A Hilbertian projection method for constrained level set-based topology optimisation

We present an extension of the projection method proposed by Challis et al ($\textit{Int J Solids Struct}$. Volume $\textbf{45}$(14-15) (2008) 4130-4146) for constrained level set-based topology optimisation that harnesses the Hilbertian velocity extension-regularisation framework. Our $\textit{Hilbertian projection method}$ chooses a normal velocity for the level set function as a linear combination of: 1) an orthogonal projection operator applied to the extended optimisation objective shape sensitivity; and 2) a weighted sum of orthogonal basis functions for the extended constraint shape sensitivities. This combination aims for the best possible first-order improvement of the optimisation objective in addition to first-order improvement of the constraints. Our formulation utilising basis orthogonalisation naturally handles linearly dependent constraint shape sensitivities. Furthermore, use of the Hilbertian extension-regularisation framework ensures that the resulting normal velocity is extended away from the boundary and enriched with additional regularity. Our approach is generally applicable to any topology optimisation problem to be solved in the level set framework. We consider several benchmark constrained microstructure optimisation problems and demonstrate that our method is effective with little-to-no parameter tuning. We also find that our method performs well when compared to a Hilbertian sequential linear programming method.Comment: 23 pages, 8 figure

High specific strength and stiffness structures produced using selective laser melting

Selective Laser Melting (SLM) was used to fabricate scaffolds using the titanium alloy Ti-6Al-4V. Two types of high porosity open-cell structures were manufactured: the first built from topology optimised designs with maximised stiffness, and the second from gyroid labyrinths. In mechanical compression tests the scaffolds demonstrate exceptional strength-and stiffness-to-weight ratios. In particular, for densities in the range 0.2-0.8 g/cm(3) the topology optimised scaffolds have specific strength and stiffness that are superior to those of comparable materials in the literature. In addition, the optimised scaffolds have the benefit of being elastically isotropic. The results of finite element calculations accurately match the measured stiffness of the scaffolds. Calculated strain energy distributions provide insight into how the high stiffness and strength of the optimised designs is connected to their efficient distribution of load. (C) 2014 Elsevier Ltd. All rights reserved

Fracture resistance via topology optimization

The fracture resistance of structures is optimized using the level-set method. Fracture resistance is assumed to be related to the elastic energy released by a crack propagating in a normal direction from parts of the boundary that are in tension, and is calculated using the virtual crack extension technique. The shape derivative of the fracture-resistance objective function is derived. Two illustrative two-dimensional case studies are presented: a hole in a plate subjected to biaxial strain; and a bridge fixed at both ends subjected to a single load in which the compliance and fracture resistance are jointly optimized. The structures obtained have rounded corners and more material at places where they are in tension. Based on the results, we propose that fracture resistance may be modeled more easily but less directly by including a term proportional to surface area in the objective function, in conjunction with nonlinear elasticity where the Young’s modulus in tension is lower than in compression

Physically realizable three-dimensional bone prosthesis design with interpolated microstructures

We present a new approach to designing three-dimensional, physically realizable porous femoral implants with spatially varying microstructures and effective material properties. We optimize over a simplified design domain to reduce shear stress at the bone-prosthetic interface with a constraint on the bone resorption measured using strain energy. This combination of objective and constraint aims to reduce implant failure and allows a detailed study of the implant designs obtained with a range of microstructure sets and parameters. The microstructure sets are either specified directly or constructed using shape interpolation between a finite number of microstructures optimized for multifunctional characteristics. We demonstrate that designs using varying microstructures outperform designs with a homogeneous microstructure for this femoral implant problem. Further, the choice of microstructure set has an impact on the objective values achieved and on the optimized implant designs. A proof-of-concept metal prototype fabricated via selective laser melting (SLM) demonstrates the manufacturability of designs obtained with our approach

Microstructure interpolation for macroscopic design

We present a method for multiple length scale structural optimisation. We first optimise isotropic microstructures for maximum bulk modulus at five solid fractions. Shape interpolation between these optimised microstructures produces a continuous set that smoothly varies in both geometry and mechanical properties. This smooth set is used for macroscopic optimisation via the material distribution method. The approach is computationally efficient and the geometric smoothness makes it clear how the microstructures can be transitioned between neighbouring elements. Performance comparisons are made to traditional structural optimisation for some example compliance optimisation problems. The interpolated microstructure designs are most advantageous for two dimensional problems involving multiple loading cases. In these cases, intermediate densities are utilised to more effectively distribute the load. In three dimensions, the method would be useful for a number of applications where specific microstructural requirements, such as a connected pore space, are needed within a multiple-scale design

High resolution topology optimization using graphics processing units (GPUs)

We present a Graphics Processing Unit (GPU) implementation of the level set method for topology optimization. The solution of three-dimensional topology optimization problems with millions of elements becomes computationally tractable with this GPU implementation and NVIDIA supercomputer-grade GPUs. We demonstrate this by solving the inverse homogenization problem for the design of isotropic materials with maximized bulk modulus. We trace the maximum bulk modulus optimization results to very high porosities to demonstrate the detail achievable with a high computational resolution. By utilizing a parallel GPU implementation rather than a sequential CPU implementation, similar increases in tractable computational resolution would be expected for other topology optimization problems

Explaining the competition between strength and toughness in perforated plates using computational finite fracture mechanics

We present a computational implementation of mode I finite fracture mechanics (FFM) that allows us to explore how hole shape and size affects the strength of linear elastic perforated plates. We compute the FFM predicted strength of a plate with centre crack, circle, diamond, and hexagon perforations of different sizes, as well as filleted (rounded) diamond perforations. Of the studied hole shapes, the diamond has the lowest predicted failure stress. By varying the toughness and strength material parameters, we elucidate how energy (toughness) and stress (strength) considerations compete for dominance in the coupled FFM failure criterion. We find that as the perforation radius goes to zero, failure is strength-dominated, while at small non-zero perforation sizes both strength and toughness play a role in determining failure. For perforation shapes with stress singularities (diamond, hexagon, centre crack), toughness dominates at larger perforation sizes, while strength strongly dominates at larger radii for circle perforations. The filleted diamond computations indicate that failure stress increases continuously as the hole shape deforms from a diamond into a circle, and so does the balance between toughness and strength in the coupled criterion. The presented results suggest new avenues for experimental work to further validate and explore the FFM coupled failure criterion. Our FFM implementation that uses Matlab and Ansys is provided as supplementary material

High specific strength and stiffness structures produced using selective laser melting

Selective Laser Melting (SLM) was used to fabricate scaffolds using the titanium alloy Ti-6Al-4V. Two types of high porosity open-cell structures were manufactured: the first built from topology optimised designs with maximised stiffness, and the second from gyroid labyrinths. In mechanical compression tests the scaffolds demonstrate exceptional strength- and stiffness-to-weight ratios. In particular, for densities in the range 0.2-0.8g/cm3 the topology optimised scaffolds have specific strength and stiffness that are superior to those of comparable materials in the literature. In addition, the optimised scaffolds have the benefit of being elastically isotropic. The results of finite element calculations accurately match the measured stiffness of the scaffolds. Calculated strain energy distributions provide insight into how the high stiffness and strength of the optimised designs is connected to their efficient distribution of load

Computationally generated cross-property bounds for stiffness and fluid permeability using topology optimization

We compute Pareto fronts that estimate the upper bounds of the bulk modulus and fluid permeability cross-property space for periodic porous materials over a range of porosities. The fronts are generated numerically using topology optimization, which is a systematic, free-form design algorithm for optimizing material layouts. The presented microstructures demonstrate the trade-off between the bulk modulus and fluid permeability achievable with a multifunctional porous material and will be useful for designers of materials for which both stiffness and permeability are important. Our results suggest that the range of achievable stiffness and permeability properties is significantly restricted when considering elastic isotropy, as compared to cubic elastic symmetry. The estimated bounds are of practical importance given the lack of microstructure-independent theoretical cross-property bounds