A Numerical Analysis of the Weak Galerkin Method for the Helmholtz Equation with High Wave Number

2017 ◽  
Vol 22 (1) ◽  
pp. 133-156 ◽  
Author(s):  
Yu Du ◽  
Zhimin Zhang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Error Estimates ◽  
Phase Error ◽  
Three Dimensions ◽  
Wave Number ◽  
Weak Galerkin ◽  
High Wave Number ◽  
Large Wave ◽  
Element Method

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.

Lower Bound of Vibration Modes Using the Node-Based Smoothed Finite Element Method (NS-FEM)

2017 ◽  
Vol 14 (04) ◽  
pp. 1750036 ◽  
Author(s):  
Guirong Liu ◽  
Meng Chen ◽  
Ming Li
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Lower Bound ◽  
Lower Bounds ◽  
Solid Mechanics ◽  
Wave Number ◽  
Vibration Modes ◽  
Large Wave ◽  
Smoothed Finite Element Method ◽  
Element Method

The smoothed finite element method (S-FEM) has been recently developed as an effective solver for solid mechanics problems. This paper represents an effective approach to compute the lower bounds of vibration modes or eigenvalues of elasto-dynamic problems, by making use of the important softening effects of node-based S-FEM (NS-FEM). We first use NS-FEM, FEM and the analytic approach to compute the eigenvalues of transverse free vibration in strings and membranes. It is found that eigenvalues by NS-FEM are always smaller than those by FEM and the analytic method. However, NS-FEM produces spurious unphysical modes because of overly soft behavior. A technique is then proposed to remove them by analyzing their vibration shapes (eigenvectors). It is observed that spurious modes with excessively large wave numbers, which are unrelated to the physical deflection shapes but related to the discretization density, therefore can be easily removed. The final results of NS-FEM become the lower bound of eigenvalues and the accuracy can be improved via mesh refinement. And NS-FEM solutions (softer) are more reliable, because the large wave number component can be used as an indicator, which is available in FEM (stiffer), on the quality of the numerical solutions. The proposed NS-FEM procedure offers a viable and practical computational means to effectively compute the lower bounds of eigenvalues for solid mechanics problems.

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

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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