Level Set Method For Topology Optimization Research Articles

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.

An efficient MATLAB code for stress-constrained level set topology optimization method

An efficient MATLAB code for stress-constrained level set topology optimization method

A novel discrete adjoint-based level set topology optimization method in B-spline space

A novel discrete adjoint-based level set topology optimization method in B-spline space

Accessibility constraints in structural optimization via distance functions

This paper is concerned with a geometric constraint, the so-called accessibility constraint, for shape and topology optimization of structures built by additive manufacturing. The motivation comes from the use of sacrificial supports to maintain a structure, submitted to intense thermal residual stresses during its building process. Once the building stage is finished, the supports are no longer useful and should be removed. However, such a removal can be very difficult or even impossible if the supports are hidden deep inside the complex geometry of the structure. A rule of thumb for evaluating the ease of support removal is to ask that the contact zone between the structure and its supports can be accessed from the exterior by a straight line which does not cross another part of the structure. It mimicks the possibility to cut the head of the supports attached to the structure with some cutting tool. The present work gives a new mathematical way to evaluate such an accessibility constraint, which is based on distance functions, solutions of eikonal equations. The main advantage is the possibility of computing shape derivatives of such a criterion with respect to both the structure and the support. We numerically demonstrate in 2D and 3D that, in the context of the level-set method for topology optimization, such an approach allows us to optimize simultaneously the mechanical performance of a structure and the accessibility of its building supports, guaranteeing its manufacturability.

Multi-material topology optimization of coated structures using level set method

The coated structures are widely applied in engineering, whose coating material covers the surface to protect substrate structure or realize certain functionalities. Currently, the coated structures are generally filled with uniform material. Moreover, the multi-material infill structures can further improve their performances. Therefore, this paper proposes a level set topology optimization method to design the coated structures with multiple infill materials. The coating material with uniform thickness is required to attach the outside of the substrate structure, which can be seen as a manufacturing constraint. In particular, we demonstrate how to construct the multi-material infill structures and formulate the multi-material interpolation model by combining multiple level set functions. Besides, the shape derivatives for the multi-material optimization model of the coated structures are derived for the first time. Finally, numerical examples confirm that the coated structures with multiple infill materials can be effectively obtained by using the proposed method.

Level set-based topology optimization for the design of labyrinthine acoustic metamaterials

We present a topology optimization method for the design of labyrinthine metamaterials with a large path length for acoustic wave propagation. An objective function maximizes the pressure gradient in an air-filled region at a low frequency, and a level set-based optimization method enables the optimal design of the labyrinthine structure with high computational efficiency owing to the concept of topological derivative. First, we explain the design settings of labyrinthine metamaterials. Next, the objective functions to obtain labyrinthine structures are introduced, and the corresponding topological derivatives are obtained. The optimization problem is formulated in the framework of the level set topology optimization method, and two-dimensional numerical examples are provided to demonstrate the validity of the proposed method. Based on the frequency responses of the effective parameters and dispersion analysis, the optimized designs are characterized by a negative refractive index over a wide frequency interval, which can be controlled by a parameter introduced in the objective function.

A finite element level-set method for stress-based topology optimization of plate structures

In this paper, we present a stress-based topology optimization of plate structures by using a finite element level-set (LS) method. Since stress constraints are independent with LS-based design variables, the singularity phenomenon arising in topology optimization under stress constraints can be resolved without using any relaxation technique. The finite element method is used for solving LS-based equations of design variables and the state and adjoint equations of plate structures with arbitrary complex geometries and boundaries of the design domain. To obtain stable finite element solutions without any oscillation, both unsteady LS convection-diffusion equation and LS re-initialization convection-reaction equation are solved using a stabilized least-squares finite element method.

Level set topology optimization of load carrying battery packs

In this study, the optimum design of load carrying battery packs that can also exchange heat with a coolant is presented. The level set topology optimization method is used as the design tool. The flow is modeled using the Darcy potential flow, and the heat transfer is modeled using the convection-diffusion equation. The displacement field is computed using a thermo-elasticity model. A multi-objective optimization is used that minimizes a linear combination of structural compliance, average temperature of the battery cells, and the pressure drop across the coolant. As numerical examples, we present a design study of a load carrying battery pack module that is placed inside an aircraft wing. The efficiency of the optimized designs are demonstrated by showcasing their multifunctional nature—they reduce the temperature in the battery cells by exchanging the heat generated with a coolant all the while sustaining a given load.

Topology Optimization Considering Porosity Defects in Metal Additive Manufacturing

In this paper, considering the porosity defects of Additive Manufacturing (AM), a level set topology optimization method for AM with porosity constraints is proposed. The concept of topological sensitivity is used to formulate a global porosity constraint function in the proposed method, and a level set topology optimization model considering porosity defects is obtained. To improve the robustness of the algorithm, the topology optimization model is solved in two phases. At first, the classical level set method without the porosity constraint is used to initially optimize the structure. During this process, the hole nucleation method combining bi-directional evolutionary structural optimization (BESO) and the topological sensitivity is used. Secondly, the topology optimization considering the effects of porosity is implemented on the preliminary optimization results. After performing the two-step optimization, a robust structure that alleviates the harmful impact of porosity defects is obtained. Finally, the robustness and effectiveness of the proposed method are validated by several two-dimensional numerical examples.

Level-set topology optimization for effective control of transient conductive heat response using eigenvalue

Designing thermally efficient devices for an instantaneous heating condition is a challenging task, since a structure that simultaneously satisfies a lower peak temperature as well as a slower thermal response time must be achieved. In order to design a structure for both of the requirements, a transient analysis-based structural topology optimization method can be usually employed. However, the overall thermal response time of a heat conduction system can be modelled simply via the fundamental thermal eigenvalue, which is a physical quantity. This suggests that a thermal eigenvalue analysis can be adopted instead of transient analysis for an effective design of the structure controlling its thermal response time. In this study, we aim at developing a level-set topology optimization method to effectively optimize the overall response time of heat conduction systems using the thermal eigenvalue. The proposed level-set thermal eigenvalue topology optimization updates the structural boundary based on the common fixed grid approach, and the adjoint variable method is applied to compute shape sensitivity. Numerical examples clearly show the controllability of overall heat response time through the thermal eigenvalue, and especially, it is demonstrated that the proposed method is useful in designing an effective cooling structure for an instantaneous heating.

A Hole Nucleation Method Combining BESO and Topological Sensitivity for Level Set Topology Optimization.

As is known to all, the incapacity to nucleate holes automatically in the design domain is one of the main issues of the classical level set topology optimization method. To solve the issue of hole nucleation, this paper employs the bi-directional evolutionary structural optimization (BESO) method based on the material removal scheme and the frequently used topological sensitivity and proposes the combining BESO and topological sensitivity (CBT) method for level set topology optimization. This method can replace the existing hole nucleation method for level set topology optimization. First, the topological sensitivity is combined with BESO, and the BESO method based on topological sensitivity is proposed. Second, the method is integrated into level set topology optimization to solve the issue of hole nucleation. Two sensitivity thresholds are defined depending on the evolutionary volume ratio and boundary topological sensitivity, respectively, and the smaller one is used as the sensitivity threshold for hole nucleation. The material is removed from the design domain to nucleate holes based on this threshold. Three classical two-dimensional numerical examples are used to validate the proposed hole nucleation method.

Surface segmentation design using a weighting level set topology optimization method for large radio telescope antennas

Surface segmentation design is a conceptual problem in the structural design of large radio telescope antennas. Before performing surface segmentation optimization, a weighting level set topology optimization method is proposed for the antenna backup supporting structure from the viewpoint of continuum topology optimization. With the help of weighting level set topology optimization, a clear and compact backup supporting structure is acquired. Taking the optimized topology backup supporting structure as the design basis, a surface segmentation optimization scheme is presented and extended to the full aperture. By employing this method in the conceptual surface segmentation design of two large radio telescope antenna applications, the effectiveness of this method is easily illustrated.

A discrete adjoint based level set topology optimization method for stress constraints

This paper proposes a new methodology for computing boundary sensitivities in level set topology optimization using the discrete adjoint method. The adjoint equations are constructed using the discretized governing field equations. The objective function is differentiated with respect to the boundary point movement for computing boundary sensitivities using the discrete adjoint equations. For this purpose, we present a novel approach where we perturb the boundary implicitly by locally modifying the level set function around a given boundary point. These local perturbations are combined with the derivatives of the objective function with respect to the volume fractions of individual elements to compute boundary sensitivities. This enables the circumvention of smoothing or interpolation methods typically used in level set topology optimization to compute sensitivities; and improves the accuracy of the sensitivities and the convergence characteristics. We demonstrate the effectiveness of our method in the context of stress minimization and stress constrained topology optimization problems for orthogonal bracket design under multiple load cases.

Parameterized level set method for structural topology optimization based on the Cosserat elasticity

When describing the mechanical behavior of some engineering materials, such as composites, grains, biological materials and cellular solids, the Cosserat continuum theory has more powerful capabilities compared with the classical Cauchy elasticity since an additional local rotation of point and its counterpart (couple stress) are considered in the Cosserat elasticity to represent the material microscale effects. In this paper, a parameterized level set topology optimization method is developed based on the Cosserat elasticity for the minimum compliance problem of the Cosserat solids. The influence of material characteristic length and Cosserat shear modulus on the optimized structure is investigated in detail. It can be found that the microstructural constants in the Cosserat elasticity have a significant impact on the optimized topology configurations. In addition, the minimum feature size and the geometric complexity of the optimized structure can be controlled implicitly by adjusting the parameters of the characteristic length and Cosserat shear modulus easily. Furthermore, the optimized structure obtained by the developed Cosserat elasticity based parameterized level set method will degenerate to the result by using the classical Cauchy elasticity based parameterized level set method when the Cosserat shear modulus approaches zero. A parameterized level set topology optimization method is developed based on the Cosserat elasticity for the optimization of minimum compliance of the structures with micropolar materials. The influence of characteristic length and Cosserat shear modulus of the micropolar material on the optimized structure has been investigated in detail. It can be found that the minimum feature size and the geometric complexity of the optimized structure can be controlled implicitly by adjusting the parameters of the characteristic length and Cosserat shear modulus easily.

Level set topology optimization for design-dependent pressure loads using the reproducing kernel particle method

This paper presents a level set topology optimization method in combination with the reproducing kernel particle method (RKPM) for the design of structures subjected to design-dependent pressure loads. RKPM allows for arbitrary particle placement in discretization and approximation of unknowns. This attractive property in combination with the implicit boundary representation given by the level set method provides an effective framework to handle the design-dependent loads by moving the particles on the pressure boundary without the need of remeshing or special numerical treatments. Moreover, the reproducing kernel (RK) smooth approximation allows for the Young’s modulus to be interpolated using the RK shape functions. This is another advantage of the proposed method as it leads to a smooth Young’s modulus distribution for smooth boundary sensitivity calculation which yields a better convergence. Numerical results show good agreement with those in the literature.

A filter-based level set topology optimization method using a 62-line MATLAB code

The present paper proposes a fast and easy to implement level set topology optimization method that is able to adjust the complexity of resulting configurations. While the existing methods solve a large system of linear equations, the proposed method applies a density filter to the level set function in order to smoothen the optimized configurations. Considering the proposed method, the minimum total potential energy problem and optimum design problem of compliant mechanism will be studied in the present study. In topology optimization methods, the topological derivative determines the criteria of removing/adding material from/to the domain of definition of the problem. We utilize the definition of topological derivative to measure the sensitivity of a given objective function when an infinite small hole is inserted at an arbitrary location of the domain. It is shown that a first-order stress field appears in the topological derivative. Moreover, a comprehensive Airy stress function is proposed in order to solve the boundary value problem that governs the appeared first-order stress field. Finally, a simple MATLAB implementation is provided to confirm the effectiveness and utility of the proposed method.

Coupled Aerostructural Level Set Topology Optimization of Aircraft Wing Boxes

In this paper, the wing box of an aircraft is optimized using a level set topology optimization method and considering aerostructural coupling. A new level set method is developed where the design domain and the finite element analysis domain are represented by two separate meshes, and coordinates and sensitivities are transformed from one mesh to the other mesh using an isoparametric map. This method is applied to minimize the compliance and tip displacement of a tapered and swept wing box subjected to aerodynamic loads. The wing box is modeled using 1.47 million isoparametric hexahedral elements, whereas the skin (which is nondesignable) is modeled using shell elements. The results show that the optimizer exploits the aerostructural coupling properties and tailors the lift distribution to yield stiffer wing boxes. Moreover, the results also show that ignoring aerostructural coupling leads to suboptimal designs.

Level set topology optimization of cooling channels using the Darcy flow model

The level set topology optimization method for 2D and 3D cooling channels, considering convective heat transfer for high Reynolds number flows, is presented in this paper. The Darcy potential flow, which is a low-fidelity linear flow model, is used to simulate the flow using the finite element method. The resulting velocity field is used in a convection-diffusion model to simulate the heat transfer using the finite element method. A linear combination of the pressure drop and the average temperature is considered as the objective function, which is minimized subject to a volume constraint and a maximum length scale constraint. The results show that the pressure drop and the average temperature are conflicting criteria, and the trade-off between the two criteria is investigated. We perform a verification study by comparing the Darcy flow field of the obtained optimum designs with that of a high fidelity turbulence model. The verification study shows that there exists a reasonable agreement between the Darcy and the turbulent flow field for narrow channels. Therefore, by restricting the design space to narrow channels, we optimize the cooling performance and sufficiently capture the turbulent flow physics using the low-fidelity Darcy flow model. Finally, we show an example in 3D where we optimize the cooling channel topology that conforms to the surface of a sphere.

Design of Architected Materials for Thermoelastic Macrostructures Using Level Set Method

A level set topology optimization method is introduced and used to design periodic architected materials optimized for the maximum macrostructural stiffness considering thermoelasticity. The design variables are defined at the microscopic scale and updated by minimizing the total structural compliance induced by mechanical and thermal expansion loads at the macroscopic scale. The two scales are coupled by the effective elasticity tensor calculated through the homogenization theory. A decomposition method is constructed to formulate several subproblems from the original optimization problem, enabling the efficient solution of this otherwise computationally expensive problem, especially when the number of material subdomains is large. The proposed method is demonstrated through several numerical examples. It is shown that the macrostructural geometry and boundary conditions have a significant impact on the optimized material designs when thermoelastic effects are considered. Porous material with well-designed microstructure is preferred over solid material when the thermal load is nonzero. Moreover, when a larger number of material microstructures is allowed in optimization, the overall performance is improved due to the expanded design space.

Topology Optimization Design of Boat Boom Structure Based on Level Set Method

The marine boom is optimized by a level set topology optimization method based on the reaction diffusion equation. In order to overcome the characteristics that the boom is difficult to be topology optimized, the periodic topology optimization theory is introduced to divide the boom into several cycles, and two typical conditions of maximum amplitude and minimum amplitude are selected respectively. Finally, the optimized boom structure is processed and remodelled and finite element analysis is performed, which still meets the original performance index.