Experimental and computational analysis of energy absorption characteristics of three biomimetic lattice structures under compression

The objective of this study is to evaluate the mechanical properties and energy absorption characteristics of the gyroid, dual-lattice and spinodoid structures, as biomimetic lattices, through finite element analysis and experimental characterisation. As part of the study, gyroid and dual-lattice structures at 10% volume fraction were 3D-printed using an elastic resin, and mechanically tested under uniaxial compression. Computational models were calibrated to the observed experimental data and the response of higher volume fraction structures were simulated in an explicit finite element solver. Stress-strain data of groups of lattices at different volume fractions were studied and energy absorption parameters including total energy absorbed per unit volume, energy absorption efficiency and onset of densification strain were calculated. Also, the structures were characterized into bending-dominant and stretch-dominant structures, according to their nodal connectivity and Gibson-and-Ashby's law. The results of the study showed that the dual-lattice is capable of absorbing more energy at each volume fraction cohort. However, gyroid structures showed higher energy absorption efficiency and the onset of densification at higher strains. The spinodoid structure was found to be the poorest structure in terms of energy absorption, specifically at low volume fractions. Also, the results showed that the dual-lattice was a stretch dominated structure, while the gyroid structure was a bending dominated structure, which may be a reason that it is a better candidate for energy absorption applications.

Bayesian group testing with dilution effects

A Bayesian framework for group testing under dilution effects has been developed, using lattice-based models. This work has particular relevance given the pressing public health need to enhance testing capacity for coronavirus disease 2019 and future pandemics, and the need for wide-scale and repeated testing for surveillance under constantly varying conditions. The proposed Bayesian approach allows for dilution effects in group testing and for general test response distributions beyond just binary outcomes. It is shown that even under strong dilution effects, an intuitive group testing selection rule that relies on the model order structure, referred to as the Bayesian halving algorithm, has attractive optimal convergence properties. Analogous look-ahead rules that can reduce the number of stages in classification by selecting several pooled tests at a time are proposed and evaluated as well. Group testing is demonstrated to provide great savings over individual testing in the number of tests needed, even for moderately high prevalence levels. However, there is a trade-off with higher number of testing stages, and increased variability. A web-based calculator is introduced to assist in weighing these factors and to guide decisions on when and how to pool under various conditions. High-performance distributed computing methods have also been implemented for considering larger pool sizes, when savings from group testing can be even more dramatic.

Data registration for multi-method qualification of additive manufactured components

This work refines surface registration methods for metrological datasets to improve the multi-method qualification accuracy of additively manufactured (AM) lattices. Datasets acquired from X-ray computed tomography and a coordinate measurement machine of an AM lattice were aligned using derived geometry datum features based on a theoretical supplemental surface definition, which has been established in recent draft standards, but has had limited examination using complex AM structures. A refined sampling registration approach for lattice geometry based on spatially-dependent subsampling is derived and shown to statistically decrease variation between measurement sources. This importance of well-defined sampling practice and definition is highlighted. The applicability of this approach for multi-method qualification of complex AM parts is discussed. This work lays the foundation of utilizing specifications under consideration in a new standard with possible verification techniques that can be employed.

Lonely Points in Simplices

Given a lattice L ⊆ Z m and a subset A ⊆ R m , we say that a point in A is lonely if it is not equivalent modulo L to another point of A. We are interested in identifying lonely points for specific choices of L when A is a dilated standard simplex, and in conditions on L which ensure that the number of lonely points is unbounded as the simplex dilation goes to infinity.

Programming "Atomic Substitution" in Alloy Colloidal Crystals Using DNA

Although examples of colloidal crystal analogues to metal alloys have been reported, general routes for preparing 3D analogues to random substitutional alloys do not exist. Here, we use the programmability of DNA (length and sequence) to match nanoparticle component sizes, define parent lattice symmetry and substitutional order, and achieve faceted crystal habits. We synthesized substitutional alloy colloidal crystals with either ordered or random arrangements of two components (Au and Fe₃O₄ nanoparticles) within an otherwise identical parent lattice and crystal habit, confirmed via scanning electron microscopy and small-angle X-ray scattering. Energy dispersive X-ray spectroscopy reveals information regarding composition and local order, while the magnetic properties of Fe₃O₄ nanoparticles can direct different structural outcomes for different alloys in an applied magnetic field. This work constitutes a platform for independently defining substitution within multicomponent colloidal crystals, a capability that will expand the scope of functional materials that can be realized through programmable assembly.

Simulated tissue growth in tetragonal lattices with mechanical stiffness tuned for bone tissue engineering

Bone tissue engineering approaches have recently begun considering 3D printed lattices as viable scaffold solutions due to their highly tunable geometries and mechanical efficiency. However, scaffold design remains challenging due to the numerous biological and mechanical trade-offs related to lattice geometry. Here, we investigate novel tetragonal unit cell designs by independently adjusting unit cell height and width to find scaffolds with improved tissue growth while maintaining suitable scaffold mechanical properties for bone tissue engineering. Lattice tissue growth behavior is evaluated using a curvature-based growth model while elastic modulus is evaluated with finite element analysis. Computationally efficient modeling approaches are implemented to facilitate bulk analysis of lattice design trade-offs using design maps for biological and mechanical functionalities in relation to unit cell height and width for two contrasting unit cell topologies. Newly designed tetragonal lattices demonstrate higher tissue growth per unit volume and advantageous stiffness in preferred directions compared to cubically symmetric unit cells. When lattice beam diameter is fixed to 200 μm, Tetra and BC-Tetra lattices with elastic moduli of 200 MPa-400 MPa are compared for squashed, cubic, and stretched topologies. Squashed Tetra lattices demonstrated higher growth rates and growth densities compared to symmetrically cubic lattices. BC-Tetra lattices with the same range of elastic moduli show squashed lattices tend to achieve higher growth rates, whereas stretched lattices promote higher growth density. The results suggest tetragonal unit cells provide favorable properties for biological and mechanical tailoring, therefore enabling new strategies for diverse patient needs and applications in regenerative medicine.

Design procedure for triply periodic minimal surface based biomimetic scaffolds

Cellular additively manufactured metallic structures for load-bearing scaffolds in the context of bone tissue engineering (BTE) have emerged as promising candidates. Due to many advantages in terms of morphology, stiffness, strength and permeability compared to conventional truss structures, lattices based on triply periodic minimal surfaces (TPMS) have recently attracted increasing interest for this purpose. In addition, the finite element method (FEM) has been proven to be suitable for accurately predicting the deformation behavior as well as the mechanical properties of geometric structures after appropriate parameter validation based on experimental data. Numerous publications have examined many individual aspects, but conceptual design procedures that consider at least the essential requirements for cortical and trabecular bone simultaneously are still rare. Therefore, this paper presents a numerical approach to first determine the actual admissible design spaces for a choice of TPMS based lattices with respect to key parameters and then weight them with respect to further benefit parameters. The admissible design spaces are limited by pore size, strut size and volume fraction, and the subsequent weighting is based on Young's modulus, cell size and surface area. Additively manufactured beta-Ti-42Nb with a strain stiffness of 60.5GPa is assumed as material. In total, the procedure considers twelve lattice types, consisting of six different TPMS, each as network solid and as sheet solid. The method is used for concrete prediction of suitable TPMS based lattices for cortical bone and trabecular bone. For cortical bone a lattice based on the Schwarz Primitive sheet solid with 67.572μm pore size, 0.5445 volume fraction and 18.758GPa Young's modulus shows to be the best choice. For trabecular bone a lattice based on the Schoen Gyroid network solid with 401.39μm pore size, 0.3 volume fraction and 4.6835GPa Young's modulus is the identified lattice. Finally, a model for a long bone scaffold is generated from these two lattices using functional grading methods in terms of volume fraction, cell size and TPMS type. In particular, the presented procedure allows an efficient estimation for a likely suitable biometric TPMS-based scaffolds. In addition to medical applications, however, the method can also be transferred to numerous other applications in mechanical, civil and electrical engineering.

Formalizing the LLL Basis Reduction Algorithm and the LLL Factorization Algorithm in Isabelle/HOL

The LLL basis reduction algorithm was the first polynomial-time algorithm to compute a reduced basis of a given lattice, and hence also a short vector in the lattice. It approximates an NP-hard problem where the approximation quality solely depends on the dimension of the lattice, but not the lattice itself. The algorithm has applications in number theory, computer algebra and cryptography. In this paper, we provide an implementation of the LLL algorithm. Both its soundness and its polynomial running-time have been verified using Isabelle/HOL. Our implementation is nearly as fast as an implementation in a commercial computer algebra system, and its efficiency can be further increased by connecting it with fast untrusted lattice reduction algorithms and certifying their output. We additionally integrate one application of LLL, namely a verified factorization algorithm for univariate integer polynomials which runs in polynomial time.

Reducing MRI susceptibility artefacts in implants using additively manufactured porous Ti-6Al-4V structures

Magnetic Resonance Imaging (MRI) is critical in diagnosing post-operative complications following implant surgery and imaging anatomy adjacent to implants. Increasing field strengths and use of gradient-echo sequences have highlighted difficulties from susceptibility artefacts in scan data. Artefacts manifest around metal implants, including those made from titanium alloys, making detection of complications (e.g. bleeding, infection) difficult and hindering imaging of surrounding structures such as the brain or inner ear. Existing research focusses on post-processing and unorthodox scan sequences to better capture data around these devices. This study proposes a complementary up-stream design approach using lightweight structures produced via additive manufacturing (AM). Strategic implant mass reduction presents a potential tool in managing artefacts. Uniform specimens of Ti-6Al-4V structures, including lattices, were produced using the AM process, selective laser melting, with various unit cell designs and relative densities (3.1%-96.7%). Samples, submerged in water, were imaged in a 3T MRI system using clinically relevant sequences. Artefacts were quantified by image analysis revealing a strong linear relationship (RR² = 0.99) between severity and relative sample density. Likewise, distortion due to slice selection errors showed a squared relationship (RR² = 0.92) with sample density. Unique artefact features were identified surrounding honeycomb samples suggesting a complex relationship exists for larger unit cells. To demonstrate clinical utility, a honeycomb design was applied to a representative cranioplasty. Analysis revealed 10% artefact reduction compared to traditional solid material illustrating the feasibility of this approach. This study provides a basis to strategically design implants to reduce MRI artefacts and improve post-operative diagnosis capability. STATEMENT OF SIGNIFICANCE: MRI susceptibility artefacts surrounding metal implants present a clinical challenge for the diagnosis of post-operative complications relating to the implant itself or underlying anatomy. In this study for the first time we demonstrate that additive manufacturing may be exploited to create lattice structures that predictably reduce MRI image artefact severity surrounding titanium alloy implants. Specifically, a direct correlation of artefact severity, both total signal loss and distortion, with the relative material density of these functionalised materials has been demonstrated within clinically relevant MRI sequences. This approach opens the door for strategic implant design, utilising this structurally functionalised material, that may improve post-operative patient outcomes and compliments existing efforts in this area which focus on data acquisition and post-processing methods.

Functionally graded and multi-morphology sheet TPMS lattices: Design, manufacturing, and mechanical properties

Functionally graded and multi-morphology lattices are gaining increased attention recently in the tissue engineering research community because of the ability to control their physical, mechanical and geometrical properties spatially. In this work, relative density grading, cell size grading, and multi-morphology (lattice type grading) are mechanically investigated for sheet-based lattices with topologies based on triply periodic minimal surfaces (TPMS), namely; the Schoen Gyroid, and Schwarz Diamond minimal surfaces. To investigate the role of loading direction on the exhibited deformation mechanism, tests were performed parallel and perpendicular to the grading direction. For relative density grading, testing parallel to grading direction exhibited a layer-by-layer deformation mechanism with a lower Young's Modulus as compared to samples tested perpendicular to grading direction which exhibited a shear band deformation. Moreover, multi-morphology lattices exhibited a shift in deformation mechanism from layer-by-layer to the formation of a shear band at the interface between the different TPMS morphologies when tested parallel and perpendicular to hybridization direction, respectively. FE analysis revealed that sheet-networks multi-morphology lattices exhibit higher elastic properties as compared to solid-networks multi-morphology lattices.

Synthetic bone: Design by additive manufacturing

A broad range of synthetic trabecular-like metallic lattices are 3D printed, to study the extra design freedom conferred by this new manufacturing process. The aim is to propose new conceptual types of implant structures for superior bio-mechanical matching and osseo-integration: synthetic bone. The target designs are 3D printed in Ti-6Al-4V alloy using a laser-bed process. Systematic evaluation is then carried out: (i) their accuracy is characterised at high spatial resolution using computed X-ray tomography, to assess manufacturing robustness with respect to the original geometrical design intent and (ii) the mechanical properties - stiffness and strength - are experimentally measured, evaluated, and compared. Finally, this new knowledge is synthesised in a conceptual framework to allow the construction of so-called implant design maps, to define the processing conditions of bone tailored substitutes, with focus on spine fusion devices. The design criteria emphasise the bone stiffness-matching, preferred range of pore structure for bone in-growth, manufacturability of the device and choice of inherent materials properties which are needed for durable implants. Examples of the use of such maps are given with focus on spine fusion devices, emphasising the stiffness-matching, osseo-integration properties and choice of inherent materials properties which are needed for durable implants. STATEMENT OF SIGNIFICANCE: We present a conceptual bio-engineering design methodology for new biomedical lattices produced by additive manufacturing, which addresses some of the critical points in currently existing porous implant materials. Amongst others: (i) feasibility and accuracy of manufacturing, (ii) design to the elastic properties of bone, and (iii) sensible pores sizes for osseointegration. This has inspired new and novel geometrical latticed designs which aim at improving the properties of intervertebral fusion devices. In their fundamental form, these structures are here fabricated and tested. When integrated into medical devices, these concepts could offer superior medical outcomes.

Stereolithographic 3D Printing-Based Hierarchically Cellular Lattices for High-Performance Quasi-Solid Supercapacitor

3D printing-based supercapacitors have been extensively explored, yet the rigid rheological requirement for corresponding ink preparation significantly limits the manufacturing of true 3D architecture in achieving superior energy storage. We proposed the stereolithographic technique to fabricate the metallic composite lattices with octet-truss arrangement by using electroless plating and engineering the 3D hierarchically porous graphene onto the scaffolds to build the hierarchically cellular lattices in quasi-solid supercapacitor application. The supercapacitor device that is composed of composite lattices span several pore size orders from nm to mm holds promising behavior on the areal capacitance (57.75 mF cm^-2), rate capability (70% retention, 2-40 mA cm^-2), and long lifespan (96% after 5000 cycles), as well as superior energy density of 0.008 mWh cm^-2, which are comparable to the state-of-the-art carbon-based supercapacitor. By synergistically combining this facile stereolithographic 3D printing technology with the hierarchically porous graphene architecture, we provide a novel route of manufacturing energy storage device as well as new insight into building other high-performance functional electronics.

Bounding basis reduction properties

The paper describes improved analysis techniques for basis reduction that allow one to prove strong complexity bounds and reduced basis guarantees for traditional reduction algorithms and some of their variants. This is achieved by a careful exploitation of the linear equations and inequalities relating various bit sizes before and after one or more reduction steps.

Finding shortest lattice vectors faster using quantum search

By applying a quantum search algorithm to various heuristic and provable sieve algorithms from the literature, we obtain improved asymptotic quantum results for solving the shortest vector problem on lattices. With quantum computers we can provably find a shortest vector in time 2 1.799 n + o ( n ) , improving upon the classical time complexities of 2 2.465 n + o ( n ) of Pujol and Stehlé and the 2 2 n + o ( n ) of Micciancio and Voulgaris, while heuristically we expect to find a shortest vector in time 2 0.268 n + o ( n ) , improving upon the classical time complexity of 2 0.298 n + o ( n ) of Laarhoven and De Weger. These quantum complexities will be an important guide for the selection of parameters for post-quantum cryptosystems based on the hardness of the shortest vector problem.

Analysis of the width-[Formula: see text] non-adjacent form in conjunction with hyperelliptic curve cryptography and with lattices

In this work the number of occurrences of a fixed non-zero digit in the width-[Formula: see text] non-adjacent forms of all elements of a lattice in some region (e.g. a ball) is analysed. As bases, expanding endomorphisms with eigenvalues of the same absolute value are allowed. Applications of the main result are on numeral systems with an algebraic integer as base. Those come from efficient scalar multiplication methods (Frobenius-and-add methods) in hyperelliptic curves cryptography, and the result is needed for analysing the running time of such algorithms. The counting result itself is an asymptotic formula, where its main term coincides with the full block length analysis. In its second order term a periodic fluctuation is exhibited. The proof follows Delange's method.