The immersed boundary method: A SIMPLE approach

Kirill Goncharuk, Oz Oshri, Yuri Feldman

Research output: Contribution to journalArticlepeer-review


A novel formulation of the direct forcing immersed boundary (IB) method, that treats it as an integral part of a SIMPLE method is presented for the simulation of incompressible flows. The incompressibility and no-slip kinematic constraints are treated implicitly as distributed Lagrange multipliers and are fully coupled with each other by combining them into a single Poisson-body forces system of equations that constitutes a regularized saddle point problem. A physically justified approximation of the system, resembling a technique typical of the penalty method, is carried out to facilitate the solution. By utilizing the Schur complement approach, the approximated system is decomposed into two separate systems of equations, allowing us to compute the values for the volumetric force and pressure corrections. The first system, which is characterized by a small (only O(10)) value of the condition number, is conveniently solved by the BiCgStab method [1], converging within 2-3 iterations, while the second system is addressed by the direct TPF solver [2], characterized by O(N4/3) complexity. The entire methodology is designed to be highly portable, which facilitates the use of any available solver that was designed to simulate incompressible flows governed by the Helmholtz and Laplace operators but could not benefit from the immersed boundary formalism. The capabilities of the developed methodology applied to the simulation of representative shear- and buoyancy-driven confined flows developing in the presence of stationary immersed bodies are demonstrated. A further application of the developed approach to moving boundary and two-way coupled fluid-structure interaction problems is discussed.

Original languageEnglish
Article number112148
JournalJournal of Computational Physics
StatePublished - 15 Aug 2023


  • Distributed Lagrange multiplier
  • Implicit immersed boundary method
  • Schur complement

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Modelling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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