Piecewise Linear Reconstruction Research Articles

An edge-based interface tracking (EBIT) method for multiphase-flow simulation with surface tension

We present a novel Front-Tracking method, the Edge-Based Interface Tracking (EBIT) method for multiphase flow simulations. In the EBIT method, the markers are located on the grid edges and the interface can be reconstructed without storing the connectivity of the markers. This feature makes the process of marker addition or removal easier than in the traditional Front-Tracking method. The EBIT method also allows almost automatic parallelization due to the lack of explicit connectivity.In a previous journal article we have presented the kinematic part of the EBIT method, that includes the algorithms for piecewise linear reconstruction and advection of the interface. Here, we complete the presentation of the EBIT method and combine the kinematic algorithm with a Navier–Stokes solver. A circle fit is now implemented to improve the accuracy of mass conservation in the reconstruction phase. Furthermore, to identify the reference phase and to distinguish ambiguous topological configurations, we introduce a new feature: the Color Vertex. For the coupling with the Navier–Stokes equations, we first calculate volume fractions from the position of the markers and the Color Vertex, then viscosity and density fields from the computed volume fractions and finally surface tension stresses with the Height-Function method. In addition, an automatic topology change algorithm is implemented into the EBIT method, making it possible the simulation of more complex flows. The two-dimensional version of the EBIT method has been implemented in the free Basilisk platform, and validated with seven standard test cases: stagnation flow, translation with uniform velocity, single vortex, Zalesak's disk, capillary wave, Rayleigh-Taylor instability and rising bubble. The results are compared with those obtained with the Volume-of-Fluid (VOF) method already implemented in Basilisk.

Isosurface extraction for piecewise-linear reconstruction of complex interfaces on arbitrary grids

In the field of volume of fluid type schemes, there is still great room for improvement in the accuracy of the methods used to reconstruct interfaces, especially when highly complex interfaces are involved. Isosurface extraction is exploited in a series of piecewise-linear interface reconstruction methods to simulate complex interface dynamic problems on arbitrary grids, and high-order surface patches are applied to achieve significantly improved accuracy. Isosurface extraction is also used to choose a convenient set of neighboring cells, hereafter called reconstruction stencil, for the reconstruction of highly curved interfaces when using well-known linearity-preserving methods. The relevance of the proper selection of the reconstruction stencil is highlighted, showing that the convergence of formal second-order methods can deteriorate when inappropriate reconstruction stencils are used. The accuracy, computational efficiency and performance of the implemented piecewise-linear interface reconstruction methods coupled with different reconstruction stencils are thoroughly analyzed for the reconstruction of interfaces in static conditions and dynamic simulations that produce highly deformed interfaces by using an accurate unsplit geometric advection method and imposing prescribed velocity fields.

An efficient second-order cell-centered Lagrangian discontinuous Galerkin method for two-dimensional elastic-plastic flows

An efficient second-order cell-centered Lagrangian discontinuous Galerkin (DG) method for solving two-dimensional (2D) elastic-plastic flows with the hypo-elastic constitutive model and von Mises yield condition is presented. First, starting from the governing equations of conserved quantities in the Euler framework, the integral weak formulation of them in the Lagrangian framework is derived. Next, the DG method is used for spatial discretization of both the weak formulation of conserved quantities and the evolution equation of deviatoric stress tensor. The Taylor basis functions defined in the reference coordinates provide the piecewise polynomial expansion of the variables, including the conserved quantities and the deviatoric stress tensor. The vertex velocities and Cauchy stress tensor on the edges are computed using a nodal solver equipped with a variant of Li's new Harten-Lax-van Leer-contact approximate Riemann solver [Li et al., “An HLLC-type approximate Riemann solver for two-dimensional elastic-perfectly plastic model,” J. Comput. Phys. 448, 110675 (2022)], in which the longitudinal wave velocity in the plastic state is modified. Then the vertex velocities and Cauchy stress tensor on the edges are used to compute numerical fluxes. A second-order total variation diminishing Runge–Kutta scheme is used for time discretization of both the governing equations of conserved quantities and the evolution equation of deviatoric stress tensor. After solving the evolution equation of deviatoric stress tensor, a radial return algorithm is performed at the Gauss points of each element according to the von Mises yield condition. And then the coefficients of the DG expansion for the deviatoric stress tensor on each element are modified by a least squares procedure using the deviatoric stress tensors at these Gauss points. To achieve second-order accuracy, the least squares procedure is used for piecewise linear reconstruction of conserved quantities and the deviatoric stress tensor, and the Barth–Jespersen limiter is used to suppress the nonphysical numerical oscillation near the discontinuities. After that, the coefficients of the DG expansion are modified through L2 projection using the reconstructed polynomials. Finally, a second-order cell-centered Lagrangian DG scheme is established. Several tests demonstrate that the new scheme achieves second-order accuracy with good robustness, and that the DG method of updating the deviatoric stress tensor has comparable accuracy and much higher efficiency with mesh refinement compared with previous works.

Comparison of methods for curvature estimation from volume fractions

This paper evaluates and compares the accuracy and robustness of curvature estimation methods for three-dimensional interfaces represented implicitly by discrete volume fractions on a Cartesian mesh. The height function (HF) method is compared to three paraboloid fitting methods: fitting to the piecewise linear interface reconstruction centroids (PC), fitting to the piecewise linear interface reconstruction volumetrically (PV), and volumetrically fitting (VF) the paraboloid directly to the volume fraction field. A shape-independent test of the curvature estimation is introduced through the use of randomly oriented and shaped paraboloids as reference interfaces. The coupling of the curvature estimation, interface transport, and Navier–Stokes surface tension force is evaluated through the measurement of spurious velocities in simulations of stationary and translating droplets. These studies find that while the curvature error from the VF method converges with second-order accuracy as with the HF method for static interfaces represented by exact volume fractions, the PV method best balances low curvature errors with low computational cost when errors are introduced in the volume fraction field either artificially or through the interface advection and reconstruction steps.

Automated Estimation of Aquifer Parameters from Arbitrary-Rate Pumping Tests in Python and MATLAB.

Inspired by the analysis by Mishra et al. (2012) of variable pumping rate tests using piecewise-linear reconstructions of the pumping history, this article contains a derivation of the convolutional form of pumping tests in which the pumping history may take any possible form. The solution is very similar to the classical Theis (1935) equation but uses the Green's function for a pumped aquifer given by taking the time derivative of the well function . This eliminates one integration inside another and renders the convolution including the pumping history about as computationally demanding as calculating the well function alone, so that the convolution can be completed using handy mathematical software. It also allows nonlinear well losses, and because an easily-computed deterministic model exists for all data points and pumping history, an objective function may include all data, so that errors are reduced in calculating any nonlinear-well losses. In addition, data from multiple observation wells may be used simultaneously in the inversion. We provide codes in MATLAB and Python to solve for drawdown resulting from an arbitrary pumping history and compute the optimal aquifer parameters to fit the data. We find that the subtleties in parameter dependencies and constructing an appropriate objective function have a substantial effect on the interpreted parameters. Furthermore, the optimization from step-drawdown tests is typically nonunique and strongly suggests that a Bayesian inversion should be used to fully estimate the joint probability density of the parameter vector.

A horizontal refined piecewise linear interface reconstruction (HOPLIRE) method based on the VOSET method for capturing the two-phase interface

The interface reconstruction method is significant for the numerical research of the two-phase problem, but the implementation of popular ones is not easy enough. In this paper, a horizontal refined piecewise linear interface reconstruction (HOPLIRE) method with a straightforward implementation procedure is proposed for the incompressible flow, and the idea of that is to reconstruct the two-phase interface in one grid cell with many horizontal refined pieces. Only one situation without complex classifications needs to be considered for reconstructing the interface, and the volume fraction flux through the grid cell face can be easily obtained to solve the VOF function by a simple four fundamental operations of arithmetic. In addition to the advantages of simple implementation for the HOPLIRE method, the accuracy of the reconstructed vapor–liquid interface is also high. The proximity between the HOPLIRE method and the popular piecewise linear interface reconstruction (PLIC) method increases with the divided piece number of the HOPLIRE method, and 30 is recommended based on calculation accuracies of LS function, interface direction, and curvature. Besides, the good performances of the VOSET-HOPLIRE method in the calculation accuracy and efficiency are verified by comparing with VOF-PLIC and VOSET-PLIC methods by some classical two-phase problems. Currently, the HOPLIRE method is only presented based on a two-dimensional problem under a structured mesh. In the future, this methodology will be improved to cover three-dimensional problems and unstructured meshes.

A flux globalization based well-balanced path-conservative central-upwind scheme for the shallow water flows in channels

We develop a flux globalization based well-balanced (WB) path-conservative central-upwind (PCCU) scheme for the one-dimensional shallow water flows in channels. Challenges in developing numerical methods for the studied system are mainly related to the presence of nonconservative terms modeling the flow when the channel width and bottom topography are discontinuous. We use the path-conservative technique to treat these nonconservative product terms and implement this technique within the flux globalization framework, for which the friction and aforementioned nonconservative terms are incorporated into the global flux: This results in a quasi-conservative system, which is numerically solved using the Riemann-problem-solver-free central-upwind scheme. The WB property of the resulting scheme (that is, its ability to exactly preserve both still- and moving-water equilibria at the discrete level) is ensured by performing piecewise linear reconstruction for the equilibrium variables rather than the conservative variables, and then evaluating the global flux using the obtained point values of the equilibrium quantities. The robustness and excellent performance of the proposed flux globalization based WB PCCU scheme are demonstrated in several numerical examples with both continuous and discontinuous channel width and bottom topography. In these examples, we clearly demonstrate the advantage of the proposed scheme over its simpler counterparts.

Rotated-hybrid Riemann solver for all-speed flows

The contribution deals with numerical solution of compressible flows modeled by the Euler or the Navier–Stokes equations. Addressed flow regimes range from very low Mach number flows to hypersonic flows. The numerical solution is obtained by the finite volume method based on the rotated-hybrid Riemann solver method, where two different Riemann solvers are used in different directions. The combination of the HLL and the HLLC schemes leads to carbuncle-free numerical method. Both the HLL and the HLLC schemes are further modified for improved accuracy and stability in cases with very low Mach number flows. Second order accuracy in space is achieved by the piecewise linear WENO reconstruction while accuracy in time is maintained by the explicit two-stage TVD Runge–Kutta method. The proposed method is validated on many test cases including solution of subsonic flow over an airfoil with M=0.01 or hypersonic flow past circular cylinder with M=20.

Moving water equilibria preserving nonstaggered central scheme for open‐channel flows

In this paper, we investigate a well‐balanced and positive‐preserving nonstaggered central scheme, which has second‐order accuracy on both time and spatial scales, for open‐channel flows with the variable channel width and the nonflat bottom. We perform piecewise linear reconstructions of the conserved variables and energy as well as discretize the source term using the property that the energy remains constant, so that the complex source term and the flux can be precisely balanced so as to maintain the steady state. The scheme also ensures that the cross‐sectional wet area is positive by introducing a draining time‐step technique. Numerical experiments demonstrate that the scheme is capable of accurately maintaining both the still steady‐state solutions and the moving steady‐state solutions, simultaneously. Moreover, the scheme has the ability to accurately capture small perturbations of the moving steady‐state solution and avoids generating spurious oscillations. It is also capable of showing that the scheme is positive‐preserving and robust in solving the dam‐break problem.

Flux globalization based well-balanced path-conservative central-upwind scheme for two-layer thermal rotating shallow water equations

We develop a flux globalization based well-balanced path-conservative central-upwind scheme for the two-layer thermal rotating shallow water (TRSW) equations, which arise in both oceanography and atmospheric sciences. As it occurs in all of the two-layer shallow water models, the studied two-layer TRSW system loses hyperbolicity due to the well-known Kelvin-Helmholtz type instabilities arising when the vertical velocity shear between the layers becomes sufficiently large. Therefore, prior to developing the numerical method, we discuss the hyperbolicity criterion. Besides the hyperbolicity loss, there are additional challenges in the development of numerical methods for the studied system. They are related to the presence of nonconservative terms modeling layer interaction and also to a quite complicated structure of steady-states solutions a good well-balanced numerical method should be able to exactly preserve. In order to treat the nonconservative product terms, we use the path-conservative technique, which is implemented within the flux globalization framework: the source and nonconservative terms are incorporated into the fluxes, which results in a quasi-conservative system, which is then numerically solved using the Riemann-problem-solver-free central-upwind scheme. A well-balanced property of the resulting scheme is ensured by performing piecewise linear reconstruction for the equilibrium rather than the conservative variables, by development of special quadratures required for the flux globalization procedure, and by switching off a part of the numerical diffusion when the computed solution is near or at thermo-geostrophic equilibria. The advantages and excellent performance of the proposed scheme are demonstrated on a number of numerical examples.

Multirate Shading with Piecewise Interpolatory Approximation

AbstractEvaluating shading functions on geometry surfaces dominates the rendering computation. A high‐quality but time‐consuming estimate is usually achieved with a dense sampling rate for pixels or sub‐pixels. In this paper, we leverage sparsely sampled points on vertices of dynamically‐generated subdivision surfaces to approximate the ground‐truth shading signal by piecewise linear reconstruction. To control the introduced interpolation error at runtime, we analytically derive an L∞ error bound and compute the optimal subdivision surfaces based on a user‐specified error threshold. We apply our analysis on multiple shading functions including Lambertian, Blinn‐Phong, Microfacet BRDF and also extend it to handle textures, yielding easy‐to‐compute formulas. To validate our derivation, we design a forward multirate shading algorithm powered by hardware tessellator that moves shading computation at pixels to the vertices of subdivision triangles on the fly. We show our approach significantly reduces the sampling rates on various test cases, reaching a speedup ratio of 134% ~ 283% compared to dense per‐pixel shading in current graphics hardware.

A Piecewise-Linear Reconstruction to Reduce the Dissipation of the HLL Method in Solving the Gas Dynamics Equations

A Piecewise-Linear Reconstruction to Reduce the Dissipation of the HLL Method in Solving the Gas Dynamics Equations

Discrete unified gas-kinetic scheme for the conservative Allen-Cahn equation.

In this paper, two discrete unified gas-kinetic scheme (DUGKS) methods with piecewise-parabolic flux reconstruction are presented for the conservative Allen-Cahn equation(CACE). One includes a temporal derivative of the order parameter in the force term while the other does not include temporal derivative in the force term but results in a modified CACE with additional terms. In the context of DUGKS, the continuum equationsrecovered from the piecewise-linear and piecewise-parabolic reconstructions for the fluxes at cell faces are subsequently derived. It is proved that the resulting equationwith the piecewise-linear reconstruction is a first-order approximation to the discrete velocity kinetic equationdue to the presence of the force term and the nonconservation property of the momentum of the collision model. To guarantee second-order accuracy of DUGKS, the piecewise-parabolic reconstruction for numerical flux is proposed. To validate the accuracy of the present DUGKS with the proposed flux evaluation, several benchmark problems, including the diagonal translation of a circular interface, the rotation of a Zalesak disk and the deformation of a circular interface, have been simulated. Numerical results show that the accuracy of both proposed DUGKS methods is almost comparable and improved compared with the DUGKS with linear flux reconstruction scheme.

NUMERICAL SIMULATION OF TWO-PHASE FLOW

This article presents the results of a multiphase flow numerical calculations. The numerical modeling method used in this work includes a volume of fluid model and immersed boundary method for studying the flow structure. The simulation was carried out on a structured Cartesian adaptive grid, where the immersed boundary is a circular pipe. The volume of the liquid, based on the piecewise linear interface reconstruction, allows us to determine the liquid-liquid boundary. The Navier-Stokes equations are discretized over the entire domain using a finite-difference scheme. The possibilities of the hybrid model (Volume of Fluid model – VOF and the immersed boundary method) are demonstrated by examples in which complex topological changes in the boundary occur. The general methodology has passed a thorough series of verification tests, the results of which are presented in this paper. An application for calculating the water flow is presented.

A computational study of thermally induced secondary atomization in multicomponent droplets

This study presents computational simulations of multicomponent and multiphase flows to reproduce the physical phenomena in the secondary atomization of a droplet induced by a hot temperature environment. The computational fluid dynamics model is based on the geometric volume of fluid method, with piecewise linear interface calculation reconstruction for accurate determination of the curvature and evaporation fluxes at the interface. The purpose of the model was to faithfully reproduce complex physical processes, such as internal gas cavity formation, liquid–vapour interface instability, cavity collapse and liquid jet ejection, and the pinch-off of a secondary droplet, leading to the microexplosion phenomenon that greatly enhances the evaporation rate of non-volatile liquid droplets. The solver was validated against the analytical solution in benchmark cases, and experimental data with bicomponent droplets reported in the literature. The developed model was used to predict the atomization of heavy fuel oil exposed at high temperatures under microgravity conditions. Different atomization regimes were identified, depending on the initial size of the internal bubbles. While small bubbles led to simple gas ejections, cavity collapse caused the larger bubbles to produce a jet formation. When the ratio between the bubble and droplet volumes was bigger than 0.7, microexplosions occurred. The results were found to be consistent with cases of bubble burst on flat surfaces, showing a strong dependence on the Ohnesorge number ( $Oh$ ). Key observable quantities, particularly jet velocity and bubble cap drainage velocity, were found to agree with correlations reported in other studies. The similarities were also supported by studies extending over a wide range of simulations (4000 cases) at different $Oh$ . An inversion in the dependence of the jet velocity on $Oh$ (above a critical value $Oh_c$ ) was observed.

Using piecewise-linear reconstruction to constructing a low-dissipation HLL method for numerical solution of hydrodynamics equation

Using piecewise-linear reconstruction to constructing a low-dissipation HLL method for numerical solution of hydrodynamics equation

Parallel unstructured finite volume lattice Boltzmann method for high-speed viscid compressible flows

Based on the double distribution function Boltzmann-BGK equations, a cell-centered finite volume lattice Boltzmann method on unstructured grids for high-speed viscid compressible flows is presented. In the equations, the particle distribution function is introduced on the basis of the D2Q17 circular function, and its corresponding total energy distribution function is adopted. In the proposed method, the advective term is evaluated by Roe’s flux-difference splitting scheme, and a limiter is used to prevent the generation of oscillations. The distribution functions on the interface are calculated by piecewise linear reconstruction, in which the gradient is computed by the least-squares approach. In order to do large-scale simulations, a parallel algorithm is illustrated. The present method is validated by a flow around the NACA0012 airfoil and a flow past a circular cylinder at high Mach numbers. The results agree well with the published results, which demonstrate that the present method is an efficient numerical method for high-speed viscid compressible flows. The parallel performance results show that the proposed parallel algorithm achieves 90% parallel efficiency on 4800 cores for a problem with [Formula: see text] unstructured triangle cells, which shows the potential to perform fast and high-fidelity simulations of large-scale high-speed viscid compressible flows in complicated computational domains.

A cell-centered implicit-explicit Lagrangian scheme for a unified model of nonlinear continuum mechanics on unstructured meshes

A cell-centered implicit-explicit updated Lagrangian finite volume scheme on unstructured grids is proposed for a unified first-order hyperbolic formulation of continuum fluid and solid mechanics, namely the Godunov-Peshkov-Romenski (GPR) model. The scheme provably respects the stiff relaxation limits of the continuous model at the fully discrete level, thus it is asymptotic preserving. Furthermore, the GCL is satisfied by a compatible discretization that makes use of a nodal solver to compute vertex-based fluxes that are used both for the motion of the computational mesh as well as for the time evolution of the governing PDEs. Second order of accuracy in space is achieved using a TVD piecewise linear reconstruction, while an implicit-explicit (IMEX) Runge-Kutta time discretization allows the scheme to obtain higher accuracy also in time. Particular care is devoted to the design of a stiff ODE solver, based on approximate analytical solutions of the governing equations, that plays a crucial role when the visco-plastic limit of the model is approached. We demonstrate the accuracy and robustness of the scheme on a wide spectrum of material responses covered by the unified continuum model that includes inviscid hydrodynamics, viscous heat conducting fluids, elastic and elasto-plastic solids in multidimensional settings.

A 3D cell-centered ADER MOOD Finite Volume method for solving updated Lagrangian hyperelasticity on unstructured grids

In this paper, we present a conservative cell-centered Lagrangian Finite Volume scheme for solving the hyperelasticity equations on unstructured multidimensional grids. The starting point of the present approach is the cell-centered FV discretization named EUCCLHYD and introduced in the context of Lagrangian hydrodynamics. Here, it is combined with the a posteriori Multidimensional Optimal Order Detection (MOOD) limiting strategy to ensure robustness and stability at shock waves with piecewise linear spatial reconstruction. The ADER (Arbitrary high order schemes using DERivatives) approach is adopted to obtain second-order of accuracy in time. This strategy has been successfully tested in a hydrodynamics context and the present work aims at extending it to the case of hyperelasticity. Here, the hyperelasticity equations are written in the updated Lagrangian framework and the dedicated Lagrangian numerical scheme is derived in terms of nodal solver, Geometrical Conservation Law (GCL) compliance, subcell forces and compatible discretization. The Lagrangian numerical method is implemented in 3D under MPI parallelization framework allowing to handle genuinely large meshes. A relatively large set of numerical test cases is presented to assess the ability of the method to achieve effective second order of accuracy on smooth flows, maintaining an essentially non-oscillatory behavior and general robustness across discontinuities and ensuring at least physical admissibility of the solution where appropriate. Pure elastic neo-Hookean and non-linear materials are considered for our benchmark test problems in 2D and 3D. These test cases feature material bending, impact, compression, non-linear deformation and further bouncing/detaching motions.

An accuracy comparison of piecewise linear reconstruction techniques for the characteristic finite volume method for two‐dimensional convection‐diffusion equation

AbstractIn this paper, we apply the characteristic finite volume method (CFVM) for solving a convection‐diffusion problem on two‐dimensional triangular grids. The finite volume method is used to discretize the equation while the finite element method is applied to estimate the gradient quantities at cell faces. The numerical analysis of the convergence has been implemented for the CFVM in one‐dimension. The approximate L2 norm of the error is derived to determine the errors for the approximate solution. The accuracy of four piecewise linear reconstruction techniques, namely, Frink, Holmes‐Connell, Green‐Gauss, and least‐squares methods are investigated on structured triangular grids. Numerical evidence shows that the least‐squares method is the most accurate of all methods for smooth initial condition problems. For discontinuous initial condition problems, the Frink and the Holmes‐Connell methods give a spurious oscillating solution in the vicinity of the discontinuity upstream of the discontinuity, and the Green‐Gauss and least‐squares methods give a spurious oscillating solution in the vicinity of the discontinuity downstream of the discontinuity. Moreover, the amplitude of the oscillation could be amplified on the finer grid sizes.