Level Set Equation Research Articles

The Level-Set Method for Multi-Material Wet Etching and Non-Planar Selective Epitaxy

We present numerical methods to enable accurate and robust level-set based simulation of anisotropic wet etching and non-planar epitaxy for semiconductor fabrication. These fabrication techniques are characterized by highly crystal orientation-dependent etch/growth rates, which lead to non-convex Hamiltonians in their description by the level-set equation. As a consequence, instable surface propagation may emerge, leading to unphysical results. We propose a calibration-free Stencil Lax-Friedrichs scheme and an advanced adaptive time-stepping approach, tailored to the level-set speed functions associated with anisotropic etching and epitaxy. The scheme calculates the numerical dissipation based on information about the local geometry and the nature of the etch rates/growth function, which enables an optimized trade-off between overly rounding of sharp geometric features and stable surface propagation. Furthermore, we introduce the deposition top layer method, which allows for robust handling of multiple material regions in non-planar epitaxy simulations. Both methods are demonstrated in a prototypical implementation, which is used to validate the capability and accuracy of our approaches. In particular, two-dimensional wet etching and three-dimensional epitaxy simulations are performed and characteristic geometry parameters are compared to the ideally expected values, showing robustness and high accuracy.

A variational level set methodology without reinitialization for the prediction of equilibrium interfaces over arbitrary solid surfaces

A robust numerical methodology to predict equilibrium interfaces over arbitrary solid surfaces is developed. The kernel of the proposed method is the distance regularized level set equations (DRLSE) with techniques to incorporate the no-penetration and volume-conservation constraints. In this framework, we avoid reinitialization that is typically used in traditional level set methods. This allows for a more efficient algorithm since only one advection equation is solved, and avoids numerical error associated with the re-distancing step. A novel surface tension distribution, based on harmonic mean, is prescribed such that the zero level set has the correct liquid-solid surface tension value. This leads to a more accurate prediction of the triple contact point location. The method uses second-order central difference schemes which facilitates easy parallel implementation, and is validated by comparing to traditional level set methods for canonical problems. The application of the method in the context of Gibbs free energy minimization, to obtain liquid-air interfaces is validated against existing analytical solutions. The capability of the methodology to predict equilibrium shapes over both structured and realistic rough surfaces is demonstrated.

Selection of the optimal heat treatment conditions using advanced phase transformation models

The paper focuses on the problem of optimal cooling cycle selection for steel sheets after hot rolling in order to obtain a given phase composition using complex numerical model. The proposed model driven by diffusion of carbon with the moving boundary (Stefan problem) was designed and implemented on the basis of Level Set equations coupled with diffusion equation. Application of implicit method for interface evolution allows to overcome the limitations associated with the complicated representation of material microstructure, but also in a natural way reflects the impact of front curvature and influence of chemical forces on the velocity of the interface. The article discusses the details of the model as well as the applied optimization technique. Three test scenarios are analyzed and the results are discussed.

The parameterized level set method for structural topology optimization with shape sensitivity constraint factor

In recent years, the parameterized level set method (PLSM) has attracted widespread attention for its good stability, high efficiency and the smooth result of topology optimization compared with the conventional level set method. In the PLSM, the radial basis functions (RBFs) are often used to perform interpolation fitting for the conventional level set equation, thereby transforming the iteratively updating partial differential equation (PDE) into ordinary differential equations (ODEs). Hence, the RBFs play a key role in improving efficiency, accuracy and stability of the numerical computation in the PLSM for structural topology optimization, which can describe the structural topology and its change in the optimization process. In particular, the compactly supported radial basis function (CS-RBF) has been widely used in the PLSM for structural topology optimization because it enjoys considerable advantages. In this work, based on the CS-RBF, we propose a PLSM for structural topology optimization by adding the shape sensitivity constraint factor to control the step length in the iterations while updating the design variables with the method of moving asymptote (MMA). With the shape sensitivity constraint factor, the updating step length is changeable and controllable in the iterative process of MMA algorithm so as to increase the optimization speed. Therefore, the efficiency and stability of structural topology optimization can be improved by this method. The feasibility and effectiveness of this method are demonstrated by several typical numerical examples involving topology optimization of single-material and multi-material structures.

Numerical simulation of the growth and interaction of vapour bubbles in superheated liquid jets

Operating fluids for steering and propulsion of orbital manoeuvring systems are to be changed from toxic substances to environmentally less harmful alternatives. Liquid oxygen (LOX) can be used as oxidizer but the near vacuum conditions of outer space lead to a fast expansion into a superheated state when LOX is injected into the combustion chamber. The disintegration of the liquid jet is driven by bubble nucleation and growth, which is called flash boiling. These processes typically occur at small length and time scales and need to be modelled in macroscopic simulations of the entire combustion chamber. The goal of the present work is a quantification of bubble growth rates and bubble-bubble interactions that can be used to improve the existing submodels needed for full-scale simulations of the entire thruster. We use direct numerical simulations (DNS) of bubble clusters and compare the resulting growth rates to standard models for single bubble growth. The DNS solver is based on a discontinuous Galerkin approach and combined to a level-set equation to transport the interface between liquid and vapour. A modified HLLC Riemann solver accounts for the phase transfer. The computations show that vapour bubbles grow more slowly in the center of a jet than at its surface. The bubble radii exponentially decrease with distance from the liquid jet interface and growth rates are reduced by more than 70% in the center of the jet such that their volumetric expansion can be neglected for the computation of the jet expansion. The reduced growth can be associated with the interactions of the pressure fields surrounding the bubbles as the liquid pressure increases due to bubble growth and evaporation. The degree of superheat is locally reduced and bubbles grow at a smaller rate. The growth rates of individual bubbles can be parameterised with the local degree superheat, which may serve as a potential sub-scale model. These findings hold for various operating conditions that are characteristic for LOX flash evaporation under vacuum conditions.

Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms

<p style='text-indent:20px;'>We study a level-set mean curvature flow equation with driving and source terms, and establish convergence results on the asymptotic behavior of solutions as time goes to infinity under some additional assumptions. We also study the associated stationary problem in details in a particular case, and establish Alexandrov's theorem in two dimensions in the viscosity sense, which is of independent interest.

Interface-Capturing Method for Free-Surface Plunging and Breaking Waves

AbstractThis paper presents an interface-capturing method for two-phase flows with large-amplitude free-surface motion. The method is developed by embedding the level-set equation in the momentum b...

Numerical investigation of a fully coupled micro-macro model for mineral dissolution and precipitation

Mineral dissolution and precipitation alter a porous medium’s structure and its bulk properties. Due to the medium’s heterogeneity and lack in dynamic pore-scale measurements, there has been an increasing interest in comprehensive models accessing such phenomena on the macroscale without disregarding available pore-scale information. Such micro-macro models may be derived from detailed pore-scale models applying upscaling techniques and comprise several levels of couplings. Our model consists of transport equations at the scale of the porous medium (macroscale) while taking the processes of convection and diffusion into account. They include averaged time- and space-dependent coefficient functions which are in turn explicitly computed by means of auxiliary cell problems (microscale). Structural changes due to dissolution and precipitation reactions result in a time- and space-dependent domain on which cell problems are defined. The interface between the mineral and the fluid, and consequently the explicit geometric structure, is characterized by means of a level set. Here, information from the transport equations’ solutions is taken into account (micro-macroscale). A numerical scheme is introduced which enables evaluating such complex settings. For the level-set equation, an upwind scheme by Rouy and Tourin is applied. An extended finite element method is used for the evaluation of the cell problems while the transport equations are solved applying mixed finite elements. Ultimately, we investigate the potentially degenerating bulk properties of the medium such as porosity and effective diffusion. Moreover, we apply our approach to the dissolution of an array of calcite grains in the micro-macro context and validate our numerical scheme.

A high-order discontinuous Galerkin method for level set problems on polygonal meshes

We propose and analyze discontinuous Galerkin schemes for solving level set equations on polygonal meshes. For linear equations, we assume that the velocity is given only on the cell surface. Velocity inside mesh cells is approximated using virtual element projectors on polynomial spaces. For nonlinear equations, we use the Taylor expansion to approximate the normalized solution gradient. We analyze the new schemes for a set of typical level set problems using square and polygonal meshes. The numerical results indicate great potential for using polygonal meshes in applications.

Multilayer Airfoil Ice Accretion Simulations Using a Level-Set Method with B-Spline Representation

This paper presents a partial differential equation approach to model the icing front evolution on airfoils via the level-set equation, offering a high-fidelity alternative to common algebraic approaches. Furthermore, the automation of the multilayer icing process is achieved by using B-spline curves to improve the geometry representation, which is usually limited to linear segments. The B-splines are generated using a curve approximation algorithm based on an error-controlled knot insertion together with a least-square minimization process, allowing an adequate shape representation of the ice irregular features. The evolution of the level-set interface depends on the so-called icing velocity, which is computed locally in the field from the propagated surface ice mass accretion rate solution. Additionally, an explicit swept volume tracking method controls actively the global conservation of the ice volume. A verification phase is performed on analytical cases to highlight the characteristics of the methods and discretization schemes. Multilayer icing results are then presented on two cases and compared with experimental data providing a reasonable global mass error level in each case. Overall, the method shows great potential to model the ice geometry evolution in a globally conservative manner while ensuring a smooth geometry representation that improves multilayer automation.

Study of Droplet Shadow Zone of Aircraft Wing with Diffusion Effects

The aircraft deicing and detection tasks are complicated by the revealed existence of a shadow zone behind the wings, which contains no droplets due to the wing shielding effect. The effects of this so-called droplet shadow zone on the droplet distribution along the cylinder and the wing are theoretically and experimentally explored in this study. The computational domain of droplet motion is subdivided into a droplet flow zone and a droplet shadow zone. The interface between the two zones is captured by the level set method. The content of liquid water in the droplet shadow zone is zero. Outside the droplet shadow zone, the droplet trajectories are predicted using the Eulerian two-phase flow model. Fick’s law of diffusion is introduced into the Eulerian and level set equations. On this basis, a method for capturing the droplet trajectories and interface with an account of the diffusion effect is proposed. Besides, a series of experiments is carried out in the icing wind tunnel of Nanjing University of Aeronautics and Astronautics. The calculation and experimental results indicate that the proposed method can accurately predict the droplet shadow, and the calculated values of droplet collection coefficient also agree well with the experimental ones.

A new and improved cut-cell-based sharp-interface method for simulating compressible fluid elastic–perfectly plastic solid interaction

In this work, a new integrated, conservative and consistent cut-cell-based sharp-interface method is developed to simulate compressible multi-material problems with various interfaces. In the proposed method, the material interfaces are represented by quadratic-curve cut faces evolved with the help of the level set equation. Multiple level set functions are applied to address multiple interfaces. The system is mainly developed in Eulerian coordinates with the cut faces moving in a Lagrangian manner, thus, it benefits from a Lagrangian moving interface in terms of accuracy and handles large deformations more conveniently than those works in a Lagrangian framework. Clear improvements in robustness and accuracy over the modified ghost fluid method can be observed in various 2D test cases.

Numerical investigation of non-isothermal viscoelastic filling process by a coupled finite element and discontinuous Galerkin method

In this paper, we study the challenging non-isothermal polymer filling process for the cavity with irregular inserts via a coupled finite element method (FEM) and discontinuous Galerkin (DG) method. The non-isothermal eXtended Pom-Pom (XPP) model is utilized to suitably describe the viscoelastic behavior of the branched polymer melt. As for the energy equations, we decouple the convective and diffusive terms to get two sub-equations. It is also able to receive several sub-equations from the momentum equation via splitting scheme. Then the hyperbolic and elliptic type sub-equations are solved via DG and FEM, respectively. The level set method is employed to capture the complicated transient free surface. Due to the pure convective character of the level set and XPP constitutive equations, the Runge-Kutta Discontinous Galerkin (RKDG) is used to deal with them. In this coupled algorithm, it is able to take full advantages of FEM and DG and also avoid the stabilization terms. The non-isothermal filling process of the rectangle cavity with a circular cone insert is first investigated to illustrate the feasibility and validity of the coupled method. Moreover, a more complex mould cavity, the socket with five irregular inserts, is studied as an application case to demonstrate the robustness and practicability of our coupled method. The influences of the inlet melt temperature on the stress and stretch are all analyzed.

Risk-optimal path planning in stochastic dynamic environments

We combine decision theory with fundamental stochastic time-optimal path planning to develop partial-differential-equations-based schemes for risk-optimal path planning in uncertain, strong and dynamic flows. The path planning proceeds in three steps: (i) predict the probability distribution of environmental flows, (ii) compute the distribution of exact time-optimal paths for the above flow distribution by solving stochastic dynamically orthogonal level set equations, and (iii) compute the risk of being suboptimal given the uncertain time-optimal path predictions and determine the plan that minimizes the risk. We showcase our theory and schemes by planning risk-optimal paths of unmanned and/or autonomous vehicles in illustrative idealized canonical flow scenarios commonly encountered in the coastal oceans and urban environments. The step-by-step procedure for computing the risk-optimal paths is presented and the key properties of the risk-optimal paths are analyzed.

Hysteretic heat transfer study of liquid–liquid two-phase flow in a T-junction microchannel

Liquid–liquid two-phase flow in microchannels is capable of boosting the heat removal rate in cooling processes. Formation of different two-phase flow patterns which affect the heat transfer rate is numerically investigated here in a T-junction containing water-oil flow. For this purpose, the finite element method (FEM) is applied to solve the unsteady two-phase Navier–Stokes equations along with the level set (LS) equation in order to capture the interface between phases. It is shown that the two-phase flow pattern in microchannels depends on the flow initial condition which causes hysteresis effect in two-phase flow. In this study, the hysteresis is observed in flow pattern and consequently in the heat transfer rate. The effect of wall contact angle on the hydrodynamics and heat transfer in the microchannel is investigated to gain useful insight into the hysteresis phenomenon. It is observed that the hysteresis is significant in super-hydrophilic microchannels, while it disappears at the contact angle of 75°. The effect of water to oil flow rate ratio (Qwat/Qoil) on the heat transfer is also studied. The flow rate ratio has a negligible effect on the Nusselt number (Nu) in the dripping regime, while the Nu decreases with an increase of Qwat/Qoil in the co-flow regime. The thickness of the oil film, velocity, and temperature distribution are studied in the co-flow regime. It is revealed that the normalized slip velocity reduces at higher values of Qwat/Qoil, which causes a reduction in the averaged Nu. In dripping regimes, higher flow rate ratios lead to a more frequent generation of droplet/slugs at a smaller size. The passage of the slugs or droplets increases the local Nu. Larger droplets generated at lower flow rate ratios cause a larger increase in the local Nu than smaller droplets. The temperature and velocity field around the droplets are also illustrated to investigate the heat transfer improvement. The generated vortex at the tip of the oil jet causes an increase in the velocity and Nu on the water side.

A constitutive level-set model for shape memory polymers and shape memory polymeric composites

This work proposes a new constitutive model for shape memory polymers and shape memory polymeric composites under the use of level-set functions as additional variables of state. The model regards shape memory polymers as inhomogeneous bodies consisting of two different phases, an active (rubbery elastic) phase and a frozen (glassy elastic) phase. Introducing the level-set function provides the potential to describe the interface separating the two distinct phases in an implicit manner. Under proper thermodynamical arguments, an appropriate evolution equation is derived that provides the tool to describe phase transformations in shape memory polymers. Furthermore, an extended model version is developed that applies in shape memory polymeric composites by introducing two (or more) level-set functions so as to represent implicitly three (or more) material phases capturing the behavior of multi-shape memory polymers. The level-set constitutive equations are formulated in three dimensions, although the one-dimensional case is adopted and analyzed thoroughly. The reproduction of the shape memory thermomechanical cycles of polymers and polymeric composites provides validity and credibility to the current model.

Single vapour bubble growth under flash boiling conditions using a modified HLLC Riemann solver

Common fuel-oxidizer combinations in orbital manoeuvring systems consist of toxic substances but will be replaced in the future. Liquid oxygen (LOX) is one potential oxidizer but it rapidly superheats under the initial low-pressure conditions in the combustion chamber. Bubble nucleation and growth dominate the efficient disintegration of the liquid jet, which is called flash boiling. An investigation of the small scale bubble dynamics will help to improve existing models for the break-up of the LOX jet and the mixing with the fuel. Direct numerical simulations (DNS) can be used to analyse the underlying processes if the solver handles phase transition effects at extreme ambient conditions. In this paper we use a fully compressible discontinuous Galerkin solver combined to a level-set equation and to a modified HLLC Riemann solver. The main goal of our investigation is to assess closure conditions for the mass transfer models which accurately represent the physics of vapour bubble growth. We couple three models for the estimation of vaporisation mass fluxes to the interface Riemann solver. The Hertz-Knudsen relation and a kinetic relation predict the volumetric expansion well but cannot represent the instantaneous mass flux. A sub-grid scale heat flux model predicts the mass flux qualitatively better but the volumetric bubble expansion matches only at late time intervals. The dependency of the calibrated coefficients on the physical conditions is similar for the different vaporisation models.

A level-set model for mass transfer in bubbly flows

A level-set model is presented for simulating mass transfer or heat transfer in two-phase flows. The Navier-Stokes equations and mass transfer (or heat transfer) equation are discretized using a finite volume method on a collocated unstructured mesh, whereas a multiple marker level-set approach is used for interface capturing in bubble swarms. This method avoids the numerical coalescence of the fluid particles, whereas the mass conservation issue inherent to standard level-set methods is circumvented. Furthermore, unstructured flux-limiter schemes are used to discretize the convective term of momentum transport equation, level-set equations, and chemical species concentration equation, to avoid numerical oscillations around discontinuities, and to minimize the numerical diffusion. A convection-diffusion-reaction equation is used as a mathematical model for the chemical species mass transfer at the continuous phase. Because the mathematical analogy between dilute mass transfer and heat transfer, the same numerical model is applicable to solve both phenomena. The capabilities of this model are proved for the diffusion of chemical species from a sphere, external mass transfer in the buoyancy-driven motion of single bubbles and bubble swarms. Results are extensively validated by comparison with analytical solutions and empirical correlations from the literature.

An obstacle problem arising in large exponent limit of power mean curvature flow equation

We study limit behavior for the level-set power mean curvature flow equation as the exponent tends to infinity. Under Lipschitz continuity, quasiconvexity, and coercivity of the initial condition, we show that the limit of the viscosity solutions can be characterized as the minimal supersolution of an obstacle problem involving the 1 1 -Laplacian. Such behavior is closely related to applications of power mean curvature flow in image denoising. We also discuss analogous behavior for other evolution equations with related applications.

Numerical study of binary droplets collision in the main collision regimes

Direct numerical simulation of binary droplets collision is done using a conservative level-set method. The Navier-Stokes and level-set equations are solved using a finite-volume method on collocated grids. A novel lamella stabilization approach is introduced to numerically resolve the thin lamella film appeared during a broad range of collision regimes. This direction-independent method proves to be numerically efficient and accurate compared with experimental data. When the droplets collide, the fluid between them is pushed outward, leaving a thin gas layer bounded by the surface of two droplets. This layer progressively gets thinner and depending on the collision regime, may rupture resulting in coalescence of the droplets or may linger resulting in bouncing-off the droplets. Embedded ghost-nodes layer makes it possible to mimic both bouncing and coalescence phenomena of the droplets collision. The numerical tools introduced are validated and verified against different experimental results for a wide range of collision regimes. A very good agreement is observed between the results of this paper and experimental data available in the literature. A detailed study of the energy budget for different shares of kinetic and dissipation energies inside of the droplet and matrix, in addition to the surface tension energy for studied cases, is provided. Supplementary quantitative values of viscous dissipation rate inside of the matrix and droplet, and also the radial expansion of the droplet are presented as well.