In this study, in order to address the immersed boundary condition, which was the critical issue regarding the fictitious domain method, two new strategies for addressing the numerical integral related to the immersed boundary condition were introduced. In the first strategy, the constraint was set to live everywhere, but only equaled the desired values in the area outside the needed domain. As to the second strategy, a boundary region was conceptually generated to replace the immersed boundary. An additional function, k(x), was added as a weight function to validate this replacement. Both of these strategies transfer boundary integrals to domain integrals that all computations can be finished by using the mesh generated for the fictitious domain. In addition, in order to deal with large scale problems, a modified iterative algorithm was proposed.
Three different types of problems were studied to evaluate the capability of these two strategies. It is shown that both of these two strategies are capable of addressing problems with only one variable. However, the study of the Stokes problem indicates the second strategy is a superior choice to deal with problems with multiple variables. Finally, from the Navier-Stokes flow problem, it is concluded that the second strategy can solve large scale flow problems with complex geometries.
