Interpolation-Based Immersed Finite Element and Isogeometric Analysis of Incompressible Fluid Flow
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In this talk, we will present a new approach to immersed finite element and isogeometric analysis of incompressible fluid flow. In our approach, finite element or spline basis functions defined on a non-body-fitted background mesh are first interpolated onto a Lagrange basis defined on a body-fitted integration mesh, and these background basis function approximations are then employed for immersed finite element or isogeometric analysis. Much like spline basis functions can be represented element-wise in terms of Bernstein shape functions in body-fitted isogeometric analysis, the background basis function approximations in our approach can be represented in terms of Lagrange shape functions over each element in the body-fitted integration mesh using Lagrange extraction operators. Consequently, one can transform a classical finite element analysis code into an immersed finite element or isogeometric analysis code with minimal implementation effort using our approach. We combine this new interpolation-based immersed approach with SUPG, PSPG, and grad-div stabilization and Nitsche’s method for weak enforcement of boundary conditions to tackle challenging incompressible fluid flow problems of engineering interest. We will illustrate the efficacy of this new methodology using a number of example problems in our talk, and we will discuss how we how to implement this methodology within the popular open-source finite element analysis platform FEniCS.
