A kernel-free boundary integral method for elliptic PDEs on a doubly connected domain

Cao, Yue; Xie, Yaning; Krishnamurthy, Mahesh; Li, Shuwang; Ying, Wenjun

doi:10.1007/s10665-022-10233-8

Cited by 7 publications

(3 citation statements)

References 34 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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show abstract

“…It evaluates boundary and volume integrals indirectly by a Cartesian grid-based method, thus possessing the following two most prominent features: i) it does not require the explicit expressions of Green's function or special quadratures formulas to directly calculate integrals, especially nearly singular or hyper-singular boundary integrals, so that the dependence on the kernel can be completely eliminated in practice; ii) it reformulates the boundary value problems as the Fredholm BIE of the second kind, helping to eliminate the ill-conditioning property of the original problems so that the number of Krylov subspace iterations is essentially independent of the discretization parameter or system dimension. The KFBI method has been developed to be a general method for two-dimensional elliptic PDEs [45,46,47,44,42,43,6], but for three-dimensional problems, it is now under intensive development.…”

Section: Introductionmentioning

confidence: 99%

Kernel Free Boundary Integral Method for 3D Stokes and Navier Equations on Irregular Domains

Zhao¹,

Hua²,

Ying³

2023

Preprint

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A second-order accurate kernel-free boundary integral method is presented for Stokes and Navier boundary value problems on three-dimensional irregular domains. It solves equations in the framework of boundary integral equations, whose corresponding discrete forms are well-conditioned and solved by the GMRES method. A notable feature of this approach is that the boundary or volume integrals encountered in BIEs are indirectly evaluated by a Cartesian grid-based method, which includes discretizing corresponding simple interface problems with a MAC scheme, correcting discrete linear systems to reduce large local truncation errors near the interface, solving the modified system by a CG method together with an FFT-based Poisson solver. No extra work or special quadratures are required to deal with singular or hyper-singular boundary integrals and the dependence on the analytical expressions of Green's functions for the integral kernels is completely eliminated. Numerical results are given to demonstrate the efficiency and accuracy of the Cartesian grid-based method.

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Kernel free boundary integral method for 3D incompressible flow and linear elasticity equations on irregular domains

Zhao

Dong

Ying

2023

Computer Methods in Applied Mechanics and Engineering

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A kernel-free boundary integral method for elliptic PDEs on a doubly connected domain

Cited by 7 publications

References 34 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

Kernel Free Boundary Integral Method for 3D Stokes and Navier Equations on Irregular Domains

Kernel free boundary integral method for 3D incompressible flow and linear elasticity equations on irregular domains

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