Three-dimensional Topology Optimization Research Articles

An enhanced proportional topology optimization method with new density filtering weight function for the minimum compliance problem

This article proposes an enhanced proportional topology optimization (EPTO) method to solve the structural topology optimization problem of minimizing compliance under material volume constraints. In the proposed method, a new filtering weight function based on the improved Heaviside threshold function is adopted to filter element density during the optimization process. The optimization process of the EPTO method consists of an inner loop and an outer loop. In the inner loop, density distribution is modified in combination with a new filtering weight function. By locally averaging the weighted element compliance, an improved density distribution function in the inner loop is put forward to make the topology configuration of the optimized structure more reasonable. In addition, the projection method is used to reduce the number of intermediate density elements, thereby obtaining optimization results with clear boundaries. In the outer loop, a new termination criterion is adopted, which terminates the optimization process by determining that the relative error of the objective function in several consecutive iterations is less than the specified value. The effectiveness and efficiency of the proposed method are demonstrated through several numerical examples involving two-dimensional (2D) and three-dimensional (3D) topology optimization problems. The results of numerical examples show that the proposed method can not only accelerate the convergence of optimization iterations, but also obtain optimized structures with smaller objective function values and better topology configurations. HIGHLIGHTS A new density filtering weight function is proposed based on an improved Heaviside threshold function. An improved inner loop density distribution formula is proposed by combining a new filtering weight function. Using projection based methods to suppress the appearance of intermediate density elements. Verify the effectiveness of the proposed algorithm by comparing the optimization results with existing algorithms such as top88 and PTO.

Topological derivative based sensitivity analysis for three-dimensional discrete variable topology optimization

This study introduces a novel Topological Derivative-based Sensitivity Analysis (TDSA) methodology for three-dimensional (3D) discrete variable topology optimization. Recently, the authors pointed out that the discrete variable sensitivity can be related to the topological derivative, and thus can be rationally approximated by the specially customized topological derivative for plane stress problems. However, the 3D discrete variable sensitivity requires the 3D topological derivative with element-shaped (e.g., unit cube) domain perturbation, whose analytical solution is not available in the literature. This paper proposes a unified parameter fitting framework to calculate the 3D topological derivative with arbitrary shape 3D domain perturbation. The formula for spherical hole perturbation obtained through this parameter fitting framework is identical to the well-known analytical topological derivative formula. Further, the 3D discrete variable sensitivity can be easily obtained through element-shaped domain perturbation. With the recently proposed Sequential Approximate Integer Programming (SAIP), numerical examples demonstrate that TDSA is precise not only for the self-adjoint 3D minimum compliance problems but also for the non-adjoint 3D compliant mechanism design problems. In sum, numerical examples have been conducted by feeding the derived sensitivity to the SAIP optimization algorithms, and the correctness of the sensitivity information has been well demonstrated. This research further solidifies the foundation of rational discrete variable sensitivity analysis in 3D topology optimization.

Thermo-electrochemical level-set topology optimization of a heat exchanger for lithium-ion batteries for electric vertical take-off and landing vehicles

Developing electric vertical take-off and landing vehicles (eVTOL) that can meet the demanding power and energy requirements entails significant challenges, one of which is due to the weight of the battery packs. To address this challenge, optimization techniques can be employed to achieve lightweight designs while satisfying thermal criteria. This study focuses on optimizing a battery heat exchanger housing a high-energy-densitycylindrical cell using the level-set topology optimization method. To accurately account for heat generation from battery electrochemistry, we investigate both a high-fidelity, the Doyle Fuller Newman (DFN) model, and a low-fidelity electrochemical model, the Single Particle Model (SPM), which are compared to experimental results for an eVTOL flight profile. The novelty of the proposed approach resides in the integration of the electrochemical models within a three-dimensional unsteady thermo-electrochemical topology optimization framework. The battery heat exchanger is optimized considering the heat generated by the batteries at the material scale due to the system power requirements. The heat generated by the battery is incorporated as a source term in an unsteady heat conduction finite element model, forming the basis of the optimization process. Our objective is to minimize the integrated thermal compliance over time while satisfying a volume constraint, employing the level-set method. The SPM proves competent in predicting the voltage profile but underestimates the temperature increase. On the other hand, the DFN model accurately predicts both the temperature increase and the voltage profile, making it suitable for the thermal analysis of cells in eVTOL vehicles. Surprisingly, steady-state optimization turns out to be sufficient to generate optimized topologies that perform similarly to transient cases for the case investigated but at a reduced cost. By integrating electrochemical modeling, level-set topology optimization, and heat transfer analysis, our study contributes to the development of lightweight and thermally efficient battery heat exchangers for eVTOL vehicles, which can be extended to battery packs. Importantly, the presented methodology is versatile and can be applied to different battery chemistries, form factors, and power profiles.

Equivariant neural operators for gradient-consistent topology optimization

Abstract Most traditional methods for solving partial differential equations (PDEs) require the costly solving of large linear systems. Neural operators (NOs) offer remarkable speed-ups over classical numerical PDE solvers. Here, we conduct the first exploration and comparison of NOs for three-dimensional topology optimization. Specifically, we propose replacing the PDE solver within the popular Solid Isotropic Material with Penalization (SIMP) algorithm, which is its main computational bottleneck. For this, the NO not only needs to solve the PDE with sufficient accuracy but also has the additional challenge of providing accurate gradients which are necessary for SIMP’s density updates. To realize this, we do three things: (i) We introduce a novel loss term to promote gradient-consistency. (ii) We guarantee equivariance in our NOs to increase the physical correctness of predictions. (iii) We introduce a novel NO architecture called U-Net Fourier neural operator (U-Net FNO), which combines the multi-resolution properties of U-Nets with the Fourier neural operator (FNO)’s focus on local features in frequency space. In our experiments we demonstrate that the inclusion of the novel gradient loss term is necessary to obtain good results. Furthermore, enforcing group equivariance greatly improves the quality of predictions, especially on small training datasets. Finally, we show that in our experiments the U-Net FNO outperforms both a standard U-Net, as well as other FNO methods.

Coupling of bond-based peridynamics and continuous density-based topology optimization methods for effective design of three-dimensional structures with discontinuities

AbstractThis study proposes continuous density-based three-dimensional topology optimization (TO) approaches developed by coupling the peridynamic theory (PD) with optimality criteria (OC) and proportional approach (PROP). These frameworks, abbreviated as PD-OC-TO and PD-PROP-TO, can be practically utilized to enhance the fracture toughness of the structures during the optimization process by taking critical regions into account as pre-defined cracks. Breaking the non-local interactions (bonds) between relevant PD particles enables us to readily model cracks. Utilizing this advantage, we solve several benchmark optimization problems including different numbers, positions, and alignments of the cracks. The major differences between the proposed methods are examined by comparing optimum topologies for various cracked scenarios. Moreover, the mechanical behaviour of the optimized structures is investigated under dynamic loads to prove the significant improvements achieved by the present approach in the final designs. The results of dynamic analyses reveal the viability of both PD-TO methods for increasing the fracture toughness of the structure in the optimization stage. Overall, the proposed approach is confirmed as a superior design and optimization tool for future engineering structures. Graphical abstract

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.

Multi-objective topology optimization of heat transfer surface using level-set method and adaptive mesh refinement in OpenFOAM

The present study proposes a new efficient and robust algorithm for multi-objectives topology optimization of heat transfer surfaces to achieve heat transfer enhancement with a less pressure drop penalty based on a continuous adjoint approach. It is achieved with a customized OpenFOAM solver, which is based on a volume penalization method for solving a steady and laminar flow around iso-thermal solid objects with arbitrary geometries. The fluid-solid interface is captured by a level-set function combined with a newly proposed robust reinitialization scheme ensuring that the interface diffusion is always kept within a single local grid spacing. Adaptive mesh refinement is applied in near-wall regions automatically detected by the level-set function to keep high resolution locally, thereby reduces the overall computational cost for the forward and adjoint analyses. The developed solver is first validated in a drag reduction problem of a flow around a two-dimensional cylinder at the Reynolds numbers of 10 and 40 by comparing reference data. Then, the proposed scheme is extended to heat transfer problems in a two-dimensional flow at the Prandtl number of 0.7 and 6.9. Finally, three-dimensional topology optimization for multi-objective problems is considered for cost functionals with different weights for the total drag and heat transfer. Among various solutions obtained on the Pareto front, 4.0% of heat transfer enhancement with 12.6% drag reduction is achieved at the Reynolds number of 10 and the Prandtl number of 6.9. Moreover, the optimization of a staggered pin-fin array demonstrates that the optimal shapes and arrangement of the fins strongly depend on the number of rows from the inlet. Specifically, the pin-fins in the first and third rows extended in the upstream direction further enhance heat transfer, while the fins in the second row vanish to reduce pressure loss.

Structural topology optimization of three-dimensional multi-material composite structures with finite deformation

In this work, an explicit three-dimensional (3D) topology optimization approach is presented for multi-material composite structures accounting the finite deformation effect. The proposed method employs different sets of 3D Moving Morphable Voids (MMVs) to identify each phase material, resulting in explicit geometric descriptions of the optimized composite structures and a reduction in the number of design variables. The decoupling between the topology description and finite element analysis of composite structures enables the removal of redundant degrees of freedom, thereby mitigating the convergence issue of finite deformation analysis caused by low-density elements and leading to significant computational savings. Numerical experiments of composite structures with two- and three-phase materials are optimized to validate the accuracy and efficiency of the proposed method. The results demonstrate that, under finite deformation, the distribution of each phase material in optimized 3D composite structures is significantly affected by the amplitude of external loads, and the optimized layout could be quite different from its counterpart under small deformation assumption.

Three-Dimensional Material Mask Overlay Topology Optimization Approach With Truncated Octahedron Elements

Abstract This paper presents a 3D material mask overlay topology optimization approach using truncated octahedron elements and spheroidal masks. Truncated octahedron elements provide face connectivity between two juxtaposed elements, thus eliminating singular solutions inherently. A novel meshing scheme with Tetra-Kai-Decaheral or TKD (generic case of truncated octahedron) elements is proposed. The scheme is extended to parameterized generic-shaped domains. Various benefits of implementing the elements are also highlighted, and the corresponding finite element is introduced. Spheroidal negative masks are employed to determine the material within the elements. Seven design variables define each mask. A material density formulation is proposed, and sensitivity analysis for gradient-based optimization is developed. fminconmatlab function is used for the optimization. The efficacy and success of the approach are demonstrated by solving structures and compliant mechanism design problems. Compliance is minimized for the former, whereas a multi-criteria arising due to flexibility and stiffness measures is extremized for optimizing the mechanisms. Convergence of the optimization is smooth. The volume constraint is satisfied and remains active at the end of the optimization.

Three-dimensional topology optimization of natural convection using double multiple-relaxation-time lattice Boltzmann method

A topology optimization method based on the double multiple-relaxation-time (MRT) lattice Boltzmann method and level-set method is developed for 3D natural convection heat transfer with the adjoint double MRT model rigorously derived. The double MRT model with better numerical stability enables the optimization at higher Grashof number (Gr) and larger thermal conductivity ratio. Besides, the model is well-suited for large-scale 3D optimization problems since the ideal linear scaling can be nearly achieved for the whole optimization solution. The forward and optimization models are validated individually by typical problems with relative errors as 2 % and 5 % respectively. Physically reasonable heat sink designs in the shape of “thermal tree” or “thermal flower” are obtained, which can outperform the convectional pin fin and rectangular straight fin with at least 17.8 % lower temperature with Gr ranging from 5.9 × 103 to 1.6 × 106, mainly owing to better organization of fluid flow. Parametric studies find that increasing Gr or decreasing the thermal conductivity ratio results in more contracted heat sinks, while decreasing the penalty factor or increasing the spatial resolution leads to smoother design surface. Anisotropic regularization provides more flexibility in shape control, and adding more solid will enrich the structural details.

A novel numerical manifold method and its application in parameterized LSM-based structural topology optimization

In this paper, a high-efficiency structural topology optimization framework based on combination of numerical manifold method (NMM) and parameterized level set method (PLSM) is established. The NMM uses two cover systems to discretize the model, which can accurately represent the complex boundary of the design domain. A new numerical manifold element has been derived for the application of NMM in PLSM-based optimization. To obtain enhanced accuracy for structural analysis, a multilevel subdivision technique for numerical manifold element generation is presented, and the related numerical integration scheme coupled with the interpolation point selection strategy is provided. For the proposed topology optimization, the new update method of element stiffness matrix is exclusively formulated as well as the calculation method of volume fraction. Some representative structural optimization problems demonstrate that the proposed structural topology optimization method is very effective for two-dimensional (2D) and three-dimensional (3D) structural topology optimization problems.

A parallel geometric multigrid method for adaptive topology optimization

This work presents an efficient parallel geometric multigrid (GMG) implementation for preconditioning Krylov subspace methods solving differential equations using non-conforming meshes for discretization. The approach does not constrain such meshes to the typical multiscale grids used by Cartesian hierarchical grid methods, such as octree-based approaches. It calculates the restriction and interpolation operators for grid transferring between the non-conforming hierarchical meshes of the cycle scheme. Using non-Cartesian grids in topology optimization, we reduce the mesh size discretizing only the design domain and keeping the geometry of boundaries in the final design. We validate the GMG method operating on non-conforming meshes using an adaptive density-based topology optimization method, which coarsens the finite elements dynamically following a weak material estimation criterion. The GMG method requires the generation of the hierarchical non-conforming meshes dynamically from the one used by the adaptive topology optimization to analyze to the one coarsening all the mesh elements until the coarsest level of the mesh hierarchy. We evaluate the performance of the adaptive topology optimization using the GMG preconditioner operating on non-conforming meshes using topology optimization on a fine-conforming mesh as the reference. We also test the strong and weak scaling of the parallel GMG preconditioner with two three-dimensional topology optimization problems using adaptivity, showing the computational advantages of the proposed method.

Three-dimensional plasticity-based topology optimization with smoothed finite element analysis

AbstractThis paper presents a novel plasticity-based formulation for three-dimensional (3D) topology optimization of continuum structures. The proposed formulation addresses the optimization problem by combining mixed rigid-plastic analysis with density-based topology optimization, resulting in a volume minimization approach. Unlike conventional stress-constrained topology optimization methods that rely on linear elastic structural analysis, our developed formulation focuses on enhancing the loading capacity of the designed structures based on the plastic limit theory, leading to more cost-effective designs. To improve computational efficiency, we employ the smoothed finite element technique in our proposed method, enabling the utilization of linear tetrahedral elements for 3D mesh refinement. Moreover, the final formulation of our developed method can be efficiently solved using the advanced primal–dual interior point method, eliminating the need for a separate nonlinear finite element structural analysis. Numerical examples are presented to demonstrate the effectiveness of the proposed approach in offering enhanced design possibilities for continuum structures.

A Problem-Independent Machine Learning (PIML) enhanced substructure-based approach for large-scale structural analysis and topology optimization of linear elastic structures

Structural analysis for structures composed of highly heterogeneous materials which often involves the solution of large-scale linear algebraic equations is very time-consuming even within the linear elastic regime. Furthermore, the tremendous computational cost of iterative large-scale finite element analysis also prevents the widespread use of topology optimization as a powerful design tool especially when the desired design resolution is very high. In order to break the bottleneck hindering the efficient solutions of large-scale structural analysis and design optimization problems, in the present article, a general machine learning (ML) enhanced substructure-based framework is proposed. The essential idea is resorting to the classical substructure-based finite element analysis approach and establishing an implicit mapping between the parameters characterizing the material distribution within a substructure and the corresponding condensed stiffness matrix/numerical shape functions through offline trained deep neural networks. In contrast to most of the existing ML enhanced approaches, the proposed framework is truly independent of the forms of structural geometry, boundary condition and external load, and can be applied to solve various boundary value problems governing by the same type of partial differential equation once the offline training is completed. Armed with modern artificial intelligence techniques, the proposed approach in some sense revives the classical substructure approach in finite element analysis. Compared with the traditional paradigm, it can achieve 104–105 times solution efficiency for tested large-scale examples with satisfactory accuracy. The effectiveness of the approach was also validated for the non-adjoint topology optimization problems, i.e., 3D compliant mechanism design. Finally, to demonstrate the proposed approach’s capability in dealing with extremely large-scale three-dimensional problems, a three-dimensional topology optimization problem with about 109 design variables and 3×109 degrees of freedom (Dofs) is solved on a laptop without resorting to any parallel computing techniques.

Peridynamics topology optimization of three-dimensional structures with surface cracks for additive manufacturing

Additive manufacturing (AM) is an effective approach to fabricating intricate shapes obtained from topology optimization (TO). However, it may cause undesired manufacturing-induced defects/cracks due to high thermal residual stresses. This study proposes a PeriDynamics-enabled three-dimensional Topology Optimization method (PD-TO) for designing structures by considering surface cracks for the AM processes. The PD-TO approach employs a bi-directional evolutionary structural optimization method and uses particle discretization of geometry for mechanical analysis. Crack surfaces are generated by breaking three-dimensional nonlocal interactions of the particles, and thus, during the optimization process, complex multiple structural discontinuities can be diligently modeled. First, the proposed approach is validated by solving benchmark problems without cracks. For each benchmark geometry, the PD-TO analysis is then performed by considering different positions and numbers (single/multiple) of cracks. These analyses extensively investigate and demonstrate the effects of a priori knowledge of residual stress-induced damages/cracks on the optimum topology for additive manufacturing. Besides, the smoothing operation is applied to the optimum designs to transform voxel shapes into AM-friendly smooth surfaces. These geometries are manufactured by an extrusion-based AM process to demonstrate the practical engineering application of the proposed method. Finally, the comparison of numerical results is also supported by the experimental tests conducted on the optimized topologies. Overall, it is confirmed that the PD-TO approach is a viable and accurate optimization tool for additive manufacturing considering possible process-induced damages.

Three-dimensional topology optimization of Ni-YSZ anode for solid oxide fuel cells via multiphase level-set method

In the present study, numerical model is developed to optimize the 3D multiphase electrode microstructure for SOFC, e.g. nickel-yttria-stabilized zirconia (Ni-YSZ) anode. During the optimization, Ni-YSZ microstructure is deformed in order to maximize the total reaction current. Multiphase level-set (MLS) model is applied to represent the Ni-YSZ anode microstructure, and adjoint method is applied for the sensitivity analysis. In addition, in order to improve the computational accuracy and to make the optimized microstructure manufacturable, a radius constraint is imposed to ensure that the local particle radii are kept larger than a target value (Rtarget), where the interface is updated only when the local radius value is larger than Rtarget. It is also confirmed that the optimized microstructure and performance of the Ni-YSZ anode are independent of the initial structures and the phase division schemes. According to the optimization results, the optimized electrochemical performances are improved, and the particle and pore radii of the microstructure successfully remained larger than the target values. The optimized Ni-YSZ microstructures are composed of Ni particles embedded into a YSZ scaffold which has a pillar-like shape along the anode thickness direction.

Three-dimensional topology optimization considering overhang constraints with B-spline parameterization

In this paper, a B-spline parameterization based topology optimization method is presented to design 3D self-supporting structures for additive manufacturing (AM). B-spline parameterization being independent of the FE mesh is used to represent the density field characterizing the material layout of the structure. Heaviside projection and Boolean operations are applied to ensure the obtention of a clear topology configuration within a free-form design domain. Three specific constraints are formulated for AM: (i). The overhang angle constraint imposed on the structural boundaries in an integral form. (ii). The pyramid constraint aiming to avoid undesired inverted-cone features by constraining the values of density field at detection points uniformly distributed in the B-spline parameterized domain. (iii). The boundary gradient constraint imposed to deal with V-shaped cross-section features appearing on the boundary of the design domain by constraining the corresponding direction of the gradient of the B-spline density field. Meanwhile, the minimum feature size control is employed to deal with V-shaped cross-section features inside the design domain. Representative examples are dealt with to demonstrate the effectiveness of the proposed method. Numerical results show that the proposed method is effective to solve 3D problems.

Regional sustainable development impact through sustainable bridge optimization

This paper describes a regional optimal model curve equation to measure and calculate sustainable development assessment impact data related to infrastructure construction in any world region. The article uses a literature review and a case study as research methods—the literature review analyses the importance, practical significance, and current research status of this field. The case study application establishes a scientific algorithm program and a three-dimensional structural topology optimization interactive environment research model. The optimality of the influence equation curve and the sustainable development influence curve of China’s large-scale cable-stayed bridges and regional infrastructure is analysed. This research will fill a gap by solving construction industries' tedious and complicated work and sustainable development assessment. Simultaneously, it will provide a theoretical basis and scientific calculation methods for governments and countries to formulate relevant laws and regulations and study regional climate effects.

An efficient and easy-to-extend Matlab code of the Moving Morphable Component (MMC) method for three-dimensional topology optimization

Explicit topology optimization methods have received ever-increasing interest in recent years. In particular, a 188-line Matlab code of the two-dimensional (2D) Moving Morphable Component (MMC)-based topology optimization method was released by Zhang et al. (Struct Multidiscip Optim 53(6):1243-1260, 2016). The present work aims to propose an efficient and easy-to-extend 256-line Matlab code of the MMC method for three-dimensional (3D) topology optimization implementing some new numerical techniques. To be specific, by virtue of the function aggregation technique, accurate sensitivity analysis, which is also easy-to-extend to other problems, is achieved. Besides, based on an efficient identification algorithm for load transmission path, the degrees of freedoms (DOFs) not belonging to the load transmission path are removed in finite element analysis (FEA), which significantly accelerates the optimization process. As a result, compared to the corresponding 188-line 2D code, the performance of the optimization results, the computational efficiency of FEA, and the convergence rate and the robustness of optimization process are greatly improved. For the sake of completeness, a refined 218-line Matlab code implementing the 2D-MMC method is also provided.

The Influence of Uncertain Loading on Topology-Optimized Designs

The design of structures using topology optimization can improve the structural performance and save material, in turn reducing costs. Using a framework of large-scale, three-dimensional topology optimization implemented by the authors in an open-source multiphysical software, we investigate the influence of uncertain loading on the optimized design. Direct differentiation is used to reveal the relationship between displacements and applied force, giving an efficient and effective tool to postprocess optimized topologies. The developed methodology for the assessment of the sensitivity with respect to applied forces is explored using two three-dimensional examples: the classic MBB cantilever and a cableway pylon. The advantages and limitations of this method are discussed.