Parametric shape and topology structure optimization with radial basis functions and level set method.

Lui, Fung Yee.Thesis (M.Phil.)--Chinese University of Hong Kong, 2008.Includes bibliographical references (leaves 83-92).Abstracts in English and Chinese.Acknowledgement --- p.iiiAbbreviation --- p.xiiChapter 1 --- Introduction --- p.1Chapter 1.1 --- Background --- p.1Chapter 1.2 --- Related Work --- p.6Chapter 1.2.1 --- Parametric Optimization Method and Radial Basis Functions --- p.6Chapter 1.3 --- Contribution and Organization of the Dissertation --- p.7Chapter 2 --- Level Set Method for Structure Shape and Topology Optimization --- p.8Chapter 2.1 --- Primary Ideas of Shape and Topology Optimization --- p.8Chapter 2.2 --- Level Set models of implicit moving boundaries --- p.11Chapter 2.2.1 --- Representation of the Boundary via Level Set Method --- p.11Chapter 2.2.2 --- Hamilton-Jacobin Equations --- p.13Chapter 2.3 --- Numerical Techniques --- p.13Chapter 2.3.1 --- Sign-distance function --- p.14Chapter 2.3.2 --- Discrete Computational Scheme --- p.14Chapter 2.3.3 --- Level Set Surface Re-initialization --- p.16Chapter 2.3.4 --- Velocity Extension --- p.16Chapter 3 --- Structure Topology Optimization with Discrete Level Sets --- p.18Chapter 3.1 --- A Level Set Method for Structural Shape and Topology Optimization --- p.18Chapter 3.1.1 --- Problem Definition --- p.18Chapter 3.2 --- Shape Derivative: an Engineering-oriented Deduction --- p.21Chapter 3.2.1 --- Sensitivity Analysis --- p.23Chapter 3.2.2 --- Optimization Algorithm --- p.28Chapter 3.3 --- Limitations of Discrete Level Set Method --- p.30Chapter 4 --- RBF based Parametric Level Set Method --- p.32Chapter 4.1 --- Introduction --- p.32Chapter 4.2 --- Radial Basis Functions Modeling --- p.33Chapter 4.2.1 --- Inverse Multiquadric (IMQ) Radial Basis Functions --- p.38Chapter 4.3 --- Parameterized Level Set Method in Structure Topology Optimization --- p.39Chapter 4.4 --- Parametric Shape and Topology Structure Optimization Method with Radial Basis Functions --- p.42Chapter 4.4.1 --- Changing Coefficient Method --- p.43Chapter 4.4.2 --- Moving Knot Method --- p.45Chapter 4.4.3 --- Combination of Changing Coefficient and Moving Knot method --- p.46Chapter 4.5 --- Numerical Implementation --- p.48Chapter 4.5.1 --- Sensitivity Calculation --- p.48Chapter 4.5.2 --- Optimization Algorithms --- p.49Chapter 4.5.3 --- Numerical Examples --- p.52Chapter 4.6 --- Summary --- p.65Chapter 5 --- Conclusion and Future Work --- p.80Chapter 5.1 --- Conclusion --- p.80Chapter 5.2 --- Future Work --- p.81Bibliography --- p.8

Dynamic multiscale topology optimization for multi-regional micro-structured cellular composites

© 2018 Elsevier Ltd In this paper, a new dynamic multiscale topology optimization method for cellular composites with multi-regional material microstructures is proposed to improve the structural performance. Firstly, a free-material distribution optimization method (FMDO) is developed to generate the overall configuration for the discrete element densities distributed within a multi-regional pattern. The macrostructure is divided into several sub regions, and each of them consists of a number of elements but with the same densities. Secondly, a dynamic topology optimization formulation is developed to perform the concurrent design of the macrostructure and material microstructures, subject to the multi-regional distributed element densities. A parametric level set method is employed to optimize the topologies of the macrostructure and material microstructures, with the effective macroscopic properties evaluated by the homogenization. In the numerical implementation, the quasi-static Ritz vector (QSRV) method is incorporated into the finite element analysis so as to reduce the computational cost in numerical analysis, and some kinematical connectors are introduced to make sure the connectivity between adjacent material microstructures. Finally, 2D and 3D numerical examples are tested to demonstrate the effectiveness of the proposed dynamic multiscale topology optimization method for the material-structural composites

A Novel Discrete Adjoint-based Level Set Topology Optimization Method in B-spline Space

This paper presents a novel computational scheme for sensitivity analysis of the velocity field in the level set method using the discrete adjoint method. The velocity field is represented in B-spline space, and the adjoint equations are constructed based on the discretized governing equations. The key contribution of this work is the demonstration that the velocity field in the level set method can be entirely obtained from the discrete adjoint method. This eliminates the need for shape sensitivity analysis, which is commonly used in standard level set methods. The results demonstrate the effectiveness of the approach in producing optimized results for stress and linearized buckling problems. Overall, the proposed method has the potential to simplify the way in which topology optimization problems using level set methods are solved, and has significant implications for the design of a broad range of engineering applications

Study of optimal design of 3D mechanical metamaterials

This thesis aims to develop and extend numerical methods for solving the elastic problem in 2D to a 3D solution. The primary objective is to investigate and identify the modifications and extensions necessary to adapt existing techniques used in 2D to a 3D framework. The focus is on developing a robust and efficient numerical method that accurately models the behavior of elastic materials in three dimensions. The study encompasses several objectives to achieve a comprehensive understanding of 3D material design and topology optimization in the context of mechanical metamaterials. Firstly, an introduction to topology optimization is provided, including the formulation and regularization of the problem, and an explanation of density-based and the Level Set method. The thesis further investigates the SWAN repository’s code, an object-oriented Matlab software, assessing its applicability for conducting simulations. The thesis explores the optimization of both normal materials and metamaterials microstructures. For normal materials, a comparison is made between different optimization approaches, specifically the MMA optimizer utilizing a densitybased method and the Null Space optimizer employing a level set-based method. Additionally, the impact of varying final volume fractions on the optimization outcomes is studied. This investigation provides valuable insights into the influence of different parameter variations on the resulting microstructures and optimization performance. Furthermore, the study focuses on metamaterials microstructures and explores their optimization using the Null Space optimizer, and different α and β values are employed to examine their effects on the final design. The optimization process is also conducted for different final volume fractions to evaluate the influence of volume fraction on metamaterial performance. This study on material design and topology optimization has yielded several important conclusions: the simulations showcased the relationship between dimensionality and convergence speed, with 2D simulations demonstrating faster convergence compared to 3D simulations; analysis of parameters such as the cost function and the number of iterations has been conducted comparing different optimizers, it has been highlighted the challenges and unique considerations involved in optimizing metamaterials; and, overall, the research has contributed to the understanding of optimization processes and the generation of innovative material configurations

Development of New Global Optimization Algorithms Using Stochastic Level Set Method with Application in: Topology Optimization, Path Planning and Image Processing

A unique mathematical tool is developed to deal with global optimization of a set of engineering problems. These include image processing, mechanical topology optimization, and optimal path planning in a variational framework, as well as some benchmark problems in parameter optimization. The optimization tool in these applications is based on the level set theory by which an evolving contour converges toward the optimum solution. Depending upon the application, the objective function is defined, and then the level set theory is used for optimization. Level set theory, as a member of active contour methods, is an extension of the steepest descent method in conventional parameter optimization to the variational framework. It intrinsically suffers from trapping in local solutions, a common drawback of gradient based optimization methods. In this thesis, methods are developed to deal with this drawbacks of the level set approach. By investigating the current global optimization methods, one can conclude that these methods usually cannot be extended to the variational framework; or if they can, the computational costs become drastically expensive. To cope with this complexity, a global optimization algorithm is first developed in parameter space and compared with the existing methods. This method is called "Spiral Bacterial Foraging Optimization" (SBFO) method because it is inspired by the aggregation process of a particular bacterium called, Dictyostelium Discoideum. Regardless of the real phenomenon behind the SBFO, it leads to new ideas in developing global optimization methods. According to these ideas, an effective global optimization method should have i) a stochastic operator, and/or ii) a multi-agent structure. These two properties are very common in the existing global optimization methods. To improve the computational time and costs, the algorithm may include gradient-based approaches to increase the convergence speed. This property is particularly available in SBFO and it is the basis on which SBFO can be extended to variational framework. To mitigate the computational costs of the algorithm, use of the gradient based approaches can be helpful. Therefore, SBFO as a multi-agent stochastic gradient based structure can be extended to multi-agent stochastic level set method. In three steps, the variational set up is formulated: i) A single stochastic level set method, called "Active Contours with Stochastic Fronts" (ACSF), ii) Multi-agent stochastic level set method (MSLSM), and iii) Stochastic level set method without gradient such as E-ARC algorithm. For image processing applications, the first two steps have been implemented and show significant improvement in the results. As expected, a multi agent structure is more accurate in terms of ability to find the global solution but it is much more computationally expensive. According to the results, if one uses an initial level set with enough holes in its topology, a single stochastic level set method can achieve almost the same level of accuracy as a multi-agent structure can obtain. Therefore, for a topology optimization problem for which a high level of calculations (at each iteration a finite element model should be solved) is required, only ACSF with initial guess with multiple holes is implemented. In some applications, such as optimal path planning, objective functions are usually very complicated; finding a closed-form equation for the objective function and its gradient is therefore impossible or sometimes very computationally expensive. In these situations, the level set theory and its extensions cannot be directly employed. As a result, the Evolving Arc algorithm that is inspired by "Electric Arc" in nature, is proposed. The results show that it can be a good solution for either unconstrained or constrained problems. Finally, a rigorous convergence analysis for SBFO and ACSF is presented that is new amongst global optimization methods in both parameter and variational framework

Level set topology optimization with meshfree methods for design-dependent multiphysics problems

This thesis aims to develop computational tools for multiphysics problems in topology optimization with particular focus on design-dependent surface physics. This is a challenging class of problems governed by interface conditions and loadings. Fluid-structure interaction (FSI) is a typical example. Level set topology optimization (LSTO) possesses an advantage over traditional density-based methods since it provides clearly defined boundaries. However, maintaining this crisp boundary representation onto the computational model for the analysis is not straightforward especially with fixed grid methodologies. On the other hand, remeshing the structure at each iteration directly addresses the problem, it poses additional difficulties because of the need to ensure good quality meshes at each iteration. In this thesis a meshfree level set topology optimization methodology based on the reproducing kernel particle method (RKPM) is developed to ensure the well-defined geometrical representation of the structural boundary is transferred onto the computational domain by placing RKPM particles along the boundary. In this way, the difficulties associated with fixed grid LSTO methods and remeshing-based approaches are avoided. The methodology is first validated for purely hydrostatic pressure, which is a very simple case of design-dependent physics. The obtained results are validated through comparison with the literature. Different integration schemes, particle distributions and continuity orders are also explored to pin down the best balance between accuracy and efficiency. The development of the LSTO-RKPM methodology is later extended to fluidstructure interactions. To accomplish this, the LSTO-RKPM methodology is further combined with the modified immersed finite element method (mIFEM). The coupling of the different methods is illustrated through the analysis of transient FSI examples. For the optimization, the simplified case of steady-state FSI is assumed. The applicability of the methodology is illustrated through examples and compared with the literature. For the sensitivity analysis, a particle-based discrete adjoint methodology for the level set topology optimization method is presented. Additionally, an algorithm for identifying and removing free-floating volumes of solid material into the fluid domain is explained

A topology optimized model based on the level-set method for porous bone scaffolds

In a well-designed porous sca old, there is a need for a convenient balance between biological compatibility and mechanical functionality such that the porosity permits tissue regeneration and solid structure carries the load in the best possible way. In this thesis, a design framework that allows for the optimal design and analysis of a bone sca old which undergoes tissue regeneration according to a self healing model is proposed. Computational models are implemented using COMSOL Multiphysics which provides the opportunity to build an FEA (Finite Element Analysis) model where boundary value problems from di erent disciplines and mathematical equations are coupled and studied at the same time to reach an optimally performing tissue. To simulate the self healing process a mechano-regulatory model is developed mimicking tissue regeneration. As the topology optimization design method, level-set method is employed, where the design process starts with an initial geometry, that satisfies physical constraints. At each time step, this geometry is improved based on sensitivity analysis results until convergence is reached. Results of both the mechanoregulatory and the topology optimization methods validate well-known benchmark design problems in literature. Finite element method integrated to the level-set based topology optimization is proven to be among the most computationally e cient and generic design tool for solving non-intuitive tissue engineering problems. Hence, the proposed design framework, when implemented with corresponding physical models, is equally applicable to other hard and soft tissue designs