Parameter-free shape optimization, i.e., parameterizing the nodal coordinates of a finite element discretization, offers many advantages over traditional shape optimization techniques, e.g., increased design freedom and ease of implementation. A disadvantage of this method is the difficulty of imposing geometric design requirements. Here, we propose differentiable constraints used to enforce minimum feature size, preserve required CAD-like features, and reduce the dimensionality of the optimized design, e.g., to 2 1/2 dimensions. This paper also considers practical improvements to our recent parameter-free optimization algorithm. Shape optimization problems are solved to demonstrate various combinations of design constraints, gain insight into appropriate problem parameter choices, and demonstrate scalability of the methods.

This paper presents a level-set topology optimization approach that uses conformal meshes for the analysis of the displacement field. The structure’s boundary is represented by the iso-contour of a level-set field discretized on a fixed background design mesh. The conformal mesh is updated for each design iteration via a PDE based mesh morphing process that identifies the set of facets in the background mesh that are homeomorphic to the boundary and relaxes the homeomorphic mesh to conform to the structure’s boundary and ensure high element quality. The conformal mesh allows for a more accurate computation of the response versus density and some level-set based methods which interpolate material properties using the volume fraction. Numerical examples illustrate the proposed approach by optimizing linear-elastic two- and three-dimensional structures, wherein insight into the performance of the mesh morphing process is provided. The examples also highlight the scalability of the approach.

Liquid crystal elastomers (LCEs) are responsive materials that can undergo large reversible deformations upon exposure to external stimuli, such as electrical and thermal fields. Controlling the alignment of their liquid crystals mesogens to achieve desired shape changes unlocks a new design paradigm that is unavailable when using traditional materials. While experimental measurements can provide valuable insights into their behavior, computational analysis is essential to exploit their full potential. Accurate simulation is not, however, the end goal; rather, it is the means to achieve their optimal design. Such design optimization problems are best solved with algorithms that require gradients, i.e., sensitivities, of the cost and constraint functions with respect to the design parameters, to efficiently traverse the design space. In this work, a nonlinear LCE model and adjoint sensitivity analysis are implemented in a scalable and flexible finite element-based open source framework and integrated into a gradient-based design optimization tool. To display the versatility of the computational framework, LCE design problems that optimize both the material, i.e., liquid crystal orientation, and structural shape to reach a target actuated shapes or maximize energy absorption are solved. Multiple parameterizations, customized to address fabrication limitations, are investigated in both 2D and 3D. The case studies are followed by a discussion on the simulation and design optimization hurdles, as well as potential avenues for improving the robustness of similar computational frameworks for applications of interest.

The use of node coordinates as design variables in shape optimization offers a larger design space than computer-aided design (CAD)-based shape parameterizations. It also allows for the optimization of legacy designs, i.e., a finite element mesh from an existing design can be readily optimized to meet new performance requirements without involving a CAD model. However, it is well known that the node coordinate parameterization method is fraught with numerical difficulties, which makes it impractical to use. This has led to several of “parameter-free” shape optimization methods that seek the advantages and avoid the pitfalls of the naïve node coordinate parameterization method. These methods come in two main varieties: sensitivity filtering (or gradient smoothing) and consistent filtering. The latter is analogous to the density filter method used in topology optimization (TO). Herein, we use the PDE filter from TO and energy-based filters to implement consistent shape optimization filtering schemes easily and efficiently. Numerical experiments demonstrate that consistent methods are more robust than sensitivity filtering methods.

Due to advances in modern manufacturing, there is an increased need for accurate and efficient simulation capability for microarchitected cellular structures. It is quite expensive to simulate a large cellular structure using a fully resolved finite element method, even on a modern supercomputer. A new method is proposed that is of intermediate complexity, it is more efficient than a fully resolved finite element simulation, but more general than other approximations such as beam theory or homogenization. The proposed method is based on reduced order modeling and is compatible with any finite element simulation code that supports higher-order basis functions.

designing and analyzing photonic crystals, and a periodic density filter is implemented to impose a minimum feature size for manufacturability considerations.

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.

Machine learning models can be used to predict physical quantities like homogenized elasticity stiffness tensors, which must always be symmetric positive definite (SPD) based on conservation arguments. Two datasets of homogenized elasticity tensors of lattice materials are presented as examples, where it is desired to obtain models that map unit cell geometric and material parameters to their homogenized stiffness. Fitting a model to SPD data does not guarantee the model's predictions will remain SPD. Existing Cholsesky factorization and Eigendecomposition schemes are abstracted in this work as transformation layers which enforce the SPD condition. These layers can be included in many popular machine learning models to enforce SPD behavior. This work investigates the effects that different positivity functions have on the layers and how their inclusion affects model accuracy. Commonly used models are considered, including polynomials, radial basis functions, and neural networks. Ultimately it is shown that a single SPD layer improves the model's average prediction accuracy.

The design of photonic crystals with complete bandgaps has recently received considerable research focus for numerous reasons. This work leverages well-known nonlinear programming techniques to alleviate the non-smoothness caused by degenerate eigenvalues such that topology optimization problems can be solved with the open-source IPOPT software. A fully-vectorial plane wave expansion technique is used with an iterative eigensolver to efficiently predict dispersion properties of candidate structures. Nonlinear programming is employed to solve the inverse problem of designing three-dimensional periodic structures that exhibit complete two-dimensional (2D) and three-dimensional (3D) photonic bandgaps. Mesh refinement is performed to alleviate the large computational burden of designing and analyzing photonic crystals, and a periodic density filter is implemented to impose a minimum feature size for manufacturability considerations.

Using three design fields we develop an optimization environment that can simultaneously optimize material, shape and topology. We use the implicit representation of the boundaries with level-set functions that define the shape and topology. Differentiable R-functions allow us to combine these shapes and topology descriptions with Boolean operations. Additionally, we incorporate design dependent-stiffness materials with another design field. Notably, this framework accommodates design dependent loads, has the ability to introduce holes, and ensures the satisfaction of optimality criteria. It builds upon the fictitious domain, ersatz material, material interpolation and level-set methods. It also borrows from parameterized density-based topology optimization methods. Since analytical sensitivities can be computed, we use efficient nonlinear programming algorithms to update the design instead of the Hamilton–Jacobi’s scheme of level-set methods. We illustrate the features of our framework by designing a cantilever beam with octet truss microlattice, a dam with design-dependent loads, and a composite clevis plate.

Fatigue crack growth in Ti-6Al-4V has been explored through the combined use of high-energy x-ray diffraction (HEXD) analysis, electron backscatter diffraction analysis, and fractography to draw a connection between the driving forces and micromechanisms of fatigue crack growth. Samples were prepared from a forging with two configurations, viz. with the crack plane either perpendicular or parallel to the primary extension direction of the forging. Crack growth was examined in situ by mapping the lattice strain fields using HEXD for loading at R = 0.1 and R = 0.7. The crack growth as assessed by HEXD was contrasted with predictions of linear elastic fracture mechanics. The variation in the lattice strain field with crystallographic orientation was assessed.At the microscale, a clear correlation between the availability of c-axes along the loading direction and facet formation was found.

Topology optimization is extended to the design of multi-body mechanisms. A Gaussian function is used to parameterize the location of inter-body connections, thereby enabling the optimization of both mass distribution and inter-body connectivity simultaneously. The potential for large rigid-body rotation necessitates the use of a geometrically nonlinear finite element analysis to properly model mechanism response. The unknown displacement field is calculated with a Newton–Raphson iterative scheme. An adjoint analysis is performed to efficiently compute the design sensitivities used in the gradient-based optimization procedure. The proposed technique is demonstrated on the design of multi-body grippers and force/displacement inverters.

A reduced-order model (ROM) of mould heat transfer in the continuous casting of steel is presented and demonstrated for several commercial mould geometries. The ROM relies on a physically-based solution of the one-dimensional heat-conduction equation. The up-front cost of the ROM is a single three-dimensional finite-element calculation of a small representative portion of the exact mould geometry, which is used to calibrate the geometric parameters of the ROM. Specifically, the ROM exactly matches the average hot and cold surface temperatures of the mould as computed in the snapshot model. Other features of the ROM, namely predictions of the cooling water temperature change and thermocouple temperatures, are derived in a manner consistent with the one-dimensional solution. Combined with models of solidification and mould-metal interfacial phenomena, this accurate and efficient modelling tool can be applied to enable deeper insight into many different aspects of heat transfer and related phenomena in the continuous casting process. (C) 2016 Elsevier Inc. All rights reserved.

This article presents a threedimensional, transient, multiphase, turbulent numerical model intended to predict slag entrainment in metallurgical systems such as continuous casting. Experiments report the critical angular velocity of the cylinder at which oil entrainment starts to occur, and the model reasonably agrees. Mold slag entrainment, i.e., liquid mold powder being drawn into the melt, is a challenge in the production of clean steel. The literature identifies nine mechanisms1,2 of mold slag entrainment, shown in Figure 1:

This article presents a three-dimensional, transient, multiphase, turbulent numerical model of slag entrainment in continuous casting molds. The model uses explicit time marching, the volume-of-fluid method with the geometric-reconstruction scheme for multiphase phenomenona, and the k-w turbulence model. The numerical model is verified with analytical solutions, and then is validated with experiments of a rotating cylinder submerged in a tank of oil and water. These experiments report the critical angular velocity of the cylinder at which oil entrainment starts to occur, and the model reasonably agrees.