Topology/shape optimisation of axisymmetric continuum structures – a metamorphic development approach

A novel topology/shape optimisation method for axisymmetric elastic solids, based on solid modeling and FE analysis, is presented. Optimal profiles of minimum-mass axisymmetric structures are sought by growing and degenerating simple initial structures subject to response constraints. The rates of the growth and degeneration are controlled based on the current objective and constraint functions of the optimisation problem under consideration. The optimal structures are developed metamorphically in specified infinite design domains using both quadrilateral and triangular axisymmetric finite elements that are ideally suited for modeling continua involving curved boundaries.The robustness of this fully automatic method is studied and validated with the first example of seeking the optimal shape of a centrally suspended axisymmetric object with minimum strain energy caused by self-weight. Then the method is applied to a practical industrial design problem: the design of a turbine disk. The variations of load and boundary conditions caused by shape change in these problems, including the gravitational and centrifugal loads, and temperature distribution are accommodated in the optimisation procedures. Thus, the design model closely resembles the real design problem. The results demonstrate the success of the method in generating optimal but realistic solutions to practical design problems.     

Design of multi-component structural systems for optimal layout topology and joint locations

Present day topology optimization techniques for continuum structures consider the design of single structural components, while most real life engineering design problems involve multiple components or structures. It is therefore necessary to have a methodology that can address the design of multi-component systems and generate designs for the optimal layouts of individual structures and locations for interconnections. The interconnections include supports provided by the ground, joints and rigid connections like rivets, bolts and welds between components. While topology optimization of structures has been extensively researched, relatively little work has been done on optimizing the locations of the interconnections. In this research, a method to model and define domains for the interconnections has been developed. The optimization process redistributes material in the component design domains and locates the connections optimally based on an energy criterion. Some practical design examples are used to illustrate the capability of this method.     

Optimum design with imperfection sensitivity coefficients for limit point loads

A computational method is presented for finding a sequence of optimum designs of a discrete system which exhibits limit point behaviour. Optimality conditions are derived in terms of the theory of imperfection sensitivity coefficients for the limit point load factor. Only those designs of the structures which exhibit limit point behaviour are considered as feasible designs, and the design change is conceived as generating a kind of imperfection. The efficiency of the proposed algorithm will be appreciated particularly for large structures, because incremental nonlinear analysis to find the limit point load factor needs to be carried out only once for the structure of trivial initial optimum design. The sequence of optimum designs is described by piecewise Taylor series expansions with respect to the specified limit point load factor. It is shown in the examples that the proposed method is efficient and of good accuracy for a large space truss.     

Self-supporting rhombic infill structures for additive manufacturing

Recent work has demonstrated that the interior material layout of a 3D model can be designed to make a fabricated replica satisfy application-specific demands on its physical properties, such as resistance to external loads. A widely used practice to fabricate such models is by layer-based additive manufacturing (AM), which however suffers from the problem of adding and removing interior supporting structures. In this paper, we present a novel method for generating application-specific infill structures on rhombic cells so that the resultant structures can automatically satisfy manufacturing requirements on overhang-angle and wall-thickness. Additional supporting structures can be avoided entirely in our framework. To achieve this, we introduce the usage of an adaptive rhombic grid, which is built from an input surface model. Starting from the initial sparse set of rhombic cells, via numerical optimization techniques an objective function can be improved by adaptively subdividing the rhombic grid and thus adding more walls in cells. We demonstrate the effectiveness of our method for generating interior designs in the applications of improving mechanical stiffness and static stability.     

Discrete sizing/layout/topology optimization of truss structures with an advanced Jaya algorithm

Discrete optimization of truss structures is a hard computing problem with many local minima. Metaheuristic algorithms are naturally suited for discrete optimization problems as they do not require gradient information. A recently developed method called Jaya algorithm (JA) has proven itself very efficient in continuous engineering problems. Remarkably, JA has a very simple formulation and does not utilize algorithm-specific parameters. This study presents a novel JA formulation for discrete optimization of truss structures under stress and displacement constraints. The new algorithm, denoted as discrete advanced JA (DAJA), implements efficient search mechanisms for generating new trial designs including discrete sizing, layout and topology optimization variables. Besides the JA’s basic concept of moving towards the best design of the population and moving away from the worst design, DAJA tries to form a set of descent directions in the neighborhood of each candidate designs thus generating high quality trial designs that are very likely to improve current population. Results collected in seven benchmark problems clearly demonstrate the superiority of DAJA over other state-of-the-art metaheuristic algorithms and multi-stage continuous–discrete optimization formulations.     

Design of heterogeneous turbine blade

Constantly rising operating pressure and temperature in turbine drivers push the material capabilities of turbine blades to the limit. The recent development of heterogeneous objects by layered manufacturing offers new potentials for the turbine blades. In heterogeneous turbine blades, multiple materials can be synthesized to provide better properties than any single material. A critical task of such synthesis in turbine blade design is an effective design method that allows a designer to design geometry and material composition simultaneously.This paper presents a new approach for turbine blade design, which ties B-spline representation of a turbine blade to a physics (diffusion) process. In this approach, designers can control both geometry and material composition. Meanwhile, material properties are directly conceivable to the designers during the design process. The designer's role is enhanced from merely interpreting the optimization result to explicitly controlling both material composition and geometry according to the acquired experience (material property constraints).The mathematical formulation of the approach includes three steps: using B-spline to represent the turbine blade, using diffusion equation to generate material composition variation, using finite element method to solve the constrained diffusion equation. The implementation and examples are presented to validate the effectiveness of this approach for heterogeneous turbine blade design.     

Implementing a structural continuity constraint and a halting method for the topology optimization of energy absorbers

This study investigates topology optimization of energy absorbing structures in which material damage is accounted for in the optimization process. The optimization objective is to design the lightest structures that are able to absorb the required mechanical energy. A structural continuity constraint check is introduced that is able to detect when no feasible load path remains in the finite element model, usually as a result of large scale fracture. This assures that designs do not fail when loaded under the conditions prescribed in the design requirements. This continuity constraint check is automated and requires no intervention from the analyst once the optimization process is initiated. Consequently, the optimization algorithm proceeds towards evolving an energy absorbing structure with the minimum structural mass that is not susceptible to global structural failure. A method is also introduced to determine when the optimization process should halt. The method identifies when the optimization method has plateaued and is no longer likely to provide improved designs if continued for further iterations. This provides the designer with a rational method to determine the necessary time to run the optimization and avoid wasting computational resources on unnecessary iterations. A case study is presented to demonstrate the use of this method.     

An explicit parameterization for casting constraints in gradient driven topology optimization

From a practical point of view it is often desirable to limit the complexity of a topology optimization design such that casting/milling type manufacturing techniques can be applied. In the context of gradient driven topology optimization this work studies how castable designs can be obtained by use of a Heaviside design parameterization in a specified casting direction. This reduces the number of design variables considerably and the approach is simple to implement.     

Continuous front propagation-based overhang control for topology optimization with additive manufacturing

Additive manufacturing enables the nearly uncompromised production of optimized topologies. However, due to the overhang limitation, some designs require a large number of supporting structures to enable manufacturing. Because these supports are costly to build and difficult to remove, it is desirable to find alternative designs that do not require support. In this work, a filter is presented that suppresses non-manufacturable regions within the topology optimization loop, resulting in designs that can be manufactured without the need for supports. The filter is based on front propagation, can be evaluated efficiently, and adjoint sensitivities are calculated with almost no additional computational cost. The filter can be applied also to unstructured meshes and the permissible degree of overhang can be freely chosen. The method is demonstrated on several compliance minimization problems in which its computational efficiency and flexibility are shown. The current applications are in 2D, and the proposed method is readily extensible to 3D.     

Development of a structural optimization strategy for the design of next generation large thermoplastic wind turbine blades

This paper presents the development of a structural optimization process for the design of future large thermoplastic wind turbine blades. The optimization process proposed in this paper consists of three optimization steps. The first step is a topology optimization of a short untwisted and non tapered section of the blade, with the inner volume used as the design domain. The second step is again a topology optimization, but on the first half of a blade to study the effect of non symmetry of the structure due to blade twist and taper. Results of this optimization step are then interpreted to build a shell model of the complete blade structure to perform composite size optimization based on a minimum mass objective subjected to constraints on deflection, composite strength and structural stability. Different blade models using ribs are then optimized and compared against conventional blade structure (box spar structure without ribs and single web structure without ribs). The use of ribs in wind turbine blade structures, which is more adapted to thermoplastic composite manufacturing than for thermoset composites, leads to slightly lighter blades than conventional blade structures.     

Mixed-integer second-order cone programming for global optimization of compliance of frame structure with discrete design variables

Frame structures are extensively used in mechanical, civil, and aerospace engineering. Besides generating reasonable designs of frame structures themselves, frame topology optimization may serve as a tool providing us with conceptual designs of diverse engineering structures. Due to its nonconvexity, however, most of existing approaches to frame topology optimization are local optimization methods based on nonlinear programming with continuous design variables or (meta)heuristics allowing some discrete design variables. Presented in this paper is a new global optimization approach to the frame topology optimization with discrete design variables. It is shown that the compliance minimization problem with predetermined candidate cross-sections can be formulated as a mixed-integer second-order cone programming problem. The global optimal solution is then computed with an existing solver based on a branch-and-cut algorithm. Numerical experiments are performed to examine computational efficiency of the proposed approach.     

Pareto frontier for simultaneous structural and manufacturing optimization of a composite part

Optimized design of composite structures requires simultaneous optimization of structural performance and manufacturing process. Such a challenge calls for a multi-objective optimization. Here, a generating multi-objective optimization method called normalized normal constraint method, which attains a set of optimal solutions and allows the designer to explore design alternatives before making the final decision, is coupled with a local-global search called constrained globalized bounded Nelder–Mead method. The proposed approach is applied to the design of a Z-shaped composite bracket for optimization of structural and manufacturing objectives. Comparison of the results with non-dominated sorting genetic algorithm (NSGA-II) shows that when only a small number of function evaluations are possible and a few Pareto optima are desired, the proposed method outperforms NSGA-II in terms of convergence to the true Pareto frontier. The results are validated by an enumeration search and by exploring the neighbourhood of the final solutions.     

Design of conjugate, conjoined shapes and tilings using topology optimization

We present a method for obtaining conjugate, conjoined shapes and tilings in the context of the design of structures using topology optimization. Optimal material distribution is achieved in topology optimization by setting up a selection field in the design domain to determine the presence/absence of material there. We generalize this approach in this paper by presenting a paradigm in which the material left out by the selection field is also utilised. We obtain conjugate shapes when the region chosen and the region left-out are solutions for two problems, each with a different functionality. On the other hand, if the left-out region is connected to the selected region in some pre-determined fashion for achieving a single functionality, then we get conjoined shapes. The utilization of the left-out material, gives the notion of material economy in both cases. Thus, material wastage is avoided in the practical realization of these designs using many manufacturing techniques. This is in contrast to the wastage of left-out material during manufacture of traditional topology-optimized designs. We illustrate such shapes in the case of stiff structures and compliant mechanisms. When such designs are suitably made on domains of the unit cell of a tiling, this leads to the formation of new tilings which are functionally useful. Such shapes are not only useful for their functionality and economy of material and manufacturing, but also for their aesthetic value.     

Efficient size and shape optimization of truss structures subject to stress and local buckling constraints using sequential linear programming

The advance in digital fabrication technologies and additive manufacturing allows for the fabrication of complex truss structure designs but at the same time posing challenging structural optimization problems to capitalize on this new design freedom. In response to this, an iterative approach in which Sequential Linear Programming (SLP) is used to simultaneously solve a size and shape optimization sub-problem subject to local stress and Euler buckling constraints is proposed in this work. To accomplish this, a first order Taylor expansion for the nodal movement and the buckling constraint is derived to conform to the SLP problem formulation. At each iteration a post-processing step is initiated to map a design vector to the exact buckling constraint boundary in order to facilitate the overall efficiency. The method is verified against an exact non-linear optimization problem formulation on a range of benchmark examples obtained from the literature. The results show that the proposed method produces optimized designs that are either close or identical to the solutions obtained by the non-linear problem formulation while significantly decreasing the computational time. This enables more efficient size and shape optimization of truss structures considering practical engineering constraints.     

A new thickness parameterization for Discrete Material and Thickness Optimization

In this work, a new thickness parameterization which allows for internal ply-drops without intermediate voids is introduced in the Discrete Material and Thickness Optimization (DMTO) method. With the original DMTO formulation, material had to be removed from the top in order to prevent non-physical intermediate voids in the structure. The new thickness formulation relies on a relation between density variables and ply-thicknesses rather than constitutive properties. This new formulation allows internal ply-drops which is essential for composite structures as it is common practice to cover dropped plies as to avoid delaminations. Furthermore, it is demonstrated how the new thickness formulation in some cases improves the convergence characteristics. Finally, it is also shown how solid-shell elements can be utilized within the DMTO method for structural optimization of tapered laminated composite structures.     

Optimal topology design of internal stiffeners for machine pedestal structures using biological branching phenomena

Naturally evolved biological structures exhibit the optimal characteristics of light weight, high stiffness, and high strength. Based on the growth mechanism of biological branch systems in nature, an optimization method for internal stiffener plate distribution in box structures is suggested. Under the given load and support conditions, the internal stiffener plates of machine pedestal structures grow, bifurcate, and degenerate towards the direction of maximum overall structural stiffness in accordance with the adaptive growth law. The optimal and distinct distribution of internal stiffener plates with the most effective load path is thus obtained. Based on this, a size optimization for lightweight design is conducted, in which the self-weight of the structures is taken as the design objective, and the natural vibration frequency and static stiffness in the direction that is sensitive to machining accuracy are set as constraints. Finally, an optimized structure is obtained. The effectiveness of the proposed method is verified by using a precision grinder bed as an example. The results of numerical simulation and 3D–printed model experiment indicate that both the dynamic and the static performance of the optimized structure are improved, while the structural weight is reduced by compared with the initial structure. The suggested design method provides a new solution approach for the design optimization of machine pedestal structures.     

Topology optimization considering fracture mechanics behaviors at specified locations

As a typical form of material imperfection, cracks generally cannot be avoided and are critical for load bearing capability and integrity of engineering structures. This paper presents a topology optimization method for generating structural layouts that are insensitive/sensitive as required to initial cracks at specified locations. Based on the linear elastic fracture mechanics model (LEFM), the stress intensity of initial cracks in the structure is analyzed by using singularity finite elements positioned at the crack tip to describe the near-tip stress field. In the topology optimization formulation, the J integral, as a criterion for predicting crack opening under certain loading and boundary conditions, is introduced into the objective function to be minimized or maximized. In this context, the adjoint variable sensitivity analysis scheme is derived, which enables the optimization problem to be solved with a gradient-based algorithm. Numerical examples are given to demonstrate effectiveness of the proposed method on generating structures with desired overall stiffness and fracture strength property. This method provides an applicable framework incorporating linear fracture mechanics criteria into topology optimization for conceptual design of crack insensitive or easily detachable structures for particular applications.     

Evolutionary topology optimization of continuum structures with smooth boundary representation

This paper develops an extended bi-directional evolutionary structural optimization (BESO) method for topology optimization of continuum structures with smoothed boundary representation. In contrast to conventional zigzag BESO designs and removal/addition of elements, the newly proposed evolutionary topology optimization (ETO) method, determines implicitly the smooth structural topology by a level-set function (LSF) constructed by nodal sensitivity numbers. The projection relationship between the design model and the finite element analysis (FEA) model is established. The analysis of the design model is replaced by the FEA model with various elemental volume fractions, which are determined by the auxiliary LSF. The introduction of sensitivity LSF results in intermediate volume elements along the solid-void interface of the FEA model, thus contributing to the better convergence of the optimized topology for the design model. The effectiveness and robustness of the proposed method are verified by a series of 2D and 3D topology optimization design problems including compliance minimization and natural frequency maximization. It has been shown that the developed ETO method is capable of generating a clear and smooth boundary representation; meanwhile the resultant designs are less dependent on the initial guess design and the finite element mesh resolution.     

Multiobjective layout optimization of robotic cellular manufacturing systems

This paper proposes a multiobjective layout optimization method for the conceptual design of robot cellular manufacturing systems. Robot cellular manufacturing systems utilize one or more flexible robots which can carry out a large number of operations, and can conduct flexible assemble processes. The layout design stage of such manufacturing systems is especially important since fundamental performances of the manufacturing system under consideration are determined at this stage. In this paper, the design criteria for robot cellular manufacturing system layout designs are clarified, and objective functions are formulated. Next, layout design candidates are represented using a sequence-pair scheme to avoid interference between assembly system components, and the use of dummy components is proposed to represent layout areas where components are sparse. A multiobjective genetic algorithm is then used to obtain Pareto optimal solutions for the layout optimization problems. Finally, several numerical examples are provided to illustrate the effectiveness and usefulness of the proposed method.     

Mixed-integer linear programming approach for global discrete sizing optimization of frame structures

This paper focuses on discrete sizing optimization of frame structures using commercial profile catalogs. The optimization problem is formulated as a mixed-integer linear programming (MILP) problem by including the equations of structural analysis as constraints. The internal forces of the members are taken as continuous state variables. Binary variables are used for choosing the member profiles from a catalog. Both the displacement and stress constraints are formulated such that for each member limit values can be imposed at predefined locations along the member. A valuable feature of the formulation, lacking in most contemporary approaches, is that global optimality of the solution is guaranteed by solving the MILP using branch-and-bound techniques. The method is applied to three design problems: a portal frame, a two-story frame with three load cases and a multiple-bay multiple-story frame. Performance profiles are determined to compare the MILP reformulation method with a genetic algorithm.