An Efficient, Interface-Preserving Level Set Redistancing Algorithm and Its Application to Interfacial Incompressible Fluid Flow

This paper deals with structural optimization for designing periodic structures in a hole array wavelength filter. The hole array wavelength filter that consists of metallic thin film and dielectric enables to transmit narrow bandwidth light. It is known that transmission spectrum can be changed not only by the periodicity of hole array but also by the shape of holes. For optimizing the hole shape, the level set method is used in this study. In the ordinary level set method, the boundaries are implicitly expressed by the zero level set of a scalar function, called the level set function, within a fixed mesh. Therefore, the material interpolation becomes numerically awkward within the elements across the implicit zero level set because those elements inevitably take on intermediate material properties even if the boundary of zero level set are mathematically clear. As the result, the optimization is likely to yield wrong solution. Here, a new level set optimization method incorporating Arbitrary Lagrangian Eulerian method is proposed to eliminate intermediate values on the interfaces perfectly. As a result, the proposed method can successfully perform the structural optimization of hole shape without intermediate values.

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.

Geometric models often contain small features that we would like to remove, including small extrusions and holes. Feature removal can be desirable for many reasons, including finite element simulation, analysis of feature purpose and so on. In this study, an algorithm based on level set method is proposed to remove the small features. First, we initialize the zero level set in such a way that the zero level set includes the input model. And then the zero level set evolves towards the input model based on a novel speed function. When zero level set stops moving, a simplified model without small features can be got. The method receives approval from the experimental results.

In this paper, we present a robust and accurate algorithm for interactive image segmentation. The level set method is clearly advantageous for image objects with a complex topology and fragmented appearance. Our method integrates discriminative classification models and distance transforms with the level set method to avoid local minima and better snap to true object boundaries. The level set function approximates a transformed version of pixelwise posterior probabilities of being part of a target object. The evolution of its zero level set is driven by three force terms, region force, edge field force, and curvature force. These forces are based on a probabilistic classifier and an unsigned distance transform of salient edges. We further propose a technique that improves the performance of both the probabilistic classifier and the level set method over multiple passes. It makes the final object segmentation less sensitive to user interactions. Experiments and comparisons demonstrate the effectiveness of our method.

Level set methods have been widely used in image processing and computer vision. In conventional level set formulations, the level set function typically develops irregularities during its evolution, which may cause numerical errors and eventually destroy the stability of the evolution. Therefore, a numerical remedy, called reinitialization, is typically applied to periodically replace the degraded level set function with a signed distance function. However, the practice of reinitialization not only raises serious problems as when and how it should be performed, but also affects numerical accuracy in an undesirable way. This paper proposes a new variational level set formulation in which the regularity of the level set function is intrinsically maintained during the level set evolution. The level set evolution is derived as the gradient flow that minimizes an energy functional with a distance regularization term and an external energy that drives the motion of the zero level set toward desired locations. The distance regularization term is defined with a potential function such that the derived level set evolution has a unique forward-and-backward (FAB) diffusion effect, which is able to maintain a desired shape of the level set function, particularly a signed distance profile near the zero level set. This yields a new type of level set evolution called distance regularized level set evolution (DRLSE). The distance regularization effect eliminates the need for reinitialization and thereby avoids its induced numerical errors. In contrast to complicated implementations of conventional level set formulations, a simpler and more efficient finite difference scheme can be used to implement the DRLSE formulation. DRLSE also allows the use of more general and efficient initialization of the level set function. In its numerical implementation, relatively large time steps can be used in the finite difference scheme to reduce the number of iterations, while ensuring sufficient numerical accuracy. To demonstrate the effectiveness of the DRLSE formulation, we apply it to an edge-based active contour model for image segmentation, and provide a simple narrowband implementation to greatly reduce computational cost.

With increasing computer capabilities, numerical modeling of two-phase flows has developed significantly over the last few years. Although there are two main categories, namely 'one' fluid and 'two' fluid methods, the 'one' fluid methods are more commonly used for tracking or capturing the interface between two fluids. Level set (LS), volume-of-fluid (VOF), front tracking, marker-and-cell (MAC) and lattice Boltzmann (LB) methods are all 'one' fluid methods. It is clear that there is no perfect method; each method has advantages and disadvantages which make it more appropriate for one kind of problem than for others. For instance, a LS method will accurately compute the curvature and the normal to the interface, but tends to loss mass which is physically incorrect. On the other hand, a VOF method will conserve mass up to machine precision, but the computation of the curvature and normal to the interface is not as accurate. In order to minimize the disadvantages of these methods, several authors have used two or more methods together to model two-phase flows. This is the case for the CLSVOF (Couple Level Set Volume Of Fluid) method, where LS and VOF are coupled together in order to better capture the interface. In CLSVOF, the level set function is used to compute the interface curvature and normal to the interface, while the volume of fluid function is used to capture the interface. For two-phase flows in microchannels, surface tension forces play an important role in determining the dynamics of bubbles whereas gravitational forces are generally negligible. Also it is very important to consider the interaction between the boundaries and the fluids by prescribing or computing the correct contact angle between them. The commercial CFD code FLUENT allows the use of static constant angles, or the use of User Defined Functions (UDF) to compute the dynamic contact angles. It is inappropriate to use a static contact angle to model cases involving moving contact lines. For such cases a dynamic contact angle scheme should be implemented. In this study, FLUENT was used to model adiabatic and diabatic, time dependent two-phase flows. Since FLUENT already contains a VOF method, a LS method was implemented and coupled with VOF into FLUENT via UDFs. Furthermore, since the LS function, used to compute the surface tension force, ceases to be a signed distance to the interface even after one time step, a re-initialization equation was solved after each time step. This involved using a fifth order WENO (Weighted Essentially Non Oscillatory) scheme to discretize the space derivatives (otherwise oscillations of the interface occurred), and a first order Euler method for the time integration. In another part of the study, a 3D dynamic contact angle model based on volume fraction, interface reconstruction, and experimentally available advancing and receding static contact angles was also developed and implemented into FLUENT via UDFs. Several validations for the developed CLSVOF method and dynamic contact angle model are presented in this thesis, these includes a static bubble, a bubble rising in a stagnant liquid for Morton numbers ranging from 102 to 10-11, droplet deformation due to a vortex flow field, droplets spreading over a wall under the gravity effect and droplets sliding over a wall due to gravity. These validations demonstrated the high accuracy and the stability of our methods for modeling these phenomena. A heat and mass transfer model was also implemented into the commercial CFD code FLUENT for simulating of boiling (and condensation) heat transfer. Several simulations were presented with water and R134a as working fluids. The influence of the contact angle and the wall superheat was also studied.

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.

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 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.

The design of noise-free, low-pollutant combustion devices requires the understanding and the prediction of the combustion noise. The noise originates from the coupling between perturbations in the heat release rate with the oscillating gas (acoustic velocity). Prediction of combustion noise requires the transfer function (TF) that correlates the variation in the heat release rate of the ame with the perturbation (acoustic velocity). Computational costs prohibit an e??cient calculation of the TF by simulating all combustion phenomena associated with the ame dynamics. Simpli??ed theoretical models, such as the hydrodynamic models, allow a signi??cant reduction of the computation time and capture the main features of the ame front dynamics. The fundamental assumption of the hydrodynamic models is that the Bunsen ame, which is described as a thin layer ( ame front) separating the burnt from the unburnt gas, is attached at the burner rim. Within the thin layer approximation, the computation of the TF requires the ame area and, consequenctly, the time-dependent ame front position. By de??ning the ame front as the zero-level set of a distance function, the evolution of the ame front is described by a level set equation. The existent experiments and theoretic kinematic models based on the thin layer approximation (e.g., the G-equation model) provide contradictory information on the phase of the TF. Experiments suggest a low-pass ??lter behaviour of the gain of the TF and an increase of the TF's phase up to several ?? as the frequency increases. In contrast, most of the theoretical models indicate a saturation of the phase of the TF at the level of ??/2. Although the use of a convective wave model for the ow [80] leads to a better agreement between theory and experiments, the origin of the convective wave remains little understood. The discrepancies between theory and experiments most likely reside in the simpli??cations employed in the theoretical models. To understand the behaviour of the experimental TF, here we extend the theoretical models based on the G-equation approach. In the ??rst step, we extend the models by addressing ames that have an arbitrary cone angle and a burning velocity with variable direction. To carry out this ??rst extension we analyse the evolution of a Bunsen ame in a Poiseuille ow from an initially at pro??le to a stationary conical shape. The analysis leads to the correct derivation of the boundary conditions in the extended model employed for the ame evolution in a perturbed Poiseuille ow. We demonstrate that the new model proposed here improves the description of the front close to the boundary and, hence, the behaviour of the ame response to velocity perturbations. Because recent measurements [52] suggest that a better understanding of the TF's phase behaviour requires a hydrodynamic model that accounts for the moving edge of the ame, in the second step we extend the hydrodynamic model to account for non-attached ames. We model the ame front as an open curve by using two level set functions whose zero-level sets are locally normal at the edge point. The movement of the edge along the ame front is accounted for via a novel model for the edge speed. During computer simulations of the ame kinematics with the new extended model the local orthogonality of the level sets is preserved by employing a novel algorithm. The simulations demonstrate that our extended model qualitatively accounts for the ame stabilisation, the ame front and edge kinematics, and the stand-o?? distance of the ame above the burner rim.

Topology optimization of compliant mechanism and its support through a level set method

An efficient lithographic source and mask optimization approach is developed based on localized level set methods, which is reformulated as an inverse problem by tracking the evolution of level-set functions (LSFs) embedding the level-set representation of source and mask patterns. A distance regularized level-set (DRLS) term is incorporated into the level-set formulation enabling not only stable LSF evolution and accurate computation by maintaining a signed-distance property, but also a simple and efficient narrow-band implementation. Consequently, optimization dimensionality is significantly reduced by updating the pixels in the vicinity of zero level set (narrow band) instead of all the level sets, effectively reducing computation complexity, resulting in significantly improvements in pattern fidelity convergence in terms of runtime, computation load, Euler time step and caching memory requirement which are merited by numerical simulations.

The aim of this research is to construct a structural optimization method based on a level set method, the so-called convected level set method. The basic idea of this method is that the level set function is defined as a truncated smooth function obtained using a sinus filter based on a hyperbolic tangent function. The local property of the hyperbolic tangent level set function stabilizes the computation of the advection equation used for evolving the level set function and reduces the computational cost of the reinitialization that maintains the profile of the level set function. Additionally, the low computational cost characteristic enables the use of convective reinitialization, whose basic idea is that the reinitialization is incorporated in the advection equation, avoiding the need for a separate computation. Consequently, the convected level set method can avoid the need for additional computations when conducting reinitialization. The validity of our proposed method is tested with standard structural optimization problems.

Consistent mass and momentum transport for simulating incompressible interfacial flows with large density ratios using the level set method

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.