Bridging topology optimization and additive manufacturing

Topology optimization is a technique that allows for increasingly efficient designs with minimal a priori decisions. Because of the complexity and intricacy of the solutions obtained, topology optimization was often constrained to research and theoretical studies. Additive manufacturing, a rapidly evolving field, fills the gap between topology optimization and application. Additive manufacturing has minimal limitations on the shape and complexity of the design, and is currently evolving towards new materials, higher precision and larger build sizes. Two topology optimization methods are addressed: the ground structure method and density-based topology optimization. The results obtained from these topology optimization methods require some degree of post-processing before they can be manufactured. A simple procedure is described by which output suitable for additive manufacturing can be generated. In this process, some inherent issues of the optimization technique may be magnified resulting in an unfeasible or bad product. In addition, this work aims to address some of these issues and propose methodologies by which they may be alleviated. The proposed framework has applications in a number of fields, with specific examples given from the fields of health, architecture and engineering. In addition, the generated output allows for simple communication, editing, and combination of the results into more complex designs. For the specific case of three-dimensional density-based topology optimization, a tool suitable for result inspection and generation of additive manufacturing output is also provided.     

A performance index for topology and shape optimization of plate bending problems with displacement constraints

This paper presents a performance index for topology and shape optimization of plate bending problems with displacement constraints. The performance index is developed based on the scaling design approach. This performance index is used in the Performance-Based Optimization (PBO) method for plates in bending to keep track of the performance history when inefficient material is gradually removed from the design and to identify optimal topologies and shapes from the optimization process. Several examples are provided to illustrate the validity and effectiveness of the proposed performance index for topology and shape optimization of bending plates with single and multiple displacement constraints under various loading conditions. The topology optimization and shape optimization are undertaken for the same plate in bending, and the results are evaluated by using the performance index. The proposed performance index is also employed to compare the efficiency of topologies and shapes produced by different optimization methods. It is demonstrated that the performance index developed is an effective indicator of material efficiency for bending plates. From the manufacturing and efficient point of view, the shape optimization technique is recommended for the optimization of plates in bending. Received November 27, 1998?Revised version received June 6, 1999     

On minimum length scale control in density based topology optimization

The archetypical topology optimization problem concerns designing the layout of material within a given region of space so that some performance measure is extremized. To improve manufacturability and reduce manufacturing costs, restrictions on the possible layouts may be imposed. Among such restrictions, constraining the minimum length scales of different regions of the design has a significant place. Within the density filter based topology optimization framework the most commonly used definition is that a region has a minimum length scale not less than D if any point within that region lies within a sphere with diameter D >?0 that is completely contained in the region. In this paper, we propose a variant of this minimum length scale definition for subsets of a convex (possibly bounded) domain. We show that sets with positive minimum length scale are characterized as being morphologically open. As a corollary, we find that sets where both the interior and the exterior have positive minimum length scales are characterized as being simultaneously morphologically open and (essentially) morphologically closed. For binary designs in the discretized setting, the latter translates to that the opening of the design should equal the closing of the design. To demonstrate the capability of the developed theory, we devise a method that heuristically promotes designs that are binary and have positive minimum length scales (possibly measured in different norms) on both phases for minimum compliance problems. The obtained designs are almost binary and possess minimum length scales on both phases.     

Evolutionary topology optimization of continuum structures with a global displacement control

The conventional compliance minimization of load-carrying structures does not directly deal with displacements that are of practical importance. In this paper, a global displacement control is realized through topology optimization with a global constraint that sets a displacement limit on the whole structure or certain sub-domains. A volume minimization problem is solved by an extended evolutionary topology optimization approach. The local displacement sensitivities are derived following a power-law penalization material model. The global control of displacement is realized through multiple local displacement constraints on dynamically located critical nodes. Algorithms are proposed to secure the stability and convergence of the optimization process. Through numerical examples and by comparing with conventional stiffness designs, it is demonstrated that the proposed approach is capable of effectively finding optimal solutions which satisfy the global displacement control. Such solutions are of particular importance for structural designs whose deformed shapes must comply with functioning requirements such as aerodynamic performances.     

Discrete thickness optimization via piecewise constraint penalization

Structural engineers are often constrained by cost or manufacturing considerations to select member thicknesses from a discrete set of values. Conventional, gradient-free techniques to solve these discrete problems cannot handle large problem sizes, while discrete material optimization (DMO) techniques may encounter difficulties, especially for bending-dominated problems. To resolve these issues, we propose an efficient gradient-based technique to obtain engineering solutions to the discrete thickness selection problem. The proposed technique uses a series of constraints to enforce an effective stiffness-to-mass and strength-to-mass penalty on intermediate designs. In conjunction with these constraints, we apply an exact penalty function which drives the solution towards a discrete design. We utilize a continuation approach to obtain approximate solutions to the discrete thickness selection problem by solving a sequence of relaxed continuous problems with increasing penalization. We also show how this approach can be applied to combined discrete thickness selection and topology optimization design problems. To demonstrate the effectiveness of the proposed technique, we present both compliance and stress-constrained results for in-plane and bending-dominated problems.     

Design of cellular structures for optimum efficiency of heat dissipation

Metal cellular material is a new material attractive for its light weight and potential multifunctionality. In the present paper, we study cylindrical structures made of linear metal cellular material. The outer surface of the cylindrical structure is subjected to thermal boundary condition, and cooling fluid is forced through the cylinder to remove heat through the inner cell walls. Optimum design aims at maximization of heat dissipation efficiency under prescribed flow pressure. Two classes of design variables, relative density, and local aperture distribution of cellular material are to be determined by optimization under given total material volume constraints. Although similar to the structural topology optimization concept of material distribution, our formulation results in a structure with realistic cellular material of finite-sized aperture. Numerical results for different cross-sectional shapes and thermal boundary conditions are presented. Interestingly, our present formulation leads to optimum designs for cellular structures that mimic natural biomaterials. We discuss in general the guideline for cellular structure design to maximize heat dissipation efficiency based on insights from these optimization results.     

Integrated optimal topology design and shape optimization using neural networks

In this paper, neural network- and feature-based approaches are introduced to overcome current shortcomings in the automated integration of topology design and shape optimization. The topology optimization results are reconstructed in terms of features, which consist of attributes required for automation and integration in subsequent applications. Features are defined as cost-efficient simple shapes for manufacturing. A neural network-based image-processing technique is presented to match the arbitrarily shaped holes inside the structure with predefined features. The effectiveness of the proposed approach in integrating topology design and shape optimization is demonstrated with several experimental examples.     

Fixed-mesh curvature-parameterized shape optimization of an acoustic?horn

We suggest a boundary shape optimization approach in which the optimization is carried out on the coefficients in a boundary parameterization based on a local, discrete curvature. A fixed mesh is used to numerically solve the governing equations, in which the geometry is represented through inhomogeneous coefficients, similarly as done in the material distribution approach to topology optimization. The method is applied to the optimization of an acoustic horn in two space dimensions. Numerical experiments show that this method can calculate the horn’s transmission properties as accurately as a traditional, body-fitted approach. Moreover, the use of a fixed mesh allows the optimization to create shapes that would be difficult to handle with a traditional approach that uses deformations of a body-fitted mesh. The parameterization inherently promotes smooth designs without unduly restriction of the design flexibility. The optimized, smooth horns consistently show favorable transmission properties.     

Fatigue constrained topology optimization

We present a contribution to a relatively unexplored application of topology optimization: structural topology optimization with fatigue constraints. A probability based high-cycle fatigue analysis is combined with principal stress calculations in order to find the topology with minimum mass that can withstand prescribed variable-amplitude loading conditions for a specific life time. This allows us to generate optimal conceptual designs of structural components where fatigue life is the dimensioning factor. We describe the fatigue analysis and present ideas that make it possible to separate the fatigue analysis from the topology optimization. The number of constraints is kept low as they are applied to stress clusters, which are created such that they give adequate representations of the local stresses. Optimized designs constrained by fatigue and static stresses are shown and a comparison is also made between stress constraints based on the von Mises criterion and the highest tensile principal stresses. The paper is written with focus on structural parts in the avionic industry, but the method applies to any load carrying structure, made of linear elastic isotropic material, subjected to repeated loading conditions.     

Topology optimization considering overhang constraints: Eliminating sacrificial support material in additive manufacturing through design

Additively manufactured components often require temporary support material to prevent the component from collapsing or warping during fabrication. Whether these support materials are removed chemically as in the case of many polymer additive manufacturing processes, or mechanically as in the case of (for example) Direct Metal Laser Sintering, the use of sacrificial material increases total material usage, build time, and time required in post-fabrication treatments. The goal of this work is to embed a minimum allowable self-supporting angle within the topology optimization framework such that designed components and structures may be manufactured without the use of support material. This is achieved through a series of projection operations that combine a local projection to enforce minimum length scale requirements and a support region projection to ensure a feature is adequately supported from below. The magnitude of the self-supporting angle is process dependent and is thus an input variable provided by the manufacturing or design engineer. The algorithm is demonstrated on standard minimum compliance topology optimization problems and solutions are shown to satisfy minimum length scale, overhang angle, and volume constraints, and are shown to be dependent on the allowable magnitudes of these constraints.     

Concurrent multi-scale design optimization of composite frames with manufacturing constraints

This paper presents a gradient based concurrent multi-scale design optimization method for composite frames considering specific manufacturing constraints raised from the aerospace industrial requirements. Geometrical parameters of the frame components at the macro-structural scale and the discrete fiber winding angles at the micro-material scale are introduced as the independent design variables at the two geometrical scales. The DMO (Discrete Material Optimization) approach is utilized to couple the two geometrical scales and realize the simultaneous optimization of macroscopic topology and microscopic material selection. Six kinds of manufacturing constraints are explicitly included in the optimization model as series of linear inequalities or equalities. The capabilities of the proposed optimization model are demonstrated with the example of compliance minimization, subject to constraint on the composite volume. The linear constraints and optimization problems are solved by Sequential Linear Programming (SLP) optimization algorithm with move limit strategy. Numerical results show the potential of weight saving and structural robustness design with the proposed concurrent optimization model. The multi-scale optimization model, considering specific manufacturing constraints, provides new choices for the design of the composite frame structure in aerospace and other industries.     

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.     

Topology optimization for light-trapping structure in solar cells

The limitation associated with the low optical absorption remains to be the main technical barrier that constrains the efficiency of thin–film solar cells in energy conversion. Effective design of light-trapping structure is critical to increase light absorption, which is a highly complex phenomenon governed by several competing physical processes, imposing a number of challenges to topology optimization. This paper presents a general, yet systematic approach exploiting topology optimization for designing highly efficient light-trapping structures. We first demonstrate the proposed approach using genetic algorithm (GA) based non-gradient topology optimization (NGTO), which is robust for achieving highly-efficient designs of slot-waveguide based cells with both low-permittivity and high-permittivity scattering material at single wavelength or over a broad spectrum. The optimized light-trapping structure achieves a broadband absorption efficiency of 48.1 % and more than 3-fold increase over the Yablonovitch limit. The fabrication feasibility of the optimized design is also demonstrated. Next, the gradient topology optimization (GTO) approach for designing light-trapping structure is explored based on the Solid Isotropic Material with Penalization (SIMP) method. Similar designs are obtained through both GA based NGTO and SIMP based GTO, which verifies the validity of both approaches. Insights into the application of both approaches for solving the nanophotonic design problem with optimization nonlinearity are provided.     

Unified synthesis of compact planar path-generating linkages with rigid and deformable members

A unified procedure for the synthesis of planar linkages that may take the form of rigid body, fully compliant or partially compliant mechanisms is presented. The procedure automates the selection of mechanism topology as characterized by the number and connectivity of the links as well as the nature of the connections between them, the mechanism shape as characterized by the shapes of the individual links, and the mechanism dimensions which include the locations of the joints and the cross-sectional dimensions of the links. The synthesis task is posed as a constrained optimization problem and is solved by a hybrid, elite-preserving genetic algorithm. Three examples of compact mechanisms that trace different non-smooth paths in response to a single, monotonic and bounded force input are used to illustrate the synthesis capability of the procedure. Prototypes of the designs are built and tested to verify their performance. It is observed that in all three examples, partially compliant mechanism designs offer better conformance with design intent than either rigid body or fully compliant mechanisms.     

Design of periodic elastoplastic energy dissipating microstructures

The design of periodic elastoplastic microstructures for maximum energy dissipation is carried out using topology optimization. While the topology optimization of elastic microstructures has been performed in numerous studies, microstructural design considering inelastic behavior is relatively untouched due to a number of reasons which are addressed in this study. An RVE-based multiscale model is employed for computational homogenization with periodic boundary constraints, satisfying the Hill-Mandel principle. The plastic anisotropy which may be prevalent in materials fabricated through additive manufacturing processes is considered by modeling the constitutive behavior at the microscale with Hoffman plasticity. Discretization is done using enhanced assumed strain elements to avoid locking from incompressible plastic flow under plane strain conditions and a Lagrange multiplier approach is used to enforce periodic boundary constraints in the discrete system. The design problem is formulated using a density-based parameterization in conjunction with a SIMP-like material interpolation scheme. Attention is devoted to issues such as dependence on initial design and enforcement of microstructural connectivity, and a number of optimized microstructural designs are obtained under different prescribed deformation modes.

     

Deposition path planning-integrated structural topology optimization for 3D additive manufacturing subject to self-support constraint

This paper presents a novel level set-based topology optimization implementation, which addresses two main problems of design-for-additive manufacturing (AM): the material anisotropy and the self-support manufacturability constraint. AM material anisotropy is widely recognized and taking it into account while performing structural topology optimization could more realistically evaluate the structural performance. Therefore, both build direction and in-plane raster directions are considered by the topology optimization algorithm, especially for the latter, which is calculated through deposition path planning. The self-support manufacturability constraint is addressed through a novel multi-level set modeling. The related optimization problem formulation and solution process are demonstrated in detail. It is proved by several numerical examples that the manufacturability constraints are always strictly satisfied. Marginally, the recently popular structural skeleton-based deposition paths are also employed to assist the structural topology optimization, and its characteristics are discussed.     

Structural design for desired eigenfrequencies and mode shapes using topology optimization

A technique is proposed for determining the material distribution of a structure to obtain desired eigenmode shapes for problems of maximizing the fundamental eigenfrequency. The design objective is achieved using the solid isotropic method with penalization (SIMP) for topology optimization. Weighted constraints added in bound formulation are proposed to maximize the fundamental natural frequency, which provides an easy and straightforward way to prevent mode switching in the optimization process. Aside from maximizing the fundamental frequency, a method to modify existing eigenmodes to continuously evolve and assume the same shapes as the desired modes within the optimization process is proposed. The topology layout of a structure with desired eigenmodes is obtained by adding the modal assurance criterion (MAC) as additional constraints in the bound formulation optimization. Examples are presented to illustrate the proposed method, and a potential application of the proposed technique in decoupling a mechanical system is demonstrated.