A finite element method for first order elliptic systems in three dimensions

AbstractWe study the error analysis of the weak Galerkin finite element method in [24, 38] (WG-FEM) for the Helmholtz problem with large wave number in two and three dimensions. Using a modified duality argument proposed by Zhu and Wu, we obtain the pre-asymptotic error estimates of the WG-FEM. In particular, the error estimates with explicit dependence on the wave numberkare derived. This shows that the pollution error in the brokenH1-norm is bounded byunder mesh conditionk7/2h2≤C0or (kh)2+k(kh)p+1≤C0, which coincides with the phase error of the finite element method obtained by existent dispersion analyses. Herehis the mesh size,pis the order of the approximation space andC0is a constant independent ofkandh. Furthermore, numerical tests are provided to verify the theoretical findings and to illustrate the great capability of the WG-FEM in reducing the pollution effect.

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Modelling of triaxial fracture processes in heterogeneous quasi-brittle materials using the Embedded Finite Element Method (E-FEM)

10.31224/osf.io/ecv73 ◽

2021 ◽

Author(s):

Alejandro Ortega Laborin ◽

Yann MALECOT ◽

Emmanuel ROUBIN ◽

Laurent DAUDEVILLE

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Spherical Inclusion ◽

Brittle Materials ◽

Three Dimensions ◽

Strong Discontinuity ◽

Homogeneous Material ◽

Fem Model ◽

Fracture Processes ◽

Element Method

This paper studies the use of the Embedded Finite Element Method (E-FEM) for the numerical modelling of triaxial fracture processes in non-homogeneous quasi-brittle materials. The E-FEM framework used in this study combines two kinematics enhancements: a weak discontinuity allowing the model to account for material heterogeneities and a strong discontinuity allowing the model to represent local fractures. The strong discontinuity features enriched fracture kinematics that allow the modelling of all typical fracture modes in three dimensions. A brief review is done of past work using similar enriched finite element frameworks to approach this problem. The work continues by establishing the theoretical basis of each kind of discontinuity formulation and their superposition through the Hu-Washizu variational principle. Afterwards, two groups of simulations have been done for discussing the performance of this combined E-FEM model: homogeneous simulations and simple heterogeneous simulations. Simple homogeneous material simulations aim to test the capabilities of the strong discontinuity model featuring full 3-D kinematics. Simple heterogeneous simulations show numerical applications of the model to the problem of a single spherical inclusion embedded into a homogeneous matrix. Comparisons will be made with another E-FEM model considering a single local fracture mode approach to discuss the differences on the representation of fracture physics under all explored conditions. A concluding statement is made on the benefits and complications identified for the E-FEM framework in this kind of applications.

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A MSFEM to simulate the eddy current problem in laminated iron cores in 3D

COMPEL The International Journal for Computation and Mathematics in Electrical and Electronic Engineering ◽

10.1108/compel-12-2018-0538 ◽

2019 ◽

Vol 38 (5) ◽

pp. 1667-1682 ◽

Cited By ~ 3

Author(s):

Karl Hollaus

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Eddy Current ◽

Eddy Currents ◽

Three Dimensional ◽

Harmonic Balance Method ◽

Three Dimensions ◽

Magnetic Vector Potential ◽

Content Type ◽

Element Method

Purpose The simulation of eddy currents in laminated iron cores by the finite element method (FEM) is of great interest in the design of electrical devices. Modeling each laminate by finite elements leads to extremely large nonlinear systems of equations impossible to solve with present computer resources reasonably. The purpose of this study is to show that the multiscale finite element method (MSFEM) overcomes this difficulty. Design/methodology/approach A new MSFEM approach for eddy currents of laminated nonlinear iron cores in three dimensions based on the magnetic vector potential is presented. How to construct the MSFEM approach in principal is shown. The MSFEM with the Biot–Savart field in the frequency domain, a higher-order approach, the time stepping method and with the harmonic balance method are introduced and studied. Findings Various simulations demonstrate the feasibility, efficiency and versatility of the new MSFEM. Originality/value The novel MSFEM solves true three-dimensional eddy current problems in laminated iron cores taking into account of the edge effect.

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