Topology Optimization Formulation Research Articles

Porous structures inspired by porcupine quill: multiscale design optimization approach

This paper presents a novel approach for designing a freeform bending-resistant structure from the combination of explicit discrete component-based topology optimization (TO) and the porcupine quill-inspired features. To embed the porcupine quill’s features into the TO formulations, the method involves constructing discrete components at various scales to imitate features including solid shell, stochastically distributed pores, and graded stiffeners. The components are iteratively updated, and the optimization process allows for the grading of quill-inspired features while achieving optimal structural compliance under bending loads. The proposed approach is demonstrated to be effective through the resolution of Messershmitt–Bolkow–Blohm (MBB) beam designs, parameterized studies of geometric parameters, and numerical validation of long-span and short-span quill-inspired beam designs. By examining the von Mises stress distribution, the study highlights the mitigation of material yielding at the shell region brought by the geometric features of porcupine quills, leading to the potential theory support for the bending resistance. The optimized MBB beams are manufactured using the material extrusion technique, and three-point bending tests are conducted to explore the failure mitigation capability of the quill-inspired beam under large deformation. Consequently, the study concludes that the proposed quill-inspired component-based TO approach can design a structure with excellent bending resistance according to the improved energy absorption as well as increased deformation after reaching 75% peak load.

Isogeometric topology optimization of structures using the overweight approach

In this paper, a 2D isogeometric formulation of the material distribution for structural topology optimization considering minimum weight and local stress constraints using the overweight approach is proposed. The aim of this isogeometric formulation is to provide solutions with high spatial definition using a lower number of design variables in comparison with the formulations previously developed to define the material layout. Despite of this, an important number of local stress constraints has to be considered in the solution of the problem. For this purpose, an Overweight Constraint is used to consider all of them. The structural analysis is performed by means of the Isogeometric Analysis (IGA) and the distribution of material is modeled by means of quadratic B-splines. Moreover, the optimization is addressed by means of the Sequential Linear Programming algorithm (SLP) that is driven by the information provided by a full first-order sensitivity analysis extension of the IGA formulation. Finally, the proposed formulation is tested by means of some benchmark problems, and the results show that the isogeometric formulation provides solutions with high spatial definition. A comparison with a Finite Element Method (FEM) topology optimization formulation is included.

Connectivity Constraints Ensuring Continuous Electrodes in Topology Optimization of Electroactive Polymer

Abstract Electroactive polymers (EAPs) deform when subject to an electric field, which is generated by two or more electrodes. To ensure proper function of the EAP, these electrodes are connected to a source and they are therefore required to be continuous such that no isolated islands exist. Increasing an EAP’s performance using topology optimization while ensuring electrode connectivity is the goal of this work. A topology optimization formulation is introduced where electrode connectivity is ensured using the virtual temperature method. Numerical experiments demonstrate that this is an efficient method to guarantee connectivity.

Topology optimization for flow machine rotor design considering resonance and low mass density flows

When designing structures that take part in rotating fluid flow machines, one important effect that may be taken into account is resonance, which is evaluated through the modal analysis. Under rotating conditions, the formulation and behavior of the structure changes, requiring a new formulation. Thus, this work formulates the fluid flow topology optimization for rotating resonance, while also considering a low mass density (compressible or incompressible) fluid flow. Therefore, three different effects are considered in the fluid flow-based design: from the modal analysis, from the solid structure, and from the low mass density fluid flow. The topology optimization problem is formulated for a simplified 2D model of a rotating fluid flow device, and a nodal design variable. The topology optimization is performed through an integer linear programming algorithm. Numerical examples are presented for the developed topology optimization formulation.

Topology Optimization of Shape Memory Alloy Actuators for Prescribed Two-Way Transforming Shapes

This paper proposes a new topology optimization formulation for obtaining shape memory alloy actuators which are designed with prescribed two-way transforming shapes. The actuation behaviors of shape memory alloy structures are governed by austenite-martensite phase transformations effected by thermal-mechanical loading processes; therefore, to realize the precise geometric shape variations of shape memory alloy actuators, traditional methods involve iteration processes including heuristic structural design, numerical predictions and experimental validation. Although advanced structural optimization methods such as topology optimization have been used to design three-dimensional (3D) shape memory alloy actuators, the maximization/minimization of quantities such as structural compliance or inaccurate stroke distances has usually been selected as the optimization objective to obtain feasible solutions. To bridge the gap between precise shape-morphing requirements and efficient shape memory alloy actuator designs, this paper formulates optimization criteria with quantitatively desired geometric shapes, and investigates the automatic designs of two-way prescribed shape morphing shape memory alloy structures based on the proposed topology optimization method. The super element method and adjoint method are used to derive the analytical sensitivities of the objective functions with respect to the design variables. Numerical examples demonstrate that the proposed method can obtain 3D actuator designs that have the desired two-way transforming shapes.

A compatible boundary condition-based topology optimization paradigm for static mechanical cloak design

We propose a novel compatible boundary condition-based topology optimization formulation for realizing static mechanical cloaking of linear elastic structures. The key idea is to make the actual boundary conditions along the exterior of the cloaking region consistent with the reference boundary conditions, including the displacement and the nodal force, by designing the cloak structure solely. This compatible boundary condition-based optimization paradigm is not only applicable to various shaped voids (e.g., circle, ellipse, star, Mickey, square, “DUT” logo) and diverse loading conditions (e.g., pressure-sliding, biaxial-stretching, linear-stretching, random-stretching, stretching-sliding) but also demonstrates insensitivity to element quantities and slight variations of external load, ensuring consistent and robust elastostatic cloaking performance. Additionally, adopting the Moving Morphable Voids (MMV) method guarantees the optimized structures have the merits of explicit geometric boundaries and easy fabrication. Numerical examples and experimental validation demonstrate the effectiveness of the proposed approach. The present approach can be easily extended for cloak design in three dimensions and multiphysics.

Adaptive fail-safe topology optimization using a hierarchical parallelization scheme

This work presents an efficient, flexible, and scalable strategy to implement density-based topology optimization formulation for fail-safe structural design. Such optimized designs can operate after faults modeled as loss of stiffness. However, the need for assessing the physical behavior in all the fault cases can make this problem unfeasible computationally. We combine many ingredients to exploit the different levels of parallelism of the formulation to deal successfully with this problem. We use a non-overlapping domain decomposition method to solve the failure cases using multi-core computing with distributed memory computation. These subdomains use adaptive mesh refinement (AMR) techniques to focus computing efforts on the regions of interest. A key point to evaluate the fault cases is the organization of the computing threads, especially in clusters of computers. We group the computing threads by physical computing nodes to reduce the inter-node communications, maximizing intra-node communications to mitigate bandwidth problems. Another ingredient of paramount importance is the use of computing buffers to adapt the evaluation of the fault cases to the computing resources. We test the scalability of the computational framework using a large cluster of computers.

Topology optimization of stability‐constrained structures with simple/multiple eigenvalues

AbstractThis work focuses on topology optimization formulations with linear buckling constraints wherein eigenvalues of arbitrary multiplicities can be canonically considered. The non‐differentiability of multiple eigenvalues is addressed by a mean value function which is a symmetric polynomial of the repeated eigenvalues in each cluster. This construction offers accurate control over each cluster of eigenvalues as compared to the aggregation functions such as ‐norm and Kreisselmeier–Steinhauser (K–S) function where only approximate maximum/minimum value is available. This also avoids the two‐loop optimization procedure required by the use of directional derivatives (Seyranian et al. Struct Optim. 1994;8(4):207‐227.). The spurious buckling modes issue is handled by two approaches—one with different interpolations on the initial stiffness and geometric stiffness and another with a pseudo‐mass matrix. Using the pseudo‐mass matrix, two new optimization formulations are proposed for incorporating buckling constraints together with the standard approach employing initial stiffness and geometric stiffness as two ingredients within generalized eigenvalue frameworks. Numerical results show that all three formulations can help to improve the stability of the optimized design. In addition, post‐nonlinear stability analysis on the optimized designs reveals that a higher linear buckling threshold might not lead to a higher nonlinear critical load, especially in cases when the pre‐critical response is nonlinear.

EMsFEM based concurrent topology optimization method for hierarchical structure with multiple substructures

An efficient concurrent topology optimization method based on the Extended Multiscale Finite Element Method (EMsFEM) is proposed to design the hierarchical structure with multiple substructures in this paper. Firstly, the mathematical description model for the topology of hierarchical structures and the interpolation model for material are established at both the macro and substructural levels. Then the concurrent topology optimization formulation for hierarchical structures is proposed based on EMsFEM. After that, the sensitivity analysis scheme of the objective function and constraint functions is given, based on which the Method of Moving Asymptotes (MMA) is employed to update design variables. The assumption of separation of length scales is not required in the proposed method, which avoids the disconnection between adjacent substructures. Besides, without limiting the topology and spatial distribution of substructures in advance, the concurrent topology optimization for hierarchical structures with multiple substructures can be realized using this method. The provided numerical examples verify the correctness and effectiveness of the proposed sensitivity analysis method as well as the feasibility, effectiveness and scalability of the proposed concurrent topology optimization method for hierarchical structures. Finally, the influence of the initial substructure configurations and the number of substructure types on the final configuration and mechanical performance of hierarchical structures are explored with a series of numerical examples.

Topology optimization of differentiable microstructures

Recent years have seen a growing interest in topology optimization of functionally graded microstructures, characterized by an array of microstructures with varying volume fractions. However, microstructures optimized at slightly different volume fractions do not necessarily connect well when placed adjacently. Furthermore, optimization is commonly performed on a finite set of volume fractions, limiting the number of microstructure configurations.In this paper, we introduce the concept of differentiable microstructures, which are parameterized microstructures that exhibit continuous variations in both geometry and mechanical properties. To construct such microstructures, we propose a novel formulation for topology optimization. In this approach, a series of 2-dimensional microstructures is represented using a height field, and the objective is to maximize the bulk modulus of the entire series. Through this optimization process, an initial microstructure with a small volume fraction undergoes non-uniform transformations, generating a series of microstructures with progressively increasing volume fractions. Notably, when compared to traditional uniform morphing methods, our proposed optimization approach yields a series of microstructures with bulk moduli that closely approach the theoretical limit.

A new design method for stiffened plate based on topology optimization with min-max length-scale control

Stiffened plates are widely used in engineering due to their excellent manufacturing and mechanical properties. This paper introduces a novel method for designing stiffener plates that combines the H-DGTP formulation, robust topology optimization formulation, and maximum length-scale control. In comparison to existing methods, the proposed approach not only provides a clear layout of stiffeners but also optimizes their height. Sensitivity analysis of all design variables is derived for utilization with gradient-based optimizers. The study demonstrates that the implementation of the robust filter approach enables precise control of both structural features and gap widths, effectively avoiding sharp angles. Moreover, as the maximum length approaches the minimum length, the stiffeners assume uniform thickness, which better meets engineering requirements. Numerical examples are presented to validate the effectiveness of the proposed method.

A fatigue-resistance topology optimization formulation for continua subject to general loads using rainflow counting

A fatigue-resistance topology optimization formulation for continua subject to general loads using rainflow counting

Tuning Buckling Behaviors in Magnetically Active Structures: Topology Optimization and Experimental Validation

Abstract Buckling, a phenomenon historically considered undesirable, has recently been harnessed to enable innovative functionalities in materials and structures. While approaches to achieve specific buckling behaviors are widely studied, tuning these behaviors in fabricated structures without altering their geometry remains a major challenge. Here, we introduce an inverse design approach to tune buckling behavior in magnetically active structures through the variation of applied magnetic stimuli. Our proposed magneto-mechanical topology optimization formulation not only generates the geometry and magnetization distribution of these structures but also informs how the external magnetic fields should be applied to control their buckling behaviors. By utilizing the proposed strategy, we discover magnetically active structures showcasing a broad spectrum of tunable buckling mechanisms, including programmable peak forces and buckling displacements, as well as controllable mechano- and magneto-induced bistability. Furthermore, we experimentally demonstrate that multiple unit designs can be assembled into architectures, resulting in tunable multistability and programmable buckling sequences under distinct applied magnetic fields. By employing a hybrid fabrication method, we manufacture and experimentally validate the generated designs and architectures, confirming their ability to exhibit precisely programmed and tunable buckling behaviors. This research contributes to the advancement of multifunctional materials and structures that harness buckling phenomena, unlocking transformative potential for various applications, including robotics, energy harvesting, and deployable and reconfigurable devices.

Conceptual design of compliant bone scaffolds by full-scale topology optimization

A promising new treatment for large and complex bone defects is to implant specifically designed and additively manufactured synthetic bone scaffolds. Optimizing the scaffold design can potentially improve bone in-growth and prevent under- and over-loading of the adjacent tissue. This study aims to optimize synthetic bone scaffolds over multiple-length scales using the full-scale topology optimization approach, and to assess the effectiveness of this approach as an alternative to the currently used mono- and multi-scale optimization approaches for orthopaedic applications. We present a topology optimization formulation, which is matching the scaffold's mechanical properties to the surrounding tissue in compression. The scaffold's porous structure is tuneable to achieve the desired morphological properties to enhance bone in-growth. The proposed approach is demonstrated in-silico, using PEEK, cortical bone and titanium material properties in a 2D parameter study and on 3D designs. Full-scale topology optimization indicates a design improvement of 81% compared to the multi-scale approach. Furthermore, 3D designs for PEEK and titanium are additively manufactured to test the applicability of the method. With further development, the full-scale topology optimization approach is anticipated to offer a more effective alternative for optimizing orthopaedic structures compared to the currently used multi-scale methods.

Pure-displacement formulation and bulk modulus propagation for topology optimization with pressure loads

This work proposes a pure-displacement formulation for topology optimization with pressure loads and the use of a continuous boundary propagation technique. This technique allows controlling the non-solid (fluid or void) constitutive behavior by disseminating the desired bulk modulus to holes in the solid domain. The novelty of the proposed approach is to enable the distribution of solid, fluid, and void simultaneously with one design variable field. A density-based topology optimization approach is considered, and a gradient-based interior point algorithm is used. The linear elasticity equilibrium equations are discretized by the finite element method and solved with a sparse direct solver. The sensitivities are calculated by the discrete adjoint method with automatic differentiation. The optimization problems encompass compliance minimization (in 2D and 3D) and a compliant mechanism design to maximize the output displacement. Three examples evaluate the proposed formulation: the internally-pressurized lid, the piston head, and the inverter-compliant mechanism. The results show that different designs are obtained by considering different constitutive behavior for the non-solid phase.

Influence of topology optimization parameters on the mechanical response of an additively manufactured test structure

Topology Optimization (TO) determines a material distribution within a domain under given conditions and design constraints, and generally generates complex geometries as a result. Complementary to TO, Additive Manufacturing (AM) offers the ability to fabricate complex geometries which may be difficult to manufacture using traditional techniques such as milling. AM has been used in multiple industries including the medical devices area. Hence, TO may be used to create patient-matched devices where the mechanical response is catered to a particular patient. However, during a medical device regulatory 510(k) pathway, demonstrating that worst-cases are known and tested is critical to the review process. Using TO and AM to predict worst-case designs for subsequent performance testing may be challenging and does not appear to have been thoroughly explored. Investigating the effects of TO input parameters when AM is employed may be the first step in determining the feasibility of predicting these worst-cases.In this paper, the effect of selected TO parameters on the resulting mechanical response and geometries of an AM pipe flange structure are investigated. Four different input parameters were chosen in the TO formulation: (1) penalty factor, (2) volume fraction, (3) element size, and (4) density threshold. Topology optimized designs were fabricated using PA2200 polyamide and the mechanical responses (reaction force, stress, and strain) were observed through experiments (universal testing machine and 3D Digital Image Correlation) and in silico environments (finite element analysis). In addition, 3D scanning and mass measurement were performed to inspect the geometric fidelity of the AM structures. A sensitivity analysis is performed to examine the effect of each TO parameters. The sensitivity analysis revealed mechanical responses can have non-monotonic and non-linear relationships between each tested parameter.

Level-set topology optimization for Ductile and Brittle fracture resistance using the phase-field method

This work presents a rigorous mathematical formulation for topology optimization of a macro structure undergoing ductile failure. The prediction of ductile solid materials which exhibit dominant plastic deformation is an intriguingly challenging task and plays an extremely important role in various engineering applications. Here, we rely on the phase-field approach to fracture which is a widely adopted framework for modeling and computing the fracture failure phenomena in solids. The first objective is to optimize the topology of the structure in order to minimize its mass, while accounting for structural damage. To do so, the topological phase transition function (between solid and void phases) is introduced, thus resulting in an extension of all the governing equations. Our second objective is to additionally enhance the fracture resistance of the structure. Accordingly, two different formulations are proposed. One requires only the residual force vector of the deformation field as a constraint, while in the second formulation, the residual force vector of the deformation and phase-field fracture simultaneously have been imposed. An incremental minimization principles for a class of gradient-type dissipative materials are used to derive the governing equations. Thereafter, to obtain optimal topology to enhance the structural resistance due to fracture, the level-set-based formulation is formulated. The level-set-based topology optimization is employed to seek an optimal layout with smooth and clear boundaries. Sensitivities are derived using the analytical gradient-based adjoint method to update the level-set surface for both formulations. Here, the evolution of the level-set surface is realized by the reaction–diffusion equation to maximize the strain energy of the structure while a certain volume of design domain is prescribed. Several three-dimensional numerical examples are presented to substantiate our algorithmic developments.

Topology optimization with local stress constraints and continuously varying load direction and magnitude: towards practical applications

Topology optimization problems typically consider a single load case or a small, discrete number of load cases; however, practical structures are often subjected to infinitely many load cases that may vary in intensity, location and/or direction (e.g. moving/rotating loads or uncertain fixed loads). The variability of these loads significantly influences the stress distribution in a structure and should be considered during the design. We propose a locally stress-constrained topology optimization formulation that considers loads with continuously varying direction to ensure structural integrity under more realistic loading conditions. The problem is solved using an Augmented Lagrangian method, and the continuous range of load directions is incorporated through a series of analytic expressions that enables the computation of the worst-case maximum stress over all possible load directions. Variable load intensity is also handled by controlling the magnitude of load basis vectors used to derive the worst-case load. Several two- and three-dimensional examples demonstrate that topology-optimized designs are extremely sensitive to loads that vary in direction. The designs generated by this formulation are safer, more reliable, and more suitable for real applications, because they consider realistic loading conditions.

Topology optimization for fluid flow devices modeled through the Multiple Reference Frame approach

Fluid flow devices have been algorithmically designed through optimization methods, and this task is relatively complex for flow machines. Particularly, one recent development is the use of the topology optimization method for this task, which is capable of achieving possibly non-intuitive geometries and even auxiliary structures, such as splitter blades without ever imposing them as a condition for the design. The design of the rotating part of the rotor has already been previously considered. However, in the presence of stationary parts surrounding the rotor, such as diffuser blades, the fluid flow behavior may change, meaning that it may be interesting to consider this different dynamic in the design. One possible approach for this is the Multiple Reference Frame approach, which considers the fluid flow dynamics inside the rotor being computed in the rotating reference frame, whilst the fluid flow dynamics outside the rotor is being computed in the stationary reference frame. In particular, this implies the need for some changes in the topology optimization formulation, also impacting the choice of objective functions. Therefore, here the topology optimization method is formulated for the Multiple Reference Frame approach, and a new combination for the multi-objective function is proposed. The discrete design variable configuration from the Topology Optimization of Binary Structures approach is adopted with some adjustments to account for the Multiple Reference Frame approach. The fluid flow is chosen to be laminar or turbulent. Numerical examples are presented in 2D for the rotor-diffuser in order to consider some aspects that affect the overall topology optimization design.

On topology optimization with gradient-enhanced damage: An alternative formulation based on linear physics

Numerous topology optimization formulations have been proposed in order to enhance structural resistance to material failure. A clear line can often be drawn between those methods which attempt to constrain local failure criteria and those that explicitly model the failure physics during the optimization process. In this work the former method is extended in a manner inspired by the mathematical form of typical gradient-enhanced damage models. Importantly, the proposed formulation relies on linear physics during the optimization procedure, which greatly increases its speed and robustness, both of which are essential in industrial applications where large numerical models are typically used. The size effect introduced by using such a numerical model is further investigated and select observations are provided, such as spurious “fin-like” patterns that emerge depending on the type of structure and loading conditions. Finally, the load capacity is verified for each optimized design through a post-optimization verification procedure which is unaffected by the density-based design parameterization and associated material interpolation schemes.