p-version time-discontinuous Galerkin's method for nonlinear transient heat transfer analysis and sensitivity analysis

The finite element method is a powerful tool for approximating the solution of most engineering problems. Traditionally, the finite element method uses linear approximation of field variables over each element, where numerical results converge to the exact solution by increasing the number of elements. A newer finite element method called the p-version finite element method uses higher-order elements which assume a polynomial of order p for the approximation on each element. Here, the approximation will achieve accuracy without increasing the number of elements but by increasing the value of order p.; The focus of this research is on the development of numerical analysis and sensitivity analysis equations for nonlinear transient heat transfer problems modeled by p-version time-discontinuous finite element approximation. The resulting matrix equation of the state equation is simply in the form of A(x)x = c, which represents a single step time marching scheme. The Newton-Raphson's method is used to solve the nonlinear equation. Examples of heat transfer are first provided to demonstrate the accuracy characteristics of the resultant finite element approximation. A direct differentiation approach is then used to compute the thermal sensitivities of nonlinear heat transfer problems. The results show that only minimal coding effort is required to enhance the p-version time discontinuous finite element analysis code with the sensitivity analysis capability.