Immersed Finite Element Method Research Articles

Long‐time stability and asymptotic analysis of the IFE method for the multilayer porous wall model

In this article, we study the long‐time stability and asymptotic behavior of the immersed finite element (IFE) method for the multilayer porous wall model for the drug‐eluting stents. First, with the IFE method for the spatial descretization, and the implicit Euler scheme for the temporal discretization, respectively, we deduce the global stability of fully discrete solution. Then, we investigate the asymptotic behavior of the discrete scheme which reveals that the multilayer porous wall model converges to the corresponding elliptic equation if approaches to a steady‐state in both and norms as . Finally, some numerical experiments are given to verify the theoretical predictions.

An improved immersed finite element particle-in-cell method for plasma simulation

The particle-in-cell (PIC) method has been widely used for plasma simulation, because of its noise-reduction capability and moderate computational cost. The immersed finite element (IFE) method is efficient for solving interface problems on Cartesian meshes, which is desirable for the PIC method. The combination of these two methods provides an effective tool for plasma simulation with complex interface/boundary. This paper introduces an improved IFE–PIC method that enhances the performance in both IFE and PIC aspects. For the electric field solver, we adopt the newly developed partially penalized IFE method with enhanced accuracy. For PIC implementation, we introduce a new interpolation technique to ensure the conservation of the charge. Numerical examples are provided to demonstrate the features of the improved IFE–PIC method.

An IMPES scheme for a two-phase flow in heterogeneous porous media using a structured grid

We develop a numerical scheme for a two-phase immiscible flow in heterogeneous porous media using a structured grid finite element method, which has been successfully used for the computation of various physical applications involving elliptic interface equations (Li et al., 2003, 2004; Chou et al., 2010; Kwak et al., 2010; Chang and Kwak, 2011). The proposed method is based on the implicit pressure-explicit saturation procedure. To solve the pressure equation, we use an IFEM based on the Rannacher–Turek Rannacher and Turek (1992) nonconforming space, which is a modification of the work in Kwak et al. (2010) where ‘broken’ P1 nonconforming element of Crouzeix–Raviart (1973) [1] was developed.For the Darcy velocity, we apply the mixed finite volume method studied in Chou et al. (2003) and Kwak et al. (2010) on the basis of immersed finite element method (IFEM). In this way, the Darcy velocity of the flow can be computed cheaply (locally) after we solve the pressure equation. The computed Darcy velocity is used to solve the saturation equation explicitly. The whole procedure can be implemented on a structured grid which is independent of the underlying heterogeneous porous media. In fact, we apply a new version of multigrid algorithm to solve the pressure equation for which the CPU time grows like O(N). Numerical tests for analytic problems show that our method is almost optimal in all variables. Problem with sources/sinks is also computed and it seems to work well.

An Immersed Finite Element Method for the Elasticity Problems with Displacement Jump

AbstractIn this paper, we propose a finite element method for the elasticity problems which have displacement discontinuity along the material interface using uniform grids. We modify the immersed finite element method introduced recently for the computation of interface problems having homogeneous jumps [20, 22]. Since the interface is allowed to cut through the element, we modify the standard Crouzeix-Raviart basis functions so that along the interface, the normal stress is continuous and the jump of the displacement vector is proportional to the normal stress. We construct the broken piecewise linear basis functions which are uniquely determined by these conditions. The unknowns are only associated with the edges of element, except the intersection points. Thus our scheme has fewer degrees of freedom than most of the XFEM type of methods in the existing literature [1,8,13]. Finally, we present numerical results which show optimal orders of convergence rates.

A Cartesian grid nonconforming immersed finite element method for planar elasticity interface problems

In this paper, a new nonconforming immersed finite element (IFE) method on triangular Cartesian meshes is developed for solving planar elasticity interface problems. The proposed IFE method possesses optimal approximation property for both compressible and nearly incompressible problems. Its degree of freedom is much less than those of existing finite element methods for the same problem. Moreover, the method is robust with respect to the shape of the interface and its location relative to the domain and the underlying mesh. Both theory and numerical experiments are presented to demonstrate the effectiveness of the new method. Theoretically, the unisolvent property and the consistency of the IFE space are proved. Experimentally, extensive numerical examples are given to show that the approximation orders in L2 norm and semi-H1 norm are optimal under various Lamé parameters settings and different interface geometry configurations.

Immersed finite element method for eigenvalue problem

We consider the approximation of elliptic eigenvalue problem with an interface. The main aim of this paper is to prove the stability and convergence of an immersed finite element method (IFEM) for eigenvalues using Crouzeix–Raviart P1-nonconforming approximation. We show that spectral analysis for the classical eigenvalue problem can be easily applied to our model problem. We analyze the IFEM for elliptic eigenvalue problems with an interface and derive the optimal convergence of eigenvalues. Numerical experiments demonstrate our theoretical results.

A 3D immersed finite element method with non-homogeneous interface flux jump for applications in particle-in-cell simulations of plasma–lunar surface interactions

Motivated by the need to handle complex boundary conditions efficiently and accurately in particle-in-cell (PIC) simulations, this paper presents a three-dimensional (3D) linear immersed finite element (IFE) method with non-homogeneous flux jump conditions for solving electrostatic field involving complex boundary conditions using structured meshes independent of the interface. This method treats an object boundary as part of the simulation domain and solves the electric field at the boundary as an interface problem. In order to resolve charging on a dielectric surface, a new 3D linear IFE basis function is designed for each interface element to capture the electric field jump on the interface. Numerical experiments are provided to demonstrate the optimal convergence rates in L2 and H1 norms of the IFE solution. This new IFE method is integrated into a PIC method for simulations involving charging of a complex dielectric surface in a plasma. A numerical study of plasma–surface interactions at the lunar terminator is presented to demonstrate the applicability of the new method.

Immersed finite element methods for unbounded interface problems with periodic structures

Interface problems arise in many physical and engineering simulations involving multiple materials. Periodic structures often appear in simulations with large or even unbounded domain, such as magnetostatic/electrostatic field simulations. Immersed finite element (IFE) methods are efficient tools to solve interface problems on a Cartesian mesh, which is desirable to many applications like particle-in-cell simulation of plasma physics. In this article, we develop an IFE method for an interface problem with periodic structure on an infinite domain. To cope with the periodic boundary condition, we modify the stiffness matrix of the IFE method. The new matrix is maintained symmetric positive definite, so that the linear system can be solved efficiently. Numerical examples are provided to demonstrate features of this method.

A Consistent Immersed Finite Element Method for the Interface Elasticity Problems

We propose a new scheme for elasticity problems having discontinuity in the coefficients. In the previous work (Kwak et al., 2014), the authors suggested a method for solving such problems by finite element method using nonfitted grids. The proposed method is based on theP1-nonconforming finite element methods with stabilizing terms. In this work, we modify the method by adding the consistency terms, so that the estimates of consistency terms are not necessary. We show optimal error estimates inH1and divergence norms under minimal assumptions. Various numerical experiments also show optimal rates of convergence.

A MODIFIED $P_1$ -- IMMERSED FINITE ELEMENT METHOD

In recent years, the immersed finite element methods (IFEM) introduced in [?], [?] to solve elliptic problems having an interface in the domain due to the discontinuity of coefficients are getting more attentions of researchers because of their simplicity and efficiency. Unlike the conventional finite element methods, the IFEM allows the interface cut through the interior of the element, yet after the basis functions are altered so that they satisfy the flux jump conditions, it seems to show a reasonable order of convergence. In this paper, we propose an improved version of the P1 based IFEM by adding the line integral of flux terms on each element. This technique resembles the discontinuous Galerkin (DG) method, however, our method has much less degrees of freedom than the DG methods since we use the same number of unknowns as the conventional P1 finite element method. We prove H and L error estimates which are optimal both in order and regularity. Numerical experiments were carried out for several examples, which show the robustness of our scheme. Department of Mathematical Sciences, KAIST,, 291 Daehak-ro, Yuseong-gu, Daejeon, Korea 305-701. E-mail address: kdy@kaist.ac.kr Department of Mathematical Sciences, KAIST,, 291 Daehak-ro, Yuseong-gu, Daejeon, Korea 305-701. E-mail address: cool295@kaist.ac.kr 2000 Mathematics Subject Classification. primary 65N30, secondary 74S05, 76S05 .

On the immersed boundary method: Finite element versus finite volume approach

Two immersed boundary methods (IBM) for the simulation of fluid flow problems with complex geometries are introduced: a direct-forcing immersed finite element method (IFEM) and a direct-forcing immersed finite volume method (IFVM).In the IFEM a projection approach is presented for the coupled system of time-dependent incompressible Navier–Stokes equations (NSEs) in conjunction with the IBM for solving fluid flow problems in the presence of rigid objects not represented by the underlying mesh. The IBM allows solving the flow for geometries with complex objects without the need of generating a body-fitted mesh. Dirichlet boundary constraints are satisfied applying a boundary force at the immersed body surface. Using projection and interpolation operators from the fluid volume mesh to the solid surface/volume mesh (i.e., the “immersed” boundary) and vice versa, it is possible to impose the extra constraint to the NSEs as a Lagrange multiplier in a fashion very similar to the effect that pressure has on the momentum equations to satisfy the divergence-free constraint. The IFEM approach presented shows third order accuracy in space and second order accuracy in time when the simulation results for the Taylor–Green decaying vortex are compared to the analytical solution.For the IFVM presented, a ghost-cell approach with sharp interface scheme is used to enforce the boundary condition at the fluid/solid interface. The interpolation procedure at the immersed boundary preserves the overall second order accuracy of the base solver. The developed ghost-cell method is applied on a staggered configuration with the Semi-Implicit Method for Pressure-Linked Equations Revised algorithm. Second order accuracy in space and first order accuracy in time are obtained when the Taylor–Green decaying vortex test case computed with the IFVM is compared to the analytical solution.Computations were performed using the IFEM and IFVM approaches for the two-dimensional flow over a backward-facing step, two-dimensional flow past a stationary circular cylinder, three-dimensional flow past a sphere and two-dimensional oscillating circular cylinder in a stationary square domain. The numerical results obtained with the discussed IFEM and IFVM were compared against other IBMs available in literature and simulations performed with the commercial computational fluid dynamics code STAR-CCM+/V7.04.006. The benchmark test cases showed that the numerical results obtained with the implemented immersed boundary methods are in good agreement with the predictions from STAR-CCM+ and the numerical data from the other IBMs. The immersed boundary method based on finite element approach is numerically more accurate than the IBM based on finite volume discretization. In contrast, the latter is computationally more efficient than the former.

Immersed finite elements for optimal control problems of elliptic PDEs with interfaces

This paper presents a numerical method and analysis, based on the variational discretization concept, for optimal control problems governed by elliptic PDEs with interfaces. The method uses a simple uniform mesh which is independent of the interface. Due to the jump of the coefficient across the interface, the standard linear finite element method cannot achieve optimal convergence when the uniform mesh is used. Therefore the immersed finite element method (IFEM) developed in Li et al. [20] is used to discretize the state equation required in the variational discretization approach. Optimal error estimates for the control, state and adjoint state are derived. Numerical examples are provided to confirm the theoretical results.

A new augmented immersed finite element method without using SVD interpolations

Augmented immersed interface methods have been developed recently for interface problems and problems on irregular domains including CFD applications with free boundaries and moving interfaces. In an augmented method, one or several augmented variables are introduced along the interface or boundary so that one can get efficient discretizations. The augmented variables should be chosen such that the interface or boundary conditions are satisfied. The key to the success of the augmented methods often relies on the interpolation scheme to couple the augmented variables with the governing differential equations through the interface or boundary conditions. This has been done using a least squares interpolation (under-determined) for which the singular value decomposition (SVD) is used to solve for the interpolation coefficients. In this paper, based on properties of the finite element method, a new augmented immersed finite element method (IFEM) that does not need the interpolations is proposed for elliptic interface problems that have a piecewise constant coefficient. Thus the new augmented method is more efficient and simple than the old one that uses interpolations. The method then is extended to Poisson equations on irregular domains with a Dirichlet boundary condition. Numerical experiments with arbitrary interfaces/irregular domains and large jump ratios are provided to demonstrate the accuracy and the efficiency of the new augmented methods. Numerical results also show that the number of GMRES iterations is independent of the mesh size and nearly independent of the jump in the coefficient.

Partially Penalized Immersed Finite Element Methods For Elliptic Interface Problems

This article presents new immersed finite element (IFE) methods for solving the popular second order elliptic interface problems on structured Cartesian meshes even if the involved interfaces have nontrivial geometries. These IFE methods contain extra stabilization terms introduced only at interface edges for penalizing the discontinuity in IFE functions. With the enhanced stability due to the added penalty, not only these IFE methods can be proven to have the optimal convergence rate in the H1-norm provided that the exact solution has sufficient regularity, but also numerical results indicate that their convergence rates in both the H1-norm and the L2-norm do not deteriorate when the mesh becomes finer which is a shortcoming of the classic IFE methods in some situations. Trace inequalities are established for both linear and bilinear IFE functions that are not only critical for the error analysis of these new IFE methods, but also are of a great potential to be useful in error analysis for other IFE methods.

An immersed interface method for Pennes bioheat transfer equation

We consider an immersed finite element method for solving one dimensional Pennes bioheat transfer equation with discontinuous coefficients and nonhomogenous flux jump condition. Convergence properties of the semidiscrete and fully discrete schemes are investigated in the $L^{2}$ and energy norms. By using the computed solution from the immerse finite element method, an inexpensive and effective flux recovery technique is employed to approximate flux over the whole domain. Optimal order convergence is proved for the immersed finite element approximation and its flux. Results of the simulation confirm the convergence analysis.

Numerical modeling of gas–liquid–solid interactions: Gas–liquid free surfaces interacting with deformable solids

A numerical framework to model and simulate gas–liquid–solid three-phase flows is proposed and implemented in this work. The overall algorithm adopts a non-boundary-fitted approach that avoids frequent mesh-updating procedures by defining independent meshes and explicit interfacial points to represent each phase. Here, we couple our existing solvers, the immersed finite element method (IFEM) and the connectivity-free front tracking (CFFT) method that model fluid–solid and gas–liquid interactions, respectively, for the three-phase models. A modified IFEM algorithm is derived for modeling fluid–solid interactions that accounts for the dynamics of the solid, while the CFFT is used here to simulate gas–liquid multi-fluid flows that uses explicit interfacial points to represent the gas–liquid interface and for its easy handling of interface topology changes. Instead of defining different levels simultaneously as used in level sets, an indicator function naturally couples the two methods together to represent and track each of the three phases. Several 2-D and 3-D testing cases are performed to show the robustness and capability of the coupled numerical framework in dealing with complex three-phase problems, in particular free surfaces interacting with deformable solids.

A NURBS-based immersed methodology for fluid–structure interaction

We introduce an isogeometric, immersed, and fully-implicit formulation for fluid–structure interaction (FSI). The method focuses on viscous incompressible flows and nonlinear hyperelastic incompressible solids, which are a common case in various fields, such as, for example, biomechanics. In our FSI method, we utilize an Eulerian mesh on the whole domain and a Lagrangian mesh on the solid domain. The Lagrangian mesh is arbitrarily located on top of the Eulerian mesh in a non-conforming fashion. Due to the formulation of our problem, based on the Immersed Finite Element Method (IFEM), we do not need mesh update or remeshing algorithms. The fluid–structure interface is the boundary of the Lagrangian mesh, but cuts arbitrarily the Eulerian mesh. The generalized- α method is used for time discretization and NURBS-based isogeometric analysis is employed for the spatial discretization on both fluid and solid domains. The information transfer between the two meshes is carried out using the NURBS functions, which avoids the use of the so-called discretized delta functions. The higher order and especially the higher continuity of NURBS functions allow us to deal with severe mesh distortion in the Lagrangian mesh in comparison with classical C0 linear piecewise functions as we prove numerically. Our numerical solutions attain good agreement with theoretical data for free-falling objects in two and three dimensions, which confirms the feasibility of our methodology.

Immersed Finite Element Method for Interface Problems with Algebraic Multigrid Solver

AbstractThis article is to discuss the bilinear and linear immersed finite element (IFE) solutions generated from the algebraic multigrid solver for both stationary and moving interface problems. For the numerical methods based on finite difference formulation and a structured mesh independent of the interface, the stiffness matrix of the linear system is usually not symmetric positive-definite, which demands extra efforts to design efficient multigrid methods. On the other hand, the stiffness matrix arising from the IFE methods are naturally symmetric positive-definite. Hence the IFE-AMG algorithm is proposed to solve the linear systems of the bilinear and linear IFE methods for both stationary and moving interface problems. The numerical examples demonstrate the features of the proposed algorithms, including the optimal convergence in both L2and semi-H1norms of the IFE-AMG solutions, the high efficiency with proper choice of the components and parameters of AMG, the influence of the tolerance and the smoother type of AMG on the convergence of the IFE solutions for the interface problems, and the relationship between the cost and the moving interface location.

A Symmetric and Consistent Immersed Finite Element Method for Interface Problems

The non-conforming immersed finite element method (IFEM) developed in Li et al. (Numer Math 96:61---98, 2003) for interface problems is extensively studied in this paper. The non-conforming IFEM is very much like the standard finite element method but with modified basis functions that enforce the natural jump conditions on interface elements. While the non-conforming IFEM is simple and has reasonable accuracy, it is not fully second order accurate due to the discontinuities of the modified basis functions. While the conforming IFEM also developed in Li et al. (Numer Math 96:61---98, 2003) is fully second order accurate, the implementation is more complicated. A new symmetric and consistent IFEM has been developed in this paper. The new method maintains the advantages of the non-conforming IFEM by using the same basis functions but it is symmetric, consistent, and more important, it is second order accurate. The idea is to add some correction terms to the weak form to take into account of the discontinuities in the basis functions. Optimal error estimates are derived for the new symmetric and consistent IFE method in the $$L^2$$ L 2 and $$H^1$$ H 1 norms. Numerical examples presented in this paper confirm the theoretical analysis and show that the new developed IFE method has $$O(h^2)$$ O ( h 2 ) convergence in the $$L^\infty $$ L ? norm as well.

Modeling of Soft Tissues Interacting with Fluid (Blood or Air) Using the Immersed Finite Element Method.

This paper presents some biomedical applications that involve fluid-structure interactions which are simulated using the Immersed Finite Element Method (IFEM). Here, we first review the original and enhanced IFEM methods that are suitable to model incompressible or compressible fluid that can have densities that are significantly lower than the solid, such as air. Then, three biomedical applications are studied using the IFEM. Each of the applications may require a specific set of IFEM formulation for its respective numerical stability and accuracy due to the disparities between the fluid and the solid. We show that these biomedical applications require a fully-coupled and stable numerical technique in order to produce meaningful results.