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It is quite meaningful in an engineering aspect to investigate liquid drop behaviors in two-phase flow, on which dispersive mixing or overall rheological characteristics heavily depend. There are two approaches numerically to deal with two-phase flow problems, which are the moving- and the fixed-grid methods: that is, interface tracking and interface capturing methods. In this study, we developed multiphase flow codes in both Lagrangian and Eulerian frameworks and compared each solution for the single drop problem in an axisymmetric channel flow. Deformation patterns of drops were observed by these two methods and we have discussed the characteristic features of two methods from the deformation of drop for the moving boundary problems.

A three-dimension numerical wave model (3DWAVE) has been developed to simulate free surface flows. The model solves Navier-Stokes equations for two-phase flows of air and water. The level set method is employed to track water surfaces. The model is tested for water sloshing in a 3-D confined tank. The relative error in the mass and total energy computation is less than 1%. Excellent agreements between numerical results and analytical solution are obtained for free surface calculation. The nonlinearity in the 3-D fluid sloshing is analyzed. These have laid a foundation on research of breaking waves.

In this paper, we propose an efficient active contour model for multiphase image segmentation in a variational level set formulation. By incorporating the globally convex segmentation idea and the split Bregman method into the multiphase formulation of the local and global intensity fitting energy model, our new model improved the original local and global intensity fitting energy model in the following aspects. First, we propose a new energy functional using the globally convex segmentation method to guarantee fast convergence. Second, we incorporate information from the edge into the energy functional by using a non-negative edge detector function to detect boundaries more easily. Third, instead of a constant value to control the influence of the local and global intensity fitting terms, we use a weight function varying with the locations of the image to balance the weights between the local and the global fitting terms dynamically. Lastly, the special structure of our energy functional enables us to apply the split Bregman method to minimize the energy much more efficiently. We have applied our model to synthetic images and real brain MR images with promising results. Experimental results demonstrate the efficiency and superiority of our model.

In order to improve the accuracy of image segmentation, an improved adaptive level set method is proposed based on level set evolution without re-initialization method and adaptive distance preserving level set evolution method. A new definition of weight coefficient in evolution equations is the main innovation of this paper. The improved method can detect certain object boundaries, interior and exterior contours of an object, edges of multi-objects and weak boundaries of an object by synthetic and real images numerical experiments. Numerical results show that the improved adaptive level set method has faster segmentation speed and higher segmentation accuracy compared with the previous two methods, especially in weak boundaries and edges of multi-objects segmentation problems.

A novel hybrid-fitting energy-based active contours model in the level set framework is proposed. The method fuses the local image fitting term and the global image fitting term to drive the contour evolution, and a special extra term that penalizes the deviation of the level set function from a signed distance function is also included in our method, so the complex and costly reinitialization procedure is completely eliminated. Our model can efficiently segment the images with intensity inhomogeneity no matter where the initial curve is located in the image. In its numerical implementation, two efficient numerical schemes are used to ensure the sufficient efficiency of the evolution process, one is the Lattice Boltzmann Model (LBM), which is used for breaking the restrictions on time step, the other is the Sparse Field Method (SFM), which is introduced for fast local computation. Compared with the traditional schemes, these two strategies can further shorten the time consumption of the evolution process, this allows the level set to quickly reach the true target location. The extensive and promising experimental results on numerous synthetic and real images have shown that our method can efficiently improve the image segmentation performance, in terms of accuracy, efficiency, and robustness.

We consider the reconstruction of singular surfaces from the over-determined boundary conditions of an elliptic problem. The problem arises in optical and impedance tomography, where void-like structure or cracks may be modeled as diffusion processes supported on co-dimension one surfaces. The reconstruction of such surfaces is obtained theoretically and numerically by combining a shape sensitivity analysis with a level set method. The shape sensitivity analysis is used to define a velocity field, which allows us to update the surface while decreasing a given cost function, which quantifies the error between the prediction of the forward model and the measured data. The velocity field depends on the geometry of the surface and the tangential diffusion process supported on it. The latter process is assumed to be known in this paper. The level set method is next applied to evolve the surface in the direction of the velocity field. Numerical simulations show how the surface may be reconstructed from noisy estimates of the full, or local, Neumann-to-Dirichlet map.

In this paper, we develop two efficient numerical methods for a multiscale kinetic equation in the context of crowd dynamics with emotional contagion [A. Bertozzi, J. Rosado, M. Short and L. Wang, Contagion shocks in one dimension, J. Stat. Phys.158 (2014) 647–664]. In the continuum limit, the mesoscopic kinetic equation produces a natural Eulerian limit with nonlocal interactions. However, such limit ceases to be valid when the underlying microscopic particle characteristics cross, corresponding to the blow up of the solution in the Eulerian system. One method is to couple these two situations — using Eulerian dynamics for regions without characteristic crossing and kinetic evolution for regions with characteristic crossing. For such a hybrid setting, we provide a regime indicator based on the macroscopic density and fear level, and propose an interface condition via continuity to connect these two regimes. The other method is based on a level set formulation for the continuum system. The level set equation shares similar forms as the kinetic equation, and it successfully captures the multi-valued solution in velocity, which implies that the multi-valued solution other than the viscosity solution should be the physically relevant ones for the continuum system. Numerical examples are presented to show the efficiency of these new methods.

With a different speed function, Level Set Method has been widely applied to many applications. Generally speaking, speed function may depend on many factors, such as curvature, normal direction. In this paper, we discuss a novel speed function which is only determined by neighbors' support. With enough support, zero level set can move or stop. Otherwise, it must wait for a moment before a decision to move or stop is made. In addition, an algorithm based on normal diffusion is proposed to smooth the zero level set, which can preserve the sharp feature and round corner at the same time. Experimentally, the proposed method has been successfully used for interested objection segmentation and mesh segmentation.

Scanning electron microscopy (SEM) is of great importance for studying fractal permeability. In this work, we presented a new technique, by applying the high-order upwind compact difference schemes to solve the hyperbolic conservation laws, to enhance textural differences for accurate segmentation of the SEM images. From the enhanced SEM images, the channels and pores can be obtained by using the two-stage image segmentation. Combining with the box counting method, the key parameters for evaluation of the fractal permeability such as the tortuosity fractal dimension, the pore area fractal dimension and the maximum pore area can be derived from the segmented images. Application of the technique to the SEM images of a red sandstone from south China shows remarkable enhancement of edge details, allowing the more accurate segmentation of the SEM images. Rather than the original image algorithm, the fractal permeability derived from this new approach is closer to the experimental value, especially when the magnification falls in the range of 500–600. The results evidence that our enhanced images approach may provide stronger constraints on evaluations of permeability of sandstones.

The level set method (LSM) is used in a wide variety of applications. In this paper, the authors propose an approach to detect and eliminate protruding and hollow features in triangular meshes using the technique. First, uniform grid points are generated based on a given mesh model, and these points are then clustered with respect to a user-defined threshold using LSM. Finally, the mesh model is segmented in consideration of the grouped grid points. The results of experiments performed using the proposed method indicate its suitability for the purpose at hand.

Two familiar approaches to image segmentation are the salient contour extraction approach and the closed-contour deformation approach. The former uses Gestalt laws to link individual edge elements and construct segmentation boundaries. However, it is often difficult to have both closure and precision of the boundary addressed at the same time. The latter starts with a closed contour and deforms the contour to localize the segmentation boundary more precisely whilst maintaining the closure. The approach does not have the closure problem, but how to assign a proper initial contour for it remains an open issue. In this work, we propose a scheme that puts together the two approaches to let them work complementarily. Specifically, we design a salient contour extraction process that extracts a proper initialization of the closed contours; the process looks into edge evidence and proximity to the desired segmentation boundaries. Then, a region-based active contour in a level set formulation is adopted to refine the contour position to locate the segmentation boundaries more precisely. The scheme requires neither manual input on contour initialization nor prior knowledge about the imaged scene. Experiments on extensive benchmarking image-sets are presented to illustrate the performance of the scheme.

In this paper, we propose a novel cosegmentation algorithm based on active contour model which utilizes local and global image statistics. Many localized region-based active contour models have been proposed to solve a challenging problem of the property (such as intensity, color, texture, etc.) inhomogeneities that often occurs in real images, but these models usually cannot reasonably evolve the curve in this situation that some center points along the curve are in homogeneous regions and their local regions are far away from the object. In order to overcome the difficulties we selectively enlarge the driven force of some points and introduce the edge indicator function to avoid the curve over-shrinking or over-expanding on the salient boundaries. In addition, we introduce global image statistics to better the curve evolution and try to avoid the given energy functional converging to a local minimum. Practical experiments show that our algorithm can obtain better segmentation results.

Antibody-based cell isolation using microfluidics finds widespread applications in disease diagnostics and treatment monitoring at point of care (POC) for global health. However, the lack of knowledge on underlying mechanisms of cell capture greatly limits their developments. To address this, in this study, we developed a mathematical model using a direct numerical simulation for the detachment of single leukocyte captured on a functionalized surface in a rectangular microchannel under different flow conditions. The captured leukocyte was modeled as a simple liquid drop and its deformation was tracked using a level set method. The kinetic adhesion model was used to calculate the adhesion force and analyze the detachment of single captured leukocyte. The results demonstrate that the detachment of single captured leukocyte was dependent on both the magnitude of flow rate and flow acceleration, while the latter provides more significant effects. Pressure gradient was found to represent as another critical factor promoting leukocyte detachment besides shear stress. Cytoplasmic viscosity plays a much more important role in the deformation and detachment of captured leukocyte than cortex tension. Besides, better deformability (represented as lower cytoplasmic viscosity) noteworthy accelerates leukocyte detachment. The model presented here provides an enabling tool to clarify the interaction of target cells with functional surface and could help for developing more effective POC devices for global health.

Level set method has been widely applied in the field of image segmentation. However, the level set formulation is inevitably affected by the regularization function, in-homogeneity and weak edge in the process of evolution, which often leads to the instability and inaccuracy of image segmentation results. To solve these problems, a new distance regularization term defined by a double-well potential function is proposed to satisfy more ideal characteristics of signed distance property. In addition, a novel edge indicator function is introduced to segment images with uneven intensity or weak edge. Finally, the adaptive adjustment formulas of distance regularization and area parameters are derived to alleviate the difficulty of parameter adjustment. Experimental results show that the proposed model provides better accuracy and versatility, quantitative experiment on Weizmann segmentation evaluation database achieves mean Dice score (96.87%), IoU (94.38%), Hausdorff distance (3.20mm), Recall (97.68%) and Precision (96.32%), respectively.

A structured overset grid approach coupled with Reynolds-Averaged Navier-Stokes (overset-RANS) method is presented to provide an accurate resolution of two surface ships moving with opposite velocity in viscous fluids. The RANS equations with shear stress transport (SST) k - ω model are employed to treat the viscous turbulent flows. The fully nonlinear boundary condition at the free surface is satisfied at each time step and the evolution of the free surface is achieved by using the level set method. A structured overset grid approach is used to allow flexibility in grid generation, local mesh refinement, as well as the simulation of moving objects while maintaining good grid quality. The presented overset-RANS method is demonstrated by two surface Wigley ship hulls moving with opposite velocity in still water. The simulating results illustrate the feasibility of the presented method to compute the complex viscous free surface flows interacting with many moving ships in still water or in waves.

We present a fully second order IMplicit/EXplicit (IMEX) time integration technique for solving incompressible multi-phase flow problems. A typical incompressible multi-phase flow model consists of the Navier–Stokes equations plus an interface dynamics equation (e.g., the level set equation). Our IMEX strategy is applied to such a model in the following manner. The hyperbolic terms of the Navier–Stokes equations together with the interface dynamics equation are solved explicitly (Explicit Block) making use of the well-understood explicit numerical schemes [Leveque, R. J. [1998] Finite Volume Methods for Hyperbolic Problems, “Texts in Applied Mathematics”, (Cambridge University Press); Thomas, J. W. [1999] Numerical Partial Differential Equations II (Conservation Laws and Elliptic Equations), “Texts in Applied Mathematics” (Springer-Verlag, New York)]. On the other hand, the nonhyperbolic (stiff) parts of the flow equations are solved implicitly (Implicit Block) within the framework of the Jacobian-Free Newton Krylov (JFNK) method [Knoll, D. A. and Keyes, D. E. [2004] Jacobian-free Newton Krylov methods: A survey of approaches and applications. J. Comput. Phys.193, 357–397; Saad, Y. [2003] Iterative Methods for Sparse Linear Systems (Siam); Kelley, C. T. [2003] Solving Nonlinear Equations with Newton’s Method (Siam)]. In our algorithm implementation, the explicit block is embedded in the implicit block in a way that it is always part of the nonlinear function evaluation. In this way, there exists a continuous interaction between the implicit and explicit algorithm blocks meaning that the improved solutions (in terms of time accuracy) at each nonlinear iteration are immediately felt by the explicit block and the improved explicit solutions are readily available to form the next set of nonlinear residuals. This continuous interaction between the two algorithm blocks results in an implicitly balanced algorithm in that all nonlinearities due to coupling of different time terms are converged with the desired numerical time accuracy. In other words, we obtain a self-consistent IMEX method that eliminates the possible order reductions in time convergence that is quite common in certain types of nonlinearly coupled systems. We remark that an incompressible multi-phase flow model can be a highly nonlinearly coupled system with the involvement of very stiff surface tension source terms. These kinds of flow problems are difficult to tackle numerically. In other words, highly nonlinear surface terms may remain unconverged leading to time inaccuracies or time order reductions to the first order even though the overall numerical scheme is designed as high order (second-order or higher) [Sussman, M. and Ohta, M. [2009] A stable and efficient method for treating surface tension in incompressible two-phase flow, SIAM J. Sci. Comput.31(4), 2447–2471; Zheng, W., Zhu, B., Kim, B. and Fedkiw, R. [2015] A new incompressibility discretization for a hybrid particle MAC grid representation with surface tension, J. Comput. Phys.280, 96–142]. These and few more issues are addressed in this paper. We have numerically tested our newly proposed scheme by solving several multi-phase flow settings such as an air bubble rising in water, a Rayleigh–Taylor instability problem that is initiated by placing a heavy fluid on top of a lighter one, and a droplet problem in which a water droplet hits the pool of water. Our numerical results show that we have achieved the second-order time accuracy without any order reductions. Moreover, the interfaces between the fluids are captured reasonably well.

This paper presents the development of an accurate and robust numerical modeling of instability of an interface separating two-phase system, such as liquid–gas and/or solid–gas systems. The instability of the interface can be refereed to the buoyancy and capillary effects in liquid–gas system. The governing unsteady Navier–Stokes along with the stress balance and kinematic conditions at the interface are solved separately in each fluid using the finite-volume approach for the liquid–gas system and the Hamilton–Jacobi equation for the solid–gas phase. The developed numerical model represents the surface and the body forces as boundary value conditions on the interface. The adapted approaches enable accurate modeling of fluid flows driven by either body or surface forces. The moving interface is tracked and captured using the level set function that initially defined for both fluids in the computational domain. To asses the developed numerical model and its versatility, a selection of different unsteady test cases including oscillation of a capillary wave, sloshing in a rectangular tank, the broken-dam problem involving different density fluids, simulation of air/water flow, and finally the moving interface between the solid and gas phases of solid rocket propellant combustion were examined. The latter case model allowed for the complete coupling between the gas-phase physics, the condensed-phase physics, and the unsteady nonuniform regression of either liquid or the propellant solid surfaces. The propagation of the unsteady nonplanar regression surface is described, using the Essentially-Non-Oscillatory (ENO) scheme with the aid of the level set strategy. The computational results demonstrate a remarkable capability of the developed numerical model to predict the dynamical characteristics of the liquid–gas and solid–gas flows, which is of great importance in many civilian and military industrial and engineering applications.

An algorithm has been developed to identify the position of voids in structures by coupling level set method and finite element method (FEM). The identification problem is transformed into a minimum problem whose objective function is defined as a least square form of displacement error. A perimeter constraint term is also added in the objective function to make the solution well posed. The level set is applied in the present algorithm to represent the position and geometry of the voids. The velocity field of level set function is obtained by analyzing the shape derivative of objective function. FEM based on Euler description is employed for solving the forward problem. The same fixed meshes adopted by the solution of forward problem are used for finite difference computation of the level set function. The procedure of this algorithm has been applied to the voids identification of two-dimensional (2D) and three-dimensional (3D) structures, the examples of single void and multiple voids are considered. The results indicate that the voids in structure can be identified effectively by the present algorithm and the algorithm is also stable to noise.

Thermal actuators use thermal expansion and contraction of an elastic body to produce motion at its output port. In the present study, a thermal actuator comprises an elastic body and heating/cooling devices. Such devices have a thin-layer shape and are installed on the surface of the elastic body. The design optimization of thermal actuator is a multiphysics problem, including both heat conduction and elastic deformation. The design variables include multiple types of boundaries, i.e., the temperature boundary (high temperature and low temperature) and the free boundary. In order to solve such a multiphysics optimization problem involving multiple types of boundaries, the level set-based multiple-type boundary method is employed. In the analysis for the shape derivative of the temperature boundary, the constrained variational principle is employed to explicitly include the temperature boundary condition into the weak form of heat conduction equation. Numerical examples in two dimensions are investigated.

As an implementation form of basis function, interpolation matrices (IMs) have a crucial impact on parametric level set method (PLSM)-based structural topology optimization (STO). However, there are few studies on compressing IM into triangular matrix (TM) with less storage and computation. Algorithm using LU decomposition and Algorithm using innovative asymmetric basis functions that transform the IMs of compactly supported radial basis functions (CSRBFs) into highly sparse TMs are proposed. Theoretical derivation and numerical experiments show that they effectively improve computational efficiency.