Shape optimization of hydrodynamic textured surfaces for enhancing load-carrying capacity based on level set method

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

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.

In this paper, we construct a level set method for an elliptic obstacle problem, which can be reformulated as a shape optimization problem. We provide a detailed shape sensitivity analysis for this reformulation and a stability result for the shape Hessian at the optimal shape. Using the shape sensitivities, we construct a geometric gradient flow, which can be realized in the context of level set methods. We prove the convergence of the gradient flow to an optimal shape and provide a complete analysis of the level set method in terms of viscosity solutions. To our knowledge this is the first complete analysis of a level set method for a nonlocal shape optimization problem. Finally, we discuss the implementation of the methods and illustrate its behavior through several computational experiments.

The paper presents a detailed computational study of flow patterns and performance indices in a dimpled parallel thrust bearing. The bearing consists of eight pads; the stator surface of each pad is partially textured with rectangular dimples, aiming at maximizing the load carrying capacity. The bearing tribological performance is characterized by means of Computational Fluid Dynamics (CFD) simulations, based on the numerical solution of the Navier-Stokes and energy equations for incompressible flow. Realistic boundary conditions are implemented. The effects of operating conditions and texture design are studied for the case of isothermal flow. First, for a reference texture pattern, the effects of varying operating conditions, in particular minimum film thickness (thrust load), rotational speed and feeding oil pressure are investigated. Next, the effects of varying texture geometry characteristics, in particular texture zone circumferential/radial extent, dimple depth and texture density on the bearing performance indices (load carrying capacity, friction torque and friction coefficient) are studied, for a representative operating point. For the reference texture design, the effects of varying operating conditions are further investigated, by also taking into account thermal effects. In particular, adiabatic conditions and conjugate heat transfer at the bearing pad are considered. The results of the present study indicate that parallel thrust bearings textured by proper rectangular dimples are characterized by substantial load carrying capacity levels. Thermal effects may significantly reduce load capacity, especially in the range of high speeds and high loads. Based on the present results, favorable texture designs can be assessed.

The paper presents a detailed computational study of flow patterns and performance indices in a dimpled parallel thrust bearing. The bearing consists of eight pads; the stator surface of each pad is partially textured with rectangular dimples, aiming at maximizing the load carrying capacity. The bearing tribological performance is characterized by means of computational fluid dynamics (CFD) simulations, based on the numerical solution of the Navier–Stokes and energy equations for incompressible flow. Realistic boundary conditions are implemented. The effects of operating conditions and texture design are studied for the case of isothermal flow. First, for a reference texture pattern, the effects of varying operating conditions, in particular minimum film thickness (thrust load), rotational speed and feeding oil pressure are investigated. Next, the effects of varying texture geometry characteristics, in particular texture zone circumferential/radial extent, dimple depth, and texture density on the bearing performance indices (load carrying capacity, friction torque, and friction coefficient) are studied, for a representative operating point. For the reference texture design, the effects of varying operating conditions are further investigated, by also taking into account thermal effects. In particular, adiabatic conditions and conjugate heat transfer at the bearing pad are considered. The results of the present study indicate that parallel thrust bearings textured by proper rectangular dimples are characterized by substantial load carrying capacity levels. Thermal effects may significantly reduce load capacity, especially in the range of high speeds and high loads. Based on the present results, favorable texture designs can be assessed.

Numerical optimization of texture shape for parallel surfaces under unidirectional and bidirectional sliding

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.

The aim of this paper is to provide a survey on the recent development in level set methods in inverse problems and optimal design. We give introductions on the general features of such problems involving geometries and on the general framework of the level set method. In subsequent parts we discuss shape sensitivity analysis and its relation to level set methods, various approaches on constructing optimization algorithms based on the level set approach, and special tools needed for the application of level set based optimization methods to ill-posed problems. Furthermore, we provide a review on numerical methods important in this context, and give an overview of applications treated with level set methods. Finally, we provide a discussion of the most challenging and interesting open problems in this field, that might be of interest for scientists who plan to start future research in this field.

An optimization study of trapezoidal surface texturing in slider micro-bearings, via computational fluid dynamics (CFD), is presented. The bearings are modeled as micro-channels, consisting of a moving and a stationary wall. The moving wall (rotor) is assumed smooth, while part of the stationary wall (stator) exhibits periodic dimples of trapezoidal form. The extent of the textured part of the stator and the dimple geometry are defined parametrically; thus, a wide range of texturing configurations is considered. Flow simulations are based on the numerical solution of the Navier–Stokes equations for incompressible isothermal flow. To optimize the bearing performance, an optimization problem is formulated and solved by coupling the CFD code with an optimization tool based on genetic algorithms and local search methods. Here, the design variables define the bearing geometry, while load carrying capacity is the objective function to be maximized. Optimized texturing geometries are obtained for the case of parallel bearings for several numbers of dimples, illustrating significant load carrying capacity levels. Further, these optimized texturing patterns are applied to converging bearings for different convergence ratio values; the results demonstrate that, for small and moderate convergence ratios, a substantial increase in load carrying capacity, in comparison to smooth bearings, is obtained. Finally, an optimization study performed at a high convergence ratio shows that, in comparison to the parallel slider, the optimal texturing geometry is substantially different, and that performance improvement over smooth bearings is possible even for steep sliders.

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.

A method for topology optimization of hinge-free compliant mechanisms with multiple outputs using level set method is presented in this paper. The focus of this paper is on how to prevent generating the flexible hinges during the process of topology optimization of compliant mechanisms. In the proposed method, two types of mean compliances are introduced and built in the proposed multi-objective function for topology optimization of hinge-free compliant mechanisms with multiple outputs, therefore, the spring model widely used for topology optimization of compliant mechanisms is no longer needed. Some numerical examples are presented to illustrate the validity of the proposed method.

The Strain Gradient Notation Finite Element Method (SGN-FEM) employs physically interpretable polynomials in the development of finite elements, allowing for the precise identification and subsequent elimination of parasitic shear sources, which cause shear locking. The element is corrected a priori, during development, by simply removing the spurious terms from the shear strain polynomials.The Level Set Method (LVM) is an optimization technique widely employed in topology optimization to minimize the compliance of structures. This method consistently provides clear boundary and geometry information during the optimization process, giving it inherent advantages in solving boundary and geometry-related problems.While numerous studies investigate different approaches to calculate level set optimization, there are fewer studies evaluating how the finite element technique contributes to the analysis. Considering this, this paper presents the application of Q4 SGN-FEM in optimization studies using LVM to minimize the compliance of a two-dimensional linear elastic structure. The differences obtained when using the SGN-FEM model are investigated. The parametrized level set method, as defined in Wei et al. (2018), is utilized alongside SGN-FEM to conduct the study. The model is validated through comparison with other studies, demonstrating that SGN-FEM proves to be a viable alternative for conducting optimization studies using the level set method.

This paper presents design data defining optimal freeform notch shapes for use in tensile loading conditions for a range of geometries, notably the notch aspect ratio and plate widths. Here, an iterative two-dimensional finite element gradientless shape optimization procedure was used to obtain the optimal shapes, which offer the lowest possible stress concentration, subject to the geometric constraints. Precise non-dimensional coordinates are given, for a range of notch aspect ratios and plate widths, and are presented in a tabular form that allows them to be easily used by designers. All cases have been solved with a minimum radius of curvature constraint, which results in a ‘robust’ optimal shape that has no sharp corners. The optimal shapes were found to provide up to a 26 per cent reduction in local peak stress over circular shapes of the same aspect ratio. Optimal geometries are also given for limiting notches that provide a net section stress equal to unity, at a finite notch aspect ratio. The results obtained are applicable to both the initial design of notched components and the shape reworking of existing components to remove damage.

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.

In today's society, with an increasing need for energy worldwide and at the same time a growing awareness of its negative consequences, optimizing and reducing energy consumption has become important for sustainability. The process of optimizing the appliances in large industry and household scale contributes to the minimization of the global energy consumption. In order to increase the efficiency of these appliances, various optimizations are employed, such as the shape optimization. In this thesis, a shape optimization method is used to modify the geometry of an existing heat exchanger. The heat exchanger is used in a domestic boiler and the shape optimization is a crucial step for its fast development. Currently, the shape of the heat exchanger is highly dependent on its manufacturing process. Thus, more complex shapes are investigated and their effects on the heat transfer and pressure drop are evaluated in this thesis. Two aspects need to be investigated for this heat exchanger optimization: the complex internal flow within the heat exchanger, and the performance of the optimization method as well as its resulting shapes. Since the heat exchanger consists of different flow regimes due to its complex geometry, a detailed study on these regimes (ReD = 2000, 100, and 10) is conducted prior to the optimization procedure. Furthermore, the complex geometry of this heat exchanger leads to different models used in this thesis. Therefore this thesis is divided into three parts based on the models: the three-dimensional single cylinder, the two-dimensional cylinder arrays, and the three-dimensional cylinder array. In optimizing the heat exchanger, the shape optimization method is justified as the most suitable method. Using the ANSYS Fluent's adjoint shape optimization method, the pressure drop and the heat transfer of the heat exchanger are optimized. This method has been used in the past for optimizing various geometries, yet to the author's knowledge, no detailed study on the cylinder array has been published. Consequently, a study on the adjoint parameters and its effect on the overall performance of the optimization procedure is of importance. Furthermore, best practices and recommendations for future work regarding the adjoint optimization procedure are elaborated.