Density-based Topology Optimization Research Articles

A multi-period topology and design optimization approach for district heating networks

The transition to modern, low-carbon district heating creates a growing need for scalable, automated design tools that accurately capture the spatial and temporal details of heating network operations. This paper presents an automated design approach for the optimal design of district heating networks that combines scalable density-based topology optimization with a multi-period approach. In this way, temporal variations in demand, supply, and heat losses can be taken into account while optimizing the network design based on a nonlinear physics model. The transition of the automated design approach from worst-case to multi-period shows a design progression from separate branched networks to a single integrated meshed network topology connecting all producers. These integrated topologies emerge without imposing such structures a priori. They increase network connectivity, and allow for more flexible shifting of heat loads between different producers and heat consumers, resulting in more cost-effective use of heat. In a case study, this integrated design resulted in an increase in waste heat share of 42.8 % and a subsequent reduction in project cost of 17.9 %. We show how producer unavailability can be accounted for in the automated design at the cost of a 3.1 % increase in the cost of backup capacity. The resulting optimized network designs of this approach connect multiple low temperature heat sources in a single integrated network achieving high waste heat utilization and redundancy, highlighting the applicability of the approach to next-generation district heating networks.

Topology Optimization based Integrated Design of Structure-Foundation System

This study introduces a novel topology optimization method with the goal of achieving an integrated design for offshore infrastructure. A key feature of this method is its capability to eliminate artificial boundaries between the design of the above-ground support structure and underlying soil foundation, thus facilitating a more efficient and effective design solution. In contrast to traditional stress-based topology optimization methods, the proposed approach combines limit analysis with density-based topology optimization techniques. Formulated as a volume minimization task, it targets the structure’s bearing capacity, ensuring that the external force applied is precisely matches the bearing capacity of the system obtained in the design. Furthermore, it is demonstrated that the proposed method exhibits favorable convergence properties within a prescribed number of iterations, indicating a steady progression towards an optimized solution.

A limit analysis-based topology optimisation method for geostructure design

Geostructures, vital for the progress of civilisation, often face inefficiencies and suboptimal performance due to the lack of optimisation in current designs. Achieving cost-efficiency in geostructure design involves optimising material usage while considering practical construction aspects. While size and shape optimisations are common in geostructure design, the application of topology optimisation remains underexplored. This paper addresses this gap by introducing a novel topology optimisation method for three-dimensional geostructure design. The method integrates mixed limit analysis and density-based topology optimisation theories, allowing for two-material design focused on the ultimate bearing capacity of the geostructure. The innovation resides in aligning the applied external load in the topology optimisation process with the ultimate load that the designed geostructure can sustain. The robustness of the proposed method is exemplified through its application to the design of an embankment and soil foundation, showcasing its potential to enhance the efficiency and performance of geostructures. This research contributes to the advancement of geostructure design practices, ultimately promoting sustainable and resilient infrastructure development.

Density-based topology optimization for minimizing von Mises stress using the modified optimality criteria method and demonstration by tensile testing

The authors propose a modified optimality criteria (OC) method that incorporates the concept of Newton's method for the OC method. The modified OC method has fewer arbitrary parameters set by the engineer. In this study, density-based topology optimization for minimizing von Mises stress (VMS) using a modified OC method is performed. In this problem, the value of the second-order derivative is not always positive semidefinite. Thus, the concept of the modified Newton's method is incorporated into the modified OC method. In conclusion, the performance function descends more quickly and with less dependence on parameter settings than when using the OC method. A mathematical proof of the convergence properties of the modified OC method, based on the Newton's method proof, is also presented. To encourage manufacturing, optimal designs obtained by strain energy or VMS minimization were fabricated and evaluated by tensile testing.

A novel design of multifunctional offshore floating platform structure based on topology optimization

Floating platforms are an economical option for offshore energy harvesting in deep water sea. Lightweight design and structural stability are most important concerns in the design of floating foundations under the premise of ensuring load-bearing capacity. Topology optimization has proven to be a successful method for determining the optimal material distribution with the desired stiffness, strength, and robustness of the subjected design structure. In this paper, a novel lightweight multifunctional offshore floating platform design is proposed utilizing the density-based topology optimization method for the combined utilization of wind-solar-wave energy. Firstly, based on the extreme sea conditions in the South China Sea, the minimum compliance subject to the prescribed volume percentage is formulated and an innovative platform structure is carried out by interoperating the variable density method in the ANSYS AQWA module. Secondly, the optimized floating platform structure was strength-checked based on the maximum von Mises stresses under extreme loading conditions specified in the DNV code. The total mass of the novel optimized structure is 40.82% of the primary structure. Moreover, the platform stiffness and durability have been significantly enhanced.

Topology optimization for rigid and compliant hybrid mechanisms

This paper proposes a topology optimization method integrating the variable trajectory constraints to design rigid and compliant hybrid mechanisms. The variable trajectory constraints are distributed in the design domain and are uniformly modeled by nonlinear spring model. Each of the variable trajectory constraints has an active or inactive state and different states map various mechanical properties of the nonlinear springs. The nonlinear relation between force and displacement of the spring is formulated. A new group of design variables denoting the states of the springs is introduced into the traditional density-based topology optimization model. The design variables representing the element density of the continuum structure and the states of the variable trajectory constraints are uniformly penalized to avoid intermediate values in order to obtain a physically realizable mechanism. Sensitivity analysis is performed by the adjoint equation method. To demonstrate the versatility and the generality of the proposed method, the typical numerical examples including linear, spline and circular trajectory constraints were implemented. The numerical comparisons between the nonlinear spring model and the actual trajectory constraints verify the accuracy. The proposed method expands the application of topology optimization and can be extensively employed to design the mechanisms integrating both the compliant members and rigid parts.

Development of a density-based topology optimization of homogenized lattice structures for individualized hip endoprostheses and validation using micro-FE

Prosthetic implants, particularly hip endoprostheses, often lead to stress shielding because of a mismatch in compliance between the bone and the implant material, adversely affecting the implant’s longevity and effectiveness. Therefore, this work aimed to demonstrate a computationally efficient method for density-based topology optimization of homogenized lattice structures in a patient-specific hip endoprosthesis. Thus, the root mean square error (RMSE) of the stress deviations between the physiological femur model and the optimized total hip arthroplasty (THA) model compared to an unoptimized-THA model could be reduced by 81 % and 66 % in Gruen zone (GZ) 6 and 7. However, the method relies on homogenized finite element (FE) models that only use a simplified representation of the microstructural geometry of the bone and implant. The topology-optimized hip endoprosthesis with graded lattice structures was synthesized using algorithmic design and analyzed in a virtual implanted state using micro-finite element (micro-FE) analysis to validate the optimization method. Homogenized FE and micro-FE models were compared based on averaged von Mises stresses in multiple regions of interest. A strong correlation (CCC > 0.97) was observed, indicating that optimizing homogenized lattice structures yields reliable outcomes. The graded implant was additively manufactured to ensure the topology-optimized result’s feasibility.

Efficient and exquisite concurrent optimization of hierarchical structures with non-uniform eccentric body centered cubic lattice

To obtain lightweight structures with desirable mechanical performances and to enlarge the design space, this work presents an innovative concurrent optimization methodology for the precise inverse design of micro lattices and the modeling of hierarchical structures. The inverse design of micro lattices with continuously changing anisotropic properties is achieved by offsetting the center nodes of eccentric body centered cubic lattices (EBCC) to designated locations. Simultaneously, the macro material distribution is adjusted utilizing 3D density-based topology optimization method. To enhance the computation efficiency and significantly amplify the versatility in optimization design, a Parameterized Interpolation for Anisotropic Lattice material (PIAL) is established based on elastic stiffness matrices of micro lattices with 21 independent components. Specifically, a cubic multinomial function is employed to construct a direct correlation between micro topology configurations and equivalent elasticities of the EBCC lattices. Several numerical examples of compliance optimization are carried out to confirm the validity of the proposed approach. Concurrently, Additive Manufacturing (AM) using AlSi10Mg is adopted to manufacture the obtained hierarchical structures, which exhibit well-improved mechanical performance, as confirmed by experimental testing, when compared to the classical density-graded structures.

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

Topology Optimization Method of Stamping Structures Based on the Directional Density Field.

The stamping process produces thin-walled structures that, in general, have uniform wall thickness and no enclosed cavity. However, it is difficult to satisfy the above geometric requirements with the current density-based topology optimization method, since configuring the related geometric constraints is challenging. In order to solve this problem, a topology optimization method for stamping structures based on a directional density field is proposed. Specifically, the directional density field is developed to enable the adding and removing of materials only along the stamping direction, so as to avoid internal voids and concave features. The geometric control for uniform wall thickness is realized by tuning the truncation threshold of the Heaviside projection that processes the directional density field into the 0-1 binary field. At the same time, a calibrated filter radius of the truncation thresholds will facilitate the drawing angle control of the stamping ribs. The effectiveness of the established method has been verified by a number of numerical case studies. Results show that the proposed method can perform topology optimization for stamping structures with tunable uniform thickness and drawing angle control of the ribs. No internal voids or undercuts appear in the results. The results also disclose that a constant truncation threshold increment does not guarantee uniform wall thickness, and varying the threshold increments through surface offset and polynomial fitting is necessary.

Nonlinear topology optimization on thin shells using a reduced-order elastic shell model

We present a novel numerical algorithm to perform nonlinear topology optimization on elastic thin shells. The main component of our method is a differentiable thin-shell simulator based on discrete differential geometry (DDG) discretization and the projected Newton method to solve geometrically nonlinear elasticity and its derivatives on a triangle mesh. We build a density-based topology optimization algorithm, enhanced by a density filter and a Heaviside projection scheme, to emerge and optimize topologically complex shell structures on curved surfaces. We validate our approach using standard test cases for nonlinear topology optimization and demonstrate the efficacy of our method by tackling highly nonlinear topology optimization problems by producing complex and high-resolution shell structural designs under various load conditions.

Density-based topology optimization of multi-condition peening pattern for laser peen forming

Laser peen forming (LPF) is a novel forming process with high flexibility but faces challenges in determining process conditions to produce complex geometries. This study proposes a process planning method to determine peening patterns with multiple process conditions based on topology optimization, providing an effective approach for complex shaping of LPF and liberating the high flexibility of the process. The process planning is formulated as a partial differential equation (PDE)-constrained optimization problem, employing the eigen-moment as the design variable. The formulation is discretized using numerical methods, and a continuous optimization model is derived via density-based topology optimization for efficient resolution. The ordered Heaviside interpolation is proposed specifically for LPF process planning to promote intermediate densities to the desired values. The globally convergent method of moving asymptotes (GCMMA), a well-established solver for topology optimization, is used to resolve the process planning of a flexible forming process. The process planning model designs the forming strategy of the dome and cylinder geometries using different numbers of process conditions. The results demonstrated that process planning with multiple process conditions is superior to using a consistent process condition. Finally, experiments are conducted using the optimized peening patterns to form the two geometries. The deformed results perform a high level of conformity to the desired objectives with a maximum error of 0.473 mm and 0.833 mm, respectively, validating the effectiveness of the proposed process planning method.

Density-based topology optimization with the Null Space Optimizer: a tutorial and a comparison

Density-based topology optimization with the Null Space Optimizer: a tutorial and a comparison

Evaluation of Neural Network-based Derivatives for Topology Optimization

Abstract Neural networks have gained popularity for modeling complex non-linear relationships. Their computational efficiency has led to their growing adoption in optimization methods, including topology optimization. Recently, there have been several contributions towards improving derivatives of neural network outputs, which can improve their use in gradient-based optimization. However, a comparative study has yet to be conducted on the different derivative methods for the sensitivity of the input features on the neural network outputs. This paper aims to evaluate four derivative methods: analytical neural network's Jacobian, central finite difference method, complex step method, and automatic differentiation. These methods are implemented into density-based and homogenization-based topology optimization using multilayer perceptrons (MLPs). For density-based topology optimization, the MLP approximates Young's modulus for the solid-isotropic-material-with-penalization (SIMP) model. For homogenization-based topology optimization, the MLP approximates the homogenized stiffness tensor of a representative volume element, e.g., square cell microstructure with a rectangular hole. The comparative study is performed by solving two-dimensional topology optimization problems using the sensitivity coefficients from each derivative method. Evaluation includes initial sensitivity coefficients, convergence plots, and the final topologies, compliance, and design variables. The findings demonstrate that neural network-based sensitivity coefficients are sufficient for density-based and homogenization-based topology optimization. The neural network's Jacobian, complex step method, and automatic differentiation produced identical sensitivity coefficients to working precision. The study's open-source code is provided through an included Python repository.

Density-Based Topological Optimization of 3D-Printed Casts for Fracture Treatment with Freefem Software

Abstract 3D printed plastic casts can be used for healing bone fractures. The main requirements for these cases are: they should be light, require little printing time, have good mechanical properties, and ensure proper skin ventilation. We present a density-based topology optimization algorithm for obtaining optimal cast shapes that fulfil these requirements. The algorithm uses a linear stress model and simplified boundary conditions to model the contact problems. The cast shapes were optimized against the influence of several sharp corners. The parametric studies showed that the mass of optimized casts was reduced by 20 %–25 % in comparison with original industrial casts, and the printing time is reduced by 1.4–1.7 h for the largest cast. A major model drawback is the use of 3D numerical volume to model the density distribution. The density distribution should be homogenized across the cast layer. The overhang problem should also be addressed. We also suggest that the cast producers collect more experimental data on the cast breakages for a better calibration of the numerical model.

Density-Based Topology Optimization in Method of Moments: Q-Factor Minimization

Density-Based Topology Optimization in Method of Moments: Q-Factor Minimization

Topology optimization for energy absorption of quasi-brittle structures undergoing dynamic fractures

In this paper, we introduce a topology optimization approach aimed at improving the energy absorption of quasi-brittle structures under impact loading. Our method focuses on incorporating dynamic fracture behavior throughout the impact process to maximize the energy acting on the structure. To achieve this, we integrate a dynamic phase field method into density-based topology optimization, enabling us to simulate the initiation and propagation of complex dynamic fractures. The optimization formulation aims to maximize the absorbed energy over a specified period while adhering to material volume and compliance constraints. Sensitivity analysis was originally provided to accelerate computations and optimization. Several numerical examples show that the incorporation of dynamic fracture effects leads to superior energy absorption of the optimized structure compared to static strategies or neglecting the fracture process. Moreover, the proposed scheme facilitates tailored designs for different impact loading rates.

Directional maximum length scale control in density-based topology optimization

In this paper, directional maximum length scale control for both the solid and void phases is proposed for density-based topology optimization with a lower computational cost. The method introduces porosity and material rate in the locally searched domain to achieve the length scale control for the solid and void phases, respectively. To enable directional length scale control, local rectangle and cylinder searches are utilized instead of conventional circle or cylinder searches. The computational cost, including both local search time and occupied memory, is analyzed and compared with that of conventional searches. The proposed method is based on the three-field approach, and a p-mean function is employed to aggregate the local maximum length scale control constraints into a single global constraint. The optimization formulations and sensitivity analysis of the related optimization responses are subsequently derived. Three numerical tests are conducted to demonstrate the effectiveness and potential engineering applications of the proposed method.

Multi–materials topology optimization using deep neural network for coupled thermo–mechanical problems

In this paper, a density-based topology optimization method using neural networks is used to design domains composed of multi-materials under combined thermo–mechanical loading. The neural network produces the topology density field by minimizing a loss function defined for the problem. Length scale control is conducted using a Fourier space projection. JAX, a high-performance automatic differentiation (AD) Python library is used to build an end-to-end differentiable neural network. The model can handle the sensitivity analysis automatically using the backpropagation process (automatic differentiation of loss function) and the need for solving adjoint equations manual sensitivity analysis is removed. The optimization problem is solved by minimizing the loss function of the network and other optimization algorithms such as the Method of Moving Asymptotes (MMA) are not required. The performance of the method is demonstrated through several examples and compared with those obtained from the popular Solid Isotropic Material with Penalization (SIMP) method. The network is able to handle high-resolution re-sampling, resulting in a more refined and smooth variation of optimal topologies.

Adaptive fail-safe topology optimization using a hierarchical parallelization scheme

This work presents an efficient, flexible, and scalable strategy to implement density-based topology optimization formulation for fail-safe structural design. Such optimized designs can operate after faults modeled as loss of stiffness. However, the need for assessing the physical behavior in all the fault cases can make this problem unfeasible computationally. We combine many ingredients to exploit the different levels of parallelism of the formulation to deal successfully with this problem. We use a non-overlapping domain decomposition method to solve the failure cases using multi-core computing with distributed memory computation. These subdomains use adaptive mesh refinement (AMR) techniques to focus computing efforts on the regions of interest. A key point to evaluate the fault cases is the organization of the computing threads, especially in clusters of computers. We group the computing threads by physical computing nodes to reduce the inter-node communications, maximizing intra-node communications to mitigate bandwidth problems. Another ingredient of paramount importance is the use of computing buffers to adapt the evaluation of the fault cases to the computing resources. We test the scalability of the computational framework using a large cluster of computers.