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The invention of the wheel is widely credited as a pivotal moment in human history, yet the details surrounding its discovery are shrouded in mystery. There remains no scholarly consensus on key questions such as where, how and by whom this technology was originally invented. In this study, we employ state-of-the-art techniques from computational structural mechanics to shed light on this long-standing puzzle. Based on this analysis, we propose a probable path along which the wheel evolved via a sequence of three major innovations. We also introduce an original computational design algorithm that autonomously generates a wheel-and-axle system using an evolutionary process that offers insight into the way in which the first wheels likely evolved nearly 6000 years ago. Our analysis provides new supporting evidence for the recently advanced theory that the wheel was invented by Neolithic miners harvesting copper ore from the Carpathian Mountains as early as 3900 BC. Moreover, we show how the discovery of the wheel was made possible by the unique physical features of the mine environment, whose impact was analogous to the selective environmental pressures that drive biological evolution.

We present a novel multiphysics and multimaterial computational design framework for shape-memory alloy-based smart structures. The proposed framework uses topology optimization to optimally distribute multiple material candidates within the design domain, and leverages a nonlinear phenomenological constitutive model for shape-memory alloys (SMAs), along with a coupled transient heat conduction model. In most practical scenarios, SMAs are activated by a nonuniform temperature field or a nonuniform stress field. This framework accurately captures the coupling between the phase transformation process and the evolution of the local temperature field. Thus, the resulting design framework is able to optimally tailor the two-way shape-memory effect and the superelasticity response of SMAs more precisely than previous algorithms that have relied on the assumption of a uniform temperature distribution. We present several case studies, including the design of a self-actuated bending beam and a gripper mechanism. The results show that the proposed framework can successfully produce SMA-based designs that exhibit targeted displacement trajectories and output forces. In addition, we present an example in which we enforce material-specific thermal constraints in a multimaterial design to enhance its thermal performance. In conclusion, the proposed framework provides a systematic computational approach to consider the nonlinear thermomechanical response of SMAs, thereby providing enhanced programmability of the SMA-based structure.

In this work, we introduce a method to incorporate stress considerations in the topology optimization of heterogeneous structures. More specifically, we focus on using functionally graded materials (FGMs) to produce compliant mechanism designs that are not susceptible to failure. Local material properties are achieved through interpolating between material properties of two or more base materials. Taking advantage of this method, we develop relationships between local Young’s modulus and local yield stress, and apply stress criterion within the optimization problem. A solid isotropic material with penalization (SIMP)–based method is applied where topology and local element material properties are optimized simultaneously. Sensitivities are calculated using an adjoint method and derived in detail. Stress formulations implement the von Mises stress criterion, are relaxed in void regions, and are aggregated into a global form using a p-norm function to represent the maximum stress in the structure. For stress-constrained problems, we maintain local stress control by imposing m p-norm constraints on m regions rather than a global constraint. Our method is first verified by solving the stress minimization of an L-bracket problem, and then multiple stress-constrained compliant mechanism problems are presented. Results suggest that good designs can be produced with the proposed method and that heterogeneous designs can outperform their homogeneous counterparts with respect to both mechanical advantage and reduced stress concentrations.

This work presents a new method for efficiently designing loads and supports simultaneously with material distribution in density-based topology optimization. We use a higher-order or super-Gaussian function to parameterize the shapes, locations, and orientations of mechanical loads and supports. With a distance function as an input, the super-Gaussian function projects smooth geometric shapes which can be used to model various types of boundary conditions using minimal numbers of additional design variables. As examples, we use the proposed formulation to model both concentrated and distributed loads and supports. We also model movable non-design regions of predetermined solid shapes using the same distance functions and design variables as the variable boundary conditions. Computing the design sensitivities using the adjoint sensitivity analysis method, we implement the technique in a 2D topology optimization algorithm with linear elasticity and demonstrate the improvements that the super-Gaussian projection method makes to some common benchmark problems. By allowing the optimizer to move the loads and supports throughout the design domain, the method produces significant enhancements to structures such as compliant mechanisms where the locations of the input load and fixed supports have a large effect on the magnitude of the output displacements.

We present a novel optimization framework for optimal design of structures exhibiting memory characteristics by incorporating shape memory polymers (SMPs). SMPs are a class of memory materials capable of undergoing and recovering applied deformations. A finite-element analysis incorporating the additive decomposition of small strain is implemented to analyze and predict temperature-dependent memory characteristics of SMPs. The finite element method consists of a viscoelastic material modelling combined with a temperature-dependent strain storage mechanism, giving SMPs their characteristic property. The thermo-mechanical characteristics of SMPs are exploited to actuate structural deflection to enable morphing toward a target shape. A time-dependent adjoint sensitivity formulation implemented through a recursive algorithm is used to calculate the gradients required for the topology optimization algorithm. Multimaterial topology optimization combined with the thermo-mechanical programming cycle is used to optimally distribute the active and passive SMP materials within the design domain. This allows us to tailor the response of the structures to design them with specific target displacements, by exploiting the difference in the glass-transition temperatures of the two SMP materials. Forward analysis and sensitivity calculations are combined in a PETSc-based optimization framework to enable efficient multi-functional, multimaterial structural design with controlled deformations.

This paper presents a computational framework for multimaterial topology optimization under uncertainty. We combine stochastic collocation with design sensitivity analysis to facilitate robust design optimization. The presence of uncertainty is motivated by the induced scatter in the mechanical properties of candidate materials in the additive manufacturing process. The effective elastic modulus in each finite element is obtained by an interpolation scheme which is parameterized with three distinct elastic moduli corresponding to the available design materials. The parametrization enables the SIMP-style penalization of intermediate material properties, thus ensuring convergence to a discrete manufacturable design. We consider independent random variables for the elastic modulus of different materials and generate designs that minimize the variability in the performance, namely structural compliance. We use a newly developed quadrature rule, designed quadrature , to compute statistical moments with reduced computational cost. We show our approach on numerical benchmark problems of linear elastic continua where we demonstrate the improved performance of robust designs compared with deterministic designs. We provide the MATLAB implementation of our approach.

This research applies topology optimization to create feasible functionally graded compliant mechanism designs with the aim of improving structural performance compared to traditional homogeneous compliant mechanism designs. Converged functionally graded designs will also be compared with two-material compliant mechanism designs. Structural performance is assessed with respect to mechanical/geometric advantage and stress distributions. Two design problems are presented – a gripper and a mechanical inverter. A novel modified solid isotropic material with penalization (SIMP) method is introduced for representing local element material properties in functionally graded structures. The method of moving asymptotes (MMA) is used in conjunction with adjoint sensitivity analysis to find the optimal distribution of material properties. Geometric non-linear analysis is used to solve the mechanics problem based on the Neo-Hookean model for hyperelastic materials. Functionally graded materials (FGMs) have material properties that vary based on spatial position. Here, FGMs are implemented using two different resource constraints – one on the mechanism’s volume and the other on the integral of the Young’s modulus distribution throughout the design domain. Tensile tests are performed to obtain the material properties used in the analysis. Results suggest that FGMs can achieve the desired improvements in mechanical/geometric advantage when compared to both homogeneous and two-material mechanisms.

Topology optimization (TO) is commonly applied to design the unit cells of periodic structures. For example, metamaterials, lattice structures, phononic crystals (PhC), and photonic crystals (PC) have all been previously designed via TO. Unfortunately, the optimal structures for certain design objectives, e.g., bandgaps, are often impossible to manufacture as they have disconnected regions or “islands” of solid material (ISM) that are not self-supporting. Further, designs with enclosed void space (EVS) are problematic for additive manufacturing (AM) since support material or pre-sintered powder cannot be removed after manufacturing. We present a series of constraints that may be incorporated into any TO framework to ensure structures are self-supporting without enclosed voids. Additionally, we employ homogenization-based constraints that allow the designer to tune the elastic stiffness and isotropy of the optimized design. The proposed constraints are evaluated on example microstructures and utilized in a simple optimization test problem to highlight their abilities and limitations so that guidelines for appropriate combinations of constraints may be proposed. Effective constraint combinations are demonstrated on the design of 3D photonic crystals for maximum bandgap subject to manufacturing and stiffness constraints.

The purpose of this paper is to apply stress constraints to structural topology optimization problems with design-dependent loading. A comparison of mass-constrained compliance minimization solutions and stress-constrained mass minimization solutions is also provided. Although design-dependent loading has been the subject of previous research, only compliance minimization has been studied. Stress-constrained mass minimization problems are solved in this paper, and the results are compared with those of compliance minimization problems for the same geometries and loading. A stress-relaxation technique is used to avoid the singularity in the stress constraints, and these constraints are aggregated in blocks to reduce the total number of constraints in the optimization problem. The results show that these design-dependent loading problems may converge to a local minimum when the stress constraints are enforced. The use of a continuation method where the stress-constraint aggregation parameter is gradually increased typically leads to better convergence; however, this may not always be possible. The results also show that the topologies of compliance-minimization and stress-constrained solutions are usually vastly different, and the sizing optimization of a compliance solution may not lead to an optimum.

We present a novel finite element analysis of inelastic structures containing Shape Memory Alloys (SMAs). Phenomenological constitutive models for SMAs lead to material nonlinearities, that require substantial computational effort to resolve. Finite element analysis methods, which rely on Gauss quadrature integration schemes, must solve two sets of coupled differential equations: one at the global level and the other at the local, i.e. Gauss point level. In contrast to the conventional return mapping algorithm, which solves these two sets of coupled differential equations separately using a nested Newton procedure, we propose a scheme to solve the local and global differential equations simultaneously. In the process we also derive closed-form expressions used to update the internal/constitutive state variables, and unify the popular closest-point and cutting plane methods with our formulas. Numerical testing indicates that our method allows for larger thermomechanical loading steps and provides increased computational efficiency, over the standard return mapping algorithm.