Checkerboard patterns in layout optimization

Two-Phase Approach for Fast Topology Optimization of Multi-Resonant MEMS Involving Model Order Reduction

In this work, we propose a two-phase approach for a fast topology optimization of multi-resonant MEMSs. The approach minimizes the computation effort required to achieve an optimal design. In the first step, we perform a pre-optimization using bi-directional evolutionary structural optimization (BESO). We found in previous research that BESO can achieve optimal MEMS designs in a significantly lower number of iterations when compared to classical density-based methods. However, we encountered convergence issues with BESO towards the end of the optimization. Therefore, we introduced a second, density-based optimization phase to circumvent this issue. Finally, we introduced model order reduction to reduce the optimization time further. The novel approach is benchmarked with the design task of two common multi-resonant MEMS devices: a linear gyroscope and a micromirror. We show that the two-phase approach can achieve an optimal design within 200 iterations. With the addition of MOR, the computation of the goal function can be further reduced by 50% in our examples.

Computational design of mechanical metamaterials

In the past few years, design of mechanical metamaterials has been empowered by computational tools that have allowed the community to overcome limitations of human intuition. By leveraging efficient optimization algorithms and computational physics models, it is now possible to explore vast design spaces, achieving new material functionalities with unprecedented performance. Here, we present our viewpoint on the state of the art of computational metamaterials design, discussing recent advances in topology optimization and machine learning design with respect to challenges in additive manufacturing.

Multi-Objective Topology Optimization of a Broadband Piezoelectric Energy Harvester

In recent years, topology optimization has proved itself to be state of the art in the design of mechanical structures. At the same time, energy harvesting has gained a lot of attention in research and industry. In this work, we present a novel topology optimization of a multi-resonant piezoelectric energy-harvester device. The goal is to develop a broadband design that can generate constant power output over a range of frequencies, thus enabling reliable operation under changing environmental conditions. To achieve this goal, topology optimization is implemented with a combined-objective function, which tackles both the frequency requirement and the power-output characteristic. The optimization suggests a promising design, with satisfactory frequency characteristics.

An Artificial Intelligence–Assisted Design Method for Topology Optimization without Pre-Optimized Training Data

Engineers widely use topology optimization during the initial process of product development to obtain a first possible geometry design. The state-of-the-art method is iterative calculation, which requires both time and computational power. This paper proposes an AI-assisted design method for topology optimization, which does not require any optimized data. An artificial neural network—the predictor—provides the designs on the basis of boundary conditions and degree of filling as input data. In the training phase, the so-called evaluators evaluate the generated geometries on the basis of random input data with respect to given criteria. The results of those evaluations flow into an objective function, which is minimized by adapting the predictor’s parameters. After training, the presented AI-assisted design procedure generates geometries that are similar to those of conventional topology optimizers, but require only a fraction of the computational effort. We believe that our work could be a clue for AI-based methods that require data that are difficult to compute or unavailable.

Enhancing Design for Additive Manufacturing Workflow: Optimization, Design and Simulation Tools

In the last few decades, complex light-weight designs have been successfully produced via additive manufacturing (AM), launching a new era in the thinking–design process. In addition, current software platforms provide design tools combined with multi-scale simulations to exploit all the technology benefits. However, the literature highlights that several stages must be considered in the design for additive manufacturing (DfAM) process, and therefore, performing holistic guided-design frameworks become crucial to efficiently manage the process. In this frame, this paper aims at providing the main optimization, design, and simulation tools to minimize the number of design evaluations generated through the different workflow assessments. Furthermore, DfAM phases are described focusing on the implementation of design optimization strategies as topology optimization, lattice infill optimization, and generative design in earlier phases to maximize AM capabilities. In conclusion, the current challenges for the implementation of the workflow are hence described.

A Sequential Approach for Aerodynamic Shape Optimization with Topology Optimization of Airfoils

The objective of this work is to study the coupling of two efficient optimization techniques, Aerodynamic Shape Optimization (ASO) and Topology Optimization (TO), in 2D airfoils. To achieve such goal two open-source codes, SU2 and Calculix, are employed for ASO and TO, respectively, using the Sequential Least SQuares Programming (SLSQP) and the Bi-directional Evolutionary Structural Optimization (BESO) algorithms; the latter is well-known for allowing the addition of material in the TO which constitutes, as far as our knowledge, a novelty for this kind of application. These codes are linked by means of a script capable of reading the geometry and pressure distribution obtained from the ASO and defining the boundary conditions to be applied in the TO. The Free-Form Deformation technique is chosen for the definition of the design variables to be used in the ASO, while the densities of the inner elements are defined as design variables of the TO. As a test case, a widely used benchmark transonic airfoil, the RAE2822, is chosen here with an internal geometric constraint to simulate the wing-box of a transonic wing. First, the two optimization procedures are tested separately to gain insight and then are run in a sequential way for two test cases with available experimental data: (i) Mach 0.729 at α=2.31°; and (ii) Mach 0.730 at α=2.79°. In the ASO problem, the lift is fixed and the drag is minimized; while in the TO problem, compliance minimization is set as the objective for a prescribed volume fraction. Improvements in both aerodynamic and structural performance are found, as expected: the ASO reduced the total pressure on the airfoil surface in order to minimize drag, which resulted in lower stress values experienced by the structure.

Sensitivity Analysis of Key Formulations of Topology Optimization on an Example of Cantilever Bending Beam

Topology optimization is a modern method for optimizing the material distribution in a given space, automatically searching for the ideal design of the product. The method aims to maximize the design performance of the system regarding given conditions. In engineering practice, a given space is first described using the finite element method and, subsequently, density-based method with solid isotropic material with penalty. Then, the final shape is found using a gradient-based method, such as the optimality criteria algorithm. However, obtaining the ideal shape is highly dependent on the correct setting of numerical parameters. This paper focuses on the sensitivity analysis of key formulations of topology optimization using the implementation of mathematical programming techniques in MATLAB software. For the purposes of the study, sensitivity analysis of a simple spatial task—cantilever bending—is performed. This paper aims to present the formulations of the optimization problem—in this case, minimization of compliance. It should be noted that this paper does not present any new mathematical formulas but rather provides an introduction into the mathematical theory (including filtering methods and calculating large-size problems using the symmetry of matrices) as well as a step-by step guideline for the minimization of compliance within the density-based topology optimization and search for an optimal shape. The results can be used for complex commercial applications produced by traditional manufacturing processes or by additive manufacturing methods.

Analysis of the uniqueness and stability of solutions to problems regarding the bone-remodeling process

Simulation of the bone remodeling process is extremely important because it makes possible the structure forecast of one or several bones when anomalous situations, such as prosthesis installation, occur. Thus, it is necessary that the mathematical model to simulate the bone remodeling process be reliable; that is, the numerical solution must be stable regardless of initial density field for a phenomenological approach to model the process. For several models found in the literature, this characteristic of stability is not observed, largely due to the discontinuities present in the property values of the models (e.g., Young's modulus and Poisson's ratio). In addition, checkerboard formation and the lazy zone prevent the uniqueness of the solution. To correct these difficulties, this study proposes a set of modifications to guarantee the uniqueness and stability of the solutions, when a phenomenological approach is used. The proposed modifications are: (a) change the rate of remodeling curve in the lazy zone region and (b) create transition functions to guarantee the continuity of the expressions used to describe Young's modulus and Poisson's ratio. Moreover, the stress smoothing process controls the checkerboard formation. Numerical analysis is used to simulate the solution behavior from each proposed modification. The results show that, when all proposed modifications are applied to the three-dimensional models simulated here, it is possible to observe the tendency toward a unique solution.

Enforcing a Force–Displacement Curve of a Nonlinear Structure Using Topology Optimization with Slope Constraints

The objective of this study is to develop an optimization methodology to find a layout that traces a prescribed force–displacement curve through a topology optimization approach. To this end, we propose an objective function to minimize the difference between a prescribed force–displacement curve and the curve calculated at each iteration of the optimization process. Slope constraints are introduced to solve issues encountered when using a small number of target points. In addition, a projection filter is employed to suppress the gray region observed between the solid and void regions, which generally occurs when using a density-based filter. A recently proposed energy interpolation scheme is implemented to stabilize the instability in the nonlinear analysis, which generally results from excessive distortion in the void region when the structure is modeled on a fixed mesh in the topology optimization process. To validate the outlined methodology, several case studies with different types of nonlinearity and structural features of the obtained layouts are investigated.

Exploring conditions that make cortical bone geometry optimal for physiological loading

While physiological loading on lower long bones changes during bone development, the bone cross section either remains circular or slowly changes from nearly circular to other shapes such as oval and roughly triangular. Bone is said to be an optimal structure, where strength is maximized using the optimal distribution of bone mass (also called Wolff's law). One of the most appropriate mathematical validations of this law would be a structural optimization-based formulation where total strain energy is minimized against a mass and a space constraint. Assuming that the change in cross section during bone development and homeostasis after adulthood is direct result of the change in physiological loading, this work investigates what optimization problem formulation (collectively, design variables, objective function, constraints, loading conditions, etc.) results in mathematically optimal solutions that resemble bones under actual physiological loading. For this purpose, an advanced structural optimization-based computational model for cortical bone development and defect repair is presented. In the optimization problem, overall bone stiffness is maximized first against a mass constraint, and then also against a polar first moment of area constraint that simultaneously constrains both mass and space. The investigation is completed in two stages. The first stage is developmental stage when physiological loading on lower long bones (tibia) is a random combination of axial, bending and torsion. The topology optimization applied to this case with the area moment constraint results into circular and elliptical cross sections similar to that found in growing mouse or human. The second investigation stage is bone homeostasis reached in adulthood when the physiological loading has a fixed pattern. A drill hole defect is applied to the adult mouse bone, which would disrupt the homeostasis. The optimization applied after the defect interestingly brings the damaged section back to the original intact geometry. The results, however, show that cortical bone geometry is optimal for the physiological loading only when there is also a constraint on polar moment of area. Further numerical experiments show that application of torsion along with the gait-analysis-based physiological loading improves the results, which seems to indicate that the cortical bone geometry is optimal for some amount of torsion in addition to the gait-based physiological loading. This work has a potential to be extended to bone growth/development models and fracture healing models, where topology optimization and polar moment of area constraint have not been introduced earlier.

A three-dimensional topology optimization model for tooth-root morphology

To obtain the root of a lower incisor through structural optimization, we used two methods: optimization with Solid Isotropic Material with Penalization (SIMP) and Soft-Kill Option (SKO). The optimization was carried out in combination with a finite element analysis in Abaqus/Standard. The model geometry was based on cone-beam tomography scans of 10 adult males with healthy bone-tooth interface. Our results demonstrate that the optimization method using SIMP for minimum compliance could not adequately predict the actual root shape. The SKO method, however, provided optimization results that were comparable to the natural root form and is therefore suitable to set up the basic topology of a dental root.

Infill Optimization for Additive Manufacturing-Approaching Bone-Like Porous Structures

Porous structures such as trabecular bone are widely seen in nature. These structures are lightweight and exhibit strong mechanical properties. In this paper, we present a method to generate bone-like porous structures as lightweight infill for additive manufacturing. Our method builds upon and extends voxel-wise topology optimization. In particular, for the purpose of generating sparse yet stable structures distributed in the interior of a given shape, we propose upper bounds on the localized material volume in the proximity of each voxel in the design domain. We then aggregate the local per-voxel constraints by their p-norm into an equivalent global constraint, in order to facilitate an efficient optimization process. Implemented on a high-resolution topology optimization framework, our results demonstrate mechanically optimized, detailed porous structures which mimic those found in nature. We further show variants of the optimized structures subject to different design specifications, and we analyze the optimality and robustness of the obtained structures.

Design of complex bone internal structure using topology optimization with perimeter control

Large facial bone loss usually requires patient-specific bone implants to restore the structural integrity and functionality that also affects the appearance of each patient. Titanium alloys (e.g., Ti-6Al-4V) are typically used in the interfacial porous coatings between the implant and the surrounding bone to promote stability. There exists a property mismatch between the two that in general leads to complications such as stress-shielding. This biomechanical discrepancy is a hurdle in the design of bone replacements. To alleviate the mismatch, the internal structure of the bone replacements should match that of the bone. Topology optimization has proven to be a good technique for designing bone replacements. However, the complex internal structure of the bone is difficult to mimic using conventional topology optimization methods without additional restrictions. In this work, the complex bone internal structure is recovered using a perimeter control based topology optimization approach. By restricting the solution space by means of the perimeter, the intricate design complexity of bones can be achieved. Three different bone regions with well-known physiological loadings are selected to illustrate the method. Additionally, we found that the target perimeter value and the pattern of the initial distribution play a vital role in obtaining the natural curvatures in the bone internal structures as well as avoiding excessive island patterns.

Analogy of strain energy density based bone-remodeling algorithm and structural topology optimization

In bone-remodeling studies, it is believed that the morphology of bone is affected by its internal mechanical loads. From the 1970s, high computing power enabled quantitative studies in the simulation of bone remodeling or bone adaptation. Among them, Huiskes et al. (1987, "Adaptive Bone Remodeling Theory Applied to Prosthetic Design Analysis," J. Biomech. Eng., 20, pp. 1135-1150) proposed a strain energy density based approach to bone remodeling and used the apparent density for the characterization of internal bone morphology. The fundamental idea was that bone density would increase when strain (or strain energy density) is higher than a certain value and bone resorption would occur when the strain (or strain energy density) quantities are lower than the threshold. Several advanced algorithms were developed based on these studies in an attempt to more accurately simulate physiological bone-remodeling processes. As another approach, topology optimization originally devised in structural optimization has been also used in the computational simulation of the bone-remodeling process. The topology optimization method systematically and iteratively distributes material in a design domain, determining an optimal structure that minimizes an objective function. In this paper, we compared two seemingly different approaches in different fields-the strain energy density based bone-remodeling algorithm (biomechanical approach) and the compliance based structural topology optimization method (mechanical approach)-in terms of mathematical formulations, numerical difficulties, and behavior of their numerical solutions. Two numerical case studies were conducted to demonstrate their similarity and difference, and then the solution convergences were discussed quantitatively.

Comparison of two numerical approaches for bone remodelling

This paper addresses the "checkerboard" phenomenon, which occurs in numerical simulation of bone remodelling. It attempts to answer the question: is an element-based approach suitable for bone remodelling? Two different numerical approaches, the element-based and the node-based finite element analyses, are implemented using ABAQUS. A comparison of the numerical results demonstrates that the checkerboard phenomenon occurs only in the element-based finite element analyses; the node-based approach eradicates the checkerboard phenomenon but requires much more computational time. This study shows that it is essential to enforce the continuity of bone density across the element boundaries. As the node-based approach requires much more computational time, the first-order Adams-Bashforth integration method is introduced to reduce computational cost. The comparisons with Euler's forward method demonstrate that the first-order Adams-Bashforth method indeed enhances accuracy and reduces computational cost. This study concludes that the node-based approach with the first-order Adams-Bashforth integration scheme is to be recommended for computational bone remodelling studies.

Surface remodeling of trabecular bone using a tissue level model

We propose a two-dimensional mathematical model of trabecular bone remodeling that simulates the surface-based addition and removal of material in the actual physiological process. The model is based on a finite element representation of individual trabecular struts in which the material properties of the subtrabecular elements are constant. The remodeling stimulus is strain energy density, sensed and communicated through the osteocytic network as proposed by Mullender et al. We propose a modified osteocyte communication scheme that incorporates bone-lining cells and examines the implications of set point locations in one or the other of these two cell types. This model produces trabecular struts that align with its general loading direction. Placing the set point in the bone-lining cells rather than in the osteocytes makes the model more sensitive to changes in the other biological parameters. Introduction of a dead zone causes the model to reach a less oscillatory equilibrium in fewer iterations and produces better in-filling of trabecular strut intersections. The model gravitates to equilibrium states in which the average strain energy density is inversely proportional to the bone volume fraction to the 3.2 power.