3D hierarchical interface-enriched finite element method: Implementation and applications

Soghrati, Soheil; Ahmadian, Hossein

doi:10.1016/j.jcp.2015.06.035

Cited by 22 publications

(8 citation statements)

References 28 publications

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“…This way, for each e i , we use the map between the NURBS parametric space and the physical space. Subsequently, we use the span-wise mapping scheme described in [26] to compute the integrals appearing in (27) and (31). In this approach, the NURBS volumes constructed in an intersected element are directly used to perform the numerical integration of the weak form.…”

Section: Finite Element Formulationmentioning

confidence: 99%

“…In this method, the additional degrees of freedom (dofs) are introduced along the intersections of the material interfaces with the edges of the non-conforming mesh and linear Lagrangian basis is used as enrichment. This method has recently been extended to NURBS-based Interface-enriched Generalized Finite Method (NIGFEM) [26,27], which incorporates directly into the finite element formulation the NURBS representation of the surfaces defining the material interfaces. Non-Uniform Rational B-Splines (NURBS) [28,29] are used widely in CAD to represent complex geometries.…”

Section: Introductionmentioning

confidence: 99%

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A NURBS-based generalized finite element scheme for 3D simulation of heterogeneous materials

Safdari

Najafi

Sottos

et al. 2016

Journal of Computational Physics

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Section: Finite Element Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A NURBS-based generalized finite element scheme for 3D simulation of heterogeneous materials

Safdari

Najafi

Sottos

et al. 2016

Journal of Computational Physics

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“…Partition of unity methods (PUM), such as the generalized and extended FEMs (GFEM and XFEM, respectively) enhance the numerical solution by enriching the basis with specially constructed functions to approximate the weakly discontinuous solutions . Recent developments in XFEM and GFEM for material interface problems include the use of adaptive mesh refinement in conjunction with the level‐set method, and the interface‐enriched GFEM . The local enrichment approach was combined with the FCM as presented by Joulaian and Düster for 2‐dimensional problems .…”

Section: Introductionmentioning

confidence: 99%

“…19,20 Recent developments in XFEM and GFEM for material interface problems include the use of adaptive mesh refinement in conjunction with the level-set method, 21,22 and the interface-enriched GFEM. [23][24][25][26][27] The local enrichment approach was combined with the FCM as presented by Joulaian and Düster for 2-dimensional problems. 15 This combination leads to a significant improvement in the convergence rates of the FCM.…”

Section: Introductionmentioning

confidence: 99%

Multi‐level hp‐finite cell method for embedded interface problems with application in biomechanics

Elhaddad

Zander

Bog

et al. 2018

Numer Methods Biomed Eng

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This work presents a numerical discretization technique for solving 3-dimensional material interface problems involving complex geometry without conforming mesh generation. The finite cell method (FCM), which is a high-order fictitious domain approach, is used for the numerical approximation of the solution without a boundary-conforming mesh. Weak discontinuities at material interfaces are resolved by using separate FCM meshes for each material sub-domain and weakly enforcing the interface conditions between the different meshes. Additionally, a recently developed hierarchical hp-refinement scheme is used to locally refine the FCM meshes to resolve singularities and local solution features at the interfaces. Thereby, higher convergence rates are achievable for nonsmooth problems. A series of numerical experiments with 2- and 3-dimensional benchmark problems is presented, showing that the proposed hp-refinement scheme in conjunction with the weak enforcement of the interface conditions leads to a significant improvement of the convergence rates, even in the presence of singularities. Finally, the proposed technique is applied to simulate a vertebra-implant model. The application showcases the method's potential as an accurate simulation tool for biomechanical problems involving complex geometry, and it demonstrates its flexibility in dealing with different types of geometric description.

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“…This limitation becomes particularly challenging for simulating moving boundary problems (e.g., localized corrosion [2] and phase transition [3]), where the evolving geometry of the domain may require the adaptive refinement [4,5] or regeneration of the FE mesh throughout the solution process. Advanced mesh-independent FEMs such as the extended/generalized FEM [6,7] and the hierarchical interface-enriched FEM (HIFEM) [8,9] alleviate this limitation by adding appropriate enrichments to field/gradient discontinuous regions of the solution, which allows the use of FE meshes that are independent of the problem morphology. The combination of the generalized/extended FEM and the level set method has successfully been implemented for simulating moving boundary problems [10,11].…”

mentioning

confidence: 99%

A Green's discrete transformation meshfree method for simulating transient diffusion problems

Mai

Soghrati

Buchheit

2016

Numerical Meth Engineering

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SUMMARYThis manuscript presents the formulation and application of the Green's discrete transformation method (GDTM) for the meshfree simulation of transient diffusion problems, including those with moving boundaries. The GDTM implements a linear combination of time-dependent Green's basis functions defined on a set of source points to approximate the field in the form of a solution series. A discrete transformation is implemented to evaluate unknown coefficients of this series, which eliminates the need to use time integration schemes. We will study the optimal number and location of the GDTM source points that yield the highest level of accuracy, while maintaining a manageable condition number for the resulting linear system of equations. The optimal values of these parameters, which are inherently independent of the domain geometry, are determined such that the basis functions have appropriate features for approximating the field. A comprehensive convergence study is presented to show the precision and convergence rate of the GDTM for modeling various diffusion problems. We also demonstrate the application of this method for simulating three diffusion problems with complex and evolving morphologies: heat transfer in a turbine blade, thermal response of a porous material, and localized (pitting) corrosion in stainless steel.

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3D hierarchical interface-enriched finite element method: Implementation and applications

Cited by 22 publications

References 28 publications

A NURBS-based generalized finite element scheme for 3D simulation of heterogeneous materials

A NURBS-based generalized finite element scheme for 3D simulation of heterogeneous materials

Multi‐level hp‐finite cell method for embedded interface problems with application in biomechanics

A Green's discrete transformation meshfree method for simulating transient diffusion problems

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