In this paper, we systematically derive jump conditions for the immersed interface method [SIAM J. Numer. Anal., 31 (1994), pp. 1019–1044; SIAM J. Sci. Comput., 18 (1997), pp. 709–735] to simulate three-dimensional incompressible viscous flows subject to moving surfaces. The surfaces are represented as singular forces in the Navier–Stokes equations, which give rise to discontinuities of flow qu...

A novel Immersed-Finite-Element Particle-in-Cell (IFE-PIC) simulation tool is presented in this paper for plasma surface interaction where charged plasma particles are represented by a number of simulation particles. The Particle-in-Cell (PIC) method is one of the major particle models for plasma simulation, which utilizes a huge number of simulation particles and hence provides a first-princip...

The authors develop finite difference methods for elliptic equations of the form V. ((x)Vu(x)) + (x)u(x) f(x) in a region in one or two space dimensions. It is assumed that gt is a simple region (e.g., a rectangle) and that a uniform rectangular grid is used. The situation is studied in which there is an irregular surface F of codimension contained in fl across which , a, and f may be discontin...

reported numerical results for three examples using the immersed interface method (IIM) and the extended finite element method (X-FEM). The results presented for the IIM showed first-order accuracy for the solution and inaccurate values of the normal derivative at the interface. This was due to an error in the implementation. The purpose of this note is to present correct results using the IIM ...

Two-dimensional capsule-substrate adhesion under an imposed Stokes flow field is numerically simulated by using the immersed interface method (IIM) with the same viscosity for the fluid enclosed by the membrane and the surrounding fluid. The capsule membrane is considered to be either impermeable or permeable while the capsule membrane is assumed to be Neo-Hookean. The interaction between the c...

Peskin’s Immersed Boundary (IB) method is one of the most popular numerical methods for many years and has been applied to problems in mathematical biology, fluid mechanics, material sciences, and many other areas. Peskin’s IB method is associated with discrete delta functions. It is believed that the IB method is first order accurate in the L∞ norm. But almost none rigorous proof could be foun...

In this paper, we present a consistent and rigorous derivation of some stochastic fluid-structure interaction models based on an implicit interface formulation of the stochastic immersed boundary method. Based on the fluctuation-dissipation theorem, a proper form can be derived for the noise term to be incorporated into the deterministic hydrodynamic fluid-structure interaction models in either...

In this paper, an adaptive mesh refinement technique is developed and analyzed for the non-conforming immersed finite element (IFE) method proposed in [27]. The IFE method was developed for solving the second order elliptic boundary value problem with interfaces across which the coefficient may be discontinuous. The IFE method was based on a triangulation that does not need to fit the interface...

A computational method is presented to resolve hydrodynamic interactions acting on solid particles immersed in incompressible host fluids. In this method, boundaries between solid particles and host fluids are replaced with a continuous interface by assuming a smoothed profile. This enabled us to calculate hydrodynamic interactions both efficiently and accurately, without neglecting many-body i...

In this work, we develop a cutting method for solving problems with moving and growing interfaces in 3D. This new method is able to resolve large displacement or deformation of immersed objects by combining the Arbitrary Lagrangian-Eulerian method with only small local mesh deformation defined on the reference domain, that is decomposed into the macro-elements. The linear system of algebraic eq...