Ghost Values Research Articles

The GFMxP and the basic extrapolation of the ghost values to solve the Poisson equation for discontinuous functions

In a recent paper, a novel coding of the Ghost Fluid Method for the variable coefficient Poisson equation with discontinuous functions (named GFMxP) was proposed. A lot of numerical tests, with all the required quantities available in a analytic form, were used to demonstrate the ability of the new procedure in modeling a sharp interface and to check the accuracy order of the solutions. In practical applications, however, the real difficulty stands in the estimation of the so-called “ghost values”, that is the values at points where the function is not only unknown, but even not defined. These values allow to compute the corrective terms enabling the use of standard finite difference formulas in presence of a singularity and/or a discontinuity, and can be only determined through some extrapolation procedure, whose truthfulness is essential to achieve a reliable result. The paper deals with such a basic issue, by testing different numerical strategies and demonstrating the strict relationship between the order of the adopted fit-model, the order of the solving scheme for the Poisson equation and the accuracy of the final solution.

Dependent Ghosts Have a Reflection for Free

We introduce ghost type theory (GTT) a dependent type theory extended with a new universe for ghost data that can safely be erased when running a program but which is not proof irrelevant like with a universe of (strict) propositions. Instead, ghost data carry information that can be used in proofs or to discard impossible cases in relevant computations. Casts can be used to replace ghost values by others that are propositionally equal, but crucially these casts can be ignored for conversion without compromising soundness. We provide a type-preserving erasure procedure which gets rid of all ghost data and proofs, a step which may be used as a first step to program extraction. We give a syntactical model of GTT using a program translation akin to the parametricity translation and thus show consistency of the theory. Because it is a parametricity model, it can also be used to derive free theorems about programs using ghost code. We further extend GTT to support equality reflection and show that we can eliminate its use without the need for the usual extra axioms of function extensionality and uniqueness of identity proofs. In particular we validate the intuition that indices of inductive type—such as the length index of vectors—do not matter for computation and can safely be considered modulo theory. The results of the paper have been formalised in Coq.

A ghost-point based second order accurate finite difference method on uniform orthogonal grids for electromagnetic scattering around curved perfect electric conductors with corners

We propose a finite difference method to solve Maxwell's equations in time domain in the presence of a perfect electric conductor (PEC) that impedes the propagation of electromagnetic waves. Our method relies on the level set characterization of the interface and the PDE-based extension technique which allows us to construct fictitious ghost values inside the PEC without scrutinizing the local geometries of its boundaries. As opposed to a locally perturbed body fitting grid used in [1], we promote a much simpler uniform orthogonal grid which necessitates a new approach for the extension technique to work. A novelty of our work is the introduction of what we call guest values which are accurately estimated boundary values deliberately and carefully misplaced near the interface. We stipulate a mild requirement on the accuracy of our ghost values that they need only be locally second-order accurate. Nevertheless, the resulting accuracy of our method is second order thanks to the application of back and forth error correction and compensation, which also relaxes CFL conditions. We demonstrate the effectiveness of our approach with some numerical examples.

A directional ghost-cell immersed boundary method for low Mach number reacting flows with interphase heat and mass transfer

This paper presents a directional ghost-cell immersed boundary method for low Mach number reacting flows with general boundary conditions, extending the approach described in Chi et al. (2020) [17]. The method employs locally directional schemes for ghost value reconstruction and utilization along each discretization direction. In this manner, the boundary condition can be naturally imposed on the boundary intersection point along each coordinate direction, allowing an easy and straightforward implementation of complex boundary conditions. Using Taylor series approximation of fluid points near the immersed boundary, the boundary variable and its gradient governed by arbitrary boundary condition can be implicitly involved, leading to a reliable polynomial extrapolation for the ghost values. In this way, Dirichlet, Neumann and Robin boundary conditions are implemented for general variables with formally second-order accuracy. For reacting gas-solid flow with surface reactions, the reaction-related coefficients in Robin boundary condition lead to a complex implementation in conventional IBM techniques, while the present directional framework leads to a straightforward algorithm while preserving accuracy. The proposed method has been checked by a series of test cases with different boundary conditions, including basic flow, heat and species transport, Stefan problem, and finally two practical applications involving heat and mass transfer. The local accuracy of all boundary conditions exhibits nearly second-order convergence, as expected. While only a single solid object is considered in this first work, the same method can be simply extended to multiple objects of arbitrary shape, leading to fully resolved simulation of reactive particle-laden flows.

Solving incompressible Navier–Stokes equations on irregular domains and quadtrees by monolithic approach

We present a second-order monolithic method for solving incompressible Navier–Stokes equations on irregular domains with quadtree grids. A semi-collocated grid layout is adopted, where velocity variables are located at cell vertices, and pressure variables are located at cell centers. Compact finite difference methods with ghost values are used to discretize the advection and diffusion terms of the velocity. A pressure gradient and divergence operator on the quadtree that use compact stencils are developed. Furthermore, the proposed method is extended to cubical domains with octree grids. Numerical results demonstrate that the method is second-order convergent in L∞ norms and can handle irregular domains for various Reynolds numbers.

Imposing mixed Dirichlet-Neumann-Robin boundary conditions on irregular domains in a level set/ghost fluid based finite difference framework

In this paper, an efficient, unified finite difference method for imposing mixed Dirichlet, Neumann and Robin boundary conditions on irregular domains is proposed, leveraging on our previous work [Chai et al., J. Comput. Phys. 400 (2020): 108890]. The level set method is applied to describe the arbitrarily-shaped interface, and the ghost fluid method is utilized to address the complex discontinuities on the interface. The core of this method lies in providing required ghost values under the restriction of mixed boundary conditions, which is done in a fractional-step way. Specifically, the normal derivative is calculated in the concerned subdomain by aid of a linear polynomial reconstruction, then the normal derivative and the ghost value are successively extrapolated to the other subdomain using a linear partial differential equation approach. A series of Poisson problems with mixed boundary conditions and a heat transfer test are performed to validate the method, highlighting its convergence accuracy in the L1 and L∞ norms. The method produces second-order accurate solutions with first-order accurate gradients, and is easy to implement in multi-dimensional configurations. In summary, the method represents a promising tool for imposing mixed boundary conditions, which will be applied to practical problems in future work.

A directional ghost-cell immersed boundary method for incompressible flows

This paper describes an efficient ghost-cell immersed boundary method for incompressible flow simulations. The method employs a locally directional extrapolation scheme along all discretization directions for the ghost values. Additionally, it involves fictitious discrete boundary forcing terms instead of the ghost values in the governing equations. In this way, the boundary is represented more accurately than in prior IBMs and it is possible to fulfill the divergence-free condition. When combined with high-order spatial discretization schemes, the IBM order is reduced locally near the immersed boundary in a step-wise manner. In this way, the method delivers more compact stencils and is able to deal with sharp interfaces. By handling the distance from the ghost point to the boundary carefully, the proposed method delivers lower truncation errors than standard IBM, with clean (persistent) convergence rate and enhanced stability. The parallel implementation of this approach is straightforward. Its accuracy has been checked by considering a variety of test cases, including irregular, three-dimensional, and moving boundaries. The local accuracy for the proposed method is formally second order, and is measured to be close to this value using numerical tests.

A finite difference discretization method for heat and mass transfer with Robin boundary conditions on irregular domains

This paper proposes a finite difference discretization method for simulations of heat and mass transfer with Robin boundary conditions on irregular domains. The level set method is utilized to implicitly capture the irregular evolving interface, and the ghost fluid method to address variable discontinuities on the interface. Special care has been devoted to providing ghost values that are restricted by the Robin boundary conditions. Specifically, it is done in two steps: 1) calculate the normal derivative in cells adjacent to the interface by reconstructing a linear polynomial system; 2) successively extrapolate the normal derivative and the ghost value in the normal direction using a linear partial differential equation approach. This method produces second-order accurate solutions for both the Poisson and heat equations with Robin boundary conditions, and first-order accurate solutions for the Stefan problems. The solution gradients are of first-order accuracy, as expected. It is easy to implement in three-dimensional configurations, and can be straightforwardly generalized into higher-order variants. The method thus represents a promising tool for practical heat and mass transfer problems involving Robin boundary conditions.

An ALE/embedded boundary method for two-material flow simulations

We present a computational framework that combines an embedded boundary method and an arbitrary Lagrangian–Eulerian (ALE) method for multi-material flow computations of compressible fluids. The methodology is presented for two-material flows and one-material flow with moving boundaries, whereas the concept extends easily to multiple immiscible fluids. The proposed method is capable of handling geometrically complex material interfaces on moving grids and is suitable for multi-material flow computations in the context of ALE shock hydrodynamics. Furthermore, it can serve as the underlying framework for multi-fluid/structure interaction, where a portion of the boundary of a computational domain for a fluid mixture deforms according to the solid domain motion. Specifically, we use a variational multiscale stabilized finite element method to update the fluid state of each material, and define ghost values to enforce the transmission condition at the embedded material interface, which is captured using a level-set approach. Two different strategies to populate ghost values are considered and compared in extensive benchmark numerical tests: simple constant extrapolation and a multi-material Riemann solver, respectively.

Inverse Lax–Wendroff boundary treatment for compressible Lagrange-remap hydrodynamics on Cartesian grids

We propose a new high-order accurate numerical boundary treatment for solving hyperbolic systems of conservation laws and Euler equations using a Lagrange-remap approach on Cartesian grids in cases of physical boundaries not aligned with the mesh. The method is an adaptation of the Inverse Lax–Wendroff procedure [34–38] to the Lagrange-remap approach, which considerably alleviates the algebra. High-order accurate ghost values of conservative variables are imposed using Taylor expansions whose coefficients are found by inverting a (linear or non-linear) system which is well posed in all our examples. For 2D problems, a least-square procedure is added to prevent extrapolation instabilities. The Lagrange-remap formalism also provides a simpler fluid–structure coupling which is also described. Numerical examples are given for the linear case and Euler equations in 1D and 2D.

A cell-centred finite volume method for the Poisson problem on non-graded quadtrees with second order accurate gradients

This paper introduces a two-dimensional cell-centred finite volume discretization of the Poisson problem on adaptive Cartesian quadtree grids which exhibits second order accuracy in both the solution and its gradients, and requires no grading condition between adjacent cells. At T-junction configurations, which occur wherever resolution differs between neighboring cells, use of the standard centred difference gradient stencil requires that ghost values be constructed by interpolation. To properly recover second order accuracy in the resulting numerical gradients, prior work addressing block-structured grids and graded trees has shown that quadratic, rather than linear, interpolation is required; the gradients otherwise exhibit only first order convergence, which limits potential applications such as fluid flow. However, previous schemes fail or lose accuracy in the presence of the more complex T-junction geometries arising in the case of general non-graded quadtrees, which place no restrictions on the resolution of neighboring cells. We therefore propose novel quadratic interpolant constructions for this case that enable second order convergence by relying on stencils oriented diagonally and applied recursively as needed. The method handles complex tree topologies and large resolution jumps between neighboring cells, even along the domain boundary, and both Dirichlet and Neumann boundary conditions are supported. Numerical experiments confirm the overall second order accuracy of the method in the L∞ norm.

Resilience for Massively Parallel Multigrid Solvers

Fault tolerant massively parallel multigrid methods for elliptic partial differential equations are a step towards resilient solvers. Here, we combine domain partitioning with geometric multigrid methods to obtain fast and fault-robust solvers for three-dimensional problems. The recovery strategy is based on the redundant storage of ghost values, as they are commonly used in distributed memory parallel programs. In the case of a fault, the redundant interface values can be easily recovered, while the lost inner unknowns are recomputed approximately with recovery algorithms using multigrid cycles for solving a local Dirichlet problem. Different strategies are compared and evaluated with respect to performance, computational cost, and speedup. Especially effective are asynchronous strategies combining global solves with accelerated local recovery. By this, multiple faults can be fully compensated with respect to both the number of iterations and run-time. For illustration, we use a state-of-the-art petascale supercomputer to study failure scenarios when solving systems with up to $6 \cdot 10^{11}$ (0.6 trillion) unknowns.

Finite difference elastic wave modeling with an irregular free surface using ADER scheme

In numerical modeling of seismic wave propagation in the earth, we encounter two important issues: the free surface and the topography of the surface (i.e. irregularities). In this study, we develop a 2D finite difference solver for the elastic wave equation that combines a 4th- order ADER scheme (Arbitrary high-order accuracy using DERivatives), which is widely used in aeroacoustics, with the characteristic variable method at the free surface boundary. The idea is to treat the free surface boundary explicitly by using ghost values of the solution for points beyond the free surface to impose the physical boundary condition. The method is based on the velocity-stress formulation. The ultimate goal is to develop a numerical solver for the elastic wave equation that is stable, accurate and computationally efficient. The solver treats smooth arbitrary-shaped boundaries as simple plane boundaries. The computational cost added by treating the topography is negligible compared to flat free surface because only a small number of grid points near the boundary need to be computed. In the presence of topography, using 10 grid points per shortest shear-wavelength, the solver yields accurate results. Benchmark numerical tests using several complex models that are solved by our method and other independent accurate methods show an excellent agreement, confirming the validity of the method for modeling elastic waves with an irregular free surface.

Compact cell-centered discretization stencils at fine–coarse block structured grid interfaces

Different strategies for coupling fine–coarse grid patches are explored in the context of the adaptive mesh refinement (AMR) method. We show that applying linear interpolation to fill in the fine grid ghost values can produce a finite volume stencil of comparable accuracy to quadratic interpolation provided the cell volumes are adjusted. The volume of fine cells expands whereas the volume of neighboring coarse cells contracts. The amount by which the cells contract/expand depends on whether the interface is a face, an edge, or a corner. It is shown that quadratic or better interpolation is required when the conductivity is spatially varying, anisotropic, the refinement ratio is other than two, or when the fine–coarse interface is concave.

Finite-difference ghost-point multigrid methods on Cartesian grids for elliptic problems in arbitrary domains

In this paper we present a numerical method for solving elliptic equations in an arbitrary domain (described by a level-set function) with general boundary conditions (Dirichlet, Neumann, Robin, etc.) on Cartesian grids, using finite difference discretization and non-eliminated ghost values. A system of Ni+Ng equations in Ni+Ng unknowns is obtained by finite difference discretization on the Ni internal grid points, and second order interpolation to define the conditions for the Ng ghost values. The resulting large sparse linear system is then solved by a multigrid technique. The novelty of the papers can be summarized as follows: general strategy to discretize the boundary condition to second order both in the solution and its gradient; a relaxation of inner equations and boundary conditions by a fictitious time method, inspired by the stability conditions related to the associated time dependent problem (with a convergence proof for the first order scheme); an effective geometric multigrid, which maintains the structure of the discrete system at all grid levels. It is shown that by increasing the relaxation step of the equations associated to the boundary conditions, a convergence factor close to the optimal one is obtained. Several numerical tests, including variable coefficients, anisotropic elliptic equations, and domains with kinks, show the robustness, efficiency and accuracy of the approach.

A systematic approach for constructing higher-order immersed boundary and ghost fluid methods for fluid–structure interaction problems

A systematic approach is presented for constructing higher-order immersed boundary and ghost fluid methods for CFD in general, and fluid–structure interaction problems in particular. Such methods are gaining popularity because they simplify a number of computational issues. These range from gridding the fluid domain, to designing and implementing Eulerian-based algorithms for challenging fluid–structure applications characterized by large structural motions and deformations or topological changes. However, because they typically operate on non body-fitted grids, immersed boundary and ghost fluid methods also complicate other issues such as the treatment of wall boundary conditions in general, and fluid–structure transmission conditions in particular. These methods also tend to be at best first-order space-accurate at the immersed interfaces. In some cases, they are also provably inconsistent at these locations. A methodology is presented in this paper for addressing this issue. It is developed for inviscid flows and prescribed structural motions. For the sake of clarity, but without any loss of generality, this methodology is described in one and two dimensions. However, its extensions to flow-induced structural motions and three dimensions are straightforward. The proposed methodology leads to a departure from the current practice of populating ghost fluid values independently from the chosen spatial discretization scheme. Instead, it accounts for the pattern and properties of a preferred higher-order discretization scheme, and attributes ghost values as to preserve the formal order of spatial accuracy of this scheme. It is illustrated in this paper by its application to various finite difference and finite volume methods. Its impact is also demonstrated by one- and two-dimensional numerical experiments that confirm its theoretically proven ability to preserve higher-order spatial accuracy, including in the vicinity of the immersed interfaces.

Guidelines for Poisson Solvers on Irregular Domains with Dirichlet Boundary Conditions Using the Ghost Fluid Method

We consider the variable coefficient Poisson equation with Dirichlet boundary conditions on irregular domains. We present numerical evidence for the accuracy of the solution and its gradients for different treatments at the interface using the Ghost Fluid Method for Poisson problems of Gibou et al. (J. Comput. Phys. 176:205---227, 2002; 202:577---601, 2005). This paper is therefore intended as a guide for those interested in using the GFM for Poisson-type problems (and by consequence diffusion-like problems and Stefan-type problems) by providing the pros and cons of the different choices for defining the ghost values and locating the interface. We found that in order to obtain second-order-accurate gradients, both a quadratic (or higher order) extrapolation for defining the ghost values and a quadratic (or higher order) interpolation for finding the interface location are required. In the case where the ghost values are defined by a linear extrapolation, the gradients of the solution converge slowly (at most first order in average) and the convergence rate oscillates, even when the interface location is defined by a quadratic interpolation. The same conclusions hold true for the combination of a quadratic extrapolation for the ghost cells and a linear interpolation. The solution is second-order accurate in all cases. Defining the ghost values with quadratic extrapolations leads to a non-symmetric linear system with a worse conditioning than that of the linear extrapolation case, for which the linear system is symmetric and better conditioned. We conclude that for problems where only the solution matters, the method described by Gibou, F., Fedkiw, R., Cheng, L.-T. and Kang, M. in (J. Comput. Phys. 176:205---227, 2002) is advantageous since the linear system that needs to be inverted is symmetric. In problems where the solution gradient is needed, such as in Stefan-type problems, higher order extrapolation schemes as described by Gibou, F. and Fedkiw, R. in (J. Comput. Phys. 202:577---601, 2005) are desirable.

Multiple interacting liquids

The particle level set method has proven successful for the simulation of two separate regions (such as water and air, or fuel and products). In this paper, we propose a novel approach to extend this method to the simulation of as many regions as desired. The various regions can be liquids (or gases) of any type with differing viscosities, densities, viscoelastic properties, etc. We also propose techniques for simulating interactions between materials, whether it be simple surface tension forces or more complex chemical reactions with one material converting to another or two materials combining to form a third. We use a separate particle level set method for each region, and propose a novel projection algorithm that decodes the resulting vector of level set values providing a "dictionary" that translates between them and the standard single-valued level set representation. An additional difficulty occurs since discretization stencils (for interpolation, tracing semi-Lagrangian rays, etc.) cross region boundaries naively combining non-smooth or even discontinuous data. This has recently been addressed via ghost values, e.g. for fire or bubbles. We instead propose a new paradigm that allows one to incorporate physical jump conditions in data "on the fly," which is significantly more efficient for multiple regions especially at triple points or near boundaries with solids.

Blood pressure as a surrogate end point for hypertension.

To review relevant literature and provide opinions regarding the use of blood pressure as a surrogate measure to predict cardiovascular risk. Primary and review articles were identified by MEDLINE search (1990-January 2001) and through secondary sources. Studies and review articles that related to the interpretation of blood pressure as a surrogate measure were reviewed. Information that was relevant to this topic was included. The measurement of blood pressure is subject to numerous sources of error and bias. Patients who perform home blood pressure testing and self-report these values frequently leave out high values and add ghost values into logbooks. Additionally, analysis of recent data suggests that at any given level of blood pressure that is achieved, cardiovascular risk reduction may not be the same with different therapeutic agents. It is also now recommended that systolic blood pressure be used in preference to diastolic blood pressure to determine risk and to assess management strategies. Although 24-hour blood pressure measurements may be the best predictors of cardiovascular risk, this has not been demonstrated in a long-term morbidity trial. Blood pressure is a relatively poor surrogate measure. Unfortunately, no alternatives are available at this time. Therefore, every attempt must be made to accurately determine blood pressure and to assess risk and benefit from specific antihypertensive agents. Systolic blood pressure should be the predominant blood pressure measure used to evaluate patients, especially middle-aged and elderly individuals.

Blood Pressure as a Surrogate End Point for Hypertension

OBJECTIVE:To review relevant literature and provide opinions regarding the use of blood pressure as a surrogate measure to predict cardiovascular risk.DATA SOURCES:Primary and review articles were identified by MEDLINE search (1990–January 2001) and through secondary sources.STUDY SELECTION AND DATA EXTRACTION:Studies and review articles that related to the interpretation of blood pressure as a surrogate measure were reviewed. Information that was relevant to this topic was included.DATA SYNTHESIS:The measurement of blood pressure is subject to numerous sources of error and bias. Patients who perform home blood pressure testing and self-report these values frequently leave out high values and add ghost values into logbooks. Additionally, analysis of recent data suggests that at any given level of blood pressure that is achieved, cardiovascular risk reduction may not be the same with different therapeutic agents. It is also now recommended that systolic blood pressure be used in preference to diastolic blood pressure to determine risk and to assess management strategies. Although 24-hour blood pressure measurements may be the best predictors of cardiovascular risk, this has not been demonstrated in a long-term morbidity trial.CONCLUSIONS:Blood pressure is a relatively poor surrogate measure. Unfortunately, no alternatives are available at this time. Therefore, every attempt must be made to accurately determine blood pressure and to assess risk and benefit from specific antihypertensive agents. Systolic blood pressure should be the predominant blood pressure measure used to evaluate patients, especially middle-aged and elderly individuals.