Topology optimization analysis based on the direct coupling of the boundary element method and the level set method

This thesis aims at understanding and improving topology optimization techniques focusing on density-based level-set methods and geometrical nonlinearities. Central in this work are the numerical modeling of the mechanical response of a design and the consistency of the optimization process itself. Concerning the first topic, we investigate different means to improve the robustness of density-based numerical models including geometrical nonlinearities. The conventional approach (scaling the local material properties) can result in convergence problems due to excessive deformation in low-stiffness finite elements. To avoid excessive deformation, we combine the element connectivity parameterization method (adapting the connectivity between finite elements) with level-set-based topology optimization. Furthermore, we achieve greater robustness of analysis method using a second and improved approach called element deformation scaling. This approach eliminates the need for solving internal equilibrium equations (needed for the element connectivity parameterization method) via an explicit relation between the local internal and global external displacement field. The second focus of this thesis is on the optimization process of level-set-based topology optimization, and in particular its numerical consistency. We observe that signed-distance reinitialization of the level-set function affects the shape of a design in the optimization process. To minimize this effect, we propose a discrete level-set method that is based on an approximate Heaviside function and focusing on the implementation. Furthermore, we also propose a level-set-based topology optimization method using an exact Heaviside function and mathematical programming that effectively eliminates the need for reinitialization. We demonstrate that our density-based level-set method is closely related to conventional density-based topology optimization methods, while offering the advantage of more control over the geometrical complexity. On the other hand, we confirm that the dependence of the final result on the initial design which remains one of the big challenges for level-set-based topology optimization. The potential of the proposed level-set method is shown by applying it on problems with stress constraints and geometrical nonlinearities and performing manufacturing tolerant topology optimization. Finally, this thesis offers a review of level-set methods for structural topology optimization to identify and discuss the different approaches that are available in literature. We can distinguish between level-set methods by examination of their design parameterization, sensitivities, update procedures and regularization techniques. A level-set-based design parameterization offers the advantage of a crisp distinction between subdomains. For this reason, X-FEM approaches and conforming discretizations are an interesting option to retain the crisp nature of the level-set-based description of the design. Many level-set methods are combined with density-based numerical models and are, therefore, closely related to conventional density-based topology optimization methods. In particular, recently proposed projection methods have much in common with a level-set-based design description. The results of level-set-based TO methods often rely heavily on regularization techniques that introduce inconsistencies in the optimization process. Numerical consistency does not necessarily lead to the best search direction, but is essential to find a Karush-Kuhn-Tucker point of the discretized optimization problem.

A boundary element and level set based topology optimisation using sensitivity analysis

In conventional level set methods, the slope of the level set function needs to be well controlled to maintain the numerical stability during the topology optimization process. One common solution is to regularize the level set function to be a signed distance function, which is usually achieved by periodically implementing the so called re-initialization scheme to force the level set function to gain the desired signed distance property. However, the re-initialization scheme will bring some unwanted drawbacks to the optimization process, such as zero level set drifting, time consuming etc. In addition, re-initialization is usually implemented outside the optimization loop, which will cause convergence issues. In this paper, a distance regularization functional is introduced to the structural topology optimization objective functional to ensure the signed distance property of the level set function near the structure boundaries. This functional can also keep the level set function to be constant-value at positions far away from the structural boundaries. The radial basis function (RBF) based parameterization technique together with the mathematical programming are utilized to improve the potential capability of handling multiple constraints for the topology optimization. The combination of these two techniques makes the level set based topology optimization be capable of handling complicated multi-constrained problems with higher numerical efficiency, leaving no compromise to multiple drawbacks. To demonstrate the validity of the proposed scheme, benchmark examples on minimum compliance structural optimization are employed. This type of problem is computed by the conventional level set method with the introduced distance regularization functional, the RBF based parametric level set and at last, the distance regularized RBF based parametric level set separately to demonstrate their differences.

Topology optimization has demonstrated its effectiveness in generating lightweight and structurally efficient designs. This study focuses on refining the geometry of an automatic coupler body for trains using solid isotropic material with penalization and a level set method. These optimization methods are applied to the numerical model of the automatic coupler, and their results are compared to select the optimal design. The tensile strength of the automatic coupler is examined through numerical simulations and validated by experimental tensile tests conducted on a 1:1 scale prototype. The optimization outcomes reveal a remarkable 46.41% reduction in the mass of the automatic coupler body compared to the initial model. An evaluation of the tensile strength of the prototype demonstrates the ability of the automatic coupler to withstand the primary load without undergoing plastic deformation. Furthermore, a strong correlation is observed between the numerical and experimental results. This research contributes to advancing the design of next-generation automatic couplers, emphasizing the crucial aspects of lightweight design and structural performance.

PurposeShape optimization using the level-set method is one of the most effective automatic design tools for electromagnetic machines. While level-set method has the advantage of being able to suppress unfeasible shape, it has a weakness of being unable to handle complex topology changes such as perforate at material region. With this method, it is only possible to define simple connected topology, and it is difficult to determine the optimal shape which has holes. Therefore, it is important to efficiently expand the searching area in the optimization process with level-set method.Design/methodology/approachIn this paper, the authors introduce the newly defined hole sensitivity which is based on concept of topological derivatives, and combine it with level-set method to effectively create holes in the search process. Furthermore, they consider a variable bandwidth of gray scale, which indicates the transition width between air and magnetic body and combine it with the hole creation method described above. With these methods, the authors aim to expand the searching area in comparison with the conventional level-set method.FindingsAs a result of applying the proposed methods to a magnetic shielding problem, the multi-layered shielding which effectively reduces the magnetic flux in the target area, is successfully produced.Originality/valueThe proposed methods enable us to effectively create a hole and to expand the searching area in the topology optimization process unlike in the case of conventional level-set method.

SummaryThe parametric level set approach is an extension of the conventional level set methods for topology optimization. By parameterizing the level set function, level set methods can be directly coupled with mathematical programming to achieve better numerical robustness and computational efficiency. Moreover, the parametric level set scheme can not only inherit the primary advantages of the conventional level set methods, such as clear boundary representation and the flexibility in handling topological changes, but also alleviate some undesired features from the conventional level set methods, such as the need for reinitialization. However, in the existing radial basis function–based parametric level set method, it is difficult to identify the range of the design variables. Besides, the parametric level set evolution often struggles with large fluctuations during the optimization process. Those issues cause difficulties both in numerical stability and in material property mapping. In this paper, a cardinal basis function is constructed based on the radial basis function partition of unity collocation method to parameterize the level set function. The benefit of using cardinal basis function is that the range of the design variables can now be clearly specified as the value of the level set function. A distance regularization energy functional is also introduced, aiming to maintain the desired signed distance property during the level set evolution. With this desired feature, the level set evolution is stabilized against large fluctuations. In addition, the material properties mapped from the level set function to the finite element model can be more accurate.

In this work, an efficient topology optimization approach is proposed for a three-dimensional (3D) flexible multibody system (FMBS) undergoing both large overall motion and large deformation. The FMBS of concern is accurately modeled first via the solid element of the absolute nodal coordinate formulation (ANCF), which utilizes both nodal positions and nodal slopes as the generalized coordinates. Furthermore, the analytical formulae of the elastic force vector and the corresponding Jacobian are derived for efficient computation. To deal with the dynamics in the optimization process, the equivalent static load (ESL) method is employed to transform the topology optimization problem of dynamic response into a static one. Besides, the newly developed topology optimization method by moving morphable components (MMC) is used and reevaluated to optimize the 3D FMBS. In the MMC-based framework, a set of morphable structural components serves as the building blocks of optimization and hence greatly reduces the number of design variables. Therefore, the topology optimization approach has a potential to efficiently optimize an FMBS of large scale, especially in 3D cases. Two numerical examples are presented to validate the accuracy of the solid element of ANCF and the efficiency of the proposed optimization methodology, respectively.

A desirable feature in multidisciplinary design optimization (MDO) is to achieve an acceptable preliminary or initial design in an expeditiously fashion. One way to pursue this goal is to describe the structural boundary using a small number of design variables. In this paper we propose an approach to structural optimization that enhances the performance of multidisciplinary design tools because it greatly decreases the number of design variables assigned to structural deflnition. This approach is based on the Level Set Methods and uses the coe‐cients of the Fourier series expansion of the level set function as design variables. The global character of these coe‐cients and the fact that the level set function is real-valued provide the means to reduce the design space dimension. Within the level set framework, the structure is conflned to the region in which the level set function is negative. In this way, the zero level set of this function implicitly describes the structural boundary, which can undergo topological changes during the optimization process as the function evolves. The proposed methodology also averts successive mesh generation. We show that an interface can be described to any desirable accuracy using flnitely many Fourier modes of the signed distance to the interface. The total error is inversely proportional to the number of Fourier modes. We present two validation examples in topology optimization, and use these cases to study the efiects of the methodology parameters on the results. Finally, we propose two examples of application to the enhancement of aircraft roll maneuvers. In the flrst, the goal is to determine the layout of reinforcements in the front spar of a wing-aileron that maximizes the aileron reversal speed, subject to a constraint in the amount of reinforcing material. We also propose the design of a morphing airfoil-aileron conflguration capable of changing its mean camber to maximize the aileron efiectiveness through the activation of a system of actuators, given a restriction on power consumption.

A robust topological derivative-based multi-material optimization approach: Optimality condition and computational algorithm

A boundary element method is developed for the topology optimization problem. The topological change is captured using the level set method. The level set function which is defined by signed distance from the boundary contour is assigned to fixed grid points. Boundary elements are developed along the zero contour of the level set function. The design sensitivity analysis is performed for the boundary element equation, and then the boundary velocity is obtained. The velocity field which leads the level set function to optimal material distribution is obtained by the extension of the boundary velocity. Numerical examples show that the proposed method is valid for the topology optimization problems. Copyright © 2006 John Wiley & Sons, Ltd.

Since deterministic topology optimization (TO) does not consider the uncertainties in the structure, including materials, loading and geometric dimensions, it may provide the optimal designs with the lowest state of reliability and safety. To solve this problem, reliability-based topology optimization (RBTO) is used, which is actually a combination of TO methods with reliability-based design methods (RBDO) based on a mathematical framework and process. In this article, by considering four TO methods including moving iso-surface threshold (MIST), SIMP, evolutionary structural optimization-extended finite element (XFEM-ESO) and level-set (LS), and considering a constraint or objective function, an optimal volume fraction (Vf) is obtained for TO by the bisection method. Then, with the aid of mean volume fraction, TO is performed and its optimized results are applied by an advanced reliability analysis method, i.e. accelerated dynamical mean value (ADMV), taking into account the uncertainties and their standard deviations to extract the most probable probability point (MPP). Having the MPP and the constraint, the bisection algorithm is used again and the optimized volume fraction for the RBTO model, and thereby the optimal layout for the RBTO solution, is achieved. Several examples are presented to validate and highlight the optimization capability of the RBTO method using a structural model and the mentioned TO methods, and the results are compared together. Based on the results, it is shown that the combination of RBDO and TO approach is able to result in powerful, stable, safe, and reliable structures completely different from the TO results.

This paper presents a performance index for topology and shape optimization of plate bending problems with displacement constraints. The performance index is developed based on the scaling design approach. This performance index is used in the Performance-Based Optimization (PBO) method for plates in bending to keep track of the performance history when inefficient material is gradually removed from the design and to identify optimal topologies and shapes from the optimization process. Several examples are provided to illustrate the validity and effectiveness of the proposed performance index for topology and shape optimization of bending plates with single and multiple displacement constraints under various loading conditions. The topology optimization and shape optimization are undertaken for the same plate in bending, and the results are evaluated by using the performance index. The proposed performance index is also employed to compare the efficiency of topologies and shapes produced by different optimization methods. It is demonstrated that the performance index developed is an effective indicator of material efficiency for bending plates. From the manufacturing and efficient point of view, the shape optimization technique is recommended for the optimization of plates in bending.

This paper will give a short survey about topology optimization of structures. It is particularly focused on topological shape optimization of structures using level-set methods, including the level-set based standard methods and the level-set based alternative methods. The former often directly solve the Hamilton-Jacobi partial differential equation (H-J PDE) to obtain the boundary velocity field using Finite Differential Methods (FDM), and the later commonly employ parametric or equivalent methods to evaluate the velocity field without directly solving the H-J PDE. The unique characteristics of the level-set based topology optimization methods are discussed, and a future perspective and prospects in this research area is also included. A benchmark numerical example is used to showcase the effectiveness of the level-set based methods.

In this paper, the aileron reversal speed of a wing torsion box is maximized by reinforcing its upper skin. We propose a topology optimization approach using the spectral level set methodology. This is based on the level set methods,which represent theinterface describing thereinforcements asthe zerolevel setofafunction. Accordingto the proposed methodology, this function is discretized using its Fourier coefficients. These coefficients become the design variables of the optimization problem assigned to define the reinforcing layout. Results show that approximately the sameoptimal reinforcement layout isobtained from topologically different initialconfigurations. The two underlying ideas of the present work are to propose the spectral level set methodology as a framework for topology optimization, given its potential for reducing the design space dimension, and, considering the possibilities of new custom-made materials, to readdress the reinforcement perspective in aircraft structures without the weight penalty. Nomenclature ai,bi = problem domain boundaries corresponding to componenti ofx at = � t smoothness parameter aV = � V smoothness parameter a0 = lift curve slope bt = minimum thickness C = positive constant Cl� = derivative of the rolling moment coefficient of the rigid airplane with respect to� C �� � y;n� = twist aty produced by a unit moment at�

\nThis paper research the topological optimization based on level set method which based on the reaction diffusion equation. The paper propose a Level set topological optimization method which put the compliance as the objective function and take the volume as the main constraint, the element stress as the auxiliary constraint to regulate the volume constraint. This method not only can significantly speed up the optimization process, but can improve the precision of calculation and ensure the stability of the results when meet the stress constraint. Eventually, the proposed method can convenient the level set topological optimization to apply in engineering practice.\n