Topology and geometry optimization of grid-shells incorporating design-dependent self-weight loading

In this work is presented a material model to include self-weight loads in topology optimization. The proposed approach is focused in the optimization of lifting structures and aims not only to eliminate numerical instabilities due to the weight load but also to improve the solution quality. The algorithm is tested in a two-dimensional benchmark, which is also solved with a conservative formulation of the self-weight problem for comparison of results and validation purposes. Results of this problem shows that the method implemented improves the discreteness of the design of structures subjected to load cases similar to lifting structures and that it finds also efficient designs for conventional load cases in topology optimization problems. After validation, it is optimized the full interior body of a wing where the skin is left out of the design domain. The aerodynamic load is computed assuming a rigid wing and it is also considered the self-weight load. In a first approach, the aerodynamic load is much larger than the wing weight, simulating an airliner wing and the results shows the merits of the code implemented. Then, the wing is optimizing assuming an aerodynamic load matching the wing weight in order to simulate a flying wing aircraft configuration.

Chapter 2 - Growth Method for the Size, Topology, and Geometry Optimization of Truss Structures

Geometry optimization was recently introduced to existing truss topology optimization with global stability constraints. The resulting problems are formulated as highly nonlinear semidefinite programming problems that demand extensive computational effort to solve and have been solved only for small problem instances. The main challenge for effective computation is the positive semidefinite constraints which involve large sparse matrices. In this paper, we apply several techniques to tackle the challenge. First, we use the well-known chordal decomposition approach to replace each positive semidefinite constraint on a large sparse matrix by several positive semidefinite constraints on smaller submatrices together with suitable linking constraints. Moreover, we further improve the efficiency of the decomposition by applying a graph-based clique merging strategy to combine submatrices with significant overlap. Next, we couple these techniques with an optimization algorithm that fully exploits the structure of the smaller submatrices. As a result, we can solve much larger problems, which allows us to extend the existing single-load case to the multiple-load case, and to provide a computationally tractable approach for the latter case. Finally, we employ adaptive strategies from previous studies to solve successive problem instances, enabling the joints to navigate larger regions, and ultimately obtain further improved designs. The efficiency of the overall approach is demonstrated via computational experiments on large problem instances.

Shape optimization (with topological changes) has become a de facto standard approach for synthesizing new designs. Current approaches heavily rely on voxelized modifications of the initial domain, followed by heuristic reconstruction procedures that result in a detailed shape design. The initial domain is almost always conservatively estimated so that the design space spans a sufficiently large set of possible solutions. We propose a novel approach to shape optimization that exploits the geometric and topologic properties of medial zones and a new shape modification paradigm to synthesize shapes with topological guarantees. The medial zone is a ‘thick skeleton’ of the domain induced by prescribed boundary conditions, and represents an initial solution that is topologically valid. We modify such an initial solution by adding or subtracting ‘bubbles’ of material defined via prescribed shape functions. The proposed approach has many advantages in terms of topological guarantees, shape control, and computational cost, supports parametric optimization of free-form geometry, and can be implemented in any geometric representation that supports distance computations.

Controlling the propagation of light in the form of surface modes on miniaturized platforms is crucial for multiple applications. For dielectric multilayers that sustain Bloch surface waves at their interface to an isotropic dielectric medium, a conventional approach to manipulate them exploits shallow surface topographies fabricated on top of the truncated stack. However, such structures typically exhibit low index contrasts, making it challenging to confine, steer, and guide the Bloch surface waves. Here, we theoretically and experimentally demonstrate a device for a Bloch surface wave platform that resonantly couples light from a cavity to a straight waveguide. The structure is designed using topology optimization in a 2D geometry under the effective index approximation. In particular, the cavity–waveguide coupling efficiency of the radiation emitted by an individual source in the cavity center is optimized. The cavity is experimentally found to exhibit a narrow resonant peak that can be tuned by scaling the structure. The waveguide is shown to guide only light that resonates in the cavity. Fully three-dimensional simulations of the entire device validate the experimental observations.

A new method of simultaneous optimization of geometry and topology is presented for plane and spatial trusses. Compliance under single loading condition is minimized for specified structural volume. The difficulties due to existence of melting nodes are successfully avoided by considering force density, which is the ratio of axial force to the member length, as design variable. By using the fact that the optimal truss is statically determinate with the same absolute value of stress in existing members, the compliance and structural volume are expressed as explicit functions of force density only. After obtaining optimal cross-sectional area, nodal locations, and topology, the cross-sectional areas and nodal coordinates are further optimized using a conventional method of nonlinear programming. Accuracy of the optimal solution is verified through examples of plane trusses and a spatial truss. It is shown that various nearly optimal solutions can be found using the proposed method.

This work proposes an improved topology optimization model for optimizing continuum structures with self-weight loading conditions. A modified Solid Isotropic Material with Penalization (SIMP) model is proposed to avoid the parasitic effect. At the same time, the penalty factor of the SIMP model is increased to maintain the activeness of the prescribed volume constraint and to drive the design domain to binary distribution. The optimization objectives include minimizing the total strain energy of the design domain and minimizing the total displacement of the fixed domain. The shape optimization procedure is used to furtherly enhance structural performance. The whole optimization procedure is implemented with a two-dimensional model under the loading conditions of self-weight and external force. The classic MBB beam and self-weight arch are utilized to verify the proposed method, and conceptual layout designs of the steel structure bridge are conducted. It is proved that the proposed model is effective for topology optimization of continuum structures including self-weight. And it is found that the optimal structural topology is affected by the ratio of the external force to self-weight.

Rapid design approach for U-bend of a turbine serpentine cooling passage

PurposeThe purpose of this paper is to report the design of a lightweight tree-shaped support structure for fused deposition modeling (FDM) three-dimensional (3D) printed models when the printing path is considered as a constraint. Design/methodology/approachA hybrid of genetic algorithm (GA) and particle swarm optimization (PSO) is proposed to address the topology optimization of the tree-shaped support structures, where GA optimizes the topologies of the trees and PSO optimizes the geometry of a fixed tree-topology. Creatively, this study transforms each tree into an approximate binary tree such that GA can be applied to evolve its topology efficiently. Unlike FEM-based methods, the growth of tree branches is based on a large set of FDM 3D printing experiments. FindingsThe hybrid of GA and PSO is effective in reducing the volume of the tree supports. It is shown that the results of the proposed method lead to up to 46.71% material savings in comparison with the state-of-the-art approaches. Research limitations/implicationsThe proposed approach requires a large number of printing experiments to determine the function of the yield length of a branch in terms of a set of critical parameters. For brevity, one can print a small set of tree branches (e.g. 30) on a single platform and evaluate the function, which can be used all the time after that. The steps of GA for topology optimization and those of PSO for geometry optimization are presented in detail. Originality/valueThe proposed approach is useful for the designers and manufacturers to save materials and printing time in fabricating complex models using the FDM technique. It can be adapted to the design of support structures for other additive manufacturing techniques such as Stereolithography and selective laser melting.

A novel parameterization concept for structural truss topology optimization is presented in this article that enables the use of evolutionary algorithms in design of large-scale structures. The representational power of Boolean networks is used here to parameterize truss topology. A genetic algorithm then operates on parameters that govern the generation of truss topologies using this random network instead of operating directly on design variables. A genetic algorithm implementation is also presented that is congruent with the local rule application of the random network. The primary advantage of using a Boolean random network representation is that a relatively large number of ground structure nodes can be used, enabling successful exploration of a large-scale design space. In the classical binary representation of ground structures, the number of optimization variables increases quadratically with the number of nodes, restricting the maximum number of nodes that can be considered using a ground structure approach. The Boolean random network representation proposed here allows for the exploration of the entire topology space in a systematic way using only a linear number of variables. The number of nodes in the design domain, therefore, can be increased significantly. Truss member geometry and size optimization is performed here in a nested manner where an inner loop size optimization problem is solved for every candidate topology using sequential linear programming with move-limits. The Boolean random network and nested inner-loop optimization allows for the concurrent optimization of truss topology, geometry, and size. The effectiveness of this method is demonstrated using a planar truss design optimization benchmark problem.

Genetic algorithms (GA) are applied for the optimization of the structure of metallic clusters by the calculation of the ground-state energies from a tight-binding (Huckel) Hamiltonian. The optimum topology or graph is searche by the use of the adjacency matrix A ij as a natural coding. The initial population for N-atom clusters are generated from a representative group of fit cluster structures having N — 1 atoms by the addition Of random connections or hoppings between the Nth atom and the rest of the cluster atoms (A iN = 0 or 1). The diversity of geometries is enlarged by 20% with fully random structures. Several crossover strategies are proposed for the genetic evolution that combine the “parent” clusters while trying to preserve or transmit the physical characteristics of the parents’ topologies. The performance of the different procedures is tested. For N≤ 13, the present GA yield topological structures that are in agreement with previous geometry optimizations performed using an enumerative search (N ≤ 9) or simulated annealing Monte Carlo (10 ≤ N ≤ 13) methods. Limitations and extensions for N ≥ 14 are discussed.

The present work shows an innovative design framework for fluid Topology Optimization (TO) able to fully exploit the flexibility offered by Additive Manufacturing (AM) in the production of fluid-structure interaction systems. We present a geometry optimization method able to automatically design complex and efficient heat exchangers, adapted to maximizing fluid-structure heat transfer while minimizing turbulent flow pressure drop. The core of the method is the in-house Fluid Topology Optimization solver extended to include conjugate heat transfer problems. The TO method consists in emulating a sedimentation process inside an empty cavity in which a fluid dynamics problem is numerically solved. A design variable, in this case impermeability, is iteratively updated across the fluid dynamics domain. This mechanism leads to the formation of internal solid structures accordingly to a Lagrangian multi-objective optimization approach, adopted to include a multi-objective function. The solution of the optimization routine is the set of solidified structures, shaping the final optimized geometry. In order to match engineering applications, real conditions are implemented: an impermeability dependent thermal conductivity is included and a smoother operator is adopted to bound numerical thermal conductivity gradients across solid and fluid regions. The optimization is performed on a 3-dimensional straight duct: on the walls the temperature is constant and a coolant turbulent flow is simulated (Re 10000) inside the duct. The solver builds structures enhancing the heat transfer level between the walls of the domain and a coolant flow, by generating counter rotating vortices and complex fluid patterns. This is consistent to solution proposed in the open literature, such as v-shaped ribs, even if the geometry generated is more complex and efficient. The solution is validated with a high fidelity numerical simulation on StarCCM+, using a Detached Eddy Simulation (DES). Validation results shows higher heat transfer efficiency compared to the results present in the literature: the average Nusselt number computed on the domain walls is about 20% higher than the value obtained through experimental investigations on v-shape ribbed ducts. It is the first time that this method is applied and validated on real working conditions.

The paper concerns topology and geometry optimization of statically determinate beams with arbitrary number of supports. The optimization problem is treated as a bi-criteria one, with the objectives of minimizing the absolute maximum bending moment and the maximum deflection for a uniform gravity load. The problem is formulated and solved using the Pareto optimality concept and the lexicographic ordering of the objectives. The non-dominated sorting genetic algorithm NSGA-II and the local search method are used for the optimization in the Pareto sense, whereas the genetic algorithm and the exhaustive search method for the lexicographic optimization. Trade-offs between objectives are examined and sets of Pareto-optimal solutions are provided for different topologies. Lexicographically optimal beams are found assuming that the maximum moment is a more important criterion. Exact formulas for locations and values of the maximum deflection are given for all lexicographically optimal beams of any topology and any number of supports. Topologies with lexicographically optimal geometries are classified into equivalence classes, and specific features of these classes are discussed. A qualitative principle of the division of topologies equivalent in terms of the maximum moment into topologies better and worse in terms of the maximum deflection is found.

Evolutionary topology optimization of continuum structures including design-dependent self-weight loads

This paper presents a worst case approach for robust geometry and topology optimization of plane frames with global stability constraint. Uncertainty is assumed to exist in the nodal locations and cross‐sectional areas, and the worst values of the objective and stability constraint functions are relaxed to the quantile structural responses represented by the order statistics with given robustness and confidence levels. In order to alleviate the difficulty caused by melting nodes to some extent, the force density method is applied to an auxiliary truss model for geometry optimization of the frame, and the closely spaced nodes are merged. A method is presented for generating correlated imperfections for the nodal locations along each member, and a penalization approach is proposed for geometrical stiffness matrix to exclude superficial local buckling. It is demonstrated in the numerical examples that the result of robust optimization obtained by the proposed method is less sensitive to the uncertainty, and the stability constraint is also satisfied under uncertainty with the specified robustness and confidence levels.