Numerical methods such as the finite difference method, finitevolume method, and finite element method were originally defined on meshes of data points. On computational issues of immersed finite element methods. The extended finite element method for rigid particles in. Immersed finite element methods for parabolic equations with moving interface xiaoming he,1 tao lin,2 yanping lin,3,4 xu zhang2 1department of mathematics and statistics, missouri university of science and technology, rolla, missouri 65409 2department of mathematics, virginia tech, blacksburg, virginia 24061.
Computational mechanics, volume 65, issue 3 springer. Finite elements in computational electromagnetism 241. Recent journal of computational physics articles elsevier. Department of mathematics, and interdisciplinary program in computational. Immersedinterface finiteelement methods 475 triangular immersedinterface. Thanks to dimensional or topological model reduction techniques, small inclusions in a 3d continuum can be described as onedimensional 1d concentrated sources in order to reduce the computational cost of simulations.
Citescore values are based on citation counts in a given year e. Summary this paper focuses on the computation of statistical moments of strains and stresses in a random system model where uncertainty is modeled by a stochastic finite element. These methods, which typically use an eulerian description of the fluid and a lagrangian description of the structure, can treat thin immersed boundaries and volumetric bodies, and they can model structures that are flexible or rigid or. Apr 21, 2012 a threedimensional immersed smoothed finite element method 3d isfem using fournode tetrahedral element is proposed to solve 3d fluidstructure interaction fsi problems. The recently introduced immersed finite element ife methods 79 employ an. By continuing to use our website, you are agreeing to our use of cookies. Element method ifem, for the solution of fluidstructure interaction problems. A coupled sharp interface immersedboundaryfiniteelement. Crossref qing yang and xu zhang, discontinuous galerkin immersed finite element methods for parabolic interface problems, journal of computational and applied. Computational aeroacoustics issues and methods aiaa. In the initial attempt to employ finite element formulations with immersed methods, socalled ex tended immersed boundary method was.
The present study aimed to understand the effect of venous valve lesion on the valve cycle. A possible solution consists in enriching the finite element basis on the elements that are crossed by the immersed boundary, such that the irregularity of the solution is taken into account. Solution of low mach number aeroacoustic flows using a variational multiscale finite element formulation of the compressible navierstokes equations written in primitive variables. Nonconforming immersed finite element methods for interface. Their combined citations are counted only for the first article. In ifem, a lagrangian solid mesh moves on top of a background eulerian. Frontiers in applied mathematics includes bibliographical references and index. Immersed finite element method for fluidstructure interactions. An alternative approach has recently gained popularity where the geometry, created as a solid model in cad software, is embedded or immersed in a nonconforming background mesh for analysis. Multigrid solvers for immersed finite element methods and immersed. Communications in applied mathematics and computational. Special finite element methods for a class of second order. Interpolation functions in the immersed boundary and finite element methods.
The hybrid method is most useful when the curved geometry only occupies a small region of the computational domain. Numerical methods for fluidstructure interaction a. Modified immersed finite element method for fullycoupled fluidstructure interactions x wang, lt zhang computer methods in applied mechanics and engineering 267, 150169, 20. This paper presents an overview of the representative methods for the simulation of incompressible viscous flow with moving boundaries based on conforming or nonconforming meshes. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. Pdf a method of lines based on immersed finite elements for. Recently, following the pioneering work of peskin and coworkers on the ib method, an alternative approach, the immersed finite element method ifem was developed by zhang et al. A flowstructure interaction method has been developed which couples a sharpinterface immersed boundary method ibm with a finite element method based solid dynamics solver. Several modifications to the finite element method fem, such as the meshless methods, have been proposed for avoiding mesh generation. Superconvergence of immersed finite element methods for interface problems waixiang caoa, xu zhangb, zhimin zhanga,c abeijing computational science research center, beijing, 94, china bdepartment of mathematics and statistics, mississippi state university, mississippi state ms 39762.
Department of mechanical engineering and institute for computational and mathematical engineering, stanford university, mail code 3035, stanford, ca 94305, u. In this paper, we propose an immersed solid system iss method to efficiently treat the fluidstructure interaction fsi problems. Finite element approximation of the nonstationary navier. An analysis of the finite element method second edition. The combination of these two methods provides an effective tool for plasma. Erratum to on computational issues of immersed finite element methods j. May 17, 2012 solution of low mach number aeroacoustic flows using a variational multiscale finite element formulation of the compressible navierstokes equations written in primitive variables computer methods in applied mechanics and engineering, vol. Hybrid finite differencefinite element immersed boundary method boyce e. Finite element methods for modeling multiphase flow and. Immersed finite element method for interface problems with algebraic multigrid solver by wenqiang feng a thesis presented to the faculty of the graduate school of missouri university of science and technology in partial ful. Interpolation functions in the immersed boundary and.
One is the extended finite element method xfem and the other is the immersed finite element method ifem. A method of lines based on immersed finite elements for parabolic moving interface problems. We present a unified framework for developing and analyzing immersed finiteelement ife spaces for solving typical elliptic interface problems with. Qiao zhuang and ruchi guo, high degree discontinuous petrovgalerkin immersed finite element methods using fictitious elements for elliptic interface problems, journal of computational and applied mathematics, 10. This straightforward finite element interpolation function for unstructured grids enables us to obtain a sharper interface that yields more accurate interfacial solutions. Mixed finite element methods for coupled 3d1d fluid problems english abstract. Computational methods for complex liquidfluid interfaces. An immersed finite element method based on a locally.
An immersed smoothed finite element method for fluidstructure interaction problems article pdf available in international journal of computational methods 84. An introduction to finite element methods niko manopulo may 4, 2005 abstract the finite element methods fem are nowadays one of the most frequently used computational methods in solving scientific and engineering problems. Request pdf immersed finite element method for fluidstructure interactions in. Finite element methods are severely constrained in their ability to resolve interfaces. A group of immersed finiteelement spaces for elliptic interface problems, ima journal of numerical analysis, volume 39, issue 1, january 2019. Survey of immersed boundary approaches for finite element. Pdf an immersed smoothed finite element method for fluid. Therefore, it is suitable for moving interface problems. In the immersed continuum method, the background fluid mesh for the entire cavity which includes the space occupied by the immersed solids consists of 20. Finite element methods fem have been coupled to fd 31,32. In immersed methods, in order to satisfy energy conservation, namely, the energy input to the fluid domain from the immersed solid is the same as that from the equivalent body force, the same delta function must be used in both the distribution of the resultant nodal force and the interpolation of the solid velocity based on the surrounding fluid velocities.
Journal of computational and applied mathematics 235 2011 39533964. Griffith1 xiaoyu luo2 1departments of mathematics and biomedical engineering, carolina center for interdisciplinary applied mathematics, and mcallister heart institute, university of north carolina, chapel hill, nc, usa 2school of mathematics and statistics. An overview of numerical methods for incompressible. This method is able to eliminate the aforementioned drawbacks of the ib method by adopting parts of the work on the extended immersed boundary method eibm. The finite element method fem is the most widely used method for solving problems of engineering and mathematical models.
Erratum to on computational issues of immersed finite. Computer methods in applied mechanics and engineering 195 16, 17221749, 2006. Advances in computational mathematics, volume 43, issue 4. We further define an eulerlagrange mapping to describe the motion of the. This discrete lagrangianeulerian coupling approach is also adopted byiblike methods such as the immersed structural potential method 24 and the immersed. Immersed methods provide mathematical and computational frameworks for modeling fluidstructure systems.
Immersed methods for fluidstructure interaction annual. Hybrid finite differencefinite element immersed boundary method. Benjamin vaughan, bryan smith and david chopp, a comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources, communications in applied mathematics and computational science, 10. A consistent immersed finite element method for the interface. Superconvergence of immersed finite element methods for. On computational issues of immersed finite element methods article in journal of computational physics 2287. Immersed finite element methods for parabolic equations with. In ifem, a lagrangian solid mesh moves on top of a back. The computational domain 365 and parameters are illustrated in figure 18.
Pe281 finite element method course notes summarized by tara laforce stanford, ca 23rd may 2006 1 derivation of the method in order to derive the fundamental concepts of fem we will start by looking at an extremely simple ode and approximate it using fem. A third option is to combine unstructured and structured grids to form a hybrid method. Nov 23, 2011 qiao zhuang and ruchi guo, high degree discontinuous petrovgalerkin immersed finite element methods using fictitious elements for elliptic interface problems, journal of computational and applied mathematics, 10. Multigrid solvers for immersed finite element methods and immersed isogeometric analysis f. We consider problems on a two dimensional or threedimensional domain \omega \subset \mathbb rd. The importance of the fluidparticle interaction problem is considerable. The book covers a variety of cuttingedge experimental and computational techniques ranging from macro to meso and microscale approaches including pivotal applications. Modeling of soft tissues interacting with fluid blood or air. Modeling of soft tissues interacting with fluid blood or. In this paper, the immersed finite element method ifem is proposed for the solution of complex fluid and deformable structure interaction problems encountered in many physical models. In ifem, a lagrangian solid mesh moves on top of a background eulerian fluid mesh which spans over the entire computational domain.
A three dimensional immersed smoothed finite element method. In order to compare immersed methods with traditional mesh adaptive solution procedures, we start with a 1. However, our framework described here can be used for high order immersedinterface. The coupled method provides robust and highfidelity solution for a highly complex fsi system, such involving threedimensional flows and viscoelastic solids. Hence, pure displacementbased finite element methods are not reliable when. Superconvergence of immersed finite element methods for interface problems waixiang caoa, xu zhangb, zhimin zhanga,c abeijing computational science research center, beijing, 94, china bdepartment of mathematics and statistics, mississippi state university, mississippi state ms 39762 cdepartment of mathematics, wayne state university, detroit, mi 48202.
The immersed finite element ife method is efficient for solving interface problems on cartesian meshes, which is desirable for pic method. Partially penalized immersed finite element methods for. Article pdf available in advances in applied mathematics. Mathematical foundations of the immersed finite element. Zhang and wing kam liu 1 apr 2009 journal of computational physics, vol. Immersed finite element method for interface problems with. Immersed finite element methods for 4th order differential. Therefore, the immersed finite element ife methods, which were developed for solving interface problems on meshes independent of interface 1. Hybrid finite differencefinite element immersed boundary. Correction to the article a comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources by vaughan et al. Immersed and discontinuous finite element methods nabil chaabane abstract in this dissertation we prove the superconvergence of the minimaldissipation local discontinuous galerkin method for elliptic problems and construct optimal immersed nite element approximations and discontinuous immersed nite element methods for the stokes interface. The 3d isfem is able to determine accurately the physical deformation of the nonlinear solids placed within the incompressible viscous fluid governed by navierstokes equations. Computational methods for complex liquidfluid interfaces highlights key computational challenges involved in the twoway coupling of complex liquidfluid interfaces. This work was inspired by the pioneering work of professor t.
There are two classes of methods which belong to this type. Jul 17, 2006 2012 variational implementation of immersed finite element methods. Immersed finite element method and its applications to biological systems wk liu, y liu, d farrell, l zhang, xs wang, y fukui, n patankar, y zhang. The immersed finite element method ifem 1922 was developed to tackle this problem by representing the background viscous fluid with an unstructured finite element mesh and nonlinear finite elements for the immersed deformable solid. Tremendous attention has been paid by researchers in the past two decades. The convergence of the bilinear and linear immersed finite. A modified immersed finite element method was used to model the bloodtissue interactions in the pathological vein.
We present a unified framework for developing and analyzing immersed finite element ife spaces for solving typical elliptic interface problems with we use cookies to enhance your experience on our website. Many of these limitations stem from their inability in independently representing interface geometry from the underlying discretization. Typical problem areas of interest include structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. Many immersed boundary methods do not take into account explicitly the existence of the boundary and, as a consequence, they experience loss of accuracy. A new immersed finite element ife method is developed for secondorder ellip. Augmenting a fluid in the moving solid domain, we introduce a volumetric force to obtain the correct dynamics for both the fluid and the structure. A novel procedure, immersed smoothed finite element method immersed sfem is proposed for solving fluidstructure interaction fsi problems with moving nonlinear solids, using triangular type of mesh. The fem is a particular numerical method for solving. This success is mainly due to the fact that fem are able to reflect the original mathematical model in a very natural way. Waixiang cao, xu zhang and zhimin zhang, superconvergence of immersed finite element methods for interface problems, advances in computational mathematics, 43, 4, 795, 2017. The finite element method fem is a numerical method for solving problems of engineering and mathematical physics. Immersed finite element method and its applications to.
Since then many variants of these appealing approaches have been suggested, like embedded domain and immersed boundary methods or special implementations of the extended finite element method. Surface integral methods in computational aeroacousticsfrom the cfd nearfield. Computer methods in applied mechanics and engineering 229232, 110127. The reliability of a finite element procedure means in particular that when some geometric or material properties are changed in the mathematical model, then for a given finite element mesh the accuracy of the finite element solution does not drastically decrease. Thomas, chopp, david, leveque, randall, and li, zhilin, communications in applied mathematics and computational science, 2008. Similar to the immersed boundary method, the fluid domain is defined on a fixed eulerian grid. In these finite element formulations, independent lagrangian solid meshes are introduced to move on top of a background eulerian fluid mesh. Computational aeroacoustics issues and methods aiaa journal.
Mixed finite element methods for coupled 3d1d fluid problems. An improved immersed finte element particleincell method. In such a mesh, each point has a fixed number of predefined neighbors, and this connectivity between neighbors can be used to define mathematical operators like the derivative. The contact process between leaflets or between leaflet and sinus was evaluated using an adhesive contact method. The immersed finite element method ifem is another type of interfaceunfitted methods, where special finite element basis functions are constructed to satisfy the interface jump conditions 1,31. On computational issues of immersed finite element methods x. In the xfem, they use enrichment basis functions in addition to the standard finite element basis. Recent advances in immersed boundary, fictitious domain and. It is wellknown that coercivity and accuracy issues can arise due to arbitrarily cut cells in these approaches, but. This paper summarizes the newly developed immersed finite element method ifem and its applications to the modeling of biological systems. This is to hold for any bounded, twodimensional, piecewise smooth submanifold. Superconvergence of immersed finite element methods for interface problems. Issn 17994942 pdf issn 17994934 aalto university school of science department of mathematics and systems analysis. The particleincell pic method has been widely used for plasma simulation, because of its noisereduction capability and moderate computational cost.