Gradient-augmented Level Set Research Articles

An efficient three-dimensional multiphase fluid structure interaction model on GPU for water impact of a moving body with complex geometries

This paper presents an efficient three-dimensional (3D) multiphase fluid structure interaction (FSI) model to simulate water impact of complex moving bodies in ocean engineering. This numerical model adopts an improved ghost cell method (GCM) to enforce the boundary conditions on the moving boundaries and the gradient augmented level set (GALS) method to capture the nonlinear free surfaces. An improved iso-surface representation method is highlighted to track arbitrary moving bodies such as sharp or concave shapes. Also, a GPU-CUDA parallel framework is incorporated to greatly accelerate the computational performance. The accuracy and capability of the present model is validated by simulating 3D water entry of arbitrary free-falling bodies (a sphere, a wedge, a wedge with side plates, and a catamaran). Excellent acceleration performance of the present GPU model is demonstrated. The present results including the slamming coefficient, the acceleration, and the local pressure agree satisfactorily with the experimental data. The complex flow evolutions are well captured such as long cavity wall, the pinch-off of the cavity, and the air entrapment. Also, the air entrapment and 3D effects are confirmed to be nonnegligible for the water impact of complex bodies with enclosed sections, which can be well predicted by the present model.

Parametric studies on the water impact of one and twin free-falling wedges by a Cartesian grid multiphase flow model

The water impact of one and twin free-falling wedges is numerically investigated by a Cartesian grid multiphase flow model. The effects of the drop velocity and the gap distance on the hydrodynamic behaviors are parametrically investigated. The numerical model involves a radial basis function ghost cell method (RBFGCM) for treating moving bodies and a gradient-augmented level set method (GALS) method for capturing violent free surfaces. A case of twin wedges entering water is simulated to validate the accuracy of the present method. Good convergences are achieved. Then, the water entry of one and twin wedges in free falling is considered. The interaction mechanisms between twin wedges are discussed by comprehensively examining the variation patterns of the slamming load, the moment, the local pressure, and the fluid field. It is found that the second slamming load and the huge pressure pulse occur at the transition stage at narrow gap distances. The hydrodynamic interaction has more significant effects on the local pressure than that on the global load. In addition, distinct hydrodynamic phenomena for twin wedges entering water are observed such as the connection of the pressure contours, the extremely large jet flow, formation of the cavity, and even the ventilation.

A sharp interface multiphase flow model for two-dimensional water impact of a symmetric and asymmetric wedge

A Cartesian grid based sharp interface (SI) multiphase flow model is developed to investigate the symmetric and asymmetric impact of a wedge with sharp phase changes. A level set based ghost fluid method (GFM) with formal second-order accuracy is incorporated to maintain the jump properties across the two-phase interface with transient phase change. The arbitrary moving boundaries and free surfaces are handled by a radial basis function ghost cell method (RBFGCM) and a gradient augmented level set (GALS) method, respectively. The water impact of a symmetric wedge is simulated to demonstrate the accuracy and capability of the present SI method by comparing the present peak pressures with the experimental data. Also, satisfactory convergences of the grid size and the time step are obtained. Then, the symmetric and asymmetric water impact with different deadrise and inclination angles is investigated by the present SI model and the diffused interface (DI) model. The present SI model is demonstrated to perform well for various deadrise and inclination angles, except the nearly flat plate slamming. By contrast, the diffuse interface (DI) model can obtain reasonable solution for the relative impact angle no less than 15o. However, the peak pressure is considerably underestimated and its occurring moment lags behind the experimental data for a smaller impact angle, which is severer for a smaller impact angle. In addition, we analyzed the variation rules of the impulsive pressure, the motion response, the pressure distribution, and the free surface motion with respect to the relative impact angle. The maximum fluid force grows acceleratingly as the relative impact angle decreases, while the duration is shortened. Also, some typical impact phenomena are observed such as flow separation, jet flow, and ventilation for a small impact angle.

A Characteristic Mapping method for the two-dimensional incompressible Euler equations

We propose an efficient semi-Lagrangian method for solving the two-dimensional incompressible Euler equations with high precision on a coarse grid. This new approach evolves the flow map using a combination of the Characteristic Mapping (CM) method [1] the gradient-augmented level-set (GALS) method [2]. The flow map possesses a semigroup structure which allows for the decomposition of a long-time deformation into short-time submaps. This leads to a numerical scheme that achieves exponential resolution in linear time. Error estimates are provided and conservation properties are analysed. The computational efficiency and the high precision of the method are illustrated in the vortex merger, four-modes and random flow problems. Comparisons with the Cauchy-Lagrangian method [3] are also presented.

Numerical simulation of nonlinear sloshing in a prismatic tank by a Cartesian grid based three-dimensional multiphase flow model

This paper investigates large amplitude sloshing in a prismatic tank with or without a baffle by a Cartesian grid based three-dimensional (3D) multiphase flow model. In this model, an improved ghost cell method with a unified reconstruction scheme is developed to enforce boundary conditions on the slope walls of the tank. Arbitrary shape tanks and internal structures can be treated on the Cartesian grid, instead of the body-fitted grid. A 3D gradient-augmented level set (GALS) method is presented to capture highly nonlinear free surfaces. To validate the accuracy of the present ghost cell method, uniform flows around a sphere are simulated. An approximate second-order accuracy in space is achieved. Then, nonlinear sloshing induced by rolling excitations in the 3D prismatic tank is simulated. Satisfactory convergences of grid sizes and time steps are obtained. Also, good agreements are achieved between the present results and the experimental data for five filling water depths. In addition, highly nonlinear sloshing phenomena are captured such as plunging waves and wave beating. Furthermore, the variation of the excitation frequency with the pressure amplitude, and the effects of the baffle height on the impulsive pressure and the wave elevation are studied.

Gradient-Augmented Level Set Two-Phase Flow Method With Pretreated Reinitialization for Three-Dimensional Violent Sloshing

Abstract A three-dimensional (3D) gradient-augmented level set (GALS) two-phase flow model with a pretreated reinitialization procedure is developed to simulate violent sloshing in a cuboid tank. Based on a two-dimensional (2D) GALS method, 3D Hermite, and 3D Lagrange polynomial schemes are derived to interpolate the level set function and the velocity field at arbitrary positions over a cell, respectively. A reinitialization procedure is performed on a 3D narrow band to treat the strongly distorted interface and improve computational efficiency. In addition, an identification-correction technique is proposed and incorporated into the reinitialization procedure to treat the tiny droplet which can distort the free surface shape, even lead to computation failure. To validate the accuracy of the present GALS method and the effectiveness of the proposed identification-correction technique, a 3D velocity advection case is first simulated. The present method is validated to have better mass conservation property than the classical level set and original GALS methods. Also, distorted and thin interfaces are well captured on all grid resolutions by the present GALS method. Then, sloshing under coupled surge and sway excitation, sloshing under rotational excitation are simulated. Good agreements are obtained when the present wave and pressure results are compared with the experimental and numerical results. In addition, the highly nonlinear free surface is observed, and the relationship between the excitation frequency and the impulsive pressure is investigated.

A Cartesian grid based multiphase flow model for water impact of an arbitrary complex body

A Cartesian grid based multiphase flow model is developed to simulate water impact problems. This model is capable of simulating complex moving bodies interacting with a highly non-linear free surface such as jet flow or air cushion. A radial basis function based ghost cell method (RBFGCM) is developed to treat the arbitrary moving body on a fixed Cartesian grid. The complex moving boundary is tracked with the RBF and ghost cells are identified based on the signed function property of the RBF. To capture large deformation of the free surface, a gradient-augmented level set (GALS) method is used. Sub-grid resolution is obtained by simultaneously evolving both the level set (LS) function and its gradient information. Also, a simple distance function assignment method is developed to treat the contact boundary between the free surface and the solid surface. The accuracies of the ghost cell method (GCM) and the GALS method are validated by inline oscillation of a cylinder and horizontal sloshing cases, respectively. Then, the water impact of an arbitrary body is simulated. The cases include the water entry of a free falling multihull and the water entry of a bow-flare ship section with various roll angles. The accuracy of the proposed multiphase flow model and its capability are examined by comparing the present results to experimental and numerical results. Also, the results show that the present method can more accurately predict the slamming load in the presence of flow separation and air cushion than the smoothed particle hydrodynamics (SPH) based single-phase flow model and the boundary element method (BEM). Furthermore, the influence of roll angles on the slamming load and the free surface are studied.

Three-dimensional multicomponent vesicles: dynamics and influence of material properties.

In this work, the nonlinear hydrodynamics of a three-dimensional multicomponent vesicle in shear flow are explored. Using a volume- and area-conserving projection method coupled to a gradient-augmented level set and surface phase field method, the dynamics are systematically studied as a function of the membrane bending rigidity difference between the components, the speed of diffusion compared to the underlying shear flow, and the strength of the phase domain energy compared to the bending energy. Using a pre-segregated vesicle, three dynamics are observed: stationary phase, phase-treading, and a new dynamic called vertical banding. These regimes are very sensitive to the strength of the domain line energy, as the vertical banding regime is not observed when the line energy is larger than the bending energy. The findings demonstrate that a complete understanding of multicomponent vesicle dynamics requires that the full three-dimensional system be modeled, and show the complexity obtained when considering heterogeneous material properties.

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 distortion of the level set gradient under advection

The practice of periodically reinitializing the level set function is well established in two-phase flow applications as a way of controlling the growth of anomalies and/or numerical errors. In the present work, the underlying roots of this anomalous growth are studied, where it is established that the augmentation of the magnitude of the level set gradient (|∇ϕ|) is directly connected to the nature of the flow field; hence, it is not necessarily the result of some type of numerical error. More specifically, for a general flow field advecting the level set function, it is shown that the eigenpairs of the strain rate tensor are responsible for the rate of change of |∇ϕ| along a fluid particle trajectory. This straining action not only affects the magnitude of |∇ϕ|, but the general character of ϕ, and consequently contributes to the growth in numerical error. These numerical consequences are examined by adopting the Gradient Augmented Level Set method. Specifically, it is shown that the local error for ϕ is directly connected to the size of |∇ϕ| and to the magnitude of the second and fourth order derivatives of ϕ. These analytical findings are subsequently supported by various examples. The role of reinitialization is discussed, where it is shown that in cases where the zero level set contour has a local radius of curvature that is below the local grid resolution, reinitialization exacerbates rather than diminishes the degree of error. For other cases, where the interface is well resolved, reinitialization helps stabilize the error as intended.

Level set jet schemes for stiff advection equations: The semijet method

Many interfacial phenomena in physical and biological systems are dominated by high order geometric quantities such as curvature. Here a semi-implicit method is combined with a level set jet scheme to handle stiff nonlinear advection problems. The new method offers an improvement over the semi-implicit gradient augmented level set method previously introduced by requiring only one smoothing step when updating the level set jet function while still preserving the underlying methods higher accuracy. Sample results demonstrate that accuracy is not sacrificed while strict time step restrictions can be avoided.

Adaptive Gradient-Augmented Level Set Method with Multiresolution Error Estimation

A space-time adaptive scheme is presented for solving advection equations in two space dimensions. The gradient-augmented level set method using a semi-Lagrangian formulation with backward time integration is coupled with a point value multiresolution analysis using Hermite interpolation. Thus locally refined dyadic spatial grids are introduced which are efficiently implemented with dynamic quadtree data structures. For adaptive time integration, an embedded Runge-Kutta method is employed. The precision of the new fully adaptive method is analysed and speed up of CPU time and memory compression with respect to the uniform grid discretization are reported.

Electrohydrodynamics of Three-Dimensional Vesicles: A Numerical Approach

A three-dimensional numerical model of vesicle electrohydrodynamics in the presence of DC electric fields is presented. The vesicle membrane is modeled as a thin capacitive interface through the use of a semi-implicit, gradient-augmented level set jet scheme. The enclosed volume and surface area are conserved both locally and globally by a new Navier--Stokes projection method. The electric field calculations explicitly take into account the capacitive interface by an implicit immersed interface method formulation, which calculates the electric potential field and the transmembrane potential simultaneously. The results match well with previously published experimental, analytic, and two-dimensional computational works.

A narrow-band gradient-augmented level set method for multiphase incompressible flow

A finite-difference based numerical method for the simulation of multiphase incompressible flows is presented. The proposed algorithm adapts the gradient-augmented level set method of Nave et al. [25] for use in multiphase fluid simulations by interpolating the fluid velocity and introducing a simple reinitialization procedure. The method is carried out only along a narrow band surrounding the interface, reducing computational effort while preserving the advantages of the gradient-augmented method. Numerical results show excellent agreement with an analytical solution and available experimental data. A new experimental benchmark is introduced, using data gathered in a wedge-driven wave tank. In addition to the optimally local advection scheme and sub-grid resolution afforded by the new method, numerical comparisons with the standard level set method reveal superior volume conservation.

A gradient augmented level set method for unstructured grids

We present and test a gradient augmented level set method for unstructured grids, which has been designed to allow for easy integration in parallel flow solvers. The means to this end is a Hermite formulation, where the gradient is stored and advected as an independent variable, along with the level set function. The method uses narrow-band storage in conjunction with a Fast Marching Method, which is also designed to take advantage of the gradient augmentation. We demonstrate through standard passive advection test cases that the proposed method is able to compute interface motion with performance comparable to popular high-performing methods on structured grids.

A Semi-implicit Gradient Augmented Level Set Method

Here a semi-implicit formulation of the gradient augmented level set method is presented. By tracking both the level set and it's gradient accurate subgrid information is provided,leading to highly accurate descriptions of a moving interface. The result is a hybrid Lagrangian-Eulerian method that may be easily applied in two or three dimensions. The new approach allows for the investigation of interfaces evolving by mean curvature and by the intrinsic Laplacian of the curvature. In this work the algorithm, convergence and accuracy results are presented. Several numerical experiments in both two and three dimensions demonstrate the stability of the scheme.

A level set projection model of lipid vesicles in general flows

A new numerical method to model the dynamic behavior of lipid vesicles under general flows is presented. A gradient-augmented level set method is used to model the membrane motion. To enforce the volume- and surface-incompressibility constraints a four-step projection method is developed to integrate the full Navier–Stokes equations. This scheme is implemented on an adaptive non-graded Cartesian grid. Convergence results are presented, along with sample two-dimensional results of vesicles under various flow conditions.

A gradient-augmented level set method with an optimally local, coherent advection scheme

The level set approach represents surfaces implicitly, and advects them by evolving a level set function, which is numerically defined on an Eulerian grid. Here we present an approach that augments the level set function values by gradient information, and evolves both quantities in a fully coupled fashion. This maintains the coherence between function values and derivatives, while exploiting the extra information carried by the derivatives. The method is of comparable quality to WENO schemes, but with optimally local stencils (performing updates in time by using information from only a single adjacent grid cell). In addition, structures smaller than the grid size can be located and tracked, and the extra derivative information can be employed to obtain simple and accurate approximations to the curvature. We analyze the accuracy and the stability of the new scheme, and perform benchmark tests.