Topology Optimization Framework Research Articles

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.

Nonlinear topology optimization of flexoelectric soft dielectrics at large deformation

We propose a novel nonlinear topology optimization framework tailored for flexoelectric soft dielectrics undergoing large deformation. A numerical method based on Isogeometric analysis (IGA) is introduced to nonlinear soft dielectrics at finite strain, ensuring the C1-continuity for flexoelectric problems. We outline the process of consistent linearizations and IGA discretizations. Additionally, we introduce an innovative and efficient Strain Density Function (SDF) interpolation scheme for optimizing electromechanical coupling factors (ECFs). In this scheme, the interpolation of electromechanical and hyper-elastic energy terms is grounded in the SIMP model, while a linear material interpolation model is employed for the dielectric energy. Our numerical analysis highlights the remarkable performance of the proposed SDF interpolation scheme in nonlinear electromechanical optimization scenarios. An energy remedy scheme is applied to void regions to eliminate the instability of nonlinear optimization, effectively preventing distortion deformations in low-stiffness elements. Furthermore, we investigate the influence of size effect and larger deformation on the optimization of flexoelectric soft materials. The proposed topology optimization framework adeptly leverages the interplay between size effect and larger deformation, resulting in a notable improvement in ECFs within the optimized structures. Finally, the electromechanical coupling factors (ECFs) of the optimized structures in all the examined cases exhibit enhancements up to 9 times as compared to the reference guess designs.

Multi-Objective Topology Optimization of Conjugate Heat Transfer Using Level Sets and Anisotropic Mesh Adaptation

This study proposes a new computational framework for the multi-objective topology optimization of conjugate heat transfer systems using a continuous adjoint approach. It relies on a monolithic solver for the coupled steady-state Navier–Stokes and heat equations, which combines finite elements stabilized by the variational multi-scale method, level set representations of the fluid–solid interfaces and immersed modeling of heterogeneous materials (fluid–solid) to ensure that the proper amount of heat is exchanged to the ambient fluid by solid objects in arbitrary geometry. At each optimization iteration, anisotropic mesh adaptation is applied in near-wall regions automatically captured by the level set. This considerably cuts the computational effort associated with calling the finite element solver, in comparison to traditional topology optimization algorithms operating on isotropic grids with a comparable refinement level. Given that we operate within the constraint of a specified number of nodes in the mesh, this allows not only to improve the accuracy of interface representation and motion but also to retain the high fidelity of the numerical solutions at the grid points just adjacent to the interface. Finally, the remeshing and resolution steps both run within a highly parallel environment, which makes it possible for the proposed algorithm to tackle large-scale problems in three dimensions with several tens of millions of state degrees of freedom. The developed solver is validated first by minimizing dissipation in a flow splitter device, for which the method delivers relevant optimal designs over a wide range of volume constraints and flow rate distributions over the multiple outlet orifices but yields better accuracy compared to reference data from literature obtained using uniform meshes (in the sense that the layouts are more smooth, and the solutions are better resolved). The scheme is then applied to a two-dimensional heat transfer problem, using bi-objective cost functionals combining flow resistance and thermal recoverable power. A comprehensive parametric study reveals a complex arrangement of optimal solutions on the Pareto front, with multiple branches of symmetric and asymmetric designs, some of them previously unreported. Finally, the algorithmic developments are substantiated with several three-dimensional numerical examples tackled under fixed weights for heat transfer and flow resistance, for which we show that the optimal layouts computed at low Reynolds number, that are intrinsically relevant to a broad range of microfluidic application, can also serve as smooth solutions to high-Reynolds-number engineering problems of practical interest.

Robust topology optimization for transient dynamic response minimization

This paper presents a novel framework for topology optimization (TO) of structures subjected to uncertain transient loading. Random transient uncertain but bounded loading is modeled in the form of an ellipsoid convex model. A transformation matrix is defined with uncertainty in input parameters based on orientation and lengths of semi-major and semi-minor axes of the ellipsoid. Latin hypercube sampling is utilized to generate samples inside the normalized space. These samples are then converted to a physical space using a transformation matrix. The generalized-α method is employed to compute the dynamic response of the system subjected to time-dependent loading. A density-based TO formulation is used to minimize dynamic response. The robust topology optimization problem is defined as bi-criteria optimization to minimize both the mean and variance across all deterministic cases within the ellipsoidal bounds. The solid isotropic material with penalization method is employed for the interpolation of material stiffness. Helmholtz filter is used to stabilize the problem numerically. The optimality criteria method is used to update the design variables. The proposed scheme’s effectiveness and the efficiency of the optimized designs are checked using a set of numerical examples. For most of the cases, robust design exhibits a substantial reduction of approximately 70%–75% in standard deviation when compared with deterministic design. The results demonstrate the proposed framework’s ability to reduce structure transient response under uncertain but bounded regions.

Efficient dynamic topology optimization of 2D metamaterials based on a complementary energy formulation

The advent of additive manufacturing has revolutionized the design and development of hierarchical structures, with potential applications in compliant, auxetic, and band-gap structures. This paper presents an innovative approach to developing a dynamic Topology Optimization (TO) framework for designing printable lattice structures that exhibit specific dynamic properties. Utilizing parametrically defined filament-based unit cell structures for topology optimization, we achieve desired natural frequency bandgaps in the structures composed of these unit cells. To enhance computational efficiency, we employ a complementary energy-based formulation to (semi)analytically derive the flexibility and stiffness matrices of the unit cell structure, thus, eliminating extensive finite element discretization. Consequently, a wide variety of parametrically defined filament-based meso-structures can be mathematically explored. We apply this innovative framework specifically for band-gap maximization of 2D lattice structures. By tuning the geometry within each cell using TO, we maximize the band gap. Our results show the potential of this approach to create more efficient and effective hierarchical structures with desired band-gap properties.

Directional damping design of viscoelastic composites via topology optimization

Design of architected functional materials that possess desired damping performance is a topic of great interest in both academic research and industrial applications. This study investigates the concept of ‘directional damping’ for viscoelastic composites and develops a topology optimization framework to achieve directional damping design. Microstructures with both load-bearing capacity and dominant damping in the prescribed direction are obtained by maximizing the loss factor while imposing stiffness constraints. An optimized directional damping microstructure design exhibits a loss factor 1.8 times higher than the isotropic design in the prescribed direction. It is found that the optimized configurations with curved viscoelastic strips are similar to wavy ‘suture’ lines in the beaks of woodpeckers, which possess outstanding energy absorption performance. We have also verified the superiority of the directional damping design through numerical simulations and experiments of the microstructured composites. The experiment results show significant reduction of peak response for the directional damping design compared to the isotropic counterpart. To demonstrate the application potential of the optimized unit cell microstructures, a novel conceptual airless tire that exhibits improved stiffness and shock absorption performance is designed and validated through both numerical simulations and experiments.

Topology optimization structure design of shape memory alloy with multiple constraints

As an emerging functional material, shape memory alloy (SMA) exhibits remarkable mechanical properties and finds diverse applications across industries. This paper presents a topology optimization framework based on the bi-directional evolutionary structural optimization (BESO) method for designing SMA structures, which maximizes structural stiffness under multiple constraints of specified volume fraction, displacement, and fundamental frequency. A phenomenological constitutive model is utilized to simulate the mechanical behavior of SMA accurately. The unit virtual load method is employed to determine sensitivities. Several optimized SMA beam structures and simply-supported cube structures are designed under different thermal-mechanical loads, and their displacement, mean compliance, and fundamental frequency are evaluated throughout the optimization process. The results demonstrate that the proposed framework successfully customizes the SMA topology structure with adjustable displacement and fundamental frequency, and the optimized schemes exhibit more considerable deformation and more uniform mechanical properties than their initial counterparts. The proposed framework has higher computational efficiency than the traditional SIMP-based SMA topology optimization design method.

410 Improving Patient Outcomes through Design of Biodegradable Implants for Long Bone Fractures

OBJECTIVES/GOALS: Current long bone fracture standard of care uses inert metal intramedullary nails (IMN), 10x stiffer than femur cortex. Consequent “stress-shielding” bone loss sees >5% of patients needing revision surgery. To improve nonunion healing, we develop automated design optimization methods for biodegradable Mg alloy IMNs to control local reloading. METHODS/STUDY POPULATION: Finite element analysis (FEA) is performed on 3D bone-IMN representations to establish this study’s baseline strain states for existing inert IMN geometries within QCT-informed femoral models under simulated biomechanical loading. FEA with Mg alloy properties for same IMN designs simulate transient IMN material loss through discrete time-step models with experimental in vivo Mg corrosion rates and strain-based bone density evolution using remodeling algorithms from literature. Transient stability and strength metrics, fracture zone stress profiles under gradual reloading and manufacturing constraints are formulated through gradient-based sensitivity analysis into a topology optimization framework (TOF) incorporating a reaction-diffusion degradation model to generate IMN topologies. RESULTS/ANTICIPATED RESULTS: TOF designs for Mg alloy IMNs with transient allowable strength constraints, using safety factors to prevent IMN failure, demonstrate higher compliance than standard inert IMNs with mechanical properties closer to native cortical bone. The biodegradation model within the TOF, informed by corrosion behavior from bone-IMN FEA study, predicts how potential design evolutions affect transient strain states of the system. Thus, local fracture region stress states are controlled by the algorithm optimizing for desirable transient stiffness profiles based on a minimum variance objective of fracture zone stress compared to a target bone stress profile. Optimized IMNs with porous, high surface area features achieve 50% decrease in IMN stiffness over 6 months recovery time and complete in vivo degradation in 24 months. DISCUSSION/SIGNIFICANCE: Our TOF reduces “stress-shielding” effects via design for controlled IMN biodegradation to gradually increase fracture zone loading, stimulating remodeling and reducing current risk of post-operative fracture and surgical removal in ~15k cases/yr. in the U.S. In vitro mechanical and in vivo clinical testing is required to validate design results.

Latent Crossover for Data-Driven Multifidelity Topology Design

Abstract Topology optimization is one of the most flexible structural optimization methodologies. However, in exchange for its high level of design freedom, typical topology optimization cannot avoid multimodality, where multiple local optima exist. This study focuses on developing a gradient-free topology optimization framework to avoid being trapped in undesirable local optima. Its core is a data-driven multifidelity topology design (MFTD) method, in which the design candidates generated by solving low-fidelity topology optimization problems are updated through a deep generative model and high-fidelity evaluation. As its key component, the deep generative model compresses the original data into a low-dimensional manifold, i.e., the latent space, and randomly arranges new design candidates over the space. Although the original framework is gradient free, its randomness may lead to convergence variability and premature convergence. Inspired by a popular crossover operation of evolutionary algorithms (EAs), this study merges the data-driven MFTD framework and proposes a new crossover operation called latent crossover. We apply the proposed method to a maximum stress minimization problem in 2D structural mechanics. The results demonstrate that the latent crossover improves convergence stability compared to the original data-driven MFTD method. Furthermore, the optimized designs exhibit performance comparable to or better than that in conventional gradient-based topology optimization using the P-norm measure.

Transient topology optimization for efficient design of actively cooled microvascular materials

Microvascular materials containing internal microchannels are able to achieve multi-functionality by flowing different fluids through vasculature. Active cooling is one application to protect structural components and devices from thermal overload, which is critical to modern technology including electric vehicle battery packaging and solar panels on space probes. Creating thermally efficient vascular network designs requires state-of-the-art computational tools. Prior optimization schemes have only considered steady-state cooling, rendering a knowledge gap for time-varying heat transfer behavior. In this study, a transient topology optimization framework is presented to maximize the active-cooling performance and mitigate computational cost. Here, we optimize the channel layout so that coolant flowing within the vascular network can remove heat quickly and also provide a lower steady-state temperature. An objective function for this new transient formulation is proposed that minimizes the area beneath the average temperature versus time curve to simultaneously reduce the temperature and cooling time. The thermal response of the system is obtained through a transient Geometric Reduced Order Finite Element Model (GRO-FEM). The model is verified via a conjugate heat transfer simulation in commercial software and validated by an active-cooling experiment conducted on a 3D-printed microvascular metal. A transient sensitivity analysis is derived to provide the optimizer with analytical gradients of the objective function for further computational efficiency. Example problems are solved demonstrating the method’s ability to enhance cooling performance along with a comparison of transient versus steady-state optimization results. In this comparison, both the steady-state and transient frameworks delivered different designs with similar performance characteristics for the problems considered in this study. This latest computational framework provides a new thermal regulation toolbox for microvascular material designers.

Robust contact-constrained topology optimization considering uncertainty at the contact support

In this paper, the general framework for contact-constrained topology optimization of Strömberg and Klarbring (2010) is extended to robust topology optimization. In doing so, a linear elastic design domain is considered and the augmented Lagrangian approach is used to model the unilateral contact. For topology optimization, the design space is parametrized with the SIMP-approach and the Sigmund’s filter is applied. Additionally, the robust framework considers uncertainties at the contact support such as deviations of the geometry of the contact surface and the friction coefficient. Both uncertainties are described by the first-order second-moment method which leads to minimal additional costs. In fact, only two additional linear equations must be solved to obtain the robust objective and its gradient with respect to the design variables. Having both the objective and the gradient, the design update is computed with the method of moving asymptotes. The robust framework is applied to 2D and 3D examples to prove its scalability for real-world applications.

A Fourier neural operator-based lightweight machine learning framework for topology optimization

The combination of machine learning and topology optimization has preliminarily mitigated the enormous computational costs incurred during optimization processes. Its powerful design capabilities are warmly welcomed by many scholars, but this strategy still faces many problems, such as unclear structural boundaries and high dependence on big data. Here, we propose a new machine learning topology optimization paradigm that learns the mapping relationship between the initial and optimized Level set functions in the infinite-dimensional function space under the Fourier neural operator to truly parameterize the structural model and its mechanical information, thereby achieving purely data-driven topology optimization. A base model trained under the proposed approach could obtain a topological structure with clear and smooth structural boundaries within 0.4 seconds, making it faster than traditional methods by tens of times. In addition, a lightweight retraining strategy under the base model was proposed. Four additional numerical examples were used to demonstrate that this strategy requires only 100 or 200 new samples and 3-6 seconds of retraining time to attain prediction capabilities under different boundary conditions, different meshes (resolution), various initial level set functions, and multiple loads. In addition, we discussed all the experiments and found that most of the mechanical properties of the results were close to the ground truths, while some results were even better than those of the traditional method. Although we only used the maximum structural stiffness as an example, this method can be extended to different objective functions.

Design Improvement Using Topology Optimization for the Structural Frame Design of a 40 Ft LNG ISO Container Tank

LNG ISO tank containers are a solution for bulk liquefied natural gas (LNG) delivery to the outer islands of Indonesia that are not connected to the gas pipeline network. The design of an ISO tank frame must consider two critical parameters, strength/rigidity and weight saving, which affect the operational performance of the distribution process. The current investigation aims to numerically optimize the design of the structural frame of a 40 ft LNG ISO tank for a mini LNG carrier operation using a topology optimization framework. Two design solutions are used in the topology optimization framework: reducing the strain energy and mass retained. Mass retained was selected as the objective function to be minimized, which was assumed to be 60–80%. The proposed frame design is tested using three operational loading scenarios, including racking, lifting, and stacking tests based on the ISO 1496 standard. The convergence mesh tests were initially evaluated to obtain the appropriate mesh density in the finite element analysis (FEA). The simulation findings show that the topology optimization method of the frame design resulted in an improved design, with an increase in the strength-to-weight saving ratio. A promising result from the optimization scenario demonstrates weight savings of about 18.4–37.3%, with experienced stress below the limit criteria. It is found that decreasing mass retained causes a significant stress increase in the structural frame and ISO corner castings, especially in the stacking load. The critical recommendation in the frame design of the LNG ISO tank can be improved by eliminating the saddle support and bottom frame and increasing the thickness of the vertical frame.

Phase-field method combined with optimality criteria approach for topology optimization

This paper introduces a novel topology optimization framework by combining the phase-field method and optimality criteria approach. This novel approach allows for the automatical addition of holes in the initial shape, a capability that is not achievable with the conventional phase-field model based on boundary changes for topology optimization. The distribution of the material is changed according to the sensitivity of the optimality criteria approach to nucleate holes. After the nucleation of the new holes, they are utilized for topology optimization using the phase-field method. To demonstrate the accuracy and effectiveness of the proposed approach, we conduct several numerical examples based on the two-dimensional minimum compliance problem. The obtained numerical results show the proposed method's fast convergence compared to the conventional phase-field method.

Development of a non-uniform cellular automata framework for sizing, topology and layout optimization of truss structures

This article presents a bi-level non-uniform cellular automata (CA) algorithm for the solution of sizing, topology and layout optimization of truss structures. The non-uniform CA was successfully used in a previous study to solve the weight optimization problem of truss structures for topology and sizing (El Bouzouiki, Sedaghati, and Stiharu 2021. Computers & Structures 242: 106394). In this article, an extended version of the non-uniform CA algorithm is proposed, based on the fully stressed design approach and the distribution of strain energy within the structure, to find the optimal position of the cell’s (joint’s) coordinates. The proposed non-uniform CA algorithm can solve the minimum weight optimization problem of truss structures subjected to both stress and displacement constraints. Several benchmark problems are presented to demonstrate the efficiency and accuracy of the proposed methodology.

Towards advanced piezoelectric metamaterial design via combined topology and shape optimization

Metamaterials open up a spectrum of artificially engineered properties otherwise unreachable in conventional bulk materials. For electromechanical energy conversion systems, lightweight materials with high hydrostatic piezoelectric coupling coefficients and negative Poisson’s ratio can be obtained. Thus, in this contribution, we explore the possibilities of piezoelectric metamaterials design by employing structural optimization. More specifically, we apply a sequential framework of topology and shape optimization to design piezoelectric metamaterials with negative Poisson’s ratio for electromechanical energy conversion under uniform pressure. Topology optimization is employed to generate the initial layout, whereas shape optimization fine tunes the design and improves durability and manufacturability of the structures with the help of a curvature constraint. An embedding domain discretization (EDD) method with adaptive domain and shape refinement is utilized for an efficient and accurate computation of the state problem in the shape optimization stage. Multiple case studies are conducted to determine the importance of desired stiffness characteristics, symmetry conditions and objective formulations on the design of piezoelectric metamaterials. Results show that the obtained designs are highly dependent on the desired stiffness characteristics. Moreover, the addition of the EDD-based shape optimization step introduces significant changes to the designs, confirming the usability of the sequential framework.

Multi-Physics Three-Dimensional Component Placement and Routing Optimization Using Geometric Projection

Abstract This article presents a novel three-dimensional topology optimization framework developed for 3D spatial packaging of interconnected systems using a geometric projection method (GPM). The proposed gradient-based topology optimization method simultaneously optimizes the locations and orientations of system components (or devices) and lengths, diameters, and trajectories of interconnects to reduce the overall system volume within the prescribed 3D design domain. The optimization is subject to geometric and physics-based constraints dictated by various system specifications, suited for a wide range of transportation (aerospace or automotive), heating, ventilation, air-conditioning, and refrigeration, and other complex system applications. The system components and interconnects are represented using 3D parametric shapes such as cubes, cuboids, and cylinders. These objects are then projected onto a three-dimensional finite element mesh using the geometric projection method. Sensitivities are calculated for the objective function (bounding box volume) with various geometric and physics-based (thermal and hydraulic) constraints. Several case studies were performed with different component counts, interconnection topologies, and system boundary conditions and are presented to exhibit the capabilities of the proposed 3D multi-physics spatial packaging optimization framework.

Converting pixel‐type topology optimization results to MMC‐representation based on sparse optimization and its applications

AbstractHow to realize the switching between various topology optimization approaches such as SIMP and moving morphable component (MMC) method, is a crucial challenge in the field of structural design. In this article, a robust conversion framework is proposed to convert a pixel‐type topology optimization result to MMC representation. Based on the sparse optimization approach, the framework enables the determination of the minimum number of components with a specified shape error. This method provides an efficient bridge for these two types of geometric descriptions, and promotes the free switching between the topology optimization frameworks with pixel‐based and MMC‐based design domains. The proposed procedure contains a pre‐processing of resolution improvement, symmetry axis extraction with sparse optimization, and variational shape approximation. Two practical applications are demonstrated using the proposed framework. First, it can be applied to the intermediate results of SIMP, to achieve faster optimization convergence. Furthermore, a stress‐based shape optimization approach can be applied to the obtained MMCs, and novel progressive continuity constraints are also introduced to maintain boundary continuity. Several examples demonstrate advantages of the proposed framework.

An efficient transient dynamic topology optimization framework based on successive iteration of analysis and design

Topology optimization of transient dynamics is of great importance in engineering. However, the computational cost required by the repeated dynamic response and design sensitivity analysis during design iterations is often extremely high for large-scale problems, which significantly limits its practical application. To address this issue, we propose a new method based on successive iteration of analysis and design in conjunction with a quasi-static response-enhanced mode displacement method. The conventional mode displacement method requires an inner-loop iteration in addition to the outer-loop design updating, for finding the eigenmodes of the undamped system. In order to avoid the high computational burden caused by the nested double-loop solution, we employ the concept of successive iteration of analysis and design, which integrates approximate eigenvalue analysis and design variable updating into a single iteration loop. By this means, the eigenmodes are sequentially improved through a one-step inverse iteration in each design iteration, and simultaneous convergence of the eigenmodes and the design is sought during the optimization process. These approximate eigenmodes and the quasi-static response are used as the model reduction basis for each intermediate design, and the direct time integration of the reduced-dimensional system is then performed to calculate the structural responses of interest and their derivatives with respect to the design variables. Several numerical examples are presented to demonstrate the efficiency of the proposed method. It is shown that the method can be used to solve transient dynamic topology optimization problems up to 15 million degrees of freedom on a desktop computer at affordable time cost.

HOMOGENIZATION-BASED SPACE-TIME TOPOLOGY OPTIMIZATION OF TUNABLE MICROSTRUCTURES

A topology optimization framework is developed for smart materials with tunable microstructures. The framework addresses spatial and temporal design variables in a unified setting so as to deliver the optimal periodic microstructure with stimulus-sensitive constituents. The optimal topology allows the macroscopic response of the microstructure to track a time-dependent cyclic path in the tress-strain space with minimal error. The relevant homogenization-based variational analysis for the sensitivity-based optimization framework incorporates not only material variables but also the geometry information regarding the unit cell. Extensive numerical investigations demonstrate the ability of the developed approach to deliver optimal topologies for realizable target macroscopic paths. The error in optimization increases monotonically with the degree of unrealizability, yet the critical role of the microstructure in minimizing the error in comparison to a pure time optimization approach is demonstrated in all cases.