This paper is concerned with the mathematical structure of the immersed boundary (IB) method, which is intended for the computer simulation of fluid–structure interaction, especially in biological fluid dynamics. The IB formulation of such problems, derived here from the principle of least action, involves both Eulerian and Lagrangian variables, linked by the Dirac delta function. Spatial discretization of the IB equations is based on a fixed Cartesian mesh for the Eulerian variables, and a moving curvilinear mesh for the Lagrangian variables. The two types of variables are linked by interaction equations that involve a smoothed approximation to the Dirac delta function. Eulerian/Lagrangian identities govern the transfer of data from one mesh to the other. Temporal discretization is by a second-order Runge–Kutta method. Current and future research directions are pointed out, and applications of the IB method are briefly discussed. Introduction The immersed boundary (IB) method was introduced to study flow patterns around heart valves and has evolved into a generally useful method for problems of fluid–structure interaction. The IB method is both a mathematical formulation and a numerical scheme. The mathematical formulation employs a mixture of Eulerian and Lagrangian variables. These are related by interaction equations in which the Dirac delta function plays a prominent role. In the numerical scheme motivated by the IB formulation, the Eulerian variables are defined on a fixed Cartesian mesh, and the Lagrangian variables are defined on a curvilinear mesh that moves freely through the fixed Cartesian mesh without being constrained to adapt to it in any way at all.

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The immersed boundary method

The term “immersed boundary method” was first used in reference to a method developed by Peskin (1972) to simulate cardiac mechanics and associated blood flow. The distinguishing feature of this method was that the entire simulation was carried out on a Cartesian grid, which did not conform to the geometry of the heart, and a novel procedure was formulated for imposing the effect of the immersed boundary (IB) on the flow. Since Peskin introduced this method, numerous modifications and refinements have been proposed and a number of variants of this approach now exist. In addition, there is another class of methods, usually referred to as “Cartesian grid methods,” which were originally developed for simulating inviscid flows with complex embedded solid boundaries on Cartesian grids (Berger & Aftosmis 1998, Clarke et al. 1986, Zeeuw & Powell 1991). These methods have been extended to simulate unsteady viscous flows (Udaykumar et al. 1996, Ye et al. 1999) and thus have capabilities similar to those of IB methods. In this review, we use the term immersed boundary (IB) method to encompass all such methods that simulate viscous flows with immersed (or embedded) boundaries on grids that do not conform to the shape of these boundaries. Furthermore, this review focuses mainly on IB methods for flows with immersed solid boundaries. Application of these and related methods to problems with liquid-liquid and liquid-gas boundaries was covered in previous reviews by Anderson et al. (1998) and Scardovelli & Zaleski (1999). Consider the simulation of flow past a solid body shown in Figure 1a. The conventional approach to this would employ structured or unstructured grids that conform to the body. Generating these grids proceeds in two sequential steps. First, a surface grid covering the boundaries b is generated. This is then used as a boundary condition to generate a grid in the volume f occupied by the fluid. If a finite-difference method is employed on a structured grid, then the differential form of the governing equations is transformed to a curvilinear coordinate system aligned with the grid lines (Ferziger & Peric 1996). Because the grid conforms to the surface of the body, the transformed equations can then be discretized in the

/pdf/immersed-boundary-methods-4id27kda4o.pdf

Immersed boundary methods

Thermophysical Properties Research Literature Retrieval Guide Edited by Y. S. Touloukian, J. K. Gerritsen and N. Y. Moore Second edition, revised and expanded. Book 1: Pp. xxi + 819. Book 2: Pp.621. Book 3: Pp. ix + 1315. (New York: Plenum Press, 1967.) n.p.

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Heat and Mass Transfer

3.8.2. Temperature Distribution Measurements 4749 3.8.3. Two-Phase Visualization 4750 3.8.4. Experimental Validation 4751 3.9. Modeling the Catalyst Layer at Pore Level 4751 3.10. Summary and Outlook 4752 4. Direct Methanol Fuel Cells 4753 4.1. Technical Challenges 4754 4.1.1. Methanol Oxidation Kinetics 4754 4.1.2. Methanol Crossover 4755 4.1.3. Water Management 4755 4.1.4. Heat Management 4756 4.2. DMFC Modeling 4756 4.2.1. Needs for Modeling 4756 4.2.2. DMFC Models 4756 4.3. Experimental Diagnostics 4757 4.4. Model Validation 4758 4.5. Summary and Outlook 4760 5. Solid Oxide Fuel Cells 4760 5.1. SOFC Models 4761 5.2. Summary and Outlook 4762 6. Closing Remarks 4763 7. Acknowledgments 4763 8. References 4763

Fundamental models for fuel cell engineering.

https://jupiter.math.nctu.edu.tw/~mclai/papers/Lai_1.pdf

An Immersed Boundary Method with Formal Second-Order Accuracy and Reduced Numerical Viscosity

http://people.math.sfu.ca/~stockie/papers/ijhe10_KKMS.pdf

A parametric study of cathode catalyst layer structural parameters on the performance of a PEM fuel cell

Analysis of Stiffness in the Immersed Boundary Method and Implications for Time-Stepping Schemes

Simulating the Motion of Flexible Pulp Fibres Using the Immersed Boundary Method

We develop an adaptive method for solving one-dimensional systems of hyperbolic conservation laws that employs a high resolution Godunov-type scheme for the physical equations, in conjunction with a moving mesh PDE governing the motion of the spatial grid points. Many other moving mesh methods developed to solve hyperbolic problems use a fully implicit discretization for the coupled solution-mesh equations, and so suffer from a significant degree of numerical stiffness. We employ a semi-implicit approach that couples the moving mesh equation to an efficient, explicit solver for the physical PDE, with the resulting scheme behaving in practice as a two-step predictor-corrector method. In comparison with computations on a fixed, uniform mesh, our method exhibits more accurate resolution of discontinuities for a similar level of computational work.

/pdf/a-moving-mesh-method-for-one-dimensional-hyperbolic-4676xtaj0u.pdf

John M. Stockie

Papers

A parametric study of cathode catalyst layer structural parameters on the performance of a PEM fuel cell

Analysis of Stiffness in the Immersed Boundary Method and Implications for Time-Stepping Schemes

Simulating the Motion of Flexible Pulp Fibres Using the Immersed Boundary Method

A Moving Mesh Method for One-dimensional Hyperbolic Conservation Laws

An inverse Gaussian plume approach for estimating atmospheric pollutant emissions from multiple point sources