A numerical method for solving elasticity equations with interface involving multi-domains and triple junction points

Wang, Liqun; Hou, Songming; Shi, Liwei; Solow, James

doi:10.1016/j.amc.2014.11.072

Cited by 6 publications

(4 citation statements)

References 17 publications

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“…28 This method was extended to multi-domain and triple-junction points. 29,30 Another class of IFE approximations 31 were based on nonconforming finite elements 32 whose continuity is imposed weakly through mean values over edges/faces. An unfitted discontinuous Galerkin formulation 33 has also been developed to solve interface problems.…”

Section: Introductionmentioning

confidence: 99%

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Shape optimization with immersed interface finite element method

Kaudur

Patil

2022

Numerical Meth Engineering

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Shape Optimization using conventional interface-fitted finite element method requires either mesh morphing or remeshing, which is computationally expensive and prone to errors due to suboptimal mesh or solution interpolation. In this article, we present a shape optimization scheme which uses a four-noded rectangular immersed-interface element circumventing the need for interface-fitted mesh for finite element analysis, and mesh morphing or re-meshing for shape optimization. The analysis problem is solved using a non-conformal, Petrov-Galerkin (ncPG) formulation. We use non-conformal trial functions with conformal test functions. A fixed structured grid with a linear approximation for the interface within each element is used. We perform a bi-material patch test to confirm the consistency and convergence of the immersed-interface finite element method (IIFEM). The convergence of the displacement and stress error norms are of the same or slightly better order compared to the interface-fitted FEM. Thus, IIFEM reduces the cost of meshing without compromising the accuracy of the solutions obtained. We then perform analytical design sensitivity analysis to obtain the gradient of the global stiffness matrix with respect to shape design variables. The sensitivities are used in the gradient-based optimization formulation of the shape design problem. We present several ways to parametrize the design space. Finally, verification and case studies are presented to demonstrate the accuracy and potential of the approach.

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Section: Introductionmentioning

confidence: 99%

“…The approximation capabilities of such spaces for linear elasticity problems has been studied 28 . This method was extended to multi‐domain and triple‐junction points 29,30 . Another class of IFE approximations 31 were based on nonconforming finite elements 32 whose continuity is imposed weakly through mean values over edges/faces.…”

Section: Introductionmentioning

confidence: 99%

Shape optimization with immersed interface finite element method

Kaudur

Patil

2022

Numerical Meth Engineering

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“…In [22], the authors employed a Petrov-Galerkin type method [23] to solve multi-domain interface problems. Related research papers include [24,25]. As for IFEM for multi-domain interface problems with triple junction, we developed a piecewise linear IFEM on triangular meshes [26].…”

Section: Introductionmentioning

confidence: 99%

A bilinear partially penalized immersed finite element method for elliptic interface problems with multi-domain and triple-junction points

Chen

Hou

Zhang

2020

Results in Applied Mathematics

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In this article, we introduce a new partially penalized immersed finite element method (IFEM) for solving elliptic interface problems with multi-domains and triple-junction points. We construct new IFE functions on elements intersected with multiple interfaces or with triple-junction points to accommodate interface jump conditions. For non-homogeneous flux jump, we enrich the local approximating spaces by adding up to three local flux basis functions. Numerical experiments are carried out to show that both the Lagrange interpolations and the partial penalized IFEM solutions converge optimally in L 2 and H 1 norms.

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“…There are many possible choices. For example, we can apply the immersed boundary method (IBM) proposed in [38,39], the immersed interface method (IIM) in [26], the ghost fluid method (GFM) in [5] and an extension in [34,35], and a weak formulation solver developed in [10] and later extended to [13,9,12,11,54,58,57,56,55].…”

Section: Introductionmentioning

confidence: 99%

A weak formulation for the multiphase Stokes flow problem without body fitting grids

Ying

Hou

Leung

et al. 2018

Pure and Applied Mathematics Quarterly

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We develop an effective interface tracking method to simulate the incompressible Stokes flow with moving interfaces. The Stokes equations are first rewritten into a system of elliptic equations with singular sources which can be efficiently solved by a simple weak formulation proposed in [11]. The key idea is to first split the solution into a singular part and a regular part additively. The singular part captures the interface conditions, while the regular part approximates the equations in the whole domain, which can be solved by the standard finite element formulation. We carefully design numerical methods to interpolate the velocity to the moving interface. Numerical tests are carried out to demonstrate the accuracy and other properties of our method.

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A numerical method for solving elasticity equations with interface involving multi-domains and triple junction points

Cited by 6 publications

References 17 publications

Shape optimization with immersed interface finite element method

Shape optimization with immersed interface finite element method

A bilinear partially penalized immersed finite element method for elliptic interface problems with multi-domain and triple-junction points

A weak formulation for the multiphase Stokes flow problem without body fitting grids

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