A new overhang constraint for topology optimization of self-supporting structures in additive manufacturing

This work falls within the scope of computer-aided optimal design, and aims to integrate the topology optimization procedures and recent additive manufacturing technologies (AM). The elimination of scaffold supports at the topology optimization stage has been recognized and pursued by many authors recently. The present paper focuses on implementing a novel and specific overhang constraint that is introduced inside the topology optimization problem formulation along with the regular volume constraint. The proposed procedure joins the design and manufacturing processes into a integrated workflow where any component can directly be manufactured with no requirement of any sacrificial support material right after the topology optimization process. The overhang constraint presented in this work is defined by the maximum allowable inclination angle, where the inclination of any member is computed by the Smallest Univalue Segment Assimilating Nucleus (SUSAN), an edge detection algorithm developed in the field of image analysis and processing. Numerical results on some benchmark examples, along with the numerical performances of the proposed method, are introduced to demonstrate the capacities of the presented approach.     

Achieving stress-constrained topological design via length scale control

A new suite of computational procedures for stress-constrained continuum topology optimization is presented. In contrast to common approaches for imposing stress constraints, herein it is proposed to limit the maximum stress by controlling the length scale of the optimized design. Several procedures are formulated based on the treatment of the filter radius as a design variable. This enables to automatically manipulate the minimum length scale such that stresses are constrained to the allowable value, while the optimization is driven to minimizing compliance under a volume constraint – without any direct constraints on stresses. Numerical experiments are presented that incorporate the following : 1) Global control over the filter radius that leads to a uniform minimum length scale throughout the design; 2) Spatial variation of the filter radius that leads to local manipulation of the minimum length according to stress concentrations; and 3) Combinations of the two above. The optimized designs provide high-quality trade-offs between compliance, stress and volume. From a computational perspective, the proposed procedures are efficient and simple to implement: essentially, stress-constrained topology optimization is posed as a minimum compliance problem with additional treatment of the length scale.     

Explicit level set and density methods for topology optimization with equivalent minimum length scale constraints

The goal of this paper is to introduce local length scale control in an explicit level set method for topology optimization. The level set function is parametrized explicitly by filtering a set of nodal optimization variables. The extended finite element method (XFEM) is used to represent the non-conforming material interface on a fixed mesh of the design domain. In this framework, a minimum length scale is imposed by adopting geometric constraints that have been recently proposed for density-based topology optimization with projections filters. Besides providing local length scale control, the advantages of the modified constraints are twofold. First, the constraints provide a computationally inexpensive solution for the instabilities which often appear in level set XFEM topology optimization. Second, utilizing the same geometric constraints in both the density-based topology optimization and the level set optimization enables to perform a more unbiased comparison between both methods. These different features are illustrated in a number of well-known benchmark problems for topology optimization.

     

A simple and effective inverse projection scheme for void distribution control in topology optimization

The ability to control both the minimum size of holes and the minimum size of structural members are essential requirements in the topology optimization design process for manufacturing. This paper addresses both requirements by means of a unified approach involving mesh-independent projection techniques. An inverse projection is developed to control the minimum hole size while a standard direct projection scheme is used to control the minimum length of structural members. In addition, a heuristic scheme combining both contrasting requirements simultaneously is discussed. Two topology optimization implementations are contributed: one in which the projection (either inverse or direct) is used at each iteration; and the other in which a two-phase scheme is explored. In the first phase, the compliance minimization is carried out without any projection until convergence. In the second phase, the chosen projection scheme is applied iteratively until a solution is obtained while satisfying either the minimum member size or minimum hole size. Examples demonstrate the various features of the projection-based techniques presented.     

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.     

Minimum length-scale constraints for parameterized implicit function based topology optimization

This paper introduces explicit minimum length-scale constraint functions suitable for parameterized implicit function based topology optimization methods. Length-scale control in topology optimization has many potential benefits, such as removing numerical artifacts, mesh independent solutions, avoiding thin, or single node hinges in compliant mechanism design and meeting manufacturing constraints. Several methods have been developed to control length-scale when using density-based or signed-distance-based level-set methods. In this paper a method is introduced to control length-scale for parameterized implicit function based topology optimization. Explicit constraint functions to control the minimum length in the structure and void regions are proposed and implementation issues explored in detail. Several examples are presented to show the efficacy of the proposed method. The examples demonstrate that the method can simultaneously control minimum structure and void length-scale, design hinge free compliant mechanisms and control minimum length-scale for three dimensional structures.     

Heaviside projection based topology optimization by a PDE-filtered scalar function

This paper deals with topology optimization based on the Heaviside projection method using a scalar function as design variables. The scalar function is then regularized by a PDE based filter. Several image-processing based filtering techniques have so far been proposed for regularization or restricting the minimum length scale. They are conventionally applied to the design sensitivities rather than the design variables themselves. However, it causes discrepancies between the filtered sensitivities and the actual sensitivities that may confuse the optimization process and disturb the convergence. In this paper, we propose a Heaviside projection based topology optimization method with a scalar function that is filtered by a Helmholtz type partial differential equation. Therefore, the optimality can be strictly discussed in terms of the KKT condition. In order to demonstrate the effectiveness of the proposed method, a minimum compliance problem is solved.     

Imposing maximum length scale in topology optimization

This paper presents a technique for imposing maximum length scale on features in continuum topology optimization. The design domain is searched and local constraints prevent the formation of features that are larger than the prescribed maximum length scale. The technique is demonstrated in the context of structural and fluid topology optimization. Specifically, maximum length scale criterion is applied to (a) the solid phase in minimum compliance design to restrict the size of structural (load-carrying) members, and (b) the fluid (void) phase in minimum dissipated power problems to limit the size of flow channels. Solutions are shown to be near 0/1 (void/solid) topologies that satisfy the maximum length scale criterion. When combined with an existing minimum length scale methodology, the designer gains complete control over member sizes that can influence cost and manufacturability. Further, results suggest restricting maximum length scale may provide a means for influencing performance characteristics, such as redundancy in structural design.     

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.     

Application of topology optimization to design an electric bicycle main frame

Electric bicycle main frame is the most principal structure, connecting and supporting other various components, while bearing a variety of forces and moments. In this paper the topology optimization technology is applied to generate robust electric bicycle main frame by optimizing the material distribution subject to the constraints and dynamic loads. Geometric, mechanical and finite element models, as well as a flexible coupling dynamic model are constructed. Validity and accuracy of these models are investigated through real-life testing. By applying typical road excitation, dynamic loads of all key points are extracted. A set of forces data is extracted every 0.5?s during the whole simulation, including peak values of these forces. In order to obtain appropriate topology optimization results, the values of two crucial parameters, volume fraction and minimum member size, are discussed respectively. Then the topology optimization of multi-load case is implemented with the objective of minimizing the set of weighted compliances resulting from individual load cases. Results illustrate that element density distribution of the model is optimized with manufacturing constraints of minimum member size control and extrusion constraint. Consequently, the better frame form design of the electric bicycle is obtained. Modal analysis for the original and refined models is performed respectively to evaluate the structure stiffness. The results indicate that this optimization program is effective enough to develop a new electric bicycle frame as a reference for manufacturers.     

Miniaturization limits of piezoresistive MEMS accelerometers

We present the miniaturization limits of axially loaded piezoresistive MEMS accelerometers. Therefore we identify limiting factors on the basis of FEM-verified analytical models. To ensure a broad discussion we compare two different axially loaded topologies: first a conventional topology, which can be manufactured already today, and second a future-oriented topology utilizing nanowires. To enable a realistic comparison of the different topologies we shrink the sensor while maintaining a specific performance (e.g. sensitivity and noise) considering design limitations such as fracture of silicon and buckling. To find the minimum total sensor area under certain constraints and therefore the optimal geometric and material parameters we apply optimization techniques to our analytical models. It will be seen that the piezoresistive transducer principle for MEMS accelerometers has a promising shrink potential with minimum total sensor dimensions as low as 150 × 150 × 10 μm³ achievable by use of currently available manufacturing processes.     

Discussion on symmetry of optimum topology design

Recent study by Rozvany discussed an important topic in structural topology optimization: non-uniqueness and symmetry. This paper demonstrates that for the frame topology optimization with frequency objective or frequency constraints, the topology optimum is not symmetric even when the geometry, material distribution and ground structure are symmetric. Two benchmark examples with fundamental frequency constraints support our view. Nonlinear and non-convex constraints are contributors to non-symmetric optimum solution for the class of topology optimization problems considered here.     

Solving stress constrained problems in topology and material optimization

This article is a continuation of the paper Ko?vara and Stingl (Struct Multidisc Optim 33(4?C5):323?C335, 2007). The aim is to describe numerical techniques for the solution of topology and material optimization problems with local stress constraints. In particular, we consider the topology optimization (variable thickness sheet or ??free sizing??) and the free material optimization problems. We will present an efficient algorithm for solving large scale instances of these problems. Examples will demonstrate the efficiency of the algorithm and the importance of the local stress constraints. In particular, we will argue that in certain topology optimization problems, the addition of stress constraints must necessarily lead not only to the change of optimal topology but also optimal geometry. Contrary to that, in material optimization problems the stress singularity is treated by the change in the optimal material properties.     

A computational paradigm for multiresolution topology optimization (MTOP)

This paper presents a multiresolution topology optimization (MTOP) scheme to obtain high resolution designs with relatively low computational cost. We employ three distinct discretization levels for the topology optimization procedure: the displacement mesh (or finite element mesh) to perform the analysis, the design variable mesh to perform the optimization, and the density mesh (or density element mesh) to represent material distribution and compute the stiffness matrices. We employ a coarser discretization for finite elements and finer discretization for both density elements and design variables. A projection scheme is employed to compute the element densities from design variables and control the length scale of the material density. We demonstrate via various two- and three-dimensional numerical examples that the resolution of the design can be significantly improved without refining the finite element mesh.     

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.     

Topology optimization for microstructural design under stress constraints

This work aims at introducing stress responses within a topology optimization framework applied to the design of periodic microstructures. The emergence of novel additive manufacturing techniques fosters research towards new approaches to tailor materials properties. This paper derives a formulation to prevent the occurrence of high stress concentrations, often present in optimized microstructures. Applying macroscopic test strain fields to the material, microstructural layouts, reducing the stress level while exhibiting the best overall stiffness properties, are sought for. Equivalent stiffness properties of the designed material are predicted by numerical homogenization and considering a metallic base material for the microstructure, it is assumed that the classical Von Mises stress criterion remains valid to predict the material elastic allowable stress at the microscale. Stress constraints with arbitrary bounds are considered, assuming that a sizing optimization step could be applied to match the actual stress limits under realistic service loads. Density–based topology optimization, relying on the SIMP model, is used and the qp–approach is exploited to overcome the singularity phenomenon arising from the introduction of stress constraints with vanishing material. Optimization problems are solved using mathematical programming schemes, in particular MMA, so that a sensitivity analysis of stress responses at the microstructural level is required and performed considering the adjoint approach. Finally, the developed method is first validated with classical academic benchmarks and then illustrated with an original application: tailoring metamaterials for a museum anti–seismic stand.     

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.     

Optimizing inclusion shapes and patterns in periodic materials using Discrete Object Projection

Current topology optimization methodologies assume a monolithic, free form approach to design. Many engineered materials and structures, however, are composed of discrete, non-overlapping objects such as fiber or particle-based materials. Application of the topology optimization methodology to these types of materials therefore requires controlling the shape and interaction of designed features to ensure solutions are meaningful and physically realizable. Achieving such control on continuum domains is challenging as features form via the union of elements of like phase. A topology optimization approach is proposed herein for optimizing the size, shape, and layout of inclusion-like features in a continuum domain. The technique is based on the Heaviside Projection Method and uses multiple regularized Heaviside functions whose interaction is tailored so that the designer may restrict the minimum and maximum length scale of inclusions, and minimum spacing between inclusions. The technique is demonstrated on the design of material microstructures with enhanced elastic stiffness.     

Topology optimization of continuum structures with local and global stress constraints

Topology structural optimization problems have been usually stated in terms of a maximum stiffness (minimum compliance) approach. The objective of this type of approach is to distribute a given amount of material in a certain domain, so that the stiffness of the resulting structure is maximized (that is, the compliance, or energy of deformation, is minimized) for a given load case. Thus, the material mass is restricted to a predefined percentage of the maximum possible mass, while no stress or displacement constraints are taken into account. This paper presents a different strategy to deal with topology optimization: a minimum weight with stress constraints Finite Element formulation for the topology optimization of continuum structures. We propose two different approaches in order to take into account stress constraints in the optimization formulation. The local approach of the stress constraints imposes stress constraints at predefined points of the domain (i.e. at the central point of each element). On the contrary, the global approach only imposes one global constraint that gathers the effect of all the local constraints by means of a certain so-called aggregation function. Finally, some application examples are solved with both formulations in order to compare the obtained solutions.     

Support structure constrained topology optimization for additive manufacturing

There is significant interest today in integrating additive manufacturing (AM) and topology optimization (TO). However, TO often leads to designs that are not AM friendly. For example, topologically optimized designs may require significant amount of support structures before they can be additively manufactured, resulting in increased fabrication and clean-up costs.In this paper, we propose a TO methodology that will lead to designs requiring significantly reduced support structures. Towards this end, the concept of ‘support structure topological sensitivity’ is introduced. This is combined with performance sensitivity to result in a TO framework that maximizes performance, subject to support structure constraints. The robustness and efficiency of the proposed method is demonstrated through numerical experiments, and validated through fused deposition modeling, a popular AM process.