Topology Optimization Framework Research Articles

Adaptive isogeometric topology optimization of shell structures based on PHT-splines

This paper proposes an adaptive isogeometric topology optimization framework for shell structures by utilizing a continuous density field represented as Polynomial splines over Hierarchical T-meshes (PHT-splines). This framework ensures an exact representation of shell structures, eliminating the geometric inaccuracies commonly associated with topology optimization. In the meanwhile, the meshes used for design and analysis are refined adaptively and locally along the density boundary to achieve a smooth material layout with reduced degrees of freedom (DOF). The adaptive sensitivity filter is tailored to the characteristics of PHT-splines, where the filter radius is determined automatically and adaptively, without the need to specify parameters in advance. Numerical experiments conducted on various shell structures validate the efficacy of the proposed adaptive framework. In comparison with isogeometric topology optimization based on non-adaptive cases (i.e. B-splines), the proposed adaptive framework demonstrates a notable enhancement in computational efficiency, with a 50%−80% reduction in running time and a 30%−60% reduction in DOF. Additionally, we offer a detailed comparison between PHT-splines and other locally refinable splines in the context of shell topology optimization. Numerical experiments exhibit that the efficient and localized refinement capability of PHT-splines provides advantages in both computational efficiency and structural performance for topology optimization.

Conductive paths generation using topology optimization for worst-case thermal design in space systems

Designing thermal systems for space platforms is a challenging task due to the wide variability in thermal conditions during the orbit and the strict constraints imposed by other subsystems. Several strategies have been developed to address these challenges while minimizing the thermal control subsystem’s signature, encompassing different algorithms optimize components positions in the platform. When relocation is not possible, thermal couplings between components can be modified to enhance heat transfer and achieve more uniform temperature distribution. To design optimal thermal paths in 2D space structures, in this paper a topology optimization framework based on the Heaviside Projection Method is introduced, using the Lumped Parameter Method (LPM) and taking the radiation heat transfer into account. The algorithm’s performance is tested on a Printed Circuit Board (PCB) by designing an optimal copper layer to meet specific thermal requirements under both extreme hot and cold conditions. Results show a significant improvement in thermal compliance for all components with the addition of this high-conductivity layer. Additionally, a reduction method is introduced to transfer the material distribution from the optimization to a coarser mesh of any size, which is useful to facilitate its implementation in existing software, where a large number of degrees of freedom can be a limitation.

An isogeometric approach to dynamic structures for integrating topology optimization and optimal control at macro and micro scales

In the context of active vibration suppression, an integrated topology optimization framework is proposed for the optimal control of dynamic structures. This article formulates a new composite objective function by considering the external harmonic excitations with different excitation frequencies and the complex-valued displacement field. Within the proposed optimal control performance function, the symmetric weighting matrix, the magnitudes of which are related to the importance of the control forces, is integrated into dynamic compliance to serve the state displacement. Utilizing the non-uniform rational B-splines (NURBS) basis functions, the material interpolation model defined at control points is described by the NURBS surface, which is incorporated into the optimization formulation for optimal structural and vibration control design. The sensitivity results are deduced in terms of the pseudo-densities and the corresponding weights at control points from both macro and micro perspectives of structural dynamics. To demonstrate the effectiveness of the integrated topology optimization of dynamic structures at macro and micro scales, several numerical examples, including the topology optimization of solid beams in plane stress and Mindlin plates with 3D microstructures, are provided and discussed in terms of structural shape and topology, frequency response curve, and modal displacement.

A “poor-man’s” deformation plasticity based approach to topology optimization of elastoplastic structures

This paper presents a topology optimization framework utilizing a deformation plasticity model to approximate the isotropic hardening von-Mises incremental elastoplasticity model under monotone proportional loading. One advantage of the model is that it is based on a yield surface allowing for precise matching to uniaxial elastoplastic isotropic hardening response. The deformation plasticity model and the incremental plasticity model coincides for proportional loading and since the deformation plasticity model is path-independent, the computational cost and implementation complexity reduce significantly compared to the conventional incremental elastoplasticity. To investigate the deformation plasticity model combined with topology optimization, we compare three common elastoplastic optimization objectives: stiffness, strain energy and plastic work. The possibility to limit the peak local plastic work while maximizing the strain energy is also investigated. The consistent analytical sensitivity analysis which only requires the terminal state is derived using adjoint method. Numerical examples demonstrate that the proportionality assumption is reasonable and the deformation plasticity model combined with topology optimization is a competitive alternative to cumbersome incremental elastoplasticity.

Facilitating multidisciplinary collaboration through a versatile level-set topology optimization framework via COMSOL multiphysics

Facilitating multidisciplinary collaboration through a versatile level-set topology optimization framework via COMSOL multiphysics

A Computational Design Framework for Lubrication Interfaces With Active Micro-textures

Abstract The major goal of the present study is to develop a computational design framework for the active control of hydrodynamically lubricated interfaces. The framework ultimately delivers an electrode distribution on an elastomeric substrate such that a voltage-controlled texture may be induced on its surface. This enables the setup to attain a desired time-dependent macroscopic lubrication response. The computational framework is based on a numerically efficient two-stage design approach. In the first stage, a topology optimization framework is introduced for determining a microscopic texture and the uniform modulation of its amplitude. The objective is to attain the targeted fluid flux or frictional traction signals based on the homogenization-based macroscopic response of the texture. As a minor goal, a novel unit cell geometry optimization feature will be developed which will enable working in a design space that is as unrestricted as possible. The obtained designs are then transferred to the second stage where the electrode distribution on a soft substrate is determined along with the voltage variation that delivers the desired amplitude variation. The first stage operates in a two-dimensional setting based on the Reynolds equation whereas the second stage operates in a three-dimensional setting based on an electroelasticity formulation. The two stages are heuristically coupled by transferring the texture topology to the electrode distribution through a projection step. The viability of such an active lubrication interface design approach is demonstrated through numerous examples that methodically investigate the central features of the overall computational framework.

A CAD-oriented parallel-computing design framework for shape and topology optimization of arbitrary structures using parametric level set

Recently, the high-resolution topology optimization to promote engineering applicability has gained much more attentions. However, an accurate and highly-efficient design framework for implementing shape and topology optimization of engineering structures with integration of CAD model is still in demand. In the current work, the critical intention is to develop a CAD-oriented parallel-computing design framework for arbitrary structures, where the Parametric Level Set Method (PLSM) is employed for shape and topology optimization. Firstly, an implicit identification model is constructed for generating a signed distance field using the vertex and normal information from the ‘STL’ file of engineering structures. The signed distance field is combined with the compactly supported radial basis functions (CSRBFs) to solve the initial level set function with a parametrization. This method is applied to present all domains, including design domains, Neumann boundary domains, Dirichlet boundary domains, and non-design domains. Secondly, the CPU parallel strategy is considered for allocating partitions of structural stiffness matrix in finite element analysis to different CPU cores for the parallel-computing to save computation costs. Thirdly, a parallel-computing design formulation is developed for performing shape and topology optimization of arbitrary structures, in which the partitioned terms of all design variables and stiffness matrix are concurrently computed on each CPU core. Finally, several classic benchmarks and the critical engineering structure of Virtual Reality (VR) glass part with extremely complex geometries, are discussed to demonstrate the effectiveness and efficiency of the proposed design framework.

Incorporating interface effects into multi-material topology optimization by improving interface configuration: An energy-based approach

Interfaces between structural multi-materials generally exhibit asymmetric resistance to tension and compression. Given this interface behavior, this work suggests an energy-based approach to improve the interface configuration for multi-material topology optimization. Based on the strain spectral decomposition, we decompose the structural elastic strain energy into tensile and compressive portions. In the density-based topology optimization framework, we use the gradient-based method to track the interface between multiple materials. Then, we construct an interface-associated scalar field to penalize the tensile portion of the strain energy, causing a pseudo-degradation of the strain energy at the interface region. Finally, within limited material usages and by minimizing the linear weighted structural strain energy and its pseudo-degradation, multi-material topology optimization with improved interface configuration is achieved. Several 2D and 3D numerical examples are investigated, by which the effectiveness and robustness of the suggested approach are fairly validated.

Dynamic Resistance and Energy Absorption of Sandwich Beam via a Micro-Topology Optimization

The current research of sandwich structures under dynamic loading mainly focus on the response characteristic of structure. The micro-topology of core layers would sufficiently influence the property of sandwich structure. However, the micro deformation and topology mechanism of structural deformation and energy absorption are unclear. In this paper, based on the bi-directional evolutionary structural optimization method and periodic base cell (PBC) technology, a topology optimization frame work is proposed to optimize the core layer of sandwich beams. The objective of the present optimization problem is to maximize shear stiffness of PBC with a volume constraint. The effects of the volume fraction, filter radius, and initial PBC aspect ratio on the micro-topology of the core were discussed. The dynamic response process, core compression, and energy absorption capacity of the sandwich beams under blast impact loading were analyzed by the finite element method. The results demonstrated that the over-pressure action stage was coupled with the core compression stage. Under the same loading and mass per unit area, the sandwich beam with a 20% volume fraction core layer had the best blast resistance. The filter radius has a slight effect on the shear stiffness and blast resistances of the sandwich beams. But increasing the filter radius could slightly improve the bending stiffness. Upon changing the initial PBC aspect ratio, there are three ways for PBC evolution: The first is to change the angle between the adjacent bars, the second is to further form holes in the bars, and the third is to combine the first two ways. However, not all three ways can improve the energy absorption capacity of the structure. Changing the aspect ratio of the PBC arbitrarily may lead to worse results. More studies are necessary for further detailed optimization. This research proposes a new topology sandwich beam structure by micro-topology optimization, which has sufficient shear stiffness. The micro mechanism of structural energy absorption is clarified, it is significant for structural energy absorption design.

Topology Optimization with Explicit Components Considering Stress Constraints

Topology optimization focuses on the conceptual design of structures, characterized by a large optimization space and a significant impact on structural performance, and has been widely applied in industrial fields such as aviation and aerospace. However, most topology optimization methods prioritize structural stiffness and often overlook stress levels, which are critical factors in engineering design. In recent years, explicit topology optimization methods have been extensively developed due to their ability to produce clear boundaries and their compatibility with CAD/CAE systems. Nevertheless, research on incorporating stress constraints within the explicit topology optimization framework remains scarce. This paper is dedicated to investigating stress constraints within the explicit topology optimization framework. Due to the clear boundaries and absence of intermediate density elements in the explicit topology optimization framework, this approach avoids the challenge of stress calculation for intermediate density elements encountered in the traditional density method. This provides a natural advantage in solving topology optimization problems considering stress constraints, resulting in more accurate stress calculations. Compared with existing approaches, this paper proposes a novel component topology description function that enhances the deformability of components, improving the representation of geometric boundaries. The lower-bound Kreisselmeier–Steinhauser aggregation function is employed to manage the stress constraint, reducing the solution scale and computational burden. The effectiveness of the proposed method is demonstrated through two classic examples of topology optimization.

Topology Optimization of Hard-Magnetic Soft Phononic Structures for Wide Magnetically Tunable Band Gaps

Abstract Hard-magnetic soft materials, which exhibit finite deformation under magnetic loading, have emerged as a promising class of soft active materials for the development of phononic structures with tunable elastic wave band gap characteristics. In this paper, we present a gradient-based topology optimization framework for designing the hard-magnetic soft materials-based two-phase phononic structures with wide and magnetically tunable anti-plane shear wave band gaps. The incompressible Gent hyperelastic material model, along with the ideal hard-magnetic soft material model, is used to characterize the constitutive behavior of the hard-magnetic soft phononic structure phases. To extract the dispersion curves, an in-house finite element model in conjunction with Bloch’s theorem is employed. The method of moving asymptotes is used to iteratively update the design variables and obtain the optimal distribution of the hard-magnetic soft phases within the phononic structure unit cell. Analytical sensitivity analysis is performed to evaluate the gradient of the band gap maximization function with respect to each one of the design variables. Numerical results show that the optimized phononic structures exhibit a wide band gap width in comparison to a standard hard-magnetic soft phononic structure with a central circular inclusion, demonstrating the effectiveness of the proposed numerical framework. The numerical framework presented in this study, along with the derived conclusions, can serve as a valuable guide for the design and development of futuristic tunable wave manipulators.

Topology optimization for minimizing the mean compliance under thermo-mechanical loads using element-free Galerkin method

This paper presents a numerical model for addressing thermo-mechanical coupling problems in topology optimization, utilizing the element-free Galerkin (EFG) method. A multi-parameter density-based topology optimization framework is introduced to interpolate the thermal stress coefficient, encompassing thermal conductivity, thermal expansion coefficient, and elastic modulus. Two numerical examples are provided to investigate the model's characteristics and associated considerations, including the impact of EFG node distribution, quantity, calculation points layout, design variables, and filtering techniques on optimized results. Numerical findings indicates that the present model allows for adaptable adjustments in nodes number and distribution, calculation points layout, choice of design variables, and application of various filtering techniques while maintaining consistent background grids. Although these adjustments may affect convergence rates and final objective values, they can promise satisfactory optimization structures. Additionally, the study highlights the critical influence of filtering radius on optimization outcomes and the objective value, recommending a value of approximately 16∼20 calculation points.

An improved peridynamics topology optimization formulation for compliance minimization

This work proposes an improved peridynamics density-based topology optimization framework for compliance minimization. One of the main advantages of using a peridynamics discretization relies in the fact that it provides a consistent regularization of classical continuum mechanics into a nonlocal continuum, thus containing an inherent length scale called the horizon. Furthermore, this reformulation allows for discontinuities and is highly suitable for treating fracture and crack propagation. Partial differential equations are rewritten as integrodifferential equations and its numerical implementation can be straightforwardly done using meshfree collocation, inheriting its advantages. In the optimization formulation, Solid Isotropic Material with Penalization (SIMP) is used as interpolation for the design variables. To improve the peridynamic formulation and to evaluate the objective function in a energetically consistent manner, surface correction is implemented. Moreover, a detailed sensitivity analysis reveals an analytical expression for the objective function derivatives, different from an expression commonly used in the literature, providing an important basis for gradient-based topology optimization with peridynamics. The proposed implementation is studied with two examples illustrating different characteristics of this framework. The analytical expression for the sensitivities is validated against a reference solution, providing an improvement over the referred expression in the literature. Also, the effect of using the surface correction is evidenced. An extensive analysis of the horizon size and sensitivity filter radius indicates that the current method is mesh-independent, i.e. a sensitivity filter is redundant since peridynamics intrinsically filters length scales with the horizon. Different optimization methods are also tested for uncracked and cracked structures, demonstrating the capabilities and robustness of the proposed framework.

Concurrent multiscale topology optimization for size-dependent structures incorporating multiple-phase materials

The significant progress in additive manufacturing has resulted in the development of well-engineered microstructures with innovative topological configurations, showcasing exceptional performance across diverse fields. Topology optimization is widely recognized as an advanced design tool that provides engineers with new insights for creating lightweight structures. As integration devices continue to trend towards miniaturization, the size effect in small-scale structures remains a persistent challenge. The absence of characteristic parameters to describe this size effect hinders proper explanation through traditional multiscale topology optimization based on classical mechanics. Therefore, we propose a size-dependent topology optimization framework for concurrent multiscale design involving multiple-phase materials. The modified couple stress theory is proposed to account for the size effect in small-scale structures. The numerical findings suggest that the size effect is significantly influenced by the ratio of elastic modulus, volume fraction, and width-to-length ratio of the representative volume element as its characteristic size approaches the length of the material. These results highlight the importance of considering size-dependent effects in topology optimization for multiscale design and provide valuable insights for engineers and researchers in this field.

A [formula omitted] continuous multi-patch framework for adaptive isogeometric topology optimization of plate structures

This study proposes a novel computationally efficient methodology to perform topology optimization (TO) of fourth-order plate structures within the framework of multi-patch isogeometric analysis. This is realized by taking the multifold benefits of isogeometric PHT-Splines to (1) discretize the C1 continuous weak form of plate structures, (2) develop a C0 continuous density field for the material distribution in TO and inherently remove the need for filters, and (3) provide a hierarchical tree structure for the structural mesh to effortlessly implement an adaptive mesh refinement (AMR) strategy. Moreover, to ensure continuity between isogeometric patches, we adopt a strong C1 coupling between the boundaries. This is established by constructing new basis functions, defined as a linear combination of existing C0 functions at the patch interfaces. The density field in TO is further enhanced with a first-neighbourhood smoothening algorithm based on the Shepard function to generate printable topologies and alleviate the post-processing stages after optimization. An element-centre density, based on the control point densities of the isogeometric mesh, is used as the marking scheme for the AMR to determine the subdomains to be refined. Utilizing the Geometry Independent Field approximaTion, the design and adaptive analysis-optimization stages were independently discretized respectively through NURBS and PHT-Splines, allowing easy transfer of multi-patch geometries from industry-standard packages. Multiple numerical examples illustrate the stability of the multi-patch algorithm in optimizing the geometries effectively. The results also show considerable advantages in terms of solution accuracy such as precise field, smooth topology and computational efficiency.

Inverse design of nonlinear metasurfaces for sum frequency generation

Abstract Sum frequency generation (SFG) has multiple applications, from optical sources to imaging, where efficient conversion requires either long interaction distances or large field concentrations in a quadratic nonlinear material. Metasurfaces provide an essential avenue to enhanced SFG due to resonance with extreme field enhancements with an integrated ultrathin platform. In this work, we formulate a general theoretical framework for multi-objective topology optimization of nanopatterned metasurfaces that facilitate high-efficiency SFG and simultaneously select the emitted direction and tailor the metasurface polarization response. Based on this framework, we present novel metasurface designs showcasing ultimate flexibility in transforming the outgoing nonlinearly generated light for applications spanning from imaging to polarimetry. For example, one of our metasurfaces produces highly polarized and directional SFG emission with an efficiency of over 0.2 cm2 GW−1 in a 10 nm signal operating bandwidth.

Stress-constrained optimization of multiscale structures with parameterized microarchitectures using machine learning

A multiscale topology optimization framework for stress-constrained design is presented. Spatially varying microstructures are distributed in the macroscale where their material properties are estimated using a neural network surrogate model for homogenized constitutive relations. Meanwhile, the local stress state of each microstructure is evaluated with another neural network trained to emulate second-order homogenization. This combination of two surrogate models — one for effective properties, one for local stress evaluation — is shown to accurately and efficiently predict relevant stress values in structures with spatially varying microstructures. An augmented lagrangian approach to stress-constrained optimization is then implemented to minimize the volume of multiscale structures subjected to stress constraints in each microstructure. Several examples show that the approach can produce designs with varied microarchitectures that respect local stress constraints. As expected, the distributed microstructures cannot surpass density-based topology optimization designs in canonical volume minimization problems. Despite this, the stress-constrained design of hierarchical structures remains an important component in the development of multiphysics and multifunctional design. This work presents an effective approach to multiscale optimization where a machine learning approach to local analysis has increased the information exchange between micro- and macroscales.

Easy manufacturing constraint based topology optimization of dual-scale and dual-constituent lattice metastructure with thermal dimensional stability

Thermal dimensional stability (TDS) is a crucial issue in the development of high-end industry equipment and precision instruments that work in fluctuating thermal environments. To endow meta-structures with high load-carrying capacity and TDS functionality simultaneously, this paper proposes a topology optimization framework to optimize the topologies of lattice meta-structures at macroscopic structural and microscopic material scales concurrently. An important feature of the current optimization model is the introduction of a material concentration distribution (MCD) constraint to reduce the number of dual-constituent interfaces (DIs), which enables the easy manufacturing of the optimized structures because an additional fabrication process is required for the connection of heterogeneous lattice members. Designs for two dual-scale TDS structures, potentially employed for satellite payload platforms and the supporting structure of space telescopes, are completed. Compared with mono-scale TDS structures, the dual-scale schemes exhibit superior structural performances due to the opening of the dual-scale design space. Numerical experiments are performed to validate the ultra-low structural thermal deformations of the optimized TDS structures with values of 0.0743 μm/℃ and 0.153 μm/℃. Furthermore, the imposition of the MCD constraint significantly reduces the number of DIs from 38 to 10 within a lattice cell, enabling the easy assembly of the demonstrative 3D printing dual-constituent samples.

Freeform metasurface color router for deep submicron pixel image sensors.

Advances in imaging technologies have led to a high demand for ultracompact, high-resolution image sensors. However, color filter-based image sensors, now miniaturized to deep submicron pixel sizes, face challenges such as low signal-to-noise ratio due to fewer photons per pixel and inherent efficiency limitations from color filter arrays. Here, we demonstrate a freeform metasurface color router that achieves ultracompact pixel sizes while overcoming the efficiency limitations of conventional architectures by splitting and focusing visible light instead of filtering. This development is enabled by a fully differentiable topology optimization framework to maximize the use of the design space while ensuring fabrication feasibility and robustness to fabrication errors. The metasurface can distribute an average of 85% of incident visible light according to the Bayer pattern with a pixel size of 0.6 μm. The device and design methodology enable the compact, high-sensitivity, and high-resolution image sensors for various modern technologies and pave the way for the advanced photonic device design.

Enhancing Flow Batteries: Topology Optimization of Electrode Porosity and Shape Optimization of Cell Design

This research focuses on the improvement of porosity distribution within the electrode of an all‐vanadium redox flow battery (VRFB) and on optimizing novel cell designs. A half‐cell model, coupled with topology and shape optimization framework, is introduced. The multiobjective functional in both cases aims to minimize pressure drop while maximizing reaction rate within the cell. Topology optimization results reveal dependencies on initial value, porosity constraint, and flow rate. The distribution with lower porosity is preferred downstream of the inlet manifold. This design enhances active surface area, thus facilitating more effective conversion of incoming educts and improving mass transport of products. Compared to homogeneous electrodes, two‐part design demonstrates superior performance at specific porosity values. For combined porosities of 0.7 and 0.95, optimized distribution results in 81 % reduction in pressure drop, while conversion rate decreases by 7%. As regards various cell designs, optimization suggests a need to reconsider the vertical format of a rectangular cell. Horizontal cells are favored for nearly all porosities and flow rates. Trapezoidal and radial designs characterized by reduced downstream cross sections lead to higher pressure drops and are not preferred. This work provides further valuable insight into optimizing VRFB electrodes and challenges conventional cell design assumptions.