Immersed Interface Finite Element Methods for Elasticity Interface Problems with Non-Homogeneous Jump Conditions

In this paper,a class of new immersed interface finite element methods (IIFEM) is developed to solve elasticity interface problems with homogeneous and non-homogeneous jump conditions in two dimensions.Simple non-body-fitted meshes are used.For homogeneous jump conditions,both non-conforming and conforming basis functions are constructed in such a way that they satisfy the natural jump conditions. For non-homogeneous jump conditions,a pair of functions that satisfy the same non-homogeneous jump conditions are constructed using a level-set representation of the interface.With such a pair of functions,the discontinuities across the interface in the solution and flux are removed;and an equivalent elasticity interface problem with homogeneous jump conditions is formulated.Numerical examples are presented to demonstrate that such methods have second order convergence.

Nonlinear finite element analysis of three implant-abutment interface designs

The objective of this study was to investigate the mechanical characteristics of implant-abutment interface design in a dental implant system, using nonlinear finite element analysis （FEA） method. This finite element simulation study was applied on three commonly used commercial dental implant systems： model I, the reduced-diameter 3i implant system （West Palm Beach, FL, USA） with a hex and a 12-point double internal hexagonal connection; model II, the Semados implant system （Bego, Bremen, Germany） with combination of a conical （45° taper） and internal hexagonal connection; and model III, the Br,~nemark implant system （Nobel Biocare, Gothenburg, Sweden） with external hexagonal connection. In simulation, a force of 170 N with 45°oblique to the longitudinal axis of the implant was loaded to the top surface of the abutment. It has been found from the strength and stiffness analysis that the 3i implant system has the lowest maximum yon Mises stress, prirlcipal stress and displacement, while the Br^nemark implant system has the highest. It was concluded from our preliminary study using nonlinear FEA that the reduced-diameter 3i implant system with a hex and a 12-point double internal hexagonal connection had a better stress distribution, and produced a smaller displacement than the other two implant systems.

LOCAL AND PARALLEL FINITE ELEMENT METHOD FOR THE MIXED NAVIER-STOKES/DARCY MODEL WITH BEAVERS-JOSEPH INTERFACE CONDITIONS

In this paper, we consider the mixed Navier-Stokes/Darcy model with BeaversJoseph interface conditions. Based on two-grid discretizations, a local and parallel finite element algorithm for this mixed model is proposed and analyzed. Optimal errors are obtained and numerical experiments are presented to show the efficiency and effectiveness of the local and parallel finite element algorithm.

P_1-nonconforming triangular finite element method for elliptic and parabolic interface problems

The lowest order Pl-nonconforming triangular finite element method （FEM） for elliptic and parabolic interface problems is investigated. Under some reasonable regularity assumptions on the exact solutions, the optimal order error estimates are obtained in the broken energy norm. Finally, some numerical results are provided to verify the theoretical analysis.

Finite Element Analysis of Shrinkage in the Interface of Functionally Graded Concrete Segment Used in Shield Tunneling

In functionally graded materials (FGM), the problem of interface stability caused by the volume deformation is commonly regarded as the key factor for its performance. Based on test results, in terms of finite element method (FEM) this paper analyzed problems in the shrinkage of functionally graded material interface of shield concrete segment, which was designed and produced by the principle of functionally graded materials. In the analysis model, the total shrinkage of concrete was converted into the thermal shrinkage by means of the method of 'Equivalent Temperature Difference'. Consequently, the shrinkage stress of interface layer was calculated and compared with the bond strength of interface layer. The results indicated that the volume deformation of two-phase materials of functionally graded concrete (FGC) segment, which were the concrete cover and the concrete structure layer, showed better compatibility and the tension stress of interface layer, which was resulted from the shrinkage of concrete and calculated by ANSYS, was less than the bond strength of interface layer. Therefore, the interface stability of functionally graded concrete segment was good and the sliding deformation of interface layer would not generate.

Two-grid methods for semi-linear elliptic interface problems by immersed finite element methods

In this paper,two-grid immersed finite element (IFE) algorithms are proposed and analyzed for semi-linear interface problems with discontinuous diffusion coefficients in two dimension.Because of the advantages of finite element (FE) formulation and the simple structure of Cartesian grids,the IFE discretization is used in this paper.Two-grid schemes are formulated to linearize the FE equations.It is theoretically and numerically illustrated that the coarse space can be selected as coarse as H =O(h^1/4)(or H =O(h^1/8)),and the asymptotically optimal approximation can be achieved as the nonlinear schemes.As a result,we can settle a great majority of nonlinear equations as easy as linearized problems.In order to estimate the present two-grid algorithms,we derive the optimal error estimates of the IFE solution in the L^p norm.Numerical experiments are given to verify the theorems and indicate that the present two-grid algorithms can greatly improve the computing efficiency.

Distributed Lagrange Multiplier/Fictitious Domain Finite Element Method for a Transient Stokes Interface Problem with Jump Coefficients

The distributed Lagrange multiplier/fictitious domain(DLM/FD)-mixed finite element method is developed and analyzed in this paper for a transient Stokes interface problem with jump coefficients.The semi-and fully discrete DLM/FD-mixed finite element scheme are developed for the first time for this problem with a moving interface,where the arbitrary Lagrangian-Eulerian(ALE)technique is employed to deal with the moving and immersed subdomain.Stability and optimal convergence properties are obtained for both schemes.Numerical experiments are carried out for different scenarios of jump coefficients,and all theoretical results are validated.

Solving Interface Problems of the Helmholtz Equation by Immersed Finite Element Methods

This article reports our explorations for solving interface problems of the Helmholtz equation by immersed finite elements (IFE) on interface independent meshes. Two IFE methods are investigated: the partially penalized IFE (PPIFE) and discontinuous Galerkin IFE (DGIFE) methods. Optimal convergence rates are observed for these IFE methods once the mesh size is smaller than the optimal mesh size which is mainly dictated by the wave number. Numerical experiments also suggest that higher degree IFE methods are advantageous because of their larger optimal mesh size and higher convergence rates.

Mechanical Response of the Composite Material—Concrete Interface in FRP-Strengthened Concrete Elements:Finite Element Simulation

In structural elements strengthened with Fiber Reinforced Polymer(FRP),debonding failure modes should be taken into consideration.Under specific circumstances,they may provoke a global,premature failure of the structural element.In other cases,they should be accounted for in the modeling in order to obtain more accurate results.Despite the large amount of research work carried out in this field in the last few decades,debonding failure modes are still not fully understood.This contribution is focused on a numerical procedure designed to model the progressive loss of bond action between FRP and concrete.The two-stage procedure is integrated into incremental,finite element analysis.The proposed algorithm uses experimentally obtained slip-stress relationship.Predefined failure criteria are used to predict the local bond failure.In the reported case study,an experimental set-up widely employed to investigate debonding is modeled.Results obtained by finite element analysis are discussed.

Thermo-kinetic characteristics on stabilizing hetero-phase interface of metal matrix composites by crystal plasticity finite element method

Using dislocation-based constitutive modeling in three-dimension crystal plasticity finite element(3D CPFE)simulations,co-deformation and instability of hetero-phase interface in different material systems were herein studied for polycrystalline metal matrix composites(MMCs).Local stress and strain fields in two types of 3layer MMCs such as fcc/fcc Cu-Ag and fcc/bcc Cu-Nb have been predicted under simple compressive deformations.Accordingly,more severe strain-induced interface instability can be observed in the fcc/bcc systems than in the fcc/fcc systems upon refining to metallic nanolayered composites(MNCs).By detailed analysis of stress and strain localization,it has been demonstrated that the interface instability is always accompanied by high-stress concentration,i.e.,thermodynamic characteristics,or high strain prevention i.e.,kinetic characteristics,at the hetero-phase interface.It then follows that the thermodynamic driving forceG and the kinetic energy barrier Q during dislocation and shear banding can be adopted to classify the deformation modes,following the so-called thermo-kinetic correlation.Then by inserting a high density of high-energy interfaces into the Cu-Nb composites,such thermo-kinetic integration at the hetero-phase interface allows a successful establishment of MMCs with the high△G-high Q deformation mode,which ensures high hardening and uniform strain distri-bution,thus efficiently suppressing the shear band,stabilizing the hetero-phase interface,and obtaining an exceptional combination in strength and ductility.Such hetero-phase interface chosen by a couple of thermodynamics and kinetics can be defined as breaking the thermo-kinetic correlation and has been proposed for artificially designing MNCs.

Formulations of a hydromechanical interface element

A hydromechanical interface element is proposed for the consideration of the hydraulic-mechanical coupling effect along the interface.The fully coupled governing equations and the relevant finite element formulations are derived in detail for the interface element.All the involved matrices are of the same form as those of a solid element,which makes the incorporation of the model into a finite element program straightforward.Three examples are then numerically simulated using the interface element.Reasonable results confirm the correctness of the proposed model and motivate its application in hydromechanical contact problems in the future.

A Stable FE-FD Method for Anisotropic Parabolic PDEs with Moving Interfaces

In this paper,a new finite element and finite difference(FE-FD)method has been developed for anisotropic parabolic interface problems with a known moving interface using Cartesian meshes.In the spatial discretization,the standard P,FE discretization is applied so that the part of the coefficient matrix is symmetric positive definite,while near the interface,the maximum principle preserving immersed interface discretization is applied.In the time discretization,a modified Crank-Nicolson discretization is employed so that the hybrid FE-FD is stable and second order accurate.Correction terms are needed when the interface crosses grid lines.The moving interface is represented by the zero level set of a Lipschitz continuous function.Numerical experiments presented in this paper confirm second orderconvergence.

THE ADAPTIVE IMMERSED INTERFACE FINITE ELEMENT METHOD FOR ELASTICITY INTERFACE PROBLEMS

In this paper, we propose adaptive finite element methods with error control for solving elasticity problems with discontinuous coefficients. The meshes in the methods do not need to fit the interfaces. We establish a residual-based a posteriori error estimate which is λ- independent multiplicative constants; the Lame constant λ steers the incompressibility. The error estimators are then implemented and tested with promising numerical results which will show the competitive behavior of the adaptive algorithm.

A Well-Conditioned, Nonconforming Nitsche’s Extended Finite Element Method for Elliptic Interface Problems

In this paper,we introduce a nonconforming Nitsche’s extended finite element method(NXFEM)for elliptic interface problems on unfitted triangulation elements.The solution on each side of the interface is separately expanded in the standard nonconforming piecewise linear polynomials with the edge averages as degrees of freedom.The jump conditions on the interface and the discontinuities on the cut edges(the segment of edges cut by the interface)are weakly enforced by the Nitsche’s approach.In the method,the harmonic weighted fluxes are used and the extra stabilization terms on the interface edges and cut edges are added to guarantee the stability and the well conditioning.We prove that the convergence order of the errors in energy and L 2 norms are optimal.Moreover,the errors are independent of the position of the interface relative to the mesh and the ratio of the discontinuous coefficients.Furthermore,we prove that the condition number of the system matrix is independent of the interface position.Numerical examples are given to confirm the theoretical results.

A Method of Lines Based on Immersed Finite Elements for Parabolic Moving Interface Problems

This article extends the finite element method of lines to a parabolic initial boundary value problem whose diffusion coefficient is discontinuous across an interface that changes with respect to time.The method presented here uses immersed finite element(IFE)functions for the discretization in spatial variables that can be carried out over a fixedmesh(such as a Cartesianmesh if desired),and this featuremakes it possible to reduce the parabolic equation to a system of ordinary differential equations(ODE)through the usual semi-discretization procedure.Therefore,with a suitable choice of the ODE solver,this method can reliably and efficiently solve a parabolic moving interface problem over a fixed structured(Cartesian)mesh.Numerical examples are presented to demonstrate features of this new method.

Optimization Analysis of Sliding Electrical Contact Interface Based on Electromechanical Coupling

This work addresses the critical issue of current density distribution in the sliding electrical contact interface based on electromechanical coupling, which is essential for minimizing damage and enhancing performance. Using electromechanical coupling analysis and finite element analysis (FEA), the effects of initial contact pressure, pulse current input, and armature speed on current density are examined. Key findings indicate that optimizing the convex rail and armature structures significantly reduces peak current density, improving uniformity and reducing damage. These optimizations enhance the efficiency, accuracy, and service life of sliding electrical contact interfaces, providing a theoretical foundation for designing more durable and efficient high-current-density applications.

ANALYSIS OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR PARABOLIC INTERFACE PROBLEMS WITH NONSMOOTH INITIAL DATA

This article concerns numerical approximation of a parabolic interface problem with general L 2 initial value.The problem is discretized by a finite element method with a quasi-uniform triangulation of the domain fitting the interface,with piecewise linear approximation to the interface.The semi-discrete finite element problem is furthermore discretized in time by the k-step backward difference formula with k=1,...,6.To maintain high-order convergence in time for possibly nonsmooth L 2 initial value,we modify the standard backward difference formula at the first k−1 time levels by using a method recently developed for fractional evolution equations.An error bound of O(t−k nτk+t−1 n h 2|log h|)is established for the fully discrete finite element method for general L 2 initial data.

Retrieving Topological Information of Implicitly Represented Diffuse Interfaces with Adaptive Finite Element Discretization

We consider the finite element based computation of topological quantities of implicitly represented surfaces within a diffuse interface framework.Utilizing an adaptive finite element implementation with effective gradient recovery techniques,we discuss how the Euler number can be accurately computed directly from the numerically solved phase field functions or order parameters.Numerical examples and applications to the topological analysis of point clouds are also presented.

Time-domain Finite Element Method for Transient Electric Field and Transient Charge Density on Dielectric Interface

This paper is devoted to solving the transient electric field and transient charge density on the dielectric interface under the electroquasistatic(EQS)field conditions with high accuracy.The proposed method is suitable for both 2-D and 3-D applications.Firstly,the governing equations represented by scalar electric potential are discretized by the nodal finite element method(FEM)in space and the finite difference method in time.Secondly,the transient constrained electric field equation on the boundary(TCEFEB)is derived to calculate the normal component of the transient electric field intensities on the Dirichlet boundary and dielectric interface as well as the transient charge density on the dielectric interface.Finally,a 2-D numerical example is employed to demonstrate the validity of the proposed method.Furthermore,the comparisons of the numerical accuracy of the proposed method in this paper with the existing FEMs for electric field intensity and charge density on the dielectric interface are conducted.The results show that the numerical accuracy of the proposed method for calculating the normal component of transient electric field intensities on the Dirichlet boundary and dielectric interface as well as the transient charge density on the dielectric interface is close to that of nodal electric potential and an order of magnitude higher than those of existing FEMs.

OPTIMAL QUADRATIC NITSCHE EXTENDED FINITE ELEMENT METHOD FOR INTERFACE PROBLEM OF DIFFUSION EQUATION

In this paper, we study Nitsche extended finite element method （XFEM） for the inter- face problem of a two dimensional diffusion equation. Specifically, we study the quadratic XFEM scheme on some shape-regular family of grids and prove the optimal convergence rate of the scheme with respect to the mesh size. Main efforts are devoted onto classifying the cases of intersection between the elements and the interface and prove a weighted trace inequality for the extended finite element functions needed, and the general framework of analysing XFEM c^n be implemented then.