3D Topology Optimization Research Articles

FreeTO - Freeform 3D topology optimization using a structured mesh with smooth boundaries in Matlab

Topology optimization has revolutionized the design of structures for various applications, particularly with the advancement of additive manufacturing. However, existing open-source codes for topology optimization have limitations, such as restricted domain initialization and lack of a CAD output after optimization. A novel open-source Matlab code, FreeTO, is presented, and it addresses these limitations by enabling the initialization of 3D arbitrary geometries and providing an STL file post-optimization. FreeTO utilizes a structured mesh and a smooth-edge (boundary) algorithm to generate smooth topological boundaries. The code is demonstrated through six practical design cases, showcasing its effectiveness in compliance minimization, compliant mechanisms, and self-supporting problems. FreeTO offers a user-friendly, all-in-one topology optimization package, making it an invaluable tool for educators, researchers, and practitioners. Future developments will focus on eliminating a few geometrical deviations in the optimized topologies, incorporating speedups, and extending the code to apply to more applications.

3D topology optimization design of air natural convection heat transfer fins

This study focuses on the fin-tube heat exchanger and utilizes topology optimization methods to design a completely new fin structure. In this optimization process, complete Navier-Stokes (N-S) equations were used to describe the steady-state incompressible flow, and the Boussinesq model was employed to simulate natural convection. The flow equations were coupled with the heat convection–diffusion equation to achieve topology optimization for natural convection heat transfer. Topology optimization was conducted using density-based optimization methods, and interpolation was performed on the permeability and conductivity of the distributed materials. Given the initial fin structure, an interpolation progressive approach was adopted to obtain a new “ airfoil-shaped” optimized structure through density-based topology optimization method for natural convection. The new structure enhances the convective heat transfer by perforating the fins. The perforations are mainly concentrated in the central region of the heat exchanger and the upper half of the fins. The new structure, compared to the prototype structure, not only has a reduced volume but also exhibits a decrease in convective thermal resistance within a larger range of heat flux densities, as revealed by CFD simulations. Moreover, as the heat flux density increases, the rate of reduction in convective thermal resistance shows an upward trend for the new structure compared to the prototype structure.

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.

Large-scale 3D multiphysics topology optimization of flow–heat–structural models including an islands constraint

This article demonstrates a large-scale 3D topology optimization problem formulation for components in need of internal cooling owing to surrounding hot gas flow. A conjugate heat transfer model is used and the goal of the optimization problem is to maximize the thermal performance subject to a coolant consumption limit. As a model problem, the interior design of a gas turbine guide-vane-like geometry is considered. High-quality finite element meshes are generated automatically by means of a voxelization method. A structural compliance constraint for thermo-mechanical loads, and a constraint for the suppression of free-floating structural parts, are included in the problem formulation. For the latter, an analytical expression for the constraint limit is derived. Several numerical examples indicate that the proposed problem formulation is able to generate interesting conceptual designs for interior-vane-cooling solutions.

Matlab codes for 3D topology optimization of multi-material piezoelectric actuators and energy harvesters

Matlab codes for 3D topology optimization of multi-material piezoelectric actuators and energy harvesters

Designing a TPMS metamaterial via deep learning and topology optimization

Data-driven models that act as surrogates for computationally costly 3D topology optimization techniques are very popular because they help alleviate multiple time-consuming 3D finite element analyses during optimization. In this study, one such 3D CNN-based surrogate model for the topology optimization of Schoen’s gyroid triply periodic minimal surface unit cell is investigated. Gyroid-like unit cells are designed using a voxel algorithm and homogenization-based topology optimization codes in MATLAB. A few such optimization data are used as input–output for supervised learning of the topology-optimization process via the 3D CNN model in Python code. These models could then be used to instantaneously predict the optimized unit cell geometry for any topology parameters. The high accuracy of the model was demonstrated by a low mean square error metric and a high Dice coefficient metric. The model has the major disadvantage of running numerous costly topology optimization runs but has the advantages that the trained model can be reused for different cases of TO and that the methodology of the accelerated design of 3D metamaterials can be extended for designing any complex, computationally costly problems of metamaterials with multi-objective properties or multiscale applications. The main purpose of this paper is to provide the complete associated MATLAB and PYTHON codes for optimizing the topology of any cellular structure and predicting new topologies using deep learning for educational purposes.

3D Topology Optimization and Mesh Dependency for Redesigning Locking Compression Plates Aiming to Reduce Stress Shielding.

Current fixation plates for bone fracture treatments are built with biocompatible metallic materials such as stainless steel, titanium, and its alloys (e.g., Ti6Al4V). The stiffness mismatch between the metallic material of the plate and the host bone leads to stress shielding phenomena, bone loss, and healing deficiency. This paper explores the use of three dimensional topology-optimization, based on compliance (i.e., strain energy) minimization, reshaping the design domain of three locking compression plates (four-screw holes, six-screw holes, and eight-screw holes), considering different volume reductions (25, 45, and 75%) and loading conditions (bending, compression, torsion, and combined loads). A finite-element study was also conducted to measure the stiffness of each optimized plate. Thirty-six designs were obtained. Results showed that for a critical value of volume reductions, which depend on the load condition and number of screws, it is possible to obtain designs with lower stiffness, thereby reducing the risk of stress shielding.

Patient-specific 3D-printed Splint for Mallet Finger Injury.

Despite the frequency of mallet finger injuries, treatment options can often be costly, time-consuming, and ill-fitted. Three-dimensional (3D) printing allows for the production of highly customized and inexpensive splints, which suggests potential efficacy in the prescription of casts for musculoskeletal injuries. This study explores how the use of engineering concepts such as 3D printing and topology optimization (TO) can improve outcomes for patients. 3D printing enables the direct fabrication of the patient-specific complex shapes while utilizing finite element analysis and TO in the design of the splint allowed for the most efficient distribution of material to achieve mechanical requirements while reducing the amount of material used. The reduction in used material leads to significant improvements in weight reduction and heat dissipation, which would improve breathability and less sweating for the patient, greatly increasing comfort for the duration of their recovery.

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.

Multiscale topology optimization for the design of spatially-varying three-dimensional lattice structure

This paper introduces three dimensional (3D) topology optimization specifically tailored for designing spatially-varying primitive-cubic (CP) type lattice structures. The developed design process consists of three steps: pre-processing, main processing, and post-processing. In the pre-processing step, a surrogate model between lattice geometry variables and effective elasticity tensor is constructed using an artificial neural network. This surrogate model is essential because it enables the quick calculation of the effective elasticity tensor of lattice microstructure. In the subsequent main processing step, multiscale topology optimization is performed. During this step, lattice microstructure size and rotation design variables are optimized together with the macrostructure density design variables. Macrostructures are designed using a well-established three-field density scheme with a SIMP formulation. Formulations for spatially-varying 3D lattice microstructure are newly developed based on a unit quaternion rotation representation scheme. In the final post-processing step, a spatially-varying microstructure is restored using a de-homogenization scheme, newly developed particularly for the primitive-cubic (CP) type 3D lattice structures. The effectiveness and robustness of the proposed formulations are validated through three design examples for the compliance minimization problem. Moreover, a designed lattice structure is fabricated using an additive manufacturing machine.

Geometric Complexity Control in Topology Optimization of 3D-Printed Fiber Composites for Performance Enhancement.

This paper investigates the impact of varying the part geometric complexity and 3D printing process setup on the resulting structural load bearing capacity of fiber composites. Three levels of geometric complexity are developed through 2.5D topology optimization, 3D topology optimization, and 3D topology optimization with directional material removal. The 3D topology optimization is performed with the SIMP method and accelerated by high-performance computing. The directional material removal is realized by incorporating the advection-diffusion partial differential equation-based filter to prevent interior void or undercut in certain directions. A set of 3D printing and mechanical performance tests are performed. It is interestingly found that, the printing direction affects significantly on the result performance and if subject to the uni direction, the load-bearing capacity increases from the 2.5D samples to the 3D samples with the increased complexity, but the load-bearing capacity further increases for the 3D simplified samples due to directional material removal. Hence, it is concluded that a restricted structural complexity is suitable for topology optimization of 3D-printed fiber composites, since large area cross-sections give more degrees of design freedom to the fiber path layout and also makes the inter-layer bond of the filaments firmer.

An Efficient Code for the Multi-Material Topology Optimization of 2D/3D Continuum Structures Written in Matlab

This paper presents an efficient and compact Matlab code for 2D and 3D topology optimization of multi-materials. We extend a classical 88-line-based educational code to the multi-material problem using the mapping-based interpolation function, with which each material is modeled equally and can obtain a clear 0, 1 result for each material of the optimized structures that can be manufactured easily. A solution of topology optimization of multi-materials with minimum compliance under volume constraints is documented, including the details of the optimization model, filtering, projection, and sensitivity analysis procedures. Several numerical examples are also conducted to demonstrate the effectiveness of the code, and the influence of the parameter on the optimized results is also analyzed. Complete 2D and 3D Matlab codes are provided.

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.

Investigation on topology-optimized compressor piston by metal additive manufacturing technique: Analytical and numeric computational modeling using finite element analysis in ANSYS

Abstract Air compressors are widely used in factories to power automation systems and store energy. Several studies have been conducted on the performance of reciprocating and screw compressors. Advancements in design and manufacturing techniques, such as generative design and topology optimization, are leading to improved performance and turbomachinery growth. This work presents a methodology to design and manufacture air compressor pistons using topology optimization and metal additive manufacturing. The existing piston is converted to 3D CAD data and topology optimization is conducted to reduce material in stress concentration regions. Thermal and mechanical loads are considered in boundary conditions. The results show reduced material and improved efficiency, which is validated using ANSYS fluent. The optimized 3D model of the piston is too complex for conventional subtractive manufacturing, so laser sintering 3D printing is proposed. Honeycomb pattern infill patterns are used in 3D printing. This investigation is a step toward researching similar methods in other reciprocating compressor components such as cylinder, cylinder head, piston pins, crankshaft, and connecting rods, which will ultimately lead to improved compressor efficiency.

Parallel 3D topology optimization with multiple constraints and objectives

Parallel 3D topology optimization with multiple constraints and objectives

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.

Multi-constrained topology optimization of prefabricated joints in large-span latticed structures

In recent years, topology optimization (TO) has shown great potential in the optimal design of engineering structures. In this study, a parameterized level set (PLS) method capable of solving the compliance minimization problem with both stress and volume constraints is proposed to improve the design efficiency of nodal connections in large-span latticed structures. 3D topology optimization of assembled hub (AH) joints is conducted based on the proposed method. The performances of the optimal designs are compared with those of the traditional joints in terms of volume, stress distribution and compliance value. Different connection types and complex loading conditions are considered to satisfy the requirements of engineering practice. It is indicated that the proposed PLS method exhibits good stability and high computing efficiency during the optimization of AH joints. Compared with the study cases with a single volume constraint, the maximum von Mises stress is effectively controlled by simultaneously introducing the stress and volume constraints. In addition, the variation in the volume constraint has much greater influence on the compliance results than the stress constraint. The compliance continuously increases when the volume constraint decreases from 100% to 40%, while the difference in compliance is less than 1.5% with or without stress constraints. It is demonstrated that the performances of AH joints are significantly improved after the TO design. The reduction rates of mass are in the range of 25.2 % to 35.2 %, and the compliance value is reduced by 0.5 % to 7.6 % for all the study cases.

Towards closed-walled designs in topology optimization using selective penalization

3D topology optimization using a density approach with penalization usually produces truss-like, open-walled structures. The coarser the mesh, the smaller the volume fraction, and the faster the penalization is increased, the more pronounced this effect tends to be. However, closed-walled designs are often more efficient and have other beneficial properties. For instance, closed walls can contribute to achieving self-supporting designs for additive manufacturing that potentially require fewer sacrificial support structures than truss-like designs. This paper presents a two-step optimization procedure for generating closed-walled designs using coarse meshes. The first step takes the usual Eulerian approach of performing a SIMP-based topology optimization on a fixed mesh. To keep thin geometrical features, like walls with a thickness below element size, penalization is switched off deliberately where the formation of such features is detected. Adopting a Lagrangian description, intermediate densities still present in the optimized design are subsequently eliminated in a second step by shrinking each element according to its density. By analogy with volumetric thermal contraction, this is accomplished by solving a fictitious thermo-elastic problem where the temperature has been replaced by a density expression. The outcome is a morphed mesh with a somewhat smoothed surface and a volume close to the specified material volume limit. This body-fitted representation of the design considerably simplifies the final conversion into a manufacturable CAD-type geometry. The two-step optimization procedure is applied to a cantilever, a torsion rod, and a disk reinforcement benchmark problem. Optimized designs are closed-walled and show very good agreement to those found for much finer meshes. Problem-specific stiffness improvements over truss-like designs between 6% and almost 30% were achieved and confirmed the findings previously reported by other authors.

A 3D Structure Mapping-Based Efficient Topology Optimization Framework

Abstract It is well known that the computational cost of classic topology optimization (TO) methods increases rapidly with the size of the design problem because of the high-dimensional numerical simulation required at each iteration. Recently, the technical route of replacing the TO process with artificial neural network (ANN) models has gained popularity. These ANN models, once trained, can rapidly produce an optimized design solution for a given design specification. However, the complex mapping relationship between design specifications and corresponding optimized structures presents challenges in the construction of neural networks with good generalizability. In this paper, we propose a new design framework that uses deep learning techniques to accelerate the TO process. Specifically, we present an efficient topology optimization (ETO) framework in which structure update at each iteration is conducted on a coarse scale and a structure mapping neural network (SMapNN) is constructed to map the updated coarse structure to its corresponding fine structure. As such, fine-scale numerical simulations are replaced by coarse-scale simulations, thereby greatly reducing the computational cost. In addition, fragmentation and padding strategies are used to improve the trainability and adaptability of SMapNN, leading to a better generalizability. The efficiency and accuracy of the proposed ETO framework are verified using both benchmark and complex design tasks. It has been shown that with the SMapNN, TO designs of millions of elements can be completed within a few minutes on a personal computer.

JAX-FEM: A differentiable GPU-accelerated 3D finite element solver for automatic inverse design and mechanistic data science

This paper introduces JAX-FEM, an open-source differentiable finite element method (FEM) library. Constructed on top of Google JAX, a rising machine learning library focusing on high-performance numerical computing, JAX-FEM is implemented with pure Python while scalable to efficiently solve problems with moderate to large sizes. For example, in a 3D tensile loading problem with 7.7 million degrees of freedom, JAX-FEM with GPU achieves around 10× acceleration compared to a commercial FEM code depending on platform. Beyond efficiently solving forward problems, JAX-FEM employs the automatic differentiation technique so that inverse problems are solved in a fully automatic manner without the need to manually derive sensitivities. Examples of 3D topology optimization of nonlinear materials are shown to achieve optimal compliance. Finally, JAX-FEM is an integrated platform for machine learning-aided computational mechanics. We show an example of data-driven multi-scale computations of a composite material where JAX-FEM provides an all-in-one solution from microscopic data generation and model training to macroscopic FE computations. The source code of the library and these examples are shared with the community to facilitate computational mechanics research. Program summaryProgram Title: JAX-FEMCPC Library link to program files:https://doi.org/10.17632/hgwshjbcw6.1Developer's repository link:https://github.com/tianjuxue/jax-am/tree/main/jax_am/femLicensing provisions: GPLv3Programming language: PythonNature of problem: Implementation of the finite element method (FEM) with several appealing features that classic FEM software typically does not have: realized with pure Python; running on CPU/GPU; differentiable for solving PDE-constrained optimization problems; seamless integration with machine learning.Solution method: Our framework JAX-FEM is based on Google JAX, a high-performance numerical computing library with automatic differentiation features and supporting both CPU/GPU. Unlike classic FEM software written in Fortran or C/C++, JAX-FEM is implemented with pure Python and can easily be installed as a Python package. We demonstrate our software by solving problems including forward PDE prediction, inverse design/optimization, and data-driven analysis.