A fourth-order kernel-free boundary integral method for implicitly defined surfaces in three space dimensions

Xie, Yaning; Ying, Wenjun

doi:10.1016/j.jcp.2020.109526

Cited by 12 publications

(4 citation statements)

References 49 publications

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“…Using a Generalized Minimal Residual (GMRES), the resultant linear system is solved iteratively [31,32]. Details such as discretization of PDE, corrections for the discrete system, solutions and fast solvers of the discretized system of finite difference equations, and interpolation method of volume and boundary integrals on the boundary are presented in [16][17][18][19][20][21][22].…”

Section: Kfbim Framework For Single Boundary Magnetostatic Problemsmentioning

confidence: 99%

“…Thus, the sensitivity of KFBIM to computer errors is much lower and more accurate. KFBIM was proposed to be a general method to solve constant or variable elliptic PDEs for single or double boundaries in two or three dimensions [16][17][18][19][20][21][22]. Cartesian grid-based methods are used in KFBIM to solve the integrals, which means a body-fitted mesh is not required to solve the problems and can obtain higher accuracy on a coarser mesh when solving integrals when compared to FEM and BEM.…”

Section: Introductionmentioning

confidence: 99%

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Analytical Approach for Sharp Corner Reconstruction in the Kernel Free Boundary Integral Method during Magnetostatic Analysis for Inductor Design

Jin

Cao

et al. 2023

Energies

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It is very important to perform magnetostatic analysis accurately and efficiently when it comes to multi-objective optimization of designs of electromagnetic devices, particularly for inductors, transformers, and electric motors. A kernel free boundary integral method (KFBIM) was studied for analyzing 2D magnetostatic problems. Although KFBIM is accurate and computationally efficient, sharp corners can be a major problem for KFBIM. In this paper, an inverse discrete Fourier transform (DFT) based geometry reconstruction is explored to overcome this challenge for smoothening sharp corners. A toroidal inductor core with an airgap (C-core) is used to show the effectiveness of the proposed approach for addressing the sharp corner problem. A numerical example demonstrates that the method works for the variable coefficient PDE. In addition, magnetostatic analysis for homogeneous and nonhomogeneous material is presented for the reconstructed geometry, and results carried out using KFBIM are compared with the results of FEM analysis for the original geometry to show the differences and the potential of the proposed method.

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Section: Kfbim Framework For Single Boundary Magnetostatic Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Analytical Approach for Sharp Corner Reconstruction in the Kernel Free Boundary Integral Method during Magnetostatic Analysis for Inductor Design

Jin

Cao

et al. 2023

Energies

Self Cite

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“…Various efforts have been made recently. For singular source only interface problems, fourth order schemes for Poisson equations have been developed, see for example [25,33]; and a sixth order method if the interface has an analytic expressions [7]. A third order scheme for elliptic partial differential equations (PDEs) with piecewise constant but discontinuous coefficients in the presence of fixed interfaces in [25].…”

Section: Introductionmentioning

confidence: 99%

High order two-grid finite difference methods for interface and internal layer problems

Li¹,

Pan²,

Ruiz³

2022

Preprint

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Second order accurate Cartesian grid methods have been well developed for interface problems in the literature. However, it is challenging to develop third or higher order accurate methods for problems with curved interfaces and internal boundaries. In this paper, alternative approaches based on two different grids are developed for some interface and internal layer problems, which are different from adaptive mesh refinement (AMR) techniques. For one dimensional, or two-dimensional problems with straight interfaces or boundary layers that are parallel to one of the axes, the discussion is relatively easy. One of challenges is how to construct a fourth order compact finite difference scheme at boarder grid points that connect two meshes. A two-grid method that employs a second order discretization near the interface in the fine mesh and a fourth order discretization away from the interface in the coarse and boarder grid points is proposed. For two dimensional problems with a curved interface or an internal layer, a level set representation is utilized for which we can build a fine mesh within a tube |ϕ(x)| ≤ δh of the interface. A new super-third seven-point discretization that can guarantee the discrete maximum principle has been developed at hanging nodes. The coefficient matrices of the finite difference equations developed in this paper are M-matrices, which leads to the convergence of the finite difference schemes. Non-trivial numerical examples presented in this paper have confirmed the desired accuracy and convergence.

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“…For example, the boundary integral method (BIM) was introduced by Mayo [24,25]. Recent work by Marques et al [23] and the kernel-free boundary integral method [33,34,32] are inspired by the BIM. Some of other well established methods are the immersed boundary method [27], the immersed interface method [21,22], the explicit-jump immersed interface method [31], the ghost fluid method [12,11], the immersed boundary smooth extension method [29], the matched interface and boundary method, and other methods [8,28,13].…”

mentioning

confidence: 99%

Immersed hybrid difference methods for elliptic boundary value problems by artificial interface conditions

Jeon

Shin

2021

era

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We propose an immersed hybrid difference method for elliptic boundary value problems by artificial interface conditions. The artificial interface condition is derived by imposing the given boundary condition weakly with the penalty parameter as in the Nitsche trick and it maintains ellipticity. Then, the derived interface problems can be solved by the hybrid difference approach together with a proper virtual to real transformation. Therefore, the boundary value problems can be solved on a fixed mesh independently of geometric shapes of boundaries. Numerical tests on several types of boundary interfaces are presented to demonstrate efficiency of the suggested method.

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A fourth-order kernel-free boundary integral method for implicitly defined surfaces in three space dimensions

Cited by 12 publications

References 49 publications

Analytical Approach for Sharp Corner Reconstruction in the Kernel Free Boundary Integral Method during Magnetostatic Analysis for Inductor Design

Analytical Approach for Sharp Corner Reconstruction in the Kernel Free Boundary Integral Method during Magnetostatic Analysis for Inductor Design

High order two-grid finite difference methods for interface and internal layer problems

Immersed hybrid difference methods for elliptic boundary value problems by artificial interface conditions

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