Fast immersed interface Poisson solver for 3D unbounded problems around arbitrary geometries

doi:10.1016/j.jcp.2017.10.042

Journal of Computational Physics

Volume 354, 1 February 2018, Pages 403-416

https://doi.org/10.1016/j.jcp.2017.10.042 Get rights and content

Abstract

We present a fast and efficient Fourier-based solver for the Poisson problem around an arbitrary geometry in an unbounded 3D domain. This solver merges two rewarding approaches, the lattice Green's function method and the immersed interface method, using the Sherman-Morrison-Woodbury decomposition formula. The method is intended to be second order up to the boundary. This is verified on two potential flow benchmarks. We also further analyse the iterative process and the convergence behavior of the proposed algorithm. The method is applicable to a wide range of problems involving a Poisson equation around inner bodies, which goes well beyond the present validation on potential flows.

Introduction

Solving the Poisson equation is ubiquitous in many computational disciplines such as electromagnetism, fluid mechanics and theoretical physics. Therefore, the problem has been deeply studied and is a recurrent subject within the literature. Still, two major challenges remain: solving the problem in an unbounded framework, and taking into account the presence of a body within the computational domain. These two remaining challenges are widespread and often coupled in many applications, such as computational fluid dynamics applied to the simulation of flows past bodies, like blades and wings.

One may solve the first challenge, i.e. computing unbounded solutions, using two classes of methods. The first kind, based on James [1] and Lackner [2] ideas, solves two problems to determine coherent outer boundary conditions, as proposed by McCorquodale et al. [3], Marichal et al. [4], [5]. In this work, we use a second type of method based on the convolution of a kernel function. This convolution is performed using fast summation algorithms: Fast Multipole Methods (FMM) [6], [7], [8], [9], [10], [11], Fourier transforms (Hockney–Eastwood technique) [12], [13], [14], [15], or a combination of both [16], [17], [18] (see [19] for a comparison between the FMM and FFT-based methods). The kernel function is either built through a regularization of the Green's function [11], [13], [14], or by using the lattice Green's function [9], [10], [16], [17], [18], [20].

The second challenge, i.e. taking an immersed body and the discontinuities at its interface into account, can be solved using several methods. We will not attempt to provide an exhaustive review of all of them; instead, we highlight some of the prominent techniques. One of the earliest approach is known as the capacitance matrix method. For example, Proskurowski and Widlund [21] and O'Leary and Widlund [22] embed the irregular domain of interest into a regular one and impose the boundary condition using single layer and double layer potentials. The Helmholtz equation is then solved using iterative methods and algebraic decompositions, like the Sherman–Morrison–Woodbury formula (also known as the Woodbury formula), as proposed earlier in a similar context by George [23] and Buzbee et al. [24]. Over the same period of time, Peskin [25] introduced the Immersed Boundary (IB) method, still widely used and renowned for its robustness [18], [26], [27]. It discretizes the additional sources due to the body on several grid points using discrete delta functions, thus mollifying the body interface. Some recent developments propose a harmonic extension in the body to allow the use of a symmetric discrete delta function and therefore increase the order of convergence of the method [28]. However, this harmonic extension requires additional Poisson equation resolutions. Let us also mention the works based on the potential layer theory from Mayo [29], McKenney et al. [30], Mayo and Greenbaum [31] and a recent contribution from Askham and Cerfon [32], which is also based on the potential layer method, but reuses the harmonic expansion proposed earlier in the contexts of the IB technique [28]. In a similar way, the Boundary Element Methods (BEM) [33], [34], relies on panels and FMM acceleration to take the inner obstacle into account. They solve the integral equation associated to the panels and then compute a new Poisson equation including the additional source terms. Finally, let us present the Immersed Interface Method (IIM) [4], [35], [36], [37], [38], which uses modified Taylor series to take the inner body into account. The IIM offers a proven high order and sharp treatment of boundaries, does not require the introduction of a mollification or any extension in the body and the introduced perturbation has a tiny support around the boundary. Originally developed in a multidimensional way [35] but subject to stability issues, it has been modified to work with one-dimensional dimensions splitting [36], [37] and recently applied to 2D problems [4], [36].

However, within a regular grid framework, coupling a fast summation based solver and immersed bodies still remains an open challenge. Apart from the fast BEM methods which do not take advantage of the structured grid, none of the above techniques achieves a fast, accurate and grid-based solution of the unbounded Poisson problem around a body. For example, the Miller iteration [39], applied by Marichal et al. [4] to the IIM framework, needs to compute the outer boundary conditions iteratively, together with the inner boundary conditions: this tends to increase dramatically the computational cost. Furthermore, recent fast 3D Poisson solvers, like the FMM-based lattice Green's function convolution [16] or the FTT-based regularized kernel convolution [14], were never coupled with the immersed interface method.

In this work, we consider the inner body as a local perturbation of an unbounded body-free Poisson problem. The Sherman–Morrison–Woodbury (SMW) formula [40, p. 50] efficiently couples the Immersed Interface Method (IIM), used to impose the boundary condition, with a fast Poisson solver. This approach draws inspiration from existing works, such as the capacitance matrix method applied with a direct solver [23], [24] or the work on hybrid grid-particle methods and penalization performed by Chatelin and Poncet [41]. To be consistent in the coupling, we use lattice Green's function (LGF) kernel [16], equivalent to inverting the standard 3-points Laplacian stencil considered in the IIM.

In this paper, we first present the three different technologies (LGF, IIM and SMW) and further explain how to couple them together (Section 2). Secondly, we investigate the rate of convergence on two benchmarked potential flow applications and validate its second order behavior (Section 3). Finally, we close this work by a short conclusion and perspectives for future work.

Section snippets

Methodology

We consider the unbounded 3D scalar Poisson problem around a body, $Ω_{b}$ : $\nabla^{2} u (x) = f (x), supp (f) \in Ω ∖ Ω_{b},$ where $x \in R^{3}$ , Ω is the computational domain which encapsulates the body and the support of the source term, $f (x)$ , and the unbounded solution, $u (x)$ , decays at least as $1 / | x |$ at infinity. Without loss of generality we assume a Neumann boundary condition on $\partial Ω_{b}$ . Using a standard 3-point grid discretization for the Laplacian operator, $\nabla_{h}^{2}$ , we aim at a second order solver, which uses IIM in order to take

Validation: 3D potential flows past a body

This section validates our approach and assesses its performance on the solution of the potential flow past a body, $Ω_{b}$ . As in [47], we use the Helmholtz decomposition of the velocity, $u = U_{\infty} + \nabla \times ψ + \nabla ϕ with \nabla \cdot ψ = 0,$ where $U_{\infty} = U_{\infty} e_{z}$ represents the free-stream velocity, ψ is the stream function of the velocity induced by the bulk vorticity ( $\nabla^{2} ψ = - ω$ ) and ϕ represents the potential required to satisfy the boundary condition $u \cdot \hat{n} = 0$ on $\partial Ω_{b}$ , where $\hat{n}$ is the outgoing normal to $\partial Ω_{b}$ .

To compute the velocity field, one

Conclusion

In this work, we presented a novel way to use a fast and efficient unbounded elliptic solver in order to take an inner obstacle into account. The method proposed is based on a combination between a lattice Green's function convolution carried out by the Hockney–Eastwood method or an FMM-based algorithm and the immersed interface technique. This coupling is performed through the Sherman–Morrison–Woodbury (SMW) formula, which involves a low-rank system to solve, i.e. the boundary problem. This

Acknowledgements

We would like to acknowledge the helpful discussion with Y. Marichal (WaPT, Belgium, formerly at UCL). T. Gillis is supported by a FRIA grant from the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS). The present research also benefited from computational resources made available on the Tier-1 supercomputer of the Federation Wallonie-Bruxelles, infrastructure funded by the Walloon Region under the grant agreement no. 1117545.

References (50)

R.A. James
Solution of Poisson's equation for isolated source distributions
J. Comput. Phys.
(1977)
K. Lackner
Computation of ideal MHD equilibria
Comput. Phys. Commun.
(1976)
Y. Marichal et al.
An immersed interface solver for the 2-D unbounded Poisson equation and its application to potential flow
Comput. Fluids
(2014)
Y. Marichal et al.
Immersed interface interpolation schemes for particle–mesh methods
J. Comput. Phys.
(2016)
A. Gillman et al.
Fast and accurate numerical methods for solving elliptic difference equations defined on lattices
J. Comput. Phys.
(2010)
A. Gillman et al.
A fast solver for Poisson problems on infinite regular lattices
Am. J. Comput. Appl. Math.
(2014)
F. Vico et al.
Fast convolution with free-space Green's functions
J. Comput. Phys.
(2016)
P. Chatelain et al.
A Fourier-based elliptic solver for vortical flows with periodic and unbounded directions
J. Comput. Phys.
(2010)
M. Hejlesen et al.
A high order solver for the unbounded Poisson equation
J. Comput. Phys.
(2013)
M.M. Hejlesen et al.
A multiresolution method for solving the Poisson equation using high order regularization
J. Comput. Phys.
(2016)

S. Liska et al.

A parallel fast multipole method for elliptic difference equations

J. Comput. Phys.

(2014)

S. Liska et al.

A fast lattice Green's function method for solving viscous incompressible flows on unbounded domains

J. Comput. Phys.

(2016)

S. Liska et al.

A fast immersed boundary method for external incompressible viscous flows using lattice Green's functions

J. Comput. Phys.

(2017)

C.S. Peskin

Numerical analysis of blood flow in the heart

J. Comput. Phys.

(1977)

K. Taira et al.

The immersed boundary method: a projection approach

J. Comput. Phys.

(2007)

T. Colonius et al.

A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions

Comput. Methods Appl. Mech. Eng.

(2008)

D.B. Stein et al.

Immersed boundary smooth extension: a high-order method for solving {PDE} on arbitrary smooth domains using Fourier spectral methods

J. Comput. Phys.

(2016)

A. Mayo

The rapid evaluation of volume integrals of potential theory on general regions

J. Comput. Phys.

(1992)

A. McKenney et al.

A fast Poisson solver for complex geometries

J. Comput. Phys.

(1995)

A. Mayo et al.

Fourth order accurate evaluation of integrals in potential theory on exterior 3D regions

J. Comput. Phys.

(2007)

T. Askham et al.

An adaptive fast multipole accelerated Poisson solver for complex geometries

J. Comput. Phys.

(2017)

P. Ploumhans et al.

Vortex methods for high-resolution simulations of viscous flow past bluff bodies of general geometry

J. Comput. Phys.

(2000)

P. Ploumhans et al.

Vortex methods for direct numerical simulation of three-dimensional bluff body flows: application to the sphere at $Re = 300, 500, and 1000$

J. Comput. Phys.

(2002)

M.N. Linnick et al.

A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains

J. Comput. Phys.

(2005)

G. Miller

An iterative boundary potential method for the infinite domain Poisson problem with interior Dirichlet boundaries

J. Comput. Phys.

(2008)

Cited by (13)

A sharp immersed method for 2D flow-body interactions using the vorticity-velocity Navier-Stokes equations
2023, Journal of Computational Physics
Immersed methods discretize boundary conditions for complex geometries on background Cartesian grids. This makes such methods especially suitable for two-way coupled flow-body problems, where the body mechanics are partially driven by hydrodynamic forces. However, for the vorticity-velocity form of the Navier-Stokes equations, existing immersed geometry discretizations for two-way coupled problems only achieve first order spatial accuracy near solid boundaries. Here we introduce a sharp-interface approach based on the immersed interface method to handle the one- and two-way coupling between an incompressible flow and one or more rigid bodies using the 2D vorticity-velocity Navier-Stokes equations. Our main contributions are three-fold. First, we develop and analyze a moving boundary treatment for sharp immersed methods that can be applied to PDEs with implicitly defined boundary conditions, such as those commonly imposed on the vorticity field. Second, we develop a two-way coupling methodology for the vorticity-velocity Navier-Stokes equations based on control-volume momentum balance that does not require the pressure field. Third, we show through extensive testing and validation that our resulting flow-body solver reaches second-order accuracy for most practical scenarios, and provides significant efficiency benefits compared to a representative first-order approach.
An immersed interface method for the 2D vorticity-velocity Navier-Stokes equations with multiple bodies
2022, Journal of Computational Physics
Citation Excerpt :
In this section we review the continuous theory, and discuss the conditions under which the velocity reconstruction problem has a unique solution in multiply connected domains. This continuous formulation is then discretized using a generalization of the immersed interface Poisson solver developed by Gillis et al. [38], and the resulting algorithm is shown to achieve second order accuracy in reconstructing a velocity field on a multiply connected domain. Given a scalar vorticity field ω with compact support on a two-dimensional fluid domain Ω, the objective of the velocity reconstruction problem is to find a divergence-free velocity field Image 29 satisfying Image 30, along with no through-flow boundary conditions on solid bodies.
We present an immersed interface method for the vorticity-velocity form of the 2D Navier Stokes equations that directly addresses challenges posed by nonconvex immersed bodies, multiply connected domains, and the calculation of force distributions on immersed surfaces. The immersed interface method is re-interpreted as a polynomial extrapolation of flow quantities and boundary conditions into the immersed solid bodies, reducing computational cost and enabling simulations with nonconvex bodies that could not be discretized with previous immersed interface methods. In the flow, the vorticity transport equation is discretized using a conservative finite difference scheme and explicit Runge-Kutta time integration. The velocity reconstruction problem is transformed to a scalar Poisson equation that is discretized with conservative finite differences, and solved using an FFT-accelerated iterative algorithm. The use of conservative differencing throughout leads to exact enforcement of a discrete Kelvin's theorem, allowing for simulations with multiply connected domains and outflow boundaries that have challenged other immersed interface vortex methods. We also explore novel methods for recovering time-dependent pressure distributions on immersed bodies within a vorticity-based method and present a novel control volume formulation for recovering aerodynamic moments from only the vorticity and velocity fields. The method achieves second order spatial accuracy and third order temporal accuracy, and is validated on a variety of 2D flows in internal and free-space domains.
A FFT accelerated high order finite difference method for elliptic boundary value problems over irregular domains
2022, Journal of Computational Physics
For elliptic boundary value problems (BVPs) involving irregular domains and Robin boundary condition, no numerical method is known to deliver a fourth order convergence and $O (N \log N)$ efficiency, where N stands for the total degree-of-freedom of the system. Based on the matched interface and boundary (MIB) and fast Fourier transform (FFT) schemes, a new finite difference method is introduced for such problems, which involves two main components. First, a ray-casting MIB scheme is proposed to handle different types of boundary conditions, including Dirichlet, Neumann, Robin, and their mix combinations. By enclosing the concerned irregular domain by a large enough cubic domain, the ray-casting MIB scheme generates necessary fictitious values outside the irregular domain by imposing boundary conditions along the normal direction of the boundary, so that a high order central difference discretization of the Laplacian can be formed. Second, an augmented MIB formulation is built, in which Cartesian derivative jumps are reconstructed on the boundary as auxiliary variables. By treating such variables as unknowns, the discrete Laplacian can be efficiently inverted by the FFT algorithm, in the Schur complement solution of the augmented system. The accuracy and efficiency of the proposed augmented MIB method are numerically examined by considering various elliptic BVPs in two and three dimensions. Numerical results indicate that the new algorithm not only achieves a fourth order of accuracy in treating irregular domains and complex boundary conditions, but also maintains the FFT efficiency.
A method of immersed layers on Cartesian grids, with application to incompressible flows
2022, Journal of Computational Physics
The immersed boundary method (IBM) of Peskin (J. Comput. Phys., 1977), and derived forms such as the projection method of Taira and Colonius (J. Comput. Phys., 2007), have been useful for simulating flow physics in problems with moving interfaces on stationary grids. However, in their interface treatment, these methods do not distinguish one side from the other, but rather, apply the motion constraint to both sides, and the associated interface force is an inseparable mix of contributions from each side. In this work, we define a discrete Heaviside function, a natural companion to the familiar discrete Dirac delta function (DDF), to define a masked version of each field on the grid which, to within the error of the DDF, takes the intended value of the field on the respective sides of the interface. From this foundation we develop discrete operators and identities that are uniformly applicable to any surface geometry. We use these to develop extended forms of prototypical partial differential equations, including Poisson, convection-diffusion, and incompressible Navier-Stokes, that govern the discrete masked fields. These equations contain the familiar forcing term of the IBM, but also additional terms that regularize the jumps in field quantities onto the grid and enable us to individually specify the constraints on field behavior on each side of the interface. Drawing the connection between these terms and the layer potentials in elliptic problems, we refer to them generically as immersed layers. We demonstrate the application of the method to several representative problems, including two-dimensional incompressible flows inside a rotating cylinder and external to a rotating square.
The immersed interface method for Helmholtz equations with degenerate diffusion
2021, Mathematics and Computers in Simulation
In this paper, we consider a second-order immersed interface method for Helmholtz equations of the form $\nabla (β \nabla u) - σ u = f$ with a degenerate diffusion term $β$ . We assume that the diffusion term is discontinuous across an interface and $β$ is zero to one side of it. The method is applied to one-dimensional domains with multiple interfaces, and two-dimensional domains with circular and straight interfaces. The numerical solution is obtained by applying away from the interface the standard centered finite differences scheme and a new scheme across of the interface. Numerical results on one- and two-dimensional domains are used to compare and demonstrate the proposed numerical method’s capabilities. In all numerical experiments, the solutions of the interface problem is second order of accuracy.
Parallel implementation of a VIScous Vorticity Equation (VISVE) method in 3-D laminar flow
2021, Journal of Computational Physics
This paper presents a newly developed parallel implementation of solving the 3–D vorticity equation to fully simulate the incompressible laminar flow in the Eulerian frame. This method is designed to solve 3–D problems with irregular wall boundaries in small and compact computational domains in general shapes efficiently. The curl form of vorticity equation is discretized using the Finite Volume Method (FVM) by applying Stokes' theorem, which automatically guarantees the divergence–free condition of vorticity field at all times. The vorticity preserving velocity field is recovered by an explicit scheme without solving any linear system, and this velocity field is reprojected onto a divergence–free space by solving only one scalar velocity–potential Poisson's equation. The vorticity boundary condition is satisfied by employing a vorticity creation scheme, that can handle arbitrary wall boundary shapes. Numerical results of the flow past a 3–D NACA0012 hydrofoil with one periodic direction, the flow past a sphere, and the flow past a 3–D rectangular wing are presented to validate the scheme.

View all citing articles on Scopus

View full text

Fast immersed interface Poisson solver for 3D unbounded problems around arbitrary geometries

Abstract

Introduction

Section snippets

Methodology

Validation: 3D potential flows past a body

Conclusion

Acknowledgements

J. Comput. Phys.

Comput. Phys. Commun.

Comput. Fluids

J. Comput. Phys.

J. Comput. Phys.

Am. J. Comput. Appl. Math.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

Comput. Methods Appl. Mech. Eng.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.