A finite element (FE) based level set method is proposed for structural topology optimization problems in this thesis. The level set method has become a popular tool for structural topology optimization in recent years because of its ability to describe smooth structure boundaries and handle topological changes. There are commonly two stages in the optimization process: the stress analysis stage and the boundary evolution stage. The first stage is usually performed with the finite element method (FEM) while the second is often realized by solving the level set equation with the finite difference method (FDM). The first motivation for developing the proposed method is the desire to unify the techniques of both stages within a uniform framework. In addition, there are many problems involving irregular design domains in practice, the FEM is more powerful than the FDM in dealing with these problems. This is the second motivation for this study. / Numerical examples are involved in this thesis to illustrate the reliability of the proposed method. Problems on both regular and irregular design domains are considered and different meshes are tested and compared. / Solving the level set equation with the standard Galerkin FEM might produce unstable results because of the hyperbolic characteristic of this equation. Therefore, the streamline diffusion finite element method (SDFEM), a stabilized method, is employed to solve the level set equation. In addition to the advantage of simplicity, this method generates a system of equations with a constant, symmetric, and positive defined coefficient matrix. Furthermore, this matrix can be diagonalized by virtue of the lumping technique in structural dynamics. This makes the cost in solving and storing quite low. It is more important that the lumped coefficient matrix may help to improve the stability under some circumstance. / The accuracy of the finite element based level set method (FELSM) is compared with that of the finite difference based level set method (FDLSM). The FELSM is a first-order accurate algorithm but we prove that its accuracy is enough for structural optimization problems considered in this study. Even higher-order accurate FDLSM schemes are used, the numerical results are still the same as those obtained by FELSM. It is also shown that if the Courant-Friedreichs-Lewy (CFL) number is large, the FELSM is more robust and accurate than FDLSM. / The reinitialization equation is also solved with the SDFEM and an extra diffusion term is added to improve the stability near the boundary. We propose a criterion to select the factor of the diffusion term. Due to numerical errors and the diffusion term, boundary will drift during the process of reinitialization. To constrain the boundary from moving, a Dirichlet boundary condition is enforced. Within the framework of FEM, this enforcement can be conveniently preformed with the Lagrangian multiplier method or the penalty method. / Velocity extension is discussed in this thesis. A natural extension method and a partial differential equation (PDE)-based extension method are introduced. Some related topics, such as the "ersatz" material approach and the recovery of stresses, are discussed as well. / Xing, Xianghua. / Adviser: Michael Yu Wang. / Source: Dissertation Abstracts International, Volume: 71-01, Section: B, page: 0628. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 102-113). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese.

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

This thesis presents a framework for simulating the fluid dynamical behavior of complex moving boundary problems in a high-performance computing environment. The framework is implemented in the pELAFINT3D software package. Moving boundaries are evolved in a seamless fashion through the use of distributed narrow band level set methods and the effect of moving boundaries is incorporated into the flow solution by a novel Cartesian grid method. The proposed Cartesian grid approach builds on the concept of a ghost fluid method where boundary conditions are applied through least-squares polynomial extrapolations. The method is hybridized such that computational cells adjacent to moving boundaries change discretization schemes smoothly in time to avoid the introduction of strong oscillations in the pressure field. The hybridization is shown to have minimal effect on accuracy while significantly suppressing pressure oscillations. The computational capability of the Cartesian grid approach is enhanced with a massively parallel adaptive meshing algorithm. Local mesh enrichment is effected through the use of octree refinement, and a scalable mesh pruning algorithm is used to reduce the memory footprint of the Cartesian grid for geometries which are not well bounded by a rectangular cuboid. The computational work is kept in a well-balanced state through the use of an adaptive repartitioning strategy. The numerical scheme is validated against many benchmark problems and the composite approach is demonstrated to work well on tens of thousands of computational cores. A simulation of the closure phase of a mechanical heart valve was carried out to demonstrate the ability of the pELAFINT3D package to compute high Reynolds number flows with complex moving boundaries and a wide disparity in length scales. Finally, a novel image-to-computation algorithm was implemented to demonstrate the flexibility the current method allows in designing new applications.

The subject of this thesis is the development of an accurate, general and robust numerical method capable of predicting the flow behavior of two-phase immiscible fluids, separated by a well defined interface. In the quest of an accurate and robust numerical method for the modeling of two-phase flows, one has to keep in mind the intrinsic properties and difficulties associated with the problem: (i) those flows are mostly three-dimensional, (ii) some flows are steady, others unsteady, (iii) the interface might encounter a lot of topology changes (like merger or break-up), (iv) large jumps of density and viscosity might exist across the interface (e.g. ratio of density of 1/1000 for water and air), (v) surface tension forces may play a very important role in the interface dynamics. Hence, the influence of this force should be accurately evaluated and incorporated into the model, (vi) mass conservation is of primary importance. All these issues are addressed in this thesis, and special techniques are proposed for their treatment, which enables to construct the desired computational method. The chosen computational method combines a pressure stabilized finite element method for the Navier Stokes equations with a discontinuous Galerkin (DG) method for the level set equation. Such a combination of those two numerical methods results in a simple and effective algorithm that allows to simulate diverse flow regimes presenting large density and viscosity ratios (ratio up to 1/1000).

In bubble dynamics, an underwater bubble may evolve from being singly-connected to being toroidal. Furthermore, two or more individual bubbles may merge to form a single large bubble. These dynamics involve significant topological changes such as merging and breaking, which may not be handled well by front-tracking boundary element methods. In the level set method, topological changes are handled naturally through a higher-dimensional level set function. This makes it an attractive method for bubble simulation. In this paper, we present a method that combines the level set method and the boundary element method for the simulation of bubble dynamics. We propose a formulation for the update of a potential function in the level set context. This potential function is non-physical off the bubble surface but consistent with the physics on the bubble surface. We consider only axisymmetric cavitation bubbles in this paper. Included in the paper are some preliminary results and findings. / Singapore-MIT Alliance (SMA)

Simultaneous Localization and Mapping (SLAM) algorithms are used by autonomous robots to build or update maps of an environment while maintaining their position simultaneously. A fundamental open problem in SLAM is the e ective representation of the map in unknown, ambiguous, complex, dynamic environments. Representing such environments in a suitable manner is a complex task. Existing approaches to SLAM use map representations that store individual features (range measurements, image patches, or higher level semantic features) and their locations in the environment. The choice of how the map is represented produces limitations which in many ways are unfavourable for application in real-world scenarios. In this thesis, a new approach to SLAM is explored that rede nes sensing and robot motion as acts of deformation of a di erentiable surface. Distance elds and level set methods are utilized to de ne a parallel to the components of the SLAM estimation process and an algorithm is developed and demonstrated. The variational framework developed is capable of representing complex dynamic scenes and spatially varying uncertainty for sensor and robot models. / UOIT

This study uses the Phase-field method to simulate the transient thermal current of the metal surface heated and molten by a massing energy. The flow field uses a two-dimension module, considered with the mass conservation equation, momentum equation, energy equation and level-set equation, to solve for the distribution in whole domain, including the interface, of temperature, velocity, pressure and level-set number. We ignore the effect of concentration diffusion, but consider about the effect of heat translation on the flow field. Finally the results will display the flows of air around molten area forced by buoyancy which is caused by high temperature, and the flows in molten area forced by thermocapillary which is caused by temperature gradient.

Zugl.: Aachen, Techn. Hochsch., Diss., 2008

In this dissertation, novel variational optical-flow and active-contour methods are investigated to address challenging problems in computer vision and shape-based imaging. Starting from traditional applications of these methods in computer vision, such as object segmentation, tracking, and detection, this research subsequently applies similar active contour techniques to the realm of shape-based imaging, which is an image reconstruction technique estimating object shapes directly from physical wave measurements. In particular, the first and second part of this thesis deal with the following two physically inspired computer vision applications. Optical Flow for Vision-Based Flame Detection: Fire motion is estimated using optimal mass transport optical flow, whose motion model is inspired by the physical law of mass conservation, a governing equation for fire dynamics. The estimated motion fields are used to first detect candidate regions characterized by high motion activity, which are then tracked over time using active contours. To classify candidate regions, a neural net is trained on a set of novel motion features, which are extracted from optical flow fields of candidate regions. Coupled Photo-Geometric Object Features: Active contour models for segmentation in thermal videos are presented, which generalize the well-known Mumford-Shah functional. The diffusive nature of heat processes in thermal imagery motivates the use of Mumford-Shah-type smooth approximations for the image radiance. Mumford-Shah's isotropic smoothness constraint is generalized to anisotropic diffusion in this dissertation, where the image gradient is decomposed into components parallel and perpendicular to level set curves describing the object's boundary contour. In a limiting case, this anisotropic Mumford-Shah segmentation energy yields a one-dimensional ``photo-geometric'' representation of an object which is invariant to translation, rotation and scale. These properties allow the photo-geometric object representation to be efficiently used as a radiance feature; a recognition-segmentation active contour energy, whose shape and radiance follow a training model obtained by principal component analysis of a training set's shape and radiance features, is finally applied to tracking problems in thermal imagery. The third part of this thesis investigates a physics-driven active contour approach for shape-based imaging. Adjoint Active Contours for Shape-Based Imaging: The goal of this research is to estimate both location and shape of buried objects from surface measurements of waves scattered from the object. These objects' shapes are described by active contours: A misfit energy quantifying the discrepancy between measured and simulated wave amplitudes is minimized with respect to object shape using the adjoint state method. The minimizing active contour evolution requires numerical forward scattering solutions, which are obtained by way of the method of fundamental solutions, a meshfree collocation method. In combination with active contours being implemented as level sets, one obtains a completely meshfree algorithm; a considerable advantage over previous work in this field. With future applications in medical and geophysical imaging in mind, the method is formulated for acoustic and elastodynamic wave processes in the frequency domain.

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