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.

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.

Generalized hole nucleation through BESO for the level set based topology optimization of multi-material structures

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.

Topology optimization of natural convection using porous metal foam based on the adjoint lattice Boltzmann method and level set method

This paper deals with a new topology optimization method based on the level set method. In the proposed method, the discretized signed distance function, a kind of level set function, is used as the design variables, and these are then updated using their sensitivities. The signed distance characteristic of the design variables are maintained by performing a re-initialization at every update during the iterated optimization procedure. In this paper, a minimum mean compliance problem and a compliant mechanism design problem are formulated based on the level set method. In the formulations of these design problems, a perimeter constraint is imposed to overcome the ill-posedness of the structural optimization problem. The sensitivity analysis for the above structural optimization problems is conducted based on the adjoint variable method. The augmented Lagrangian method is incorporated to deal with multiple constraints. Finally, several numerical examples that include multiple constraints are provided to confirm the validity of the method, and it is shown that appropriate optimal structures are obtained.

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.

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.

The structural design must ensure suitable working conditions by attending for safe and economic criteria. However, the optimal solution is not easily available, because these conditions depend on the bodies’ dimensions, materials strength and structural system configuration. In this regard, topology optimization aims for achieving the optimal structural geometry, i.e. the shape that leads to the minimum requirement of material, respecting constraints related to the stress state at each material point. The present study applies an evolutionary approach for determining the optimal geometry of 2D structures using the coupling of the boundary element method (BEM) and the level set method (LSM). The proposed algorithm consists of mechanical modelling, topology optimization approach and structural reconstruction. The mechanical model is composed of singular and hyper-singular BEM algebraic equations. The topology optimization is performed through the LSM. Internal and external geometries are evolved by the LS function evaluated at its zero level. The reconstruction process concerns the remeshing. Because the structural boundary moves at each iteration, the body’s geometry change and, consequently, a new mesh has to be defined. The proposed algorithm, which is based on the direct coupling of such approaches, introduces internal cavities automatically during the optimization process, according to the intensity of Von Mises stress. The developed optimization model was applied in two benchmarks available in the literature. Good agreement was observed among the results, which demonstrates its efficiency and accuracy.

Level set-based BEM topology optimization method for maximizing total potential energy of thermal problems

The level set method (LSM), which is transplanted from the computer graphics field, has been successfully introduced into the structural topology optimization field for about two decades, but it still has not been widely applied to practical engineering problems as density-based methods do. One of the reasons is that it acts as a boundary evolution algorithm, which is not as flexible as density-based methods at controlling topology changes. In this study, a level set band method is proposed to overcome this drawback in handling topology changes in the level set framework. This scheme is proposed to improve the continuity of objective and constraint functions by incorporating one parameter, namely, level set band, to seamlessly combine LSM and density-based method to utilize their advantages. The proposed method demonstrates a flexible topology change by applying a certain size of the level set band and can converge to a clear boundary representation methodology. The method is easy to implement for improving existing LSMs and does not require the introduction of penalization or filtering factors that are prone to numerical issues. Several 2D and 3D numerical examples of compliance minimization problems are studied to illustrate the effects of the proposed method.

This paper proposes a topology optimization method for a dielectric optical cloak that provides results that are perfectly free from intermediate materials, based on a level set boundary expression and the Finite Element Method. The finite element mesh is re-generated to fit the iso-surface of the level set function at every iterative step, to remove intermediate materials, so that the obtained optimal structure consists of only two materials, the dielectric material and air. First, the level set-based topology optimization is formulated and a topology optimization algorithm is proposed for the exact level set approach. Next, design requirements for the dielectric optimal cloak device are clarified and an objective functional for the design is formulated. The proposed method is then applied to a simple numerical problem to illustrate its effectiveness.

This paper proposes a level-set based topology optimization method incorporating a boundary tracking mesh generating method and nonlinear programming. Because the boundary tracking mesh is always conformed to the structural boundary, good approximation to the boundary is maintained during optimization; therefore, structural design problems are solved completely without grayscale material. Previously, we introduced the boundary tracking mesh generating method into level-set based topology optimization and updated the design variables by solving the level-set equation. In order to adapt our previous method to general structural optimization frameworks, the incorporation of the method with nonlinear programming is investigated in this paper. To successfully incorporate nonlinear programming, the optimization problem is regularized using a double-well potential. Furthermore, the sensitivities with respect to the design variables are strictly derived to maintain consistency in mathematical programming. We expect the investigation to open up a new class of grayscale-free topology optimization. The usefulness of the proposed method is demonstrated using several numerical examples targeting two-dimensional compliant mechanism and metallic waveguide design problems. Copyright © 2014 John Wiley & Sons, Ltd.

This paper discusses a level set-based topology optimization method for the design of metallic waveguides loaded with ferrite inclusions. Ferrite materials can have frequency dependent permeability value due to the magnetic resonance phenomenon in a specific frequency range that can be altered by the magnitude of an external DC magnetic field. Thus, a rectangular waveguide loaded with periodic ferrite inserts can offer advantages in the tunability of operating frequencies and compactness. Here, the level set-based topology optimization incorporating a fictitious interface energy is used to find the configuration of ferrite materials in the waveguide that maximizes the transmission power of electromagnetic waves at prescribed frequencies. The objective function is formulated to maximize the transmission coefficient S21 at prescribed desirable frequencies. A numerical example is provided to examine the validity and utility of the presented method.

The level set method is a mathematical and computational, technique for tracking a moving interface over time. It can naturally handle topological changes such as merging or breaking interfaces. Intrinsic geometric properties of the interface, such as curvature and normal direction, are easily determined from the level set function p. There are many applications of the level set method, including kinetic crystal growth, epitaxial growth of thin films, image restoration, vortex dominated flows, and so forth. Most applications described in the growing literature on the applications of level sets advance the level set equation with explicit time integration. Hence, small CFL-respecting time steps are needed to maintain stability. In this thesis, an implicit level set method is introduced and applied to wildland firespread models, removing vulnerability to instability.