Standard Level Set Research Articles

Numerical Study of Effective Parameters on Deformation and Coalescence of Ferrofluid Droplets Under Uniform Magnetic Field

This research examines different numerical techniques for modeling ferrofluids in a non-magnetic liquid subjected to homogeneous and steady magnetic field. It particularly compares the conservative level set method with the Volume of Fluid (VOF) and Simple Coupled Level Set and Volume of Fluid (SCLSVOF) methods. By comparing these methods with experimental data, we established the superiority of the utilized method in this study. Several case studies were conducted to evaluate how a homogeneous magnetic actuation affects ferrofluid dynamics, considering parameters such as initial droplet size, surface tension coefficient, and magnetic susceptibility. Additionally, the study explored the coalescence of two falling ferrofluid droplets under the combined effects of an external magnetic field and gravity. The conservative level set method was shown to be extendable to three-dimensional environments, unlike the standard level set method. The novelty of this work lies in demonstrating that the conservative level set method is particularly effective for ferrofluids, accurately capturing their complex dynamics. The main novelty of this article lies in demonstrating the precision of Conservative Level Set Method while utilizing a reduced number of equations compared to other works. This reduction in complexity enhances the method's efficiency without compromising precision, making it especially suited for applications requiring both speed and accuracy, such as model-based control systems.

Preventing mass loss in the standard level set method: New insights from variational analyses

For decades, the computational multiphase flow community has grappled with mass loss in the level set method. Numerous solutions have been proposed, from fixing the reinitialization step to combining the level set method with other conservative schemes. However, our work reveals a more fundamental culprit: the smooth Heaviside and delta functions inherent to the standard formulation. Even if reinitialization is done exactly, i.e., the zero contour interface remains stationary, the use of smooth functions lead to violation of mass conservation. We propose a novel approach using variational analysis to incorporate a mass conservation constraint. This introduces a Lagrange multiplier that enforces overall mass balance. Notably, as the delta function sharpens, i.e., approaches the Dirac delta limit, the Lagrange multiplier approaches zero. However, the exact Lagrange multiplier method disrupts the signed distance property of the level set function. This motivates us to develop an approximate version of the Lagrange multiplier that preserves both overall mass and signed distance property of the level set function. Our framework even recovers existing mass-conserving level set methods, revealing some inconsistencies in prior analyses. We extend this approach to three-phase flows for fluid-structure interaction (FSI) simulations. We present variational equations in both immersed and non-immersed forms, demonstrating the convergence of the former formulation to the latter when the body delta function sharpens. Rigorous test problems confirm that the FSI dynamics produced by our simple, easy-to-implement immersed formulation with the approximate Lagrange multiplier method are accurate and match state-of-the-art solvers.

A hierarchical level-set numerical approach for immiscible incompressible n-phase flows ([formula omitted])

In this article, we propose an approach for interface-resolved simulations of immiscible incompressible n-phase flows (n≥3). The standard level set (LS) method can accurately simulate the motion of two phases. However, when more than two phases are considered, the interfaces cannot intersect and numerical overlaps occur, leading to unphysical spurious velocities and unrealistic distortions of the interfaces. The proposed approach organizes a group of n−1 LS functions into a hierarchy to resolve the issue of numerical overlaps. The n-phase flow field is discretized using the extended finite element method (XFEM), which adequately addresses the discontinuities occurring in the pressure and velocity fields due to distinct fluid properties and surface tension effects at the interface between the fluids. This approach is applied to three-phase and four-phase fluid dynamics, involving gas and liquids under various conditions of surface tension. The 2D numerical tests prove that the proposed numerical methods can effectively model interactions between multi-fluid interfaces, avoiding numerical overlaps.

Analisis Tingkat Kebisingan Aktivitas Jalan Trans Kalimantan Terhadap Rumah Sakit Umum Kota Palangka Raya

High economic growth also indirectly has an impact on increasing vehicle traffic activity on the highway. Noise due to traffic will continue to increase along with the increase in human activity using vehicles.Sound Level Meter (LSM) is a tool used to measure noise at a point. Specifically in this research, namely the noise level of the Trans Kalimantan Road towards the General Hospital of Palangka Raya City. From the data obtained in the research results, it shows that the noise level at the General Hospital of Palangka Raya City due to transportation activities has exceeded the quality standard limits that have been set. From the results of research and data analysis, it was found that the traffic noise level that occurred at the General Hospital of Palangka Raya City at point 1 next to the entrance to the General Hospital of Palangka Raya City was 73.70 dBA and at point 2 in front of the Emergency Room General Hospital of Palangka Raya City was 77. 29 dBA which exceeds the quality standard noise level set for hospital areas, namely 55 dBA. Based on the results of these calculations, it is recommended that the General Hospital of Palangka Raya City make a natural noise barrier, namely in the form of a fence or a thicket of plants with fairly even leaf density or an artificial noise barrier in the form of a soundproof wall, etc. so that the comfort of the hospital can be maintained.

Enhancing level set-based topology optimization with anisotropic graded meshes

We propose a new algorithm for the design of topologically optimized lightweight structures, under a minimum compliance requirement. The new process enhances a standard level set formulation in terms of computational efficiency, thanks to the employment of a computational mesh customized to the problem at hand. We pursue a twofold goal, i.e., to deliver a final layout characterized by a smooth contour and reliable mechanical properties. The smoothness of the optimized structure is ensured by the employment of an anisotropic adapted mesh, which sharply captures the material/void interface. A robust mechanical performance is guaranteed by a uniform tessellation of the internal part of the optimized configuration. A thorough numerical investigation corroborates the effectiveness of the proposed algorithm as a reliable and computationally affordable design tool, both in two- and three-dimensional contexts.

A local discontinuous Galerkin level set reinitialization with subcell stabilization on unstructured meshes

In this paper we consider a level set reinitialization technique based on a high-order, local discontinuous Galerkin method on unstructured triangular meshes. A finite volume based subcell stabilization is used to improve the nonlinear stability of the method. Instead of the standard hyperbolic level set reinitialization, the flow of time Eikonal equation is discretized to construct an approximate signed distance function. Using the Eikonal equation removes the regularization parameter in the standard approach which allows more predictable behavior and faster convergence speeds around the interface. This makes our approach very efficient especially for banded level set formulations. A set of numerical experiments including both smooth and non-smooth interfaces indicate that the method experimentally achieves design order accuracy.

Comparison of interface capturing methods for the simulation of two-phase flow in a unified low-Mach framework

This paper proposes a comparison of four popular interface capturing methods : the volume of fluid (VOF), the standard level set (SLS), the accurate conservative level set (ACLS) and the coupled level set and volume of fluid (CLSVOF). All methods are embedded into a unified low-Mach framework based on a Cartesian-grid finite-volume discretization. This framework includes a sharp transport of the interface, a well-balanced surface tension discretization and a consistent mass and momentum transport which allows capillary-driven simulations with high density ratio. The comparison relies on shared metrics for geometrical accuracy, mass and momentum conservation which exposes the weakness and strengths of each method. Finally, the versatility and capabilities of the proposed solver are demonstrated on the simulation of a 3D head-on collision of two water droplets. Overall, all methods manage to retrieve reasonable results for all test cases presented. VOF, CLSVOF and ACLS tend to artificially create little structures while SLS suffers from conservation issues in the mesh resolution limit. This study leads us to the conclusion that CLSVOF is the most promising method for two-phase flow simulations in our specific framework because of its inherent conservation properties and topology accuracy.

Effects of Selection of Inlet Perturbations, Multiphase and Turbulence Equations on Slug Flow Characteristics Using Altair® AcuSolve™

Selection of inlet perturbations, multiphase equations, and the turbulence equation may affect the development of slug flow using computational fluid dynamic simulation tools. The inlet perturbation, such as sinusoidal and random perturbations, play an essential role in inducing slug formation. Multiphase equations such as volume of fluid and level set methods are used to track and capture the gas-liquid immiscible interface. Similarly, turbulence equations such as Spalart Allmaras (SA), Detached Eddy Simulations (DES), k-omega, and k-epsilon can be used to predict the evolution of turbulence within the flow. At present, no direct comparison is available in the literature on the selection of (i) types of inlet perturbations, (ii) the choice of multiphase equations, and (iii) the turbulence equation on the development of slug flow using the Altair computational package. This article aims to compare the effects of the selection of inlet perturbations, multiphase models and turbulence equations on slug flow characteristics using Altair® AcuSolve™. The findings by Altair® simulation were compared to published experimental data and simulation works using ANSYS and STAR-CCM+. The slug flow characteristics of interest include slug morphology, a body length-to-diameter ratio, velocity, frequency, and pressure gradient. It was found that the slug flow could be developed for all combinations of settings. Although level set approach in Altair® can track fluid motion successfully, it has a limitation in modelling the convective transport of the multiphase mixture well, unlike ANSYS and STAR-CCM+. Compared to the standard level set method, the coupling of back-and-forth error compensation and correction with the level set function helps to capture the internal boundary more accurately by reducing errors caused by numerical diffusion in the transport of the level set. It was revealed that the Spalart Allmaras turbulence equation could mimic published experimental result better than DES as it produced the closest slug translational velocity. Since the frequency of the slugs for the developed models showed a good agreement with the published data, the models could be sufficient for the investigation of fluid-structure interaction.

A comparison of interface tracking methods in a low-Mach formulation

The objective of the present work is to compare the main Eulerian sharp-interface tracking methods using up-to-date numerical strategies from the literature. The study is focused on the specific case of low-Mach number formulation for the phase dynamics, and on a finite-volume cartesian-grid discretization. Volume of fluid and Level Set methods both rely on the resolution of an Eulerian transport equation that describes the interface and an additional step to prevent huge deviation from the interface representation due to numerical diffusion and dispersion in the transport step. In the VOF framework, this results in the transport of the volume fraction with a piecewise linear construction of the interface (PLIC). In the LS framework, this takes the form of a distance function transport which can be either a signed distance (here called Standard Level Set (SLS)) or an hyperbolic tangent (called the Conservative Level Set (CLS)) followed by a reinitialization step which ensures that the transported variable remains a signed distance or an hyperbolic tangent respectively. For each method, the numerical scheme used for advection and additional step are selected because of their proven accuracy and effectiveness in the literature for our specific framework. Our comparison is based on 2D and 3D canonical cases of the literature (Zalesak’s Disk rotation, vortex-in-a-box, sphere deformation). Our attention is drawn on a detailed analysis of mass conservation and transport accuracy, with the use of shared metrics for all methods.

A conservative level set method for N-phase flows with a free-energy-based surface tension model

We introduce the N-phase Conservative Level Set Free-Energy-based Surface Tension model (NCLS-FEST) for multiphase flow (N≥3). The model minimizes numerical overlaps of fluids and voids between fluids that are found in the standard Conservative Level Set (CLS) method for three fluid phases. The key differences between NCLS-FEST and CLS methods are the multiphase compression-diffusion equation to ensure conservation, an interface with constant width, a physical solution with minimal overlaps and voids between phases at each time step, and a free-energy-based surface tension model. We demonstrate that the NCLS-FEST method has accurate results for N-phase fluid flows involving complicated geometrical configurations, including triple lines in the liquid lens problem for three and four fluid phases, with accurate conservation of mass. Additionally, we show that the NCLS-FEST method can be extended to N-phase fluid-solid flow, where the solid phase is modeled by a static fluid phase.

Numerical Study of Bubble Rising Motion in a Vertical Wedge-Shaped Channel Based on a Modified Level Set Method

A modified level set method coupled with local-reinitialized process and volume-corrected method is employed to simulate bubble motions in complex channels. The volume-corrected process helps to adjust the mass loss due to bubble simulation by standard level set method and local reinitialization of level set function eliminates the numerical dissipation due to nonorthogonality of the boundary mesh. For solving the problem of two-phase flow with a complex boundary, the mesh generation is employed in the physical plane in body-fitted coordinates. All parameters in the physical domain are transformed into a computational domain, where the governing equations are discretized on collocated grids using the finite volume method and the SIMPLE algorithm to decouple the velocities and the pressure. The continuum surface force model is employed to deal with the surface tension of bubbles. By simulating bubble rising in a wedge-shaped channel, the method is demonstrated to be effective to solve the bubble motion in a complex physical plane.

Heat diffusion embedded level set evolution for infrared image segmentation

In this study, the authors present a novel level set method for infrared image segmentation. Local region-based models can fit intensity inhomogeneity partly but they are sensitive to local window scale. To deal with it, they embed an heat diffusion process in conventional level set evolution and convert heat to a part of data term in level set energy function. Besides, bias field model can extract the local intensity clustering property of the image. Therefore, the proposed method can deal with the interference of intensity inhomogeneity and complex background if appropriate seeded pixels are selected. Finally, the energy functional is minimised by a combinatorial optimal algorithm in a graph model to get a global optimal solution and accelerate the level set evolution implementation. The experiments show that the proposed method is robust to parameter setting, noise, and initial contour position. The comparisons on a large quantity of infrared image datasets with standard level set methods also demonstrate the efficiency of the proposed method.

A simple iterative geometry-based interface-preserving reinitialization for the level set method

The iterative reinitialization, as an important process for the standard level set method, is flexible and straightforward in large-scale computing but has problems of interface movement and serious mass loss. Thus a simple iterative geometry-based interface-preserving reinitialization is proposed to reconstruct a signed distance function (SDF) while fixing the interface location with the combination of the external and internal iterations. The external iteration is used to restore the nature of the SDF. And the internal iteration firstly determines displacement errors of zero-level-set interpolation points via the geometric position of the interface, then the errors are redistributed iteratively to adjacent nodes until moving zero-level-set interpolation points are driven back to their original positions. The novel approach performs excellent mass conservation and shape preservation without sacrificing the simplicity of the iterative reinitialization. The accuracy and efficiency of the proposed method are validated with typical stationary interface examples and well-known benchmark cases.

An implicit boundary integral method for computing electric potential of macromolecules in solvent

A numerical method using implicit surface representations is proposed to solve the linearized Poisson–Boltzmann equation that arises in mathematical models for the electrostatics of molecules in solvent. The proposed method uses an implicit boundary integral formulation to derive a linear system defined on Cartesian nodes in a narrowband surrounding the closed surface that separates the molecule and the solvent. The needed implicit surface is constructed from the given atomic description of the molecules, by a sequence of standard level set algorithms. A fast multipole method is applied to accelerate the solution of the linear system. A few numerical studies involving some standard test cases are presented and compared to other existing results.

Modeling of Nonlinear Crack Growth in Steel and Aluminum Alloys by the Element Free Galerkin Method

In this study, the modelling and simulation of the nonlinear fatigue crack growth phenomenon in steel and aluminium alloys has been carried out by employing the element free Galerkin method. The primary variable is approximated by the moving least square shape functions. For the consideration of the effect of various discontinuities present in the domain, appropriate enrichment functions have been added to the standard EFGM approximation. In order to keep track of the various discontinuities, the standard level set method is used. The present study includes the material and geometric nonlinearities in the mathematical formulations. The geometric nonlinearity has been modelled by employing the updated Lagrangian approach. The isotropic hardening has been used with the Ramberg-Osgood model to obtain the elasto-plastic behaviour of the material. Finally, several numerical problems have been solved by the element free Galerkin method to investigate fatigue crack growth in steel and aluminium alloys.

Analyzing Smooth and Singular Domain Perturbations in Level Set Methods

In the standard level set method, the evolution of the level set function is determined by solving the Hamilton--Jacobi equation, which is derived by considering smooth boundary perturbations of th...

Accurate prediction of complex free surface flow around a high speed craft using a single-phase level set method

This paper focuses on the analysis of a challenging free surface flow problem involving a surface vessel moving at high speeds, or planing. The investigation is performed using a general purpose high Reynolds free surface solver developed at CNR-INSEAN. The methodology is based on a second order finite volume discretization of the unsteady Reynolds-averaged Navier–Stokes equations (Di Mascio et al. in A second order Godunov—type scheme for naval hydrodynamics, Kluwer Academic/Plenum Publishers, Dordrecht, pp 253–261, 2001; Proceedings of 16th international offshore and polar engineering conference, San Francisco, CA, USA, 2006; J Mar Sci Technol 14:19–29, 2009); air/water interface dynamics is accurately modeled by a non standard level set approach (Di Mascio et al. in Comput Fluids 36(5):868–886, 2007a), known as the single-phase level set method. In this algorithm the governing equations are solved only in the water phase, whereas the numerical domain in the air phase is used for a suitable extension of the fluid dynamic variables. The level set function is used to track the free surface evolution; dynamic boundary conditions are enforced directly on the interface. This approach allows to accurately predict the evolution of the free surface even in the presence of violent breaking waves phenomena, maintaining the interface sharp, without any need to smear out the fluid properties across the two phases. This paper is aimed at the prediction of the complex free-surface flow field generated by a deep-V planing boat at medium and high Froude numbers (from 0.6 up to 1.2). In the present work, the planing hull is treated as a two-degree-of-freedom rigid object. Flow field is characterized by the presence of thin water sheets, several energetic breaking waves and plungings. The computational results include convergence of the trim angle, sinkage and resistance under grid refinement; high-quality experimental data are used for the purposes of validation, allowing to compare the hydrodynamic forces and the attitudes assumed at different velocities. A very good agreement between numerical and experimental results demonstrates the reliability of the single-phase level set approach for the predictions of high Froude numbers flows.

Gradient augmented level set method for phase change simulations

A numerical method for the simulation of two-phase flow with phase change based on the Gradient-Augmented-Level-set (GALS) strategy is presented. Sharp capturing of the vaporization process is enabled by: i) identification of the vapor–liquid interface, Γ(t), at the subgrid level, ii) discontinuous treatment of thermal physical properties (except for μ), and iii) enforcement of mass, momentum, and energy jump conditions, where the gradients of the dependent variables are obtained at Γ(t) and are consistent with their analytical expression, i.e. no local averaging is applied. Treatment of the jump in velocity and pressure at Γ(t) is achieved using the Ghost Fluid Method. The solution of the energy equation employs the sub-grid knowledge of Γ(t) to discretize the temperature Laplacian using second-order one-sided differences, i.e. the numerical stencil completely resides within each respective phase. To carefully evaluate the benefits or disadvantages of the GALS approach, the standard level set method is implemented and compared against the GALS predictions. The results show the expected trend that interface identification and transport are predicted noticeably better with GALS over the standard level set. This benefit carries over to the prediction of the Laplacian and temperature gradients in the neighborhood of the interface, which are directly linked to the calculation of the vaporization rate. However, when combining the calculation of interface transport and reinitialization with two-phase momentum and energy, the benefits of GALS are to some extent neutralized, and the causes for this behavior are identified and analyzed. Overall the additional computational costs associated with GALS are almost the same as those using the standard level set technique.

The use of a simple model in the inverse characterization of cardiac ischemic regions

Abstract In this paper, we analyze the use of simple models for solving the inverse problem in electrocardiography (IPE), which aims at recovering the heart condition from a set of remote voltages measurements. Specifically, we consider here the problem of estimating the shape, size and location of cardiac ischemic regions. The forward problem to generate the data (voltage measurements) is formulated by using the Luo–Rudy model, which provides a detailed description of the electrical behavior of cardiac cells. As for the inversion process, we use the two-current phenomenological model. The inversion procedure also incorporates a semi-automatic stage to characterize the conduction properties of the cardiac tissue. The ischemic regions are modeled by using standard level set techniques. Numerical results show that the algorithm is capable of estimating the position, size and shape of cardiac ischemic regions from noisy voltage measurements, for both 2D and 3D geometries. Our inverse procedure is benchmarked against zero-order Tikhonov regularization. This work is a proof of principle demonstrating the possibility of using simple models in the IPE towards realistic situations.

Hybrid level set phase field method for topology optimization of contact problems

The paper deals with the analysis and the numerical solution of the topology optimization of system governed by variational inequalities using the combined level set and phase field rather than the standard level set approach. The standard level set method allows to evolve a given sharp interface but is not able to generate holes unless the topological derivative is used. The phase field method indicates the position of the interface in a blurry way but is flexible in the holes generation. In the paper a two-phase topology optimization problem is formulated in terms of the modified level set function and regularized using the Cahn-Hilliard based interfacial energy term rather than the standard perimeter term. The derivative formulae of the cost functional with respect to the level set function is calculated. Modified reaction-diffusion equation updating the level set function is derived. The necessary optimality condition for this optimization problem is formulated. The finite element and finite difference methods are used to discretize the state and adjoint systems. Numerical examples are provided and discussed.