Immersed finite element methods enable the simulation of physical systems out of reach by classical finite element analysis, and they additionally streamline the development of powerful design optimization technologies. However, the development of an immersed finite element analysis code is a difficult task even for domain experts. In this talk, an alternative approach to immersed finite element analysis will be presented that overcomes this issue. In our approach, finite element basis functions defined on a non-body-fitted background mesh are interpolated onto a Lagrange basis defined on a body-fitted integration mesh, and these interpolated background basis functions are then employed for immersed finite element analysis. By construction, the interpolated background basis functions over each integration mesh element can be represented in terms of Lagrange shape functions using Lagrange extraction operators. This enables one to transform a classical finite element analysis code into an immersed finite element analysis code with minimal implementation effort. Namely, one only needs to provide the classical finite element analysis code with Lagrange extraction operators, a connectivity array relating local and global degrees of freedom, and the ability to compute the interpolated background basis function values and derivatives using the Lagrange extraction operators. One can further use these ingredients to transform a classical finite element analysis code into an immersed isogeometric analysis code. This talk will begin with a general overview of our approach, and then it will be shown how this approach can be applied to the particular coupled problem of thermoelasticity. It will be demonstrated how our approach allows one to easily approximate different physical fields (in the case of thermoelasticity, displacement and temperature) with different background meshes without excessive implementation complexity. This is desirable when different physical fields exhibit local features and gradients in differing portions of the domain. Finally, our open-source software package for generating the data structures necessary for our approach will be presented, as well as how this enables immersed finite element and isogeometric analysis of thermoelasticity within the popular open-source finite element analysis platform FEniCS.