Homotopy Methods are claimed to solve a system of nonlinear algebraic equations (NAEs) and to find solution of nonlinear PDEs without having to invert Jacobian (tangent stiffness) matrix like Newton methods.

http://techscience.com/CMES/v98n6/27167

http://www.techscience.com/cmc/v23n1/22627

https://www.depts.ttu.edu/coe/CARES/pdf/570.pdf

However, in the mentioned papers the Homotopy Methods are used for abstract NAEs or by deriving tangent stiffness matrix explicitly (analytically). So I am not clear whether it is possible to apply the Homotopy Methods to ordinary FEM process in which the tangent stiffness matrix is assembled by looping through elements. I would like to know if there is a possibility to replace Newton iteration with such methods in nonlinear FEM and how one might implement them.

Thank you in advance.