Immersed Finite Element Method Research Articles

Interface Problems-Fluid Structure Interaction: Description, Application and Review

This paper presents a critical review of numerical methods for solving a wide variety of interface problems emphasizing the immersed finite element method (IFEM). It is found in the literature that most of the researchers considered the well-known methods with some modifications, however limited number of research articles proposed new algorithms. Apart from the algorithm, this study highlights the wide range of applications of interface problems specifically in biomedical, heat-transfer and turbo-machinery. Different numerical methods for interface problems with their major finding are listed in tabulated form at the end.

Analysis of nonconforming IFE methods and a new scheme for elliptic interface problems

In this paper, an important discovery has been found for nonconforming immersed finite element (IFE) methods using the integral values on edges as degrees of freedom for solving elliptic interface problems. We show that those IFE methods without penalties are not guaranteed to converge optimally if the tangential derivative of the exact solution and the jump of the coefficient are not zero on the interface. A nontrivial counter example is also provided to support our theoretical analysis. To recover the optimal convergence rates, we develop a new nonconforming IFE method with additional terms locally on interface edges. The new method is parameter-free which removes the limitation of the conventional partially penalized IFE method. We show the IFE basis functions are unisolvent on arbitrary triangles which is not considered in the literature. Furthermore, different from multipoint Taylor expansions, we derive the optimal approximation capabilities of both the Crouzeix–Raviart and the rotated-Q1 IFE spaces via a unified approach which can handle the case of variable coefficients easily. Finally, optimal error estimates in both H1- and L2-norms are proved and confirmed with numerical experiments.

A combined immersed finite element and conservative semi-Lagrangian scheme for plasma-material interactions

In this paper, the immersed finite element (IFE) coupled with the conservative semi-Lagrangian (CSL) kinetic scheme is developed for plasma-material interactions. The proposed method (IFE-CSL) enjoys respective advantages of the IFE and CSL, i.e., the flexibility and efficiency for treating complex boundary conditions, mass conservation and being free from the Courant-Friedrichs-Lewy (CFL) condition. In the current IFE-CSL, the IFE method based on structured interface-independent meshes is developed for the spatial discretization of the field equation, which provides an accurate approach with convenient implementations to solve the field problems with irregular interface. The CSL scheme combined with an immersed method termed half-way ghost-cell is proposed for the spatial discretization of the Vlasov equation, which enables the proposed method to exactly conserve mass and conveniently handle curved boundaries. Then the IFE and CSL method are coupled via the charge density depending on full kinetic or hybrid kinetic model, where the appropriate IFE solver is developed for the linear or nonlinear Poisson equation, respectively. In the current IFE-CSL, different geometries can be treated automatically for both IFE and CSL through the specific geometric information. As a result, the proposed CSL-IFE can be conveniently and efficiently applied to simulate the plasma-material interactions with structured interface-independent meshes. Finally, several numerical experiments are performed to demonstrate the capabilities of the proposed method.

A New Immersed Finite Element Method for Two-Phase Stokes Problems Having Discontinuous Pressure

Abstract In this paper, we develop a new immersed finite element method (IFEM) for two-phase incompressible Stokes flows. We allow the interface to cut the finite elements. On the noninterface element, the standard Crouzeix–Raviart element and the P 0 {P_{0}} element pair is used. On the interface element, the basis functions developed for scalar interface problems (Kwak et al., An analysis of a broken P 1 {P_{1}} -nonconforming finite element method for interface problems, SIAM J. Numer. Anal. (2010)) are modified in such a way that the coupling between the velocity and pressure variable is different. There are two kinds of basis functions. The first kind of basis satisfies the Laplace–Young condition under the assumption of the continuity of the pressure variable. In the second kind, the velocity is of bubble type and is coupled with the discontinuous pressure, still satisfying the Laplace–Young condition. We remark that in the second kind the pressure variable has two degrees of freedom on each interface element. Therefore, our methods can handle the discontinuous pressure case. Numerical results including the case of the discontinuous pressure variable are provided. We see optimal convergence orders for all examples.

An immersed Crouzeix–Raviart finite element method in 2D and 3D based on discrete level set functions

This paper is devoted to the construction and analysis of immersed finite element (IFE) methods in three dimensions. Different from the 2D case, the points of intersection of the interface and the edges of a tetrahedron are usually not coplanar, which makes the extension of the original 2D IFE methods based on a piecewise linear approximation of the interface to the 3D case not straightforward. We address this coplanarity issue by an approach where the interface is approximated via discrete level set functions. This approach is very convenient from a computational point of view since in many practical applications the exact interface is often unknown, and only a discrete level set function is available. As this approach has also not be considered in the 2D IFE methods, in this paper we present a unified framework for both 2D and 3D cases. We consider an IFE method based on the traditional Crouzeix–Raviart element using integral values on faces as degrees of freedom. The novelty of the proposed IFE is the unisolvence of basis functions on arbitrary triangles/tetrahedrons without any angle restrictions even for anisotropic interface problems, which is advantageous over the IFE using nodal values as degrees of freedom. The optimal bounds for the IFE interpolation errors are proved on shape-regular triangulations. For the IFE method, optimal a priori error and condition number estimates are derived with constants independent of the location of the interface with respect to the unfitted mesh. The extension to anisotropic interface problems with tensor coefficients is also discussed. Numerical examples supporting the theoretical results are provided.

An enriched immersed finite element method for interface problems with nonhomogeneous jump conditions

This article presents the first higher degree immersed finite element (IFE) method with proven optimal convergence for elliptic interface problems with nonhomogeneous jump conditions. It also gives the first analysis for the condition numbers of the resulting systems including the optimal upper bounds with respect to the mesh size and its robustness with respect to small-cut interface elements. In this method, jump conditions are approximated optimally by basic IFE and enrichment IFE which are piecewise pth degree polynomial functions constructed by solving local Cauchy problems on interface elements. The proposed IFE method is based on a discontinuous Galerkin formulation on interface elements and a continuous Galerkin formulation on non-interface elements.

Error analysis of Petrov-Galerkin immersed finite element methods

This paper designs and analyzes a new and stable Petrov–Galerkin (PG) immersed finite element method (IFEM) for the second-order elliptic interface problems by introducing stabilization terms based on the classical PG-IFEM, which lacks the local positivity. The Petrov–Galerkin immersed finite element method uses the immersed finite element functions for the trial space and the standard finite element functions for the test space. Both the a priori and a posteriori error estimates of the method are analyzed in this paper. We prove the continuity and inf-sup condition and the a priori error estimate of the energy norm. The proposed a posteriori error estimator is proved to be both reliable and efficient, with both reliability and efficiency constants independent of the location of the interface. Extensive numerical results confirm the numerical scheme’s optimal convergence and indicate the robustness with respect to the interface-mesh intersection and the coefficient contrast, despite the robustness of the inf-sup constant with respect to the interface-mesh intersection has yet been theoretically proved.

Numerical study of opposed zero-net-mass-flow jet-induced erythrocyte mechanoporation

With the advantages of biosafety and efficiency, increasing attention has been paid to the devices for gene and macromolecular drug delivery based on mechanoporation. The transient pore formation on the cell membrane allows cargo transportation when the membrane areal strain is beyond the critical pore value and below the lysis tension threshold. Based on this principle, we propose a method to apply the proper fluid stress on cells moving in a microchannel under the action of zero-net-mass-flux (ZNMF) jets. In this study, an immersed finite element method (IFEM) is adopted to simulate the interaction between the cells and the fluid fields so as to investigate the cell movement and deformation in this mechanoporation system. To evaluate the efficiency of the cargo delivery, a pore integral is defined as the mean pore rate when the cell passes through the jet region. By analyzing the effects of the parameters, including the pressure gradient along the microchannel, the jet amplitude, and the jet frequency, on the pore integrals, a group of optimized parameters for cargo delivery efficiency are obtained. Additionally, the stability and safety of this system are analyzed in detail. These results are helpful in designing the mechanoporation devices and improving their efficiency of drug delivery.

Immersed Virtual Element Methods for Elliptic Interface Problems in Two Dimensions

This article presents an immersed virtual element method for solving a class of interface problems that combines the advantages of both body-fitted mesh methods and unfitted mesh methods. A background body-fitted mesh is generated initially. On those interface elements, virtual element spaces are constructed as solution spaces to local interface problems, and exact sequences can be established for these new spaces involving discontinuous coefficients. The discontinuous coefficients of interface problems are recast as Hodge star operators that are the key to project immersed virtual functions to classic immersed finite element (IFE) functions for computing numerical solutions. An a priori convergence analysis is established robust with respect to the interface location. The proposed method is capable of handling more complicated interface element configuration and provides better performance than the conventional penalty-type IFE method for the \(\mathbf{H}({\text {curl}})\)-interface problem arising from Maxwell equations. It also brings a connection between various methods such as body-fitted methods, IFE methods, virtual element methods, etc.

An immersed [formula omitted]-[formula omitted] element for Stokes interface problems and the optimal convergence analysis

This paper presents and analyzes an immersed finite element (IFE) method for solving Stokes interface problems with a piecewise constant viscosity coefficient that has a jump across the interface. In the method, the triangulation does not need to fit the interface and the IFE spaces are constructed from the traditional CR−P0 element with modifications near the interface according to the interface jump conditions. We prove that the IFE basis functions are unisolvent on arbitrary triangles without any angle conditions and the IFE spaces have the optimal approximation capabilities, although the proof is challenging due to the coupling of the velocity and the pressure. The stability and the optimal error estimates of the proposed IFE method are also derived rigorously. The constants in the error estimates are shown to be independent of the interface location relative to the triangulation. Numerical examples are provided to verify the theoretical results.

A Deep Neural Network Based on ResNet for Predicting Solutions of Poisson–Boltzmann Equation

The Poisson–Boltzmann equation (PBE) arises in various disciplines including biophysics, electrochemistry, and colloid chemistry, leading to the need for efficient and accurate simulations of PBE. However, most of the finite difference/element methods developed so far are rather complicated to implement. In this study, we develop a ResNet-based artificial neural network (ANN) to predict solutions of PBE. Our networks are robust with respect to the locations of charges and shapes of solvent–solute interfaces. To generate train and test sets, we have solved PBE using immersed finite element method (IFEM) proposed in (Kwon, I.; Kwak, D. Y. Discontinuous bubble immersed finite element method for Poisson–Boltzmann equation. Communications in Computational Physics 2019, 25, pp. 928–946). Once the proposed ANNs are trained, one can predict solutions of PBE in almost real time by a simple substitution of information of charges/interfaces into the networks. Thus, our algorithms can be used effectively in various biomolecular simulations including ion-channeling simulations and calculations of diffusion-controlled enzyme reaction rate. The performance of the ANN is reported in the result section. The comparison between IFEM-generated solutions and network-generated solutions shows that root mean squared error are below 5·10−7. Additionally, blow-ups of electrostatic potentials near the singular charge region and abrupt decreases near the interfaces are represented in a reasonable way.

A Family of Two-Grid Partially Penalized Immersed Finite Element Methods for Semi-linear Parabolic Interface Problems

A Family of Two-Grid Partially Penalized Immersed Finite Element Methods for Semi-linear Parabolic Interface Problems

Locally Conservative Immersed Finite Element Method for Elliptic Interface Problems

In this paper, we introduce a locally conservative enriched immersed finite element method (EIFEM) to tackle the elliptic problem with interface. The immersed finite element is useful for handling interface with mesh unfit with the interface. However, all the currently available method under IFEM framework may not be designed to consider the conservative flux conservation. We provide an efficient and effective remedy for this issue by introducing a local piecewise constant enrichment, which provides the locally conservative flux. We have also constructed and analyzed an auxiliary space preconditioner for the resulting system based on the application of algebraic multigrid method. The new observation in this work is that by imposing strong Dirichlet boundary condition for the standard IFEM part of EIFEM, we are able to remove the zero eigen-mode of the EIFEM system while still imposing the Dirichlet boundary condition weakly assigned to the piecewise constant enrichment part of EIFEM. A couple of issues relevant to the piecewise constant enrichment given for the mesh unfit to the interface has been discussed and clarified as well. Numerical tests are provided to confirm the theoretical development.

Computational modelling of membrane gating in capsule translocation through microchannel with variable section

As an effective physical method, cell squeezing technology based on microfluidics plays an increasingly promising role in intracellular delivery. To deepen our understanding of microfluidic chip design and optimization, it is essential to explore the underlying physics required in the generation of the appropriate tension for gating capsule membrane. In this investigation, an immersed finite element method (IFEM) has been adopted to simulate the interaction between capsule and fluid fields in microchannel with variable section. Having obtained the numerical results, the gating region on the membrane can be determined by the non-uniform tension distribution based on the critical gating membrane tension during the capsule translocates the microchannel. In addition, the gating integral, which is defined to measure the degree of gating, shows that the setting of driven pressure might be crucial for the design and optimization of the system. The numerical results demonstrate that the occurrence of secondary peak in deformed energy after the capsule passes through the confined microchannel under certain flow condition might be ascribed to the elastic recovery of the membrane. To investigate the effects of initial orientations, the translocation of an ellipsoid capsule through the channel has also been simulated numerically, which indicates that both flow shear force and compressive force due to the constrained solid wall have significant effects on membrane gating. Therefore, to improve the gating efficiency of capsule membrane, it is necessary to optimize various factors to achieve the balance among the compressive force, shear force and the translocation time.

A robust multigrid method for one dimensional immersed finite element method

AbstractIn this paper, we propose a robust multigrid method for 1D immersed finite element method (IFEM). It is shown that the multigrid method is optimal, which means that the convergence rate of the multigrid method is not only independent of the mesh size h and mesh level L, but also independent of the jump of the discontinuous coefficients. Although we only consider 1D interface method, to the best of our knowledge, this is the first attempt to give a rigorous theoretical analysis for the multigrid method for the IFEM. On the way to this goal, we also revisit the IFEM for the 1D interface problem and prove that the error estimates with respect to the L2 norm and weighted H1 semi‐norm are optimal and independent of the jump of the discontinuous coefficients. Numerical results are given to verify our theoretical findings.

A fully decoupled iterative method with three-dimensional anisotropic immersed finite elements for Kaufman-type discharge problems

In order to simulate the Kaufman-type discharge problems, a fully decoupled iterative method with anisotropic immersed finite elements on Cartesian meshes is proposed, especially for a three-dimensional (3D) non-axisymmetric anisotropic hybrid model which is more difficult than the axisymmetric or isotropic models. The classical hybrid model, which describes the important plasma distribution of the Kaufman-type discharge problems, couples several difficult equations together to form a large scale system. The 3D non-axisymmetric and anisotropic properties will further increase the complexity of this system. Hence it generally needs to be solved in the decoupled way for significantly reducing the computational cost. Based on the Particle-in-Cell Monte Carlo collision (PIC–MCC) method and the immersed finite element (IFE) method, we propose a fully decoupled iterative method for solving this complex system. The IFE method allows Cartesian meshes for general interface problems, while the traditional finite element methods require body-fitting meshes which are often unstructured. Compared with traditional finite element methods, this feature significantly improves the efficiency of the proposed 3D fully decoupled iterative method, while maintaining the optimal accuracy of the chosen finite elements. Numerical simulations of traditional Kaufman ion thruster and annular ion thruster discharge chambers are provided and compared with the corresponding lab experiment results to illustrate the features of the proposed method.

Error Estimates for an Immersed Finite Element Method for Second Order Hyperbolic Equations in Inhomogeneous Media

A group of partially penalized immersed finite element (PPIFE) methods for second-order hyperbolic interface problems were discussed in Yang (Numer Math Theor Methods Appl 11:272–298, 2018) where the author proved their optimal O(h) convergence in an energy norm under a sub-optimal piecewise $$H^3$$ regularity assumption. In this article, we reanalyze the fully discrete PPIFE method presented in Yang (2018). Utilizing the error bounds given recently in Guo et al. (Int J Numer Anal Model 16(4):575–589, 2019) for elliptic interface problems, we are able to derive optimal a-priori error bounds for this PPIFE method not only in the energy norm but also in $$L^2$$ norm under the standard piecewise $$H^2$$ regularity assumption in the space variable of the exact solution, rather than the excessive piecewise $$H^3$$ regularity. Numerical simulations for standing and travelling waves are presented, which corroboratively confirm the reported error analysis.

Long time error estimates of IFE methods for the unsteady multi-layer porous wall model

In this paper, we study the immersed finite element (IFE) method for the multi-layer porous wall model for the drug-eluting stents. Firstly, under some realistic assumptions on the initial data and the body force, we deduce the long time priori stability for the variational problem for all t∈R+. Then, based on these results, the uniform optimal error estimates for both the semi-discrete IFE method and the fully discrete Euler IFE method are proved. Finally, some numerical experiments are performed to verify the theoretical predictions.

An immersed finite element method for elliptic interface problems in three dimensions

This article presents an immersed finite element (IFE) method for solving the typical three-dimensional second order elliptic interface problem with an interface-independent Cartesian mesh. The local IFE space on each interface element consists of piecewise trilinear polynomials constructed by extending polynomials from one subelement to the whole element according to the jump conditions of the interface problem. In this space, the IFE shape functions with the Lagrange degrees of freedom can always be constructed regardless of interface location and discontinuous coefficients. The proposed IFE space is proven to have the optimal approximation capabilities to the functions satisfying the jump conditions. A group of numerical examples with representative interface geometries are presented to demonstrate features of the proposed IFE method.

A bilinear partially penalized immersed finite element method for elliptic interface problems with multi-domain and triple-junction points

In this article, we introduce a new partially penalized immersed finite element method (IFEM) for solving elliptic interface problems with multi-domain and triple-junction points. We construct new IFE functions on elements intersected with multiple interfaces or with triple-junction points to accommodate interface jump conditions. For non-homogeneous flux jump, we enrich the local approximating spaces by adding up to three local flux basis functions. Numerical experiments are carried out to show that both the Lagrange interpolations and the partial penalized IFEM solutions converge optimally in L2 and H1 norms.