency of solving associated state equilibrium partial differential equations (PDEs) (e.g., through the finite element method (FEM)). It maintains binary design variables and addresses the large-scale mixed integer nonlinear programming (MINLP) problem that arises from discretizing the design space and PDEs. The core of this framework is the integration of the generalized Benders’ decomposition and adaptive trust regions. Specifically, by formulating the master sub-problem (decomposed from the MINLP) as a multi-cut optimization problem and enabling the estimation of the upper and lower bounds of the original objective function, significant acceleration in solution convergence is achieved. The trust region radius adapts based on a merit function. To mitigate ill-conditioning due to extreme parameter values, we further introduce a parameter relaxation scheme where two parameters are relaxed in stages at different paces, improving both solution quality and efficiency. Numerical tests validate the framework’s superior performance, including minimum compliance and compliant mechanism problems in single-material and multi-material designs. We compare our results with those of other methods and demonstrate significant reductions in optimization iterations (and therefore the number of FEM analyses required) by about one order of magnitude, while maintaining comparable optimal objective function values and material layouts. As the design variables and constraints increase, the framework maintains consistent solution quality and efficiency, underscoring its good scalability. We anticipate this framework will be especially advantageous for TO applications involving substantial design variables and constraints and requiring significant computational resources for FEM analyses (or PDE solving).