Level Set Equation Research Articles

Plasma formation in ambient fluid from hypervelocity impacts

The generation of plasma from hypervelocity impacts is an active research topic due to its important science and engineering ramifications in various applications. Previous studies have mainly focused on the ionization of the solid materials that constitute the projectile and the target. In this letter, we consider impact events that occur in a fluid (e.g., gas) medium, and present a multiphysics computational modeling approach and associated analysis to predict the behavior of the dynamic fluid–solid interaction that causes the surrounding fluid to ionize. The proposed computational framework is applied to a specific case involving a system of three interacting domains: a copper rod projectile impacting onto a soda lime glass target in a neon gas environment. The impact velocity is varied between 3km/s and 6km/s in different simulations. The computational model couples the compressible inviscid Navier–Stokes equations with the Saha ionization equations. The three material interfaces formed among the projectile, the target, and the ambient gas are tracked implicitly by solving two level set equations that share the same velocity field. The mass, momentum, and energy fluxes across the interfaces are computed using the FInite Volume method with Exact two-material Riemann problems (FIVER). The simulation result reveals a region of neon gas with high velocity, temperature, pressure, and mass density, formed in the early stage of the impact mainly due to the hypersonic compression of the fluid between the projectile and the target. For impact velocities higher than 4km/s, ionization is predicted in this region.

A Novel Least-Squares Level Set Method by Using Polygonal Elements

In this study, we apply an artificial viscosity method to convert an unsteady level set (LS) convection equation into an unsteady LS convection-diffusion transport equation to stabilize the numerical solution of the convection term. Then a novel least-square polygonal finite element method is used to solve an unsteady LS convection-diffusion problem. The least-squares method provided good mathematical properties such as natural numerical diffusion and the positive definite symmetry of the resulting algebraic systems for the convection-diffusion and re-initialization equations. The proposed method is evaluated numerically in two different benchmark problems: a rigid body rotation of Zalesak’s disk, and a time-reversed single-vortex flow. In comparison with conventional triangular (T3) and quadrilateral (Q4) elements, polygonal elements are capable of providing greater flexibility in mesh generation for complicated problems as well as more accurate in solving the LS equations. In addition, the numerical results are also compared with the results which obtained from essentially non-oscillatory type formulations and particle LS methods. The results show that the proposed method completely matches the previously published results.

Malignant melanoma diagnosis applying a machine learning method based on the combination of nonlinear and texture features

Skin cancer affects people of all skin tones, including those with darker complexions. Melanomas are known as malignant tumors of skin cancer, resulting in an adverse prognosis, responsible for most deaths relating to skin cancer. Early diagnosis and treatment of skin cancer from dermoscopic images can significantly reduce mortality and save lives. In this paper, Computer-Aided Diagnosis (CAD) system based on machine learning algorithms is provided to classify various skin cancer types. The proposed method uses the Online Region-based Active Contour Model (ORACM) with a new binary level set equation and regularization operation to extract skin lesions' Region of Interest (ROI). Additionally, various combinations of different textures and nonlinear features are extracted for the ROI to show the multiple aspects of skin lesions. Several metaheuristic optimization algorithms are used to remove redundant or irrelevant features and reduce the feature space dimension. These are applied to the combination of the extracted features in which, the Non-dominated Sorting Genetic Algorithm (NSGA II) as a multi-objective optimization algorithm has the best performance. Furthermore, various machine learning algorithms including K-Nearest Neighbor (KNN), Support Vector Machine (SVM), Fitting neural network (Fit net), Feed-Forward neural network (FF net), and Pattern recognition network (Pat net) are employed for the classification. In this paper, the best-obtained evaluation parameters based on fivefold cross-validation belong to the combination of selected nonlinear indices, texture, and GLCM features with an accuracy: 92.24 %, specificity: 100 %, and sensitivity: 100 %, resulting through NSGA II and applying the pattern net classifier. Besides, the comparison between this paper's experimental results and other similar works demonstrates the proposed method's efficiency.

An interface-tracking space–time hybridizable/embedded discontinuous Galerkin method for nonlinear free-surface flows

We present a compatible space–time hybridizable/embedded discontinuous Galerkin discretization for nonlinear free-surface waves. We pose this problem in a two-fluid (liquid and gas) domain and use a time-dependent level-set function to identify the sharp interface between the two fluids. The incompressible two-fluid equations are discretized by an exactly mass conserving space–time hybridizable discontinuous Galerkin method while the level-set equation is discretized by a space–time embedded discontinuous Galerkin method. Different from alternative discontinuous Galerkin methods is that the embedded discontinuous Galerkin method results in a continuous approximation of the interface. This, in combination with the space–time framework, results in an interface-tracking method without resorting to smoothing techniques or additional mesh stabilization terms.

Micro-hole drilling of 2.5D C/SiC composite with picosecond laser: Numerical modeling and experimental validation on hole shape evolution

Micro-holes of 2.5D C/SiC composites produced by picosecond laser are widely used in aerospace, military, and automotive industries. An accurate numerical model is crucial to illustrate the formation mechanism of hole shape and ablation characteristics. This paper proposes a two-dimensional transient numerical model based on solid-liquid-gas coupling in micro-hole drilling of 2.5D C/SiC composites with a picosecond laser. In the stage of solid-liquid phase transition, original mass, energy, and momentum conservation equations are optimized by considering surface tension, gravity, and hydrostatic pressure. In the stage of liquid-gas phase change, the mass transfer during the material evaporation is considered and used as a source term to optimize the level set equation. The effectiveness of the proposed model is validated by picosecond laser micro-hole drilling experiments on 2.5D C/SiC composites. Compared with the existing research, the results show that the proposed can accurately depict hole shape evolution in picosecond laser drilling of 2.5D C/SiC composites, not only including entrance, exit diameter, and hole taper, but also the hole profile and its dynamic variation. In addition, the model effectively simulates the thermal effect phenomena, such as thermal damage, pressure concentration, and recast layer. And the thermal damage is found to be the main cause of the hole quality variation, which changes from severe to smooth along the hole entrance to exit. The research provides important support for high-quality micro-hole machining of 2.5D C/SiC composites with the picosecond laser.

Numerical simulation for liquid sloshing with baffle by the CLSVOF/IB method

The liquid sloshing problem in a partially filled rectangular tank is numerically investigated by a Coupled Level Set and Volume Of Fluid (CLSVOF) method. In the present CLSVOF method, the Level Set (LS) equation needs to be solved to advect the LS function for calculating interface curvature; the re-distancing equation is also taken into account for the LS function being maintained as a distance function. For capturing the interface, the Tangent of Hyperbola for INterface Capturing (THINC) scheme with Weighted Linear Interface Calculation (WLIC) is implemented to advect the Volume of Fluid (VOF) function. To preserve mass conservation, the interface reconstruction procedure needs to be carried out on the basis of the connection between LS and VOF functions. Rectangular and prismatic tanks are used to validate the accuracy and mass conservation of the present CLSVOF method. Then, further simulations of the problem of partially filled liquid sloshing with baffle considered are conducted by combining the CLSVOF method and the Immersed Boundary (IB) method. Finally, the influence of different baffle lengths on the slapping pressure and free-surface is discussed.

A novel method to design biomimetic, 3D printable stochastic scaffolds with controlled porosity for bone tissue engineering

Periodic cellular materials such as body-centered cubic, face-centered cubic, and triply periodic minimal surfaces, have been used to construct scaffolds for bone tissue engineering. Their use is suboptimal for reasons like stress concentration at nodes, and/or poor anisotropy. Stochastic structures can mimic the bone microarchitecture with anisotropic mechanical properties. While several methods exist for generating stochastic structures, they face limitations like being computationally expensive, complex, or only applicable in specific cases. In this work, scaffolds are created using level set equations which permit spatially controllable porosity. A 3D volume is populated with random nodes, which segment the 3D volume into subdomains. Each subdomain is occupied with a basic architecture generated through level-set equations. All the architectures in the subdomains are then smoothly integrated at sub-domain boundaries to form the stochastic scaffold. Stainless steel stochastic scaffolds with porosities from 58% to 70% were fabricated and their mechanical characteristics, as well as cell viability, was assessed. Young’s modulus of the scaffolds ranges from 0.02 to 2 GPa, in the same range as that of trabecular bone, thus, mitigating stress shielding. In-vitro assay displayed a statistically significant osteoblast growth from Day 1 to Day 3 in 58%, 61%, and 64% porosity scaffolds.

A two-grid binary level set method for structural topology optimization

A binary level set method of two-grid type is proposed for structural topology optimization. The method allows both shape and topological changes during evolution with no need for reinitialization. The Galerkin finite element method is used to discretize the linear elasticity equation, the eigenvalue problem in structure and the semi-implicit time discrete level set equation. In order to improve the efficiency of the gradient-type algorithm as well as balance the computational efforts of solvers, a nested two-grid discretization strategy is proposed. The linear elasticity or eigenvalue problem is solved on a coarse mesh, while the level set equation and the smoothing step are both discretized on a fine mesh. A variety of numerical results are given to demonstrate both the efficiency and effectiveness of the algorithm.

A parameter space approach for isogeometrical level set topology optimization

AbstractIn this article, a new isogeometrical level set topology optimization is introduced. In previous studies, the level set function is approximated by b‐splines basis functions and the control net is updated during the optimization process. Since control points are not basically on the level set surface, a discrepancy between zero level of function that represents boundaries of the structure, and zero level of its control netis appeared. In this article, the idea is solving level set equation (LSE) over the parameter space of b‐splines basis functions defined in the IGA model. Afterwards, the updated level set function is mapped into the physical space. The level set function is approximated over a grid defined in parameter space which is constantly a unit square. By doing so, control net of the IGA model and the grid for solving LSE are separated. Two well‐known radial basis functions (RBF) and reaction–diffusion (RD) based methods are employed to solve the LSE. The proposed method is applied to minimize the mean compliance when certain amount of material is used and also for weight minimization subject to local stress constraints. Several numerical examples are presented to demonstrate performance and accuracy of the proposed method.

Effective stiffness, strength, buckling and anisotropy of foams based on nine unique triple periodic minimal surfaces

Recent advancements in the field of additive manufacturing has led to an increased interest in the study of cellular architectures especially triply periodic minimal surfaces (TPMS) and their properties. In this study, 9 sheet-based TPMS architectures are studied by comparing their effective stiffness, strength, buckling and anisotropic properties. The TPMS lattices are generated using level set equations where the relative density of each lattice is varied to study the effective properties as a function of relative density. Finite element models of these lattices are built to study the elastic and plastic deformation behaviour under uniaxial, shear, hydrostatic and buckling loading conditions. Effective properties such as Young’s modulus, shear modulus, bulk modulus, Poisson’s ratio and strengths under uniaxial, shear, hydrostatic and buckling loading are computed against each other and other common lattice structures. In addition, anisotropic behaviour as well as the orientation dependence of the properties of each lattice are also studied. The geometric efficiency criterion is predominantly used to evaluate the different structures and the results obtained are consolidated in both quantitative and qualitative manner. SC lattice outperforms all the other lattices in terms of strength and stiffness efficiencies at lower and higher relative densities. On the other hand, I2 − Y** lattice shows the weakest stiffness and strength efficiency especially at higher relative densities. The current study focuses on the properties of some unique lattice architectures that have not been studied previously and shows promising efficiencies compared to the common architectures studied such as the Gyroid lattice. This study opens the door for employing these lattices in various applications and the results are very useful to multi-scale topology-optimization studies.

Modeling of the Saltwater Intrusion Using the Level Set Method. Application to Henry’s Problem

The salt intrusion phenomenon is caused by overexploitation of aquifers in coastal areas. This physical phenomenon has been the subject of numerous studies and numerous methods have been proposed, with the aim of protecting the quality of the water in these aquifers. This work proposes a two-dimensional saline intrusion model using the sharp interface approach and the level set method. It consists of a parabolic equation modeling the underground flow and a hyperbolic Equation (the level set equation) which makes it possible to track the evolution of the interface. High-order numerical schemes such as the space scheme WENO5 and the third-order time scheme TVD-RK were used for the numerical resolution of the hyperbolic equation. To limit the tightening of the contour curves of the level set function, the redistanciation or reinitialization algorithm proposed by Sussma et al. (1994) was used. To ensure the effectiveness and reliability of the proposed method, two tests relating to the standard Henry problem and the modified Henry problem were performed. Recall that Henry’s problem uses the variable density modeling approach in a confined and homogeneous aquifer. By comparing the results obtained by the level set method with reinitialization (LSMR) and those obtained by Henry (1964), and by Simpson and Clement (2004), we see in the two test cases that the level set method reproduces well the toe, the tip and the behaviour of the interface. These results correspond to the results obtained by Abarca for Henry’s problem with constant dispersion coefficients. The results obtained with LSMR, reproduced the interface with a slight spacing compared to those obtained by Henry. According to Abarca (2006), this spacing is due to the absence of the longitudinal and transversal dispersion coefficients in the model.

流体界面の一般化数理モデルの予混合火炎への適用

A unified mathematical model for fluid interface such as gas-liquid phase interface and chemical reaction surface is proposed as a viscosity solution of level-set equation, which is derived from conservative form of Allen-Cahn equation by phase field approach and is also related to an equation of arbitrary variable accompanied with order-parameter of the proposed level-set variable in the compressible fluid flow. This unified model is applied to flame approximations of pre-mixture combustion to investigate a relation between the conventional models, and is evaluated their model parameters by the machine learning using a solution of detailed chemical reaction calculation.

Mathematical study on the thickened interface model by viscosity solution of the level-set equation

Mathematical study on the thickened interface model by viscosity solution of the level-set equation

Lagrangian vs. Eulerian: An Analysis of Two Solution Methods for Free-Surface Flows and Fluid Solid Interaction Problems

As a step towards addressing a scarcity of references on this topic, we compared the Eulerian and Lagrangian Computational Fluid Dynamics (CFD) approaches for the solution of free-surface and Fluid–Solid Interaction (FSI) problems. The Eulerian approach uses the Finite Element Method (FEM) to spatially discretize the Navier–Stokes equations. The free surface is handled via the volume-of-fluid (VOF) and the level-set (LS) equations; an Immersed Boundary Method (IBM) in conjunction with the Nitsche’s technique were applied to resolve the fluid–solid coupling. For the Lagrangian approach, the smoothed particle hydrodynamics (SPH) method is the meshless discretization technique of choice; no additional equations are needed to handle free-surface or FSI coupling. We compared the two approaches for a flow around cylinder. The dam break test was used to gauge the performance for free-surface flows. Lastly, the two approaches were compared on two FSI problems—one with a floating rigid body dropped into the fluid and one with an elastic gate interacting with the flow. We conclude with a discussion of the robustness, ease of model setup, and versatility of the two approaches. The Eulerian and Lagrangian solvers used in this study are open-source and available in the public domain.

A new path planning strategy based on level set function for layered fabrication processes

In this paper, a new equidistant filling theory based on the level set function and a corresponding numerical algorithm based on the dynamic finite difference method were proposed. Firstly, a closed curve is defined as the zero-value contour of the level set function. Then, the level set equation, a partial differential equation, was built to get the level set function, and a dynamic finite difference method was proposed to solve the level set function. Secondly, three types of cross sections, simple polygon, multi-island polygon, and multi-hole polygon, were used to test the equidistant filling effect of the algorithm. The test results show that the proposed theory and algorithm in this paper solved the equidistant filling problem of these cross sections well. In addition, any complex section can be equidistant filled by the algorithm proposed. The calculation cost is significantly related to the number of the offsets, but not directly related to the section complexity. Finally, the remanufacturing of a typical crankshaft hot forging die proves that the algorithm proposed in this paper can effectively produce the remanufacturing repair path of a complex forging die. The surface of the remanufactured die is smooth, and there is no efects such as porosity and slag inclusion after machining. Compared with making a new die, the manufacturing cost can be saved more than 50%, and the efficiency of die manufacturing can be improved more than 60%. This is because the machining allowance of the die after additive manufacturing is very small, and it can be quickly repaired on site.

Effects of capillary number and flow rates on the hydrodynamics of droplet generation in two-phase cross-flow microfluidic systems

Background: The control and manipulation of the hydrodynamics of droplets primarily relate to flow governing and geometrical parameters. This study has explored the influences of capillary number (10−4≤Cac≤1) and flow rate ratio (0.1≤Qr≤10) on the hydrodynamics of droplet generation in two-phase flow through T-junction cross-flow microfluidic device.Methods: The finite element method is used to solve the Eulerian framework of a mathematical model based on mass continuity, Navier-Stokes, and conservative level set equations at fixed flow (Rec=0.1).Significant findings: Results are presented in terms of the instantaneous phase flow field, droplet size, droplet detachment time, and generation frequency as a function of governing parameters (Cac and Qr). The flow regimes namely squeezing, first transition, dripping, second transition, parallel, and jet flow are marked. In contrast to reported value of threshold Cac≈10−2, squeezing regime exists for all Cac and 2≤Qr≤10. The flow regimes are also mapped into droplets and non-droplet zones by using threshold Car which scales quadratically with Qr. The droplet length varies linearly with Qr in the squeezing regime. Both droplet size and frequency show the power-law relations with Cac and Qr in the droplet zone. Finally, predictive correlations are presented to guide the engineering and design of droplet microfluidics devices.

A conservative level-set/finite-volume method on unstructured grids based on a central interpolation scheme

In this paper, we introduce a second-order unstructured finite-volume method developed to solve a conservative level-set equation in two- and three-dimensional geometries. The characteristic function for the flow is defined at the center of each cell while the face-normal velocities are calculated at the mid-points of the corresponding cell faces. An interpolation method independent of cell shape and based on a central scheme is applied for the approximations at the cell faces. The capabilities and performance of the proposed scheme are validated using several 2D and 3D tests. Results show that the present numerical method is quite suitable for working within the smooth framework resulting from the signed distance function. Moreover, this method yields very accurate results in interface-capturing problems such as the single vortex deformation.

CFD simulation of air-sparged slug flow in the flat-sheet membrane: A concentration polarization study

In this study, CFD simulation of the air-sparged two-phase slug flow inside the flat-sheet membrane channel was performed using a novel approach. The model consists of the Level-Set equation to capture the bubble-water-membrane flow hydrodynamics and a modified convection–diffusion formulation to account for the solute transport. The effects of slug bubble motion on the variation of Concentration Polarization (CP) and permeation flux are studied in detail in the absence of biofouling and scaling. The variation of flow and concentration fields in the membrane channel, particularly near the membrane surface, is illustrated and discussed. According to the simulation results, for the two-phase slug-flow regime inside the membrane channel, the average relative error was improved from 17.82% in the previously proposed two-fluid bubbly-flow model to 1.82% in the present work.

A game-theoretic approach to dynamic boundary problems for level-set curvature flow equations and applications

This paper is devoted to a game-theoretic approach to the level-set curvature flow equation with nonlinear dynamic boundary conditions. Under the comparison principle for the dynamic boundary problem, we construct a family of deterministic discrete games, whose value functions approximate the unique viscosity solution. We also apply the game approximation to study the convexity preserving properties and the fattening phenomenon for this geometric dynamic boundary problem.

A practical simulation of a hexanitrohexaazaisowurtzitane (CL-20) sphere detonated underwater with the Taylor wave solution and modified Tait parameters

The modified ghost fluid method (MGFM) has been one of the most popular and successful algorithms for coping with the numerical calculation of multi-medium flows, especially for the interaction between strong discontinuities and material interfaces. To apply the advanced algorithm to an underwater explosion simulation, first, the uniform distribution of the state of the detonation products, which is the most generally used initial condition in an explosion simulation, is replaced by the analytic solution of the Taylor wave. The Tait equation is, then, expanded to a broader pressure coverage of up to 100 GPa to match the initial state at the discontinuity. One-dimensional Euler equations with source terms governing the explosion flow are discretized with the fifth-order weighted essentially non-oscillatory scheme in space and the third-order Runge–Kutta scheme in time. The gas–water interface is tracked with the level set equations, and the intermediate states are resolved and defined by following the MGFM. In addition to the comparative studies among diverse numerical cases, experimental data were offered as a calibration in this work. The temporal and spatial distribution characteristics of the energy and flow variables were comprehensively discussed. Studies and analysis showed that (1) the novelly achieved parameters B = 710.8 MPa and γ = 5.22 for the Tait equation of state were highly recommended for any application involving transient loads. (2) The explosion flow field produced by the Taylor wave model was closer to the nature of physical reality. (3) Without considering the details, the stationary wave model was not entirely unacceptable as an initial condition for roughly simulating an explosion effect. The most important thing was that one had to ensure that the initial energy was equivalent to the Taylor wave case.