Conventional Topology Optimization Methods Research Articles

Designing Porous Structure with Optimized Topology using Machine Learning

Biomedical engineering relies on topology optimization to refine material distribution, crucial for lightweight, high-performance prostheses and orthoses. Advanced manufacturing techniques like additive manufacturing can then be used to create these intricate designs layer by layer, ensuring precision and customization. However, conventional numerical simulation-based topology optimization methods can be time-consuming and resource-intensive, especially as the design domain expands. To overcome this issue, machine learning models are investigated for their ability to perform topology optimization. The results indicate a significant decrease in computation time, along with comparable optimization performance to conventional methods.

Mixed topology optimization: A self-guided boundary-independent approach for power sources

As the use of electrochemical devices becomes more prevalent, advanced optimization techniques, such as topology optimization, are being employed to improve their performance. Among various electrochemical systems, power sources have an intrinsic best operating point that corresponds to the maximum output power. This study proposes a mixed topology optimization approach to enhance the performance of these systems by a simultaneous modification of electrode structure and the working condition. In contrast to the conventional approaches, the proposed method focuses on enhancement of the maximum power point. Thanks to its self-guidance feature, this technique outperforms conventional topology optimization methods and eliminates the need for a prior decision on the optimization starting point. Therefore, the proposed approach has a significant potential for adapting the material distribution to the best working condition. The proposed approach enables more effective topological optimization and takes the utilization of topology optimization in power sources to the next level.

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.

Learning topology optimization process via convolutional long‐short‐term memory autoencoder‐decoder

AbstractThis article proposed an autoencoder‐decoder architecture with convolutional long‐short‐term memory (ConvLSTM) cell for the purpose of learning topology optimization iterations. The overall topology optimization process is treated as time‐series data, with each iteration as a single step. The first few steps are fed into the encoder to generate encoder embedding, which is fed into the decoder. The decoder uses the encoder embedding as input and generates the result at each future step until the end of iteration. To train the proposed neural network, a large dataset is generated by a conventional topology optimization method, that is, solid isotropic material with penalization for intermediate densities, with randomly picked boundary conditions, load conditions, and volume constraints. Unlike other deep learning models introduced before, the proposed method can learn each topology optimization step iteratively and present the full optimization path. Furthermore, the proposed method can be extended to give solutions to unseen boundary and load conditions with a significant reduction in computation cost in a little sacrifice on the performance of the optimum design.

Thermal design of microchannel heat sinks using a contour extraction based on topology optimization (CEBTO) method

The Topology Optimization (TO) has been proved to be an effective method for designing Microchannel Heat Sinks (MCHS). However, due to the relatively high computational cost and poor numerical stability, it is still challenging to design the thermo-fluid TO under high Reynolds number (Re). In this work, we propose a Contour Extraction Based on Topology Optimization (CEBTO) method to optimize the geometry of the fins based on a threshold velocity (Ut, proportional to the maximum velocity Umax) and improve the hydrothermal performance of MCHS under high Re. The 3D conjugate heat transfer numerical models are established to study the fluid flow and heat transfer characteristics of the MCHS generated by the conventional TO method and the CEBTO method. The results showed that due to the elimination of the area of stagnation zones under high Re, the CEBTO-generated heat sinks have a better interaction of fluid with the walls and more uniform velocity distribution compared to the TO-generated ones. However, the elimination of the stagnation zones area also leads to the decrease of channel width, which increases the friction loss in CEBTO-generated heat sinks. Nevertheless, the CEBTO-generated heat sinks have better overall performance compared to the TO designs owing to the smaller total entropy generation rate. For example, the CEBTO-I (Ut=28%Umax) exhibits the best overall performance with 10.1% increment of flow entropy generation rate and 7.1% reduction of thermal entropy generation rate as compared to those of TO-I (i.e., a TO-generated heat sink with alveoli-like shape), leading to the augment entropy generation number of 0.92 (Re = 1715). Finally, the CEBTO-I (Ut=28%Umax) was manufactured by micro milling technology and its hydrothermal performance was investigated experimentally to validate the accuracy of numerical simulation. The pressure drop and temperature of the CEBTO-generated heat sink in the experiment agree well with the numerical simulation.

Heat transfer augmentation in microchannel heat sink based on isogeometric topology optimization framework

This paper focuses on a design problem of branching microchannel heat sink based on solving the thermal-fluid problem in isogeometric analysis approach. A brand new topology optimization framework is proposed to determine the distribution of the microchannels, which comprises two coupled computational layers: the upper layer for channel geometry representation and the lower layer for cooling analysis. In the upper layer, several flexible components fitting two Bézier curves as the central line describe the geometry explicitly, which is quite different from the implicit and conventional level-set description ways of topology optimization, fully releasing the deformation ability of the components. By moving, rotating, deforming, bending, overlapping and merging the level-set-based components, the channel geometry gene-rates and is ready to be projected onto the lower layer for the physical field analysis purpose. The lower layer is discretized using a NURBS patch, and NURBS-based isogeometric analysis is adopted to solve the coupled thermal-fluid problem in the structure and calculate the structural sensitivity so as to drive the upper layer to iterate. Comparing with the conventional topology optimization methods, the proposed method has obvious advantages in terms of computational effectiveness and geometry description. A design of multi-layer cold plate is constructed under this framework, aiming at obtaining the best heat dissipation effect under the constraints of volume dissipation and pressure drop. In the final, simulations and experiments are performed to demonstrate these advantages of this framework.

TONR: An exploration for a novel way combining neural network with topology optimization

The rapid development of deep learning has opened a new door to the exploration of topology optimization methods. The combination of deep learning and topology optimization has become one of the hottest research fields at the moment. Different from most existing work, this paper conducts an in-depth study on the method of directly using neural networks (NN) to carry out topology optimization. Inspired by the idea from the field of “Inverting Representation of Image” and “Physics-Informed Neural Network”, a topology optimization via neural reparameterization framework (TONR) that can solve various topology optimization problems is formed. The core idea of TONR is Reparameterization, which means the update of the design variables (pseudo-density) in the conventional topology optimization method is transformed into the update of the NN’s parameters. The sensitivity analysis in the conventional topology optimization method is realized by automatic differentiation technology. With the update of NN’s parameters, the density field is optimized. Some strategies for dealing with design constraints, determining NN’s initial parameters, and accelerating training are proposed in the paper. In addition, the solution of the multi-constrained topology optimization problem is also embedded in the TONR framework. Numerical examples show that TONR can stably obtain optimized structures for different optimization problems, including the stress-constrained problem, structural natural frequency optimization problems, compliant mechanism design problems, heat conduction system design problems, and the optimization problem of hyperelastic structures. Compared with the existing methods that combine deep learning with topology optimization, TONR does not need to construct a dataset in advance and does not suffer from structural disconnection. The structures obtained by TONR can be comparable to the conventional methods.

A novel isogeometric topology optimization framework for planar compliant mechanisms

In this article, we focus on a design problem of planar compliant mechanisms within the framework of isogeometric topology optimization. An integrated model is developed to identify the optimal deformation transferring path for precise motion output. The model comprises two coupled computational layers: the upper layer for geometry representation and the lower layer for sensitivity calculation. In the upper layer, the structural geometries are described explicitly by parameterized level-set surfaces, which is quite different from the implicit description way of topology optimization, giving great advantages to improve the identification accuracy of elaborate structures (e.g., flexible hinges), which are usually encountered in compliant mechanism systems. By moving, deforming, overlapping and merging these level-set surfaces, the generated shape can be projected onto the lower layer which is discretized using a NURBS patch, and NURBS-based isogeometric analysis is adopted to calculate the structural sensitivity which is then fed back to the upper layer for driving new iteration. The proposed method has the higher ability to search the optimal topology with complex kinematic behavior with respect to the conventional topology optimization methods in terms of computational effectiveness and numerical robustness. Design formulation of compliant mechanisms is constructed under the framework of the proposed method, in which the Jacobian and stiffness matrices of compliant mechanism are optimized simultaneously to achieve kinematic and stiffness requirement respectively. Three typical benchmark problems (e.g., displacement inverter, amplifier, and redirector) are tested to demonstrate these advantages.

Generating three-dimensional structural topologies via a U-Net convolutional neural network

In this paper, we propose a deep learning based neural network to generate three-dimensional structural topologies in an efficient way. The method consists of three steps. First, conventional three dimensional solid isotropic microstructures with penalization (SIMP) method is utilized to generate datasets consisting of various domain sizes and boundary conditions. Second, a deep learning neural network based on U-Net architecture is constructed and trained by the generated datasets. Third, by feeding new cases with different domain sizes and boundary conditions into the network, near optimal results can be directly obtained without any needs of optimization iterations and finite element analysis. Compared with conventional topology optimization methods as well as recent development of machine learning approaches, our proposed method has two advantages: (1) the design boundary conditions are directly mapped with the 3D optimized structures such that no further dependency on the conventional topology optimization algorithm is required once the model is trained, and (2) topology optimization problems with variable design domain sizes can be supported instead of requiring its size to be fixed as the input of the neural network. Experiments demonstrate that once trained, our deep learning based topology optimization method can realize near optimal three dimensional topology prediction using negligible calculation cost.

Efficient topology optimization based on DOF reduction and convergence acceleration methods

This paper proposes a highly efficient topology optimization using two accelerated methods, which can reduce the degrees of freedom (DOFs) of the finite element equations and accelerate the iteration convergence of the topology optimization. For the DOF reduction, a method based on the empty elements and the displacement change during the topology iterations is presented to remove the DOFs from the finite element equations. For the convergence acceleration, a gray-scale suppression method is proposed to accelerate the polarization of design variables which accelerates the iteration convergence of the topology optimization. Three numerical examples including 2D and 3D cases are tested, and the results show that the proposed method can significantly improve the efficiency of the topology optimization and obtain the optimization results with the same accuracy. The computational time is only about 7% - 29% compared to the conventional topology optimization method.

A topology optimization method of robot lightweight design based on the finite element model of assembly and its applications.

Topology optimization is a widely used lightweight design method for structural design of the collaborative robot. In this article, a topology optimization method for the robot lightweight design is proposed based on finite element analysis of the assembly so as to get the minimized weight and to avoid the stress analysis distortion phenomenon that compared the conventional topology optimization method by adding equivalent confining forces at the analyzed part's boundary. For this method, the stress and deformation of the robot's parts are calculated based on the finite element analysis of the assembly model. Then, the structure of the parts is redesigned with the goal of minimized mass and the constraint of maximum displacement of the robot's end by topology optimization. The proposed method has the advantages of a better lightweight effect compared with the conventional one, which is demonstrated by a simple two-linkage robot lightweight design. Finally, the method is applied on a 5 degree of freedom upper-limb exoskeleton robot for lightweight design. Results show that there is a 10.4% reduction of the mass compared with the conventional method.

Transfer Learning Through Deep Learning: Application to Topology Optimization of Electric Motor

This article proposes the use of transfer learning for the deep neural network to reduce the computing cost of the topology optimization of electric motors based on a genetic algorithm (GA). The average torque and torque ripple values are shown to be accurately inferred by the transfer learning with small learning data. The individuals on the Pareto front are only evaluated by the finite-element method, while others are fast evaluated only by convolutional neural networks (CNNs). The proposed method makes it possible to reduce the computing cost to less than 15% of the conventional topology optimization method.

Method for generating manufacturable, topology-optimized parts for Laminated Layer Manufacturing

The great flexibility provided by generative manufacturing offers immense potential for the structural component optimization. However, this freedom needs to be utilized by appropriate CAx tools. In this paper, we present a novel slicing method, enabling the automatic generation of topology-optimized Laminated Layer Manufacturing (LLM) parts. With this method a part of arbitrary shape can be automatically sliced and equipped with a uniform virtual grid that is suitable for manufacturing. The optimal density distribution is then mapped onto the individual slices and locally approximated through the introduction of variably shaped cavities. By using 3D modeling techniques all LLM sheets are modeled parametrically and assembled automatically. To verify the functionality of the presented method, it is applied to two practical parts and compared with a conventional topology optimization method.

A material-field series-expansion method for topology optimization of continuum structures

This paper proposes a new topology optimization method based on a material-field topology description and a reduced series expansion. Not only does the proposed method greatly reduce the number of design variables, it also inherently avoids checkerboard patterns and the mesh dependency of conventional density-based topology optimization methods. In the present method, the structural topology is represented by a bounded material field with spatial dependency. The correlation length is introduced here to control the length-scale of the topology distribution. After approximating the bounded material field as a linear function of a reduced set of undetermined coefficients by using a series expansion, the topology optimization problem is constructed in the form of finding the optimal coefficients of the material field with the minimum structural compliance. A standard, gradient-based algorithm incorporating the design sensitivity information is then used to solve the optimization problem effectively. Several examples are given to demonstrate the effectiveness and applicability of the proposed topology optimization method.

Multistage topology optimization of induction heating apparatus in time domain electromagnetic field with magnetic nonlinearity

Purpose The purpose of this paper is to solve efficiently the topology optimization (TO) in time domain problem with magnetic nonlinearity requiring a large-scale finite element mesh. As an actual application model, the proposed method is applied to induction heating apparatus. Design/methodology/approach To achieve TO with efficient computation time, a multistage topology is proposed. This method can derive the optimum structure by repeatedly reducing the design domain and regenerating the finite element mesh. Findings It was clarified that the structure derived from proposed method can be similar to the structure derived from the conventional method, and that the computation time can be made more efficient by parameter tuning of the frequency and volume constraint value. In addition, as a time domain induction heating apparatus problem of an actual application model, an optimum topology considering magnetic nonlinearity was derived from the proposed method. Originality/value Whereas the entire design domain must be filled with small triangles in the conventional TO method, the proposed method requires finer mesh division of only the stepwise-reduced design domain. Therefore, the mesh scale is reduced, and there is a possibility that the computation time for TO can be shortened.

Topology optimization of binary microstructures involving various non-volume constraints

In this paper, we use the Topology Optimization of Binary Structures (TOBS) method recently developed by Sivapuram and Picelli (2018) for microstructural optimization. This is the first work in topology optimization addressing various non-volume microstructural constraints with discrete (0/1) design variables. The objective and constraint functions are linearized at each iteration, and the obtained linear problem is solved through Integer Linear Programming (ILP) using sensitivities computed from asymptotic homogenization. A periodic filter is used to make the optimized solutions checkerboard-free and mesh-independent. Volume minimization problems subject to elastic and thermal constraints are considered. The examples consider different sets of constraints, including bulk and shear moduli, square/cubic symmetry, isotropy, thermal conductivity and a combination of them in two and three dimensions. The non-volume constraints are treated explicitly, i.e., without the use of Lagrange multiplier/penalty as used in conventional gradient-based binary topology optimization methods (Huang and Xie, 2010). The resulting microstructures are observed to be convergent in all the examples presented and in agreement with the Hashin–Shtrikman bounds.

Design of complex bone internal structure using topology optimization with perimeter control

Large facial bone loss usually requires patient-specific bone implants to restore the structural integrity and functionality that also affects the appearance of each patient. Titanium alloys (e.g., Ti-6Al-4V) are typically used in the interfacial porous coatings between the implant and the surrounding bone to promote stability. There exists a property mismatch between the two that in general leads to complications such as stress-shielding. This biomechanical discrepancy is a hurdle in the design of bone replacements. To alleviate the mismatch, the internal structure of the bone replacements should match that of the bone. Topology optimization has proven to be a good technique for designing bone replacements. However, the complex internal structure of the bone is difficult to mimic using conventional topology optimization methods without additional restrictions. In this work, the complex bone internal structure is recovered using a perimeter control based topology optimization approach. By restricting the solution space by means of the perimeter, the intricate design complexity of bones can be achieved. Three different bone regions with well-known physiological loadings are selected to illustrate the method. Additionally, we found that the target perimeter value and the pattern of the initial distribution play a vital role in obtaining the natural curvatures in the bone internal structures as well as avoiding excessive island patterns.

Developing Topology Optimization with Additive Manufacturing Constraints in ANSYS®

Topology Optimization is an effective technique to optimize structural design and to minimize waste. Additive manufacturing (AM) is potentially a great approach to fabricate the complex organic geometric features of these optimized designs. However, topology optimization without considering manufacturing constrains may result in topologies that are expensive and difficult to manufacture. In order to satisfy the constraints imposed by AM processes, this paper proposes a topology optimization methodology using the compliance minimization approach with a constraint on overhanging surfaces. Compliance of a body is approximated through a finite element analysis procedure. To constrain the gradient and converge towards a structure without overhang surfaces, a function has been developed to estimate the overhang sensitivities. The gradient of the compliance with respect to densities are penalized using the determined overhang sensitivities. To introduce a reliable platform for research, as well as solving the resulting constrained topology optimization problems, a topology optimizer has been developed in ANSYS® using ANSYS Parametric Design Language (APDL). Using a conventional topology optimization method, the results of two different case studies solved by the developed APDL program are shown. Finally, an implementation of the proposed topology optimization methodology with AM constraints is used to solve a problem. The result is a structure that does not need support material to be manufactured.

Design optimisation for an additively manufactured automotive component

The aim of this paper is to investigate the design optimization and additive manufacture of automotive components. A Titanium brake pedal processed through Selective Laser Melting (SLM) is considered as a test case. Different design optimisation techniques have been employed including topology optimization and lattice structure design. Rather than using a conventional topology optimization method, a recently developed topology optimization method called Iso-XFEM is used in this work. This method is capable of generating high resolution topology optimised solutions using isolines/isosurfaces of a structural performance criterion and eXtended Finite Element Method (XFEM). Lattice structure design is the other technique used in this work for the design of the brake pedal. The idea is to increase the stability of the brake pedal to random loads applied to the foot pad area of the pedal. The use of lattice structures can also significantly reduce the high residual stress induced during the SLM process. The results suggest that the integration of the design optimization techniques with a metal additive manufacturing process enables development of a promising tool for producing lightweight energy efficient automotive components.

Metamodeling approach for concurrent optimization considering structural topology and its circumstances of mechanical component

Conventional topology optimization method only optimizes a single machine component structure while its circumstances are fixed. Concurrent optimization of topology and circumstances widens the effective range of topology optimization in mechanical design. We have developed a method for the concurrent optimization, which includes problem decomposition and metamodeling. However, the method requires further verification, especially, for cases where a relationship between circumstance variables and results of topology optimization becomes complex. In this research, we examine whether the method can be successfully applied to complex case if a metamodeling approach is switched. Case studies of three kinds of metamodeling techniques are performed. In the case studies, we test following metamodeling approaches: polynomial approximation, radial basis function(RBF) interpolation, polynomial approximation with spatial reduction. It shows that the proposed method can be applied even for complex cases by appropriate selection of metamodeling.