An efficient Poisson solver for complex embedded boundary domains using the multi-grid and fast multipole methods

Rapaka, Narsimha; Samtaney, Ravi

doi:10.1016/j.jcp.2020.109387

Cited by 4 publications

(4 citation statements)

References 23 publications

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“…In the DNS algorithm, the technology of directly simulating the Navier-Stokes equations using the initial value condition has matured [23]. This shows the characteristics of the Navier-Stokes equations that can solve the initial boundary value problem, the Green function, as a generalized function used to solve the non-homogeneous differential equation of the initial boundary condition [24]. The existence of its solution is unquestionable [25].…”

Section: Introductionmentioning

confidence: 99%

Surface Pressure Calculation Method of Multi-Field Coupling Mechanism under the Action of Flow Field

et al. 2023

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At present, the majority of fluid mechanisms are multi-field coupling mechanisms, and their function is also achieved in the flow field. Therefore, calculating the aerodynamic characteristics of the multi-field coupling mechanism in a symmetric flow field is very important. However, at present, the strong coupling algorithm in the domain has the problems of low accuracy and computational efficiency, which make it more difficult to solve the coupling problem. This article obtains the vector potential of the law of conservation of momentum using the tensor analysis method in a Cartesian coordinate system. Meanwhile, the generalized operator of Navier–Stokes equations and the fundamental solution of the generalized operator are obtained on this basis. Then, this article proposes the boundary integral equation of the Navier–Stokes equations by combining the fundamental solution of the Laplace equation with generalized potential theory. Based on this boundary integral equation, this article has developed a new calculation method that can help achieve integral calculation without domains, greatly reducing the problem’s difficulty. Finally, by comparing the ellipsoid example solution with the experimental results, the algorithm’s reliability in solving the incompressible problem is verified.

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Section: Introductionmentioning

confidence: 99%

Surface Pressure Calculation Method of Multi-Field Coupling Mechanism under the Action of Flow Field

et al. 2023

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“…In seeking to overcome this issue, most existing works avoid direct evaluation of (1.1) in an irregular domain; instead, one class of algorithms employ (local) volumetric PDE solvers-finite difference [19,20,21,22] and finite element [23] methods-in an embedded regular domain and layer potentials for enforcing correct boundary data. In particular, the embedded boundary integral approach of [23] solves the inhomogeneous PDE problem on a rectangular domain that embeds Ω; local low-order extrapolation for bulk forces and use of jump relations result in a second-order accurate method, which appears quite efficient.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, the embedded boundary integral approach of [23] solves the inhomogeneous PDE problem on a rectangular domain that embeds Ω; local low-order extrapolation for bulk forces and use of jump relations result in a second-order accurate method, which appears quite efficient. The method of reference [22], meanwhile, couples to geometric multigrid solvers for the bulk and is also restricted to the Poisson equation; this work in fact includes a timing comparison to the box code method of reference [24] (discussed in the next paragraph), which shows superior speed per degree of freedom. We note, however, that only first-order convergence is demonstrated in [22], with no claim made that the accuracy in the comparison example matches that of the box code with continuous extension, and we further observe that while the method implicitly requires extension of the source density outside the domain, this point is unaddressed (i.e.…”

Section: Introductionmentioning

confidence: 99%

“…The method of reference [22], meanwhile, couples to geometric multigrid solvers for the bulk and is also restricted to the Poisson equation; this work in fact includes a timing comparison to the box code method of reference [24] (discussed in the next paragraph), which shows superior speed per degree of freedom. We note, however, that only first-order convergence is demonstrated in [22], with no claim made that the accuracy in the comparison example matches that of the box code with continuous extension, and we further observe that while the method implicitly requires extension of the source density outside the domain, this point is unaddressed (i.e. known source functions are chosen that are continuous across the boundary and their values outside used).…”

Section: Introductionmentioning

confidence: 99%

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A fast, high-order scheme for evaluating volume potentials on complex 2D geometries via area-to-line integral conversion and domain mappings

Anderson¹,

Zhu²,

Veerapaneni³

2022

Preprint

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While potential theoretic techniques have received significant interest and found broad success in the solution of linear partial differential equations (PDEs) in mathematical physics, limited adoption is reported in the case of nonlinear and/or inhomogeneous problems (i.e. with distributed volumetric sources) owing to outstanding challenges in producing a particular solution on complex domains while simultaneously respecting the competing ideals of allowing complete geometric flexibility, enabling source adaptivity, and achieving optimal computational complexity. This article presents a new high-order accurate algorithm for finding a particular solution to the PDE by means of a convolution of the volumetric source function with the Green's function in complex geometries. Utilizing volumetric domain decomposition, the integral is computed over a union of regular boxes (lending the scheme compatibility with adaptive box codes) and triangular regions (which may be potentially curved near boundaries). Singular and near-singular quadrature is handled by converting integrals on volumetric regions to line integrals bounding a reference volume cell using cell mappings and elements of the Poincaré lemma, followed by leveraging existing one-dimensional near-singular and singular quadratures appropriate to the singular nature of the kernel. The scheme achieves compatibility with fast multipole methods (FMMs) and thereby optimal asymptotic complexity by coupling global rules for target-independent quadrature of smooth functions to local target-dependent singular quadrature corrections, and it relies on orthogonal polynomial systems on each cell for well-conditioned, high-order and efficient (with respect to number of required volume function evaluations) approximation of arbitrary volumetric sources. Our domain discretization scheme is naturally compatible with standard meshing software such as Gmsh, which are employed to discretize a narrow region surrounding the domain boundaries. We present 8th-order accurate results, demonstrate the success of the method with examples showing up to 12-digit accuracy on complex geometries, and, for static geometries, our numerical examples show well over 99% of evaluation time of the particular solution is spent in the FMM step.

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A block-coordinate approach of multi-level optimization with an application to physics-informed neural networks

Gratton,

Mercier,

Riccietti

et al. 2024

Comput Optim Appl

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An efficient Poisson solver for complex embedded boundary domains using the multi-grid and fast multipole methods

Cited by 4 publications

References 23 publications

Surface Pressure Calculation Method of Multi-Field Coupling Mechanism under the Action of Flow Field

Surface Pressure Calculation Method of Multi-Field Coupling Mechanism under the Action of Flow Field

A fast, high-order scheme for evaluating volume potentials on complex 2D geometries via area-to-line integral conversion and domain mappings

A block-coordinate approach of multi-level optimization with an application to physics-informed neural networks

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