Topology Optimization Approach Research Articles

Topology optimization for fatigue reserve factors

This paper describes a topology optimization approach that applies the common fatigue analysis practices of rainflow cycle counting and critical plane searches to cover both proportional and non-proportional fatigue loading conditions of metals. The existing literature on topology optimization has so far mainly considered fatigue damage under proportional loading conditions and typically uses continuous damage models to avoid the discontinuous nature of fatigue rainflow cycle counting and critical plane searches. Furthermore, previous publications often introduced heuristic schemes to scale the fatigue damage and set the move limits for the design variables rather low to avoid oscillations in the design variables and damage responses during the optimization iterations, because fatigue damage is typically highly localized. Therefore, these approaches cause many optimization iterations. Contrarily, our present approach applies the fatigue reserve factor (FRF) directly in the optimization formulation instead of the fatigue damage where FRF is a fatigue reserve factor for infinite fatigue life. The inverse FRF scales nearly linearly with the stresses. Therefore, the present approach needs no heuristic scaling for the fatigue topology optimization. The numerical implementation applies the semi-analytic adjoint sensitivity method for multiple load cases. Numerically, FRF shows more stable optimization convergence using less optimization iterations. Different FRF topology-optimized designs for a variety of fatigue damage types are validated and compared. Additionally, the optimized FRF designs are compared to both strictly stiffness optimized designs and stress strength optimized designs.

Adaptive topology optimization for enhancing resistance to brittle fracture using the phase field model

Computational cost is one of the challenges in the field of fracture resistance topology optimization. An efficient topology optimization approach is proposed for enhancing resistance to structural fracture based on the adaptive isogeometric–meshfree method. The mesh can be adaptively refined in the computational and design domains simultaneously to capture the fracture and structural boundary delicately, which significantly reduces the degree of freedom and improves the efficiency of the topology optimization problems related to crack propagation. The proposed approach naturally inherits the advantages of both the computer-aided design and continuity of the higher-order basis functions, where the geometry and the analysis models in the process of topology optimization are integrated. The problem is formulated to maximize the absorbed energy throughout the crack process using the solid isotropic material with penalization method under volume constraints. The phase field method is used for modeling crack propagation, thereby eliminating the need of an explicit representation of the crack surface and complex tracking procedures, while a delicate mesh is still required. With this approach, the external work required during the fracture process is maximized, and the crack growth is delayed remarkably. Several representative examples are demonstrated to verify the effectiveness of the present approach in fracture-resistance-enhanced topology optimization, and the computational cost shows remarkable improvement.

A smooth maximum regularization approach for robust topology optimization in the ground structure setting

A smooth maximum regularization approach for robust topology optimization in the ground structure setting

Additively manufactured conformal cooling channels through topology optimization

Cooling channels play a critical role in various casting and molding processes, impacting both the cycle time and quality of the product. As additive manufacturing technologies become increasingly prevalent, conventional straight-drilled channels are being progressively substituted by intricate cooling lines that conform to the contours of the fabricated part. This transition can lead to a significant reduction of the solidification time and temperature gradients, consequently lowering the occurrence of part defects. However, designing such channels becomes challenging as geometric complexity and manufacturing constraints increase. In this work, we present a density-based topology optimization approach to generate conformal cooling channels in molds and dies inserts. To mitigate temperature variations, the objective function is penalized using the temperature standard deviation of the insert cavity surface. A density-gradient-based constraint is further utilized to reduce the generation of overhanging structures and promote manufacturability. In particular, the use of this constraint leads to the generation of channels characterized by a teardrop-shaped cross section. The cooling efficiency of a selected optimized design is confirmed through computations using a body-fitted solver. The geometry is subsequently manufactured by Laser Powder Bed Fusion (LPBF) and experiments are conducted to compare its performance in comparison to a design featuring straight-drilled channels. The results demonstrate that the optimized geometry significantly enhances the heat extraction rate and further leads to a 43% reduction of the cavity temperature standard deviation.

Topological design of soft substrate acoustic metamaterial for mechanical tuning of sound propagation

The design of tunable acoustic waves is crucial in phononic crystals (PnCs), as acoustic waves exhibit continuous variation across diverse frequency ranges in practical applications. While there have been research efforts on tunable PnCs, existing design strategies predominantly rely on predetermined topology and scatterer shapes based on empirical experience. This reliance poses challenges in ensuring customized and reversible tuning of the bandgap. In this work, we present a customized tunable PnC design model and solution strategy based on soft substrates, which enables reversible tuning of a specific frequency/order bandgap. The designed structure consists of a soft substrate and crystalline columns dispersed in the soft substrate and air. The relative positions of the scatterers in the air are changed by stretching the substrate, thereby realizing the modulation of sound propagation. Since soft materials have both material nonlinearities and complex geometrical variations that are difficult to obtain design sensitivity information, the material-field series expansion topology optimization approach is employed to achieve custom tunable design of the bandgap. The paper presents simulation-based analyses and experimental verification of the customized bandgap opening and closing. The results demonstrate a close alignment between theoretical predictions of propagation curves and experimental findings. This concurrence serves as evidence that the soft-substrate PnC, derived through topology optimization, effectively facilitates the adjustment of bandgaps of arbitrary orders and enables switching between various acoustic functions.

Substructure-based topology optimization design method for passive constrained damping structures

Abstract This work presents a generalized substructure-based topology optimization method for passive constrained layer damping (PCLD) structures. Here, the model of PCLD structure is obtained by the Kirchhoff–Love thin plate formulation, and the whole structure is assumed to be composed of substructures with different yet connected scales and artificial lattice geometry features. Each substructure is condensed into a super-element to obtain the associated density-related matrices under the different geometry feature parameters, and the surrogate model for the stiffness and mass matrix of PCLD substructures with different densities has been particularly built. Using cubic spline interpolation, the derivatives of super-element matrices to the associated densities can be evaluated efficiently and accurately. The modal loss factor is defined as objective functions and topology optimization for the PCLD structures is formulated based on the model for PCLD plates that are described by combining the condensed substructures. Numerical examples under two lattice patterns of substructures and their corresponding physical tests show that the correctness and superiority of this substructure-based topology optimization approach for PCLD plates are verified.

Data-driven and physics-based methods to optimize structures against delamination

Composite materials and multi-material components often fail at their internal interfaces/adhesive joints, and hence special attention should be given to such catastrophic delamination events to guarantee the system’s functional requirements. So far, however, the majority of structural topology optimization problems have focused on optimal distribution of the bulk materials by considering interfaces as perfectly bonded. This motivates the introduction of optimization methods that explicitly take into account the role of the material interfaces to optimize structures against delamination. In this work, we propose a data-driven heuristic optimization approach for the identification of optimal cohesive interfaces with linearly graded fracture properties to increase the ability of the composite structure to withstand peeling. Moreover, for given cohesive interface properties, we investigate the applicability of the physics-based Solid Isotropic Material with Penalty (SIMP) topology optimization approach to optimize the internal structure of a substrate in problems where the stress field is affected by interface delamination.

A thermal model for topology optimization in additive manufacturing: Design of support structures and geometry orientation

In this study, we present an efficient Topology Optimization (TO) approach designed to optimize support structures in metal Additive Manufacturing (AM), with a particular focus on Powder Bed Fusion (PBF) technology. The developed framework uses a purely thermal formulation to identify regions within the design that are susceptible to high heat concentrations. In the proposed modeling of the AM process, we postulate that new material layers are included in a partially built design that has already cooled to a controlled temperature. This aspect provides a layer-by-layer model of AM entirely local, enabling the building process parallelization and resulting in an algorithm with superior computational efficiency. Numerical results show the robustness of the proposed strategy, with the successful incorporation of support structures beneath overhanging surfaces and their effectiveness across a wide range of geometries and orientations. Furthermore, this framework is also applied to optimize the geometry orientation within the build chamber, further enhancing its applicability in the AM context.

Quadcopter Unmanned Aerial Vehicle Structural Design Using an Integrated Approach of Topology Optimization and Additive Manufacturing

The performance of quadcopter frames, particularly in terms of weight and crash resistance, is significantly influenced by their structural design and manufacturing process. In this work, a methodology is proposed that integrates advanced principles of topology optimization (TO) and additive manufacturing (AM) techniques to optimize the frame structure for improved performance. First, an analysis is conducted to evaluate existing quadcopter frame configurations, identifying areas for improvement. Experimental evaluations of thrust and moment of motors are performed to assess the performance of the enhanced quadcopter frame, with a focus on advancing the design through computer-aided simulations of static structural analysis and impact tests. The TO technique is then employed to determine the optimal distribution of material within the frame, governed by constraints such as weight reduction and mechanical strength. The results demonstrate that the overall performance of a quadcopter frame is significantly improved by the proposed methodology, showcasing advancements in stability, weight reduction, and crashworthiness. The resulting optimized frame design is subsequently manufactured using AM methods, which offer advantages such as design flexibility and the ability to produce complex geometries. The findings of this study contribute to the field of quadcopter design and optimization by highlighting the synergies between TO and AM techniques. An avenue is offered for the development of lightweight and robust quadcopter frames, as the capabilities and performance of quadcopter systems are advanced. The insights gained from this research open up opportunities for further advancements in the design and manufacturing of UAVs.

Topology optimization method for transient heat conduction using the Lyapunov equation

This study presents a novel topology optimization approach for transient heat conduction during the transition period based on the Lyapunov equation. For the problem under consideration, traditional topology optimization methods face computational challenges due to the time dependence of the transient responses. To address this issue, this article introduces a new optimization formulation that contains a modified governing equation for transient heat conduction and a new objective function. The new formulation aims to simplify the evaluation of transient response performance by utilizing the Lyapunov equation, thereby avoiding time-consuming transient analysis during the optimization iteration process. Additionally, this study employs model reduction techniques to reduce the size of the analysis model and further enhance the efficiency of the optimization process. After comparing two classic methods, the Proper Orthogonal Decomposition (POD) method is selected for its higher efficiency. Numerical examples consider both heat conduction and temperature control problems, employing 2D and 3D models to validate the proposed method's effectiveness. The results of the point-to-point heat conduction example indicate that, compared to the conventional design, the optimized topology design obtained using the proposed method successfully reduce the heating time by approximately 35%. Furthermore, the optimized designs display clear distinguishing features from conventional designs, further emphasizing the significance of implementing transient heat conduction topology optimization.

Improving the cooling effect of proton exchange membrane fuel cells by using biomimetic capillary cooling channels based on topology optimization method

Proton exchange membrane hydrogen fuel cells (PEMFCs) are promising energy conversion devices, capable of directly converting chemical energy into electrical energy. However, the ideal operating temperature range for PEMFCs is narrow, and the intense exothermic reactions can easily lead to internal temperatures exceeding optimal levels. Thus, carefully designed cooling channels are crucial to maintain proper operating temperatures. Drawing inspiration from the heat dissipation characteristics of human capillaries on the skin, this study develops three biomimetic capillary cooling channels using a two-objective topology optimization approach. The heat transfer performance of these biomimetic cooling channels is compared against traditional parallel channels (TPC). Additionally, two new flow channel designs with varying inlet and outlet configurations are analyzed. The study examines the impact of different weight factors on the biomimetic cooling channels and uses Pareto frontiers to compare the results from varying weight factors. Findings indicate that topology-optimized biomimetic cooling channels can significantly enhance overall performance. Using a PEC = 1 for TPC as a benchmark, the biomimetic channels achieve average PECs of 1.72, 3.03, and 3.77, indicating superior performance over TPC. Furthermore, the ratio of weight factors also plays a role in the formation of cooling channels. The model with a weight factor ratio of w1:w2 = 0.6:0.4 for double inlets and double outlets arranged in opposite directions shows improved performance. Lastly, this study outlines a design guideline for PEMFC cooling channels based on biomimetic capillary structures and the appropriate ratio of the double objective functions to enhance heat dissipation performance.

Fiber Bundle Topology Optimization for Surface Flows

This paper presents a topology optimization approach for the surface flows on variable design domains. Via this approach, the matching between the pattern of a surface flow and the 2-manifold used to define the pattern can be optimized, where the 2-manifold is implicitly defined on another fixed 2-manifold named as the base manifold. The fiber bundle topology optimization approach is developed based on the description of the topological structure of the surface flow by using the differential geometry concept of the fiber bundle. The material distribution method is used to achieve the evolution of the pattern of the surface flow. The evolution of the implicit 2-manifold is realized via a homeomorphous map. The design variable of the pattern of the surface flow and that of the implicit 2-manifold are regularized by two sequentially implemented surface-PDE filters. The two surface-PDE filters are coupled, because they are defined on the implicit 2-manifold and base manifold, respectively. The surface Navier-Stokes equations, defined on the implicit 2-manifold, are used to describe the surface flow. The fiber bundle topology optimization problem is analyzed using the continuous adjoint method implemented on the first-order Sobolev space. Several numerical examples have been provided to demonstrate this approach, where the combination of the viscous dissipation and pressure drop is used as the design objective.

Adjoint-Optimized Large Dielectric Metasurface for Enhanced Purcell Factor and Directional Photon Emission.

Extracting photons efficiently from quantum sources, such as atoms, molecules, and quantum dots, is crucial for various nanophotonic systems used in quantum communication, sensing, and computation. To improve the performance of these systems, it is not only necessary to provide an environment that maximizes the number of optical modes, but it is also desirable to guide the extracted light toward specific directions. One way to achieve this goal is to use a large area metasurface that can steer the beam. Previous work has used small aperture devices that are fundamentally limited in their ability to achieve high directivity. This work proposes an adjoint-based topology optimization approach to design a large light extractor that can enhance the spontaneous decay rate of the embedded quantum transition and collimate the extracted photons. With the help of this approach, we present all-dielectric metasurfaces for a quantum transition emitting at λ = 600 nm. These metasurfaces achieve a broadband improvement of spontaneous emission compared to that in the vacuum, reaching a 10× enhancement at the design frequency. Furthermore, they can beam the extracted light into a narrow cone (±10°) along a desired direction that is predefined through their respective design process.

A multi-period topology and design optimization approach for district heating networks

The transition to modern, low-carbon district heating creates a growing need for scalable, automated design tools that accurately capture the spatial and temporal details of heating network operations. This paper presents an automated design approach for the optimal design of district heating networks that combines scalable density-based topology optimization with a multi-period approach. In this way, temporal variations in demand, supply, and heat losses can be taken into account while optimizing the network design based on a nonlinear physics model. The transition of the automated design approach from worst-case to multi-period shows a design progression from separate branched networks to a single integrated meshed network topology connecting all producers. These integrated topologies emerge without imposing such structures a priori. They increase network connectivity, and allow for more flexible shifting of heat loads between different producers and heat consumers, resulting in more cost-effective use of heat. In a case study, this integrated design resulted in an increase in waste heat share of 42.8 % and a subsequent reduction in project cost of 17.9 %. We show how producer unavailability can be accounted for in the automated design at the cost of a 3.1 % increase in the cost of backup capacity. The resulting optimized network designs of this approach connect multiple low temperature heat sources in a single integrated network achieving high waste heat utilization and redundancy, highlighting the applicability of the approach to next-generation district heating networks.

Aeroelastic analysis of a lightweight topology-optimized sandwich panel

Sandwich structures with lattice cores are novel, lightweight composite structures and are widely used in the aerospace industry. Besides, the aeroelastic behavior of sandwich panels in a supersonic flow regime still needs to be thoroughly studied. This work investigates the supersonic flutter of a sandwich panel whose core is topology-optimized. A finite element model of a sandwich panel based on the layerwise theory, coupled with the first-order piston theory, is presented. The sandwich panel core is assessed using a topology optimization approach with flutter loading constraints. The subsequent analytical homogenization scheme is developed to provide the equivalent mechanical properties of the topology-optimized panel. The modeling approach is fully validated, and the results demonstrate that the sandwich panel is capable of enlarging the flutter-free operational flight range when compared with other conventional panel designs. A parametric analysis of the topology-optimized sandwich panel regarding the critical flutter conditions is performed.

An approach for topology optimization of heat sinks for two-phase flow boiling: Part 1 – Model formulation and numerical implementation

Two-phase flow boiling in heat sinks and cold plates is attractive for various thermal management applications due to high effective heat capacity rates and high heat transfer coefficients compared to single-phase operation. In addition, multi-physics topology optimization is sought for generation of high performance thermal management components. Yet, the design of heat sinks using topology optimization has been restricted to single-phase flow physics, due to incompatibility between available two-phase flow models and traditional single-phase topology optimization approaches. Interface-capturing two-phase flow models are too computationally expensive for optimization; mixture and two-fluid models that treat the liquid and vapor phases as an interpenetrating continuum with interfacial transport phenomena captured through empirical correlations are computationally tractable for heat-sink-scale optimization. However, the penalization approaches commonly used to define the structures formed during topology optimization results in arbitrary fin and channel geometries for which a universal two-phase flow correlation is not available. To enable coupling with two-phase mixture models, this work leverages a homogenization approach to topology optimization wherein heat sink designs are represented using spatially varying microstructures with optimized dimensions. A predefined microstructure geometry is chosen for which two-phase flow correlations exist and therefore topology optimization can be performed. This work is the first to develop a topology optimization algorithm using a mixture model for simulating two-phase flow boiling to design heat sinks and cold plates. Part 1 of this study develops and implements the algorithm to investigate topology optimized heat sinks at various heat inputs, topology optimization grid sizes, and maximum vapor quality constraints. Topology optimized heat sinks designed for single-phase versus two-phase flow are compared. There are significant differences in hydraulic and thermal responses of the single-phase and two-phase designs due to high effective heat capacity rates and high heat transfer coefficients of flow boiling. A companion paper (Part 2) focuses on manufacturing, model calibration, and experimental validation of the topology optimized heat sinks under two-phase flow boiling. The algorithm demonstrated in this work extends the capabilities of topology optimization to two-phase flow physics, and thereby enables the design of various two-phase flow components such as evaporators, condensers, heat sinks, and cold plates.

An approach for topology optimization of heat sinks for two-phase flow boiling: Part 2 – Model calibration and experimental validation

Flow boiling offers increased heat transfer coefficients and effective heat capacity rates that can be leveraged by topology optimization (TO) to generate high-performance microchannel heat sinks. Topology optimization of heat sinks having two-phase flows can be accomplished using the homogenization approach wherein designs are represented as spatially varying distributions of microstructures, as demonstrated in Part 1 of this two-part study. Flow and heat transfer within the microstructures are captured using a two-phase mixture model featuring an effective porous medium formulation. However, closure of these governing equations requires empirical correlations for pressure drop and heat transfer that are specific to the operating conditions, geometry, and surface finish. Therefore, it must be demonstrated these available correlations can be successfully calibrated over a range of microstructural variations present within the homogenization framework, so as to attain the required prediction generality and accuracy needed to ensure the resulting designs achieve Pareto-optimality. This Part 2 of our study successfully demonstrates a comprehensive end-to-end process for two-phase flow model calibration, topology optimized design generation, and experimental validation of optimized performance for flow boiling heat sinks. To this end, a TO algorithm with the homogenization approach and a two-phase mixture model is implemented using available flow boiling correlations for microchannels. A set of uniform pin fin calibration samples are additively manufactured and experimentally tested under flow boiling at various flow rates and heat inputs for model calibration. All of the unknown/free coefficients in the adopted correlations are determined by minimizing the error between the model predictions and the experimental measurements using gradient-based optimization. The calibrated topology optimization algorithm is then used to generate a Pareto-optimal set of heat sinks optimized for minimum pressure drop and thermal resistance during flow boiling. Experimental characterization of these additively manufactured heat sinks, unseen during the model coefficient calibration process, reveals that the measured Pareto optimality curve matches that predicted by the topology optimization algorithm. Lastly, a heat sink design is generated for a design space involving multiple hot spots and background heating to showcase the capability of the experimentally calibrated two-phase topology optimization algorithm at handling complex boundary conditions. The optimized heat sink intelligently distributes an adequate amount of coolant flow to each of the heated regions to avoid local dry-out. This work demonstrates a complete framework for two-phase topology optimization of heat sinks through experimental calibration of flow boiling correlations to the porous medium used by the homogenization approach.

Novel designs of blade mixer impellers from the discrete element method and topology optimization

Granular mixers are essential for manufacturing products that require bulk materials as precursors. The modeling of mixers is aided by the discrete element method (DEM), which can simulate the flow of bulk material. DEM is applied in mixer design studies by varying a few design parameters and analyzing the predicted performance, but systematic design via optimization methods has yet to be performed. In this work, we utilized a combined DEM and topology optimization approach to design blade mixer impellers for improved mixing and reduced energy consumption. A DEM model was formulated from a blade mixer setup and validated with experimental particle flow data. The computational cost of the model was then reduced by employing larger particle sizes in the simulations. A binary genetic optimization algorithm was used afterwards to find the tilt angle, number, and shape of impeller blades that would improve mixing or reduce the required driving power. The high mixing impellers have four blades that each have cavities at the center and the tip. The increased blade count promotes convective mixing, while the cavities enhance shear mixing. Meanwhile, the low energy impellers have two blades that both have cavities at the center and less material at the blade tips. These minimize particle-impeller collisions, which in turn reduce the power required to drive the impeller. This work demonstrates the use of optimization methods to design the shape of impeller blades – a variable often overlooked in conventional DEM studies. Our approach may be used to systematically design other mixer types as well.

APPLICATION OF TOPOLOGY OPTIMIZATION ON A 3D-PRINTED SHELF BRACKET

In this study, the topology optimization approach was adopted to reduce the material used in manufacturing. Specifically, the mass optimization technique was deemed suitable. Mass optimization eliminates the parts that don't affect a bracket’s overall strength while under load, resulting in weight reduction and material savings. Two shelf brackets were designed to test this theory and were subjected to mass optimization. A static structural analysis of this optimized model was carried out to confirm the optimization findings. These designs were then manufactured using the 3D-printing process. The yield points were next determined by performing a uniaxial tensile test on the shelf brackets. The outcome of the tests was subsequently compared with the simulation results, and a cost analysis model was created as an output. Ultimately, a reduction of 70% in mass was achieved with acceptable structural strength. In related optimization studies, the connecting part of an unmanned aerial vehicle's landing gear has been optimized resulting in fuel savings. The theory that topology optimization may be used to make both light and stiff parts at the same time has been proven by the results of this research as well as other studies that have been done on the same topic.

Broadband metasurfaces for steering flexural waves in thin plates: A topology optimization approach

Flexural vibrations in thin plates are prevalent in practical engineering applications, and significant efforts have been dedicated to controlling these vibrations in recent years. However, steering of broadband flexural vibrations/waves in thin plates remains challenging. In this study, we propose a topology optimization-based technique to design broadband metasurfaces to steer flexural waves in plate-like structures. Broadband performance is achieved by formulating fitness functions that incorporate results at uniformly distributed sampling frequencies within the target frequency range. As an illustration, two refractive metasurfaces with a relative bandwidth of 20% and 60% are designed and characterized numerically. The first metasurface is further fabricated using 3D printing and experimentally tested under two different signal excitations. Both numerical simulations and experiments validate the efficacy of our designs. In addition, we conduct a comparative analysis with two counterparts designed through experience-based methods, highlighting the evident advantages of employing topology optimization for broadband metasurface design. This work proposes a topology optimization-based approach for designing broadband elastic metasurfaces.