The Finite Element Method with High-Order Enrichment Functions for Elastodynamic Analysis

In this work, the functionally graded magneto-electro-elastic (FG-MEE) structures were used to study mechanical performance under a thermal environment using the thermo-mechanical-electro-magnetic enriched finite element method (TMEM-EFEM). Through enhancing the traditional solution space of the finite element method (FEM), additional degrees of freedom are introduced without modifying the mesh structure. Therefore, as a class of higher-order finite element methods, field variables with high gradients can be captured without a relatively finer mesh. The accuracy of TMEM-EFEM is improved by using interpolation covers. The feasibility of TMEM-EFEM is proven by numerical examples. The proposed TMEM-EFEM is proven to be accurate under the relatively coarse mesh. Through numerical examples, the high accuracy of TMEM-EFEM is demonstrated. Therefore, TMEM-EFEM can be a good alternative for the statics study of FG-MEE structures.

An enriched finite element method with interpolation cover functions for acoustic analysis in high frequencies

Magneto-electro-elastic (MEE) composites have promising application prospects due to the rapid response to the excitation of external electric or magnetic fields. Finite element simulation of the multi-physics coupling problems in MEE-based structures is significant. The enriched finite element method (EFEM) is derived from the standard finite element method (FEM), with finite element shape functions improved by interpolation covers. Considering the complex mechanical-electro-magnetic multi-physics coupling mechanism of MEE-based structures, the mechanical-electro-magnetic coupling enriched finite element method (MEM-EFEM) was proposed. The method is well worked for investigating the mechanical-electro-magnetic multi-physics coupling issues to calculate the generalized displacement (displacement components, electric and magnetic potential) with 3-node triangular elements. In a set of numerical experiments involving the static analysis of MEE-based structures, MEM-EFEM outperforms the regular FEM. When it comes to the same solution accuracy level, the number of elements used by MEM-EFEM is lower than that of the standard FEM, suggesting that MEM-EFEM has better convergence and accuracy on this subject. Besides, the MEM-EFEM solution accuracy can be enhanced by mesh refinement. Overall, the MEM-EFEM has the potential to be used in engineering practice to solve the static responses of MEE-based structures on mechanical-electro-magnetic multi-physics coupling issues.

Enriched finite element methods have gained traction in recent years for modeling problems with material interfaces and cracks. By means of enrichment functions that incorporate a priori behavior about the solution, these methods decouple the finite element (FE) discretization from the geometric configuration of such discontinuities. Taking advantage of this greater flexibility, recent studies have proposed the adoption of Non-Uniform Rational B-Splines (NURBS) to preserve the interfaces' exact geometries throughout the analysis. In this article, we investigate NURBS-based geometries in the context of the Discontinuity-Enriched Finite Element Method (DE-FEM) based on linear field approximations. While optimal convergence is retained for problems with weak discontinuities without singularities, representing exact geometry via NURBS does not yield noticeable improvements when extracting stress intensity factors of cracked specimens. For low-order elements, we conclude that the benefits of exact geometry representation do not outweigh the increased complexity in formulation and implementation. The choice of linear FEs hinders the accuracy of the proposed formulation, suggesting that its full potential may only be unleashed by increasing the field representation order.

ABSTRACTThis paper proposes an enriched equilibrium finite element method (EFEM) for addressing crack problems within the framework of linear elastic fracture mechanics and complementary energy principle. The key ingredient of enriched EFEM is to construct the equilibrated stress field using equilibrated tractions along with asymptotic enrichment. Specifically, an auxiliary field is employed to capture the asymptotic singular behavior of the near‐tip stress field, while the traction‐based equilibrium finite element accounts for the residual terms. The remarkable property of the proposed enriched EFEM is that the stress intensity factors (SIFs) and the stress fields are directly obtained with high accuracy, even on a very coarse mesh. Moreover, in combination with the compatible solution acquired by the enriched finite element method (FEM), strict estimation of the discretization error is obtained for crack problems by dual analysis. Numerical examples are finally conducted to verify the performance of the proposed method on strict error estimation and accurate computation of the SIFs.

This paper presents a novel numerical method for simulating the transport and dispersion of pollutants in the Mediterranean sea. The governing mathematical equations consist of a barotropic ocean model with friction terms, bathymetric forces, Coriolis and wind stresses coupled to an advection–diffusion equation with anisotropic dispersion tensor and source terms. The proposed numerical solver uses a multilevel adaptive semi-Lagrangian finite element method that combines various techniques, including the modified method of characteristics, finite element discretization, coupled projection scheme based on a rotational pressure correction algorithm, and an adaptive L2-projection. The approach employs the gradient of the concentration as an error indicator for enrichment adaptations and increasing the number of quadrature points where needed without refining the mesh. The method is shown to provide accurate and efficient simulations for pollution transport in the Mediterranean sea. The proposed approach distinguishes itself from the well-established adaptive finite element methods for incompressible viscous flows by retaining the same structure and dimension of linear systems during the adaptation process.

Extended finite element modeling of deformable porous media with arbitrary interfaces

The Orthonormalized Generalized Finite Element Method–OGFEM: Efficient and stable reduction of approximation errors through multiple orthonormalized enriched basis functions

An enriched finite element method is presented to numerically solve the eigenvalue problem on electromagnetic waveguides governed by the Helmholtz equation. In this work, a highly efficient, simple and precise higher order subparametric method was developed using a 2D automated mesh generator performed with JuliaFEM. The transcendence computerized discretization code in Julia is developed for the present work. For curved waveguide structures, meshes with one side and curving higher orders are proposed with triangular elements with parabolic arcs. The technique is shown for distinct waveguide constructions, and the results are compared with the strongest numerical or analytical results available. The results demonstrate that the proposed methodology is effective and accurate for generating finite element simulations for complex structures with black holes and irregular topology due to no curvature loss. This article presents a finding cutoff frequency performed with JuliaFEM—an open-source program. Analysis results produced by commercial software are considered for the comparison and show that the calculation results between the two programs do not differ significantly. This procedure can be used to achieve the most effective transmission of energy for electromagnetic applications.

International audience

In this paper, a new enrichment scheme is proposed to model fractures and other conduits in porous media flow problems. Inserting this scheme into a partition of unity based method results in a new numerical method that does not require the mesh to honor the specific geometry of these subsurface features. The new scheme involves a specially designed integration procedure and enrichment functions, which can capture effects of local heterogeneity introduced by subsurface features on the pressure solution. The new method is also capable of modeling fractures with low as well as high conductivity. Another feature of the proposed scheme is that, even though two enrichment functions are used to model the permeability change at the two rock/fracture interfaces of a fracture, only one element partition is made for numerical integration. To demonstrate the accuracy and effectiveness of the proposed approach, production problems for wells that were stimulated or completed by longitudinal fracture, transverse fractures, and perforations are studied.

Generalized or extended finite element method (G/XFEM) models the crack by enriching functions of partition of unity type with discontinuous functions that represent well the physical behavior of the problem. However, this enrichment functions are not available for all problem types. Thus, one can use numerically-built (global-local) enrichment functions to have a better approximate procedure. This paper investigates the effects of micro-defects/inhomogeneities on a main crack behavior by modeling the micro-defects/inhomogeneities in the local problem using a two-scale G/XFEM. The global-local enrichment functions are influenced by the micro-defects/inhomogeneities from the local problem and thus change the approximate solution of the global problem with the main crack. This approach is presented in detail by solving three different linear elastic fracture mechanics problems for different cases: two plane stress and a Reissner–Mindlin plate problems. The numerical results obtained with the two-scale G/XFEM are compared with the reference solutions from the analytical, numerical solution using standard G/XFEM method and ABAQUS as well, and from the literature.

A new numerical approach for fluid-structure interaction of viscous fluid flow and flexible structures of negligible thickness (e.g. membranes, plates) on a topologically fixed fluid discretization is presented. The linear elastic and geometrically nonlinear structure is embedded [1] into the flow field described by the incompressible Navier-Stokes equations using locally enriched space-time (EST) finite elements [2], [3] to consider resulting strong and weak discontinuities in the fluid field appropriately [4]. Since the formulation of fluid, structure and coupling conditions uniformly uses velocities as unknowns and the integration of governing equations is perfomed on the deformed space-time mesh, the simulation of the strongly coupled physical domains becomes very comfortable and results in a monolithic system [5]. Numerical examples of fluid-stucture interaction processes show the ability of the developed numerical method to describe those situations of coupled systems for that common mesh moving strategies are not useful applicable and would require the introduction of time-consuming remeshing algorithms.

When modeling crack growth in composite materials, one encounters strong (cracks) and weak (material interface) discontinuities. Traditional finite element methods require the mesh to conform to the material interface and crack surface. In addition, costly remeshing is required as crack growth occurs. In this paper, the extended finite element method (XFEM) coupled with the level set method is presented to model both strong and weak discontinuities. For the material interface, an element-based enrichment function is introduced, which shows more consistent behavior than the node-based enrichment function. The XFEM crack implementation and element-based bi-material implementation are validated using analytical solutions.

Construction of discontinuous enrichment functions for enriched or generalized FEM’s for interface elliptic problems in 1D