A selective Taylor Weak statement based, quadrature-free finite element algorithm for aerodynamic flow

A new finite element computational fluid dynamics (CFD) algorithm has been developed for efficiently solving multi-dimensional compressible flow in general geometry. The algorithm employs a metric data mean (MDM) technique that enables quadrature-free finite element computing, a selective Taylor Weak Statement (S- TWS) procedure that yields diagonalized and positive-definite numerical dissipation for the partial differential equation systems of aerodynamics, and an invariant dis- sipation length (IDL) scheme that ensures just measurement/bounding of numerical dissipation in mesh. Coded jointly, the two-dimensional compressible potential flow (CPF) and Euler equations have been solved, observing stiff shock/stability. Observed in this process is also the necessity for the "boundary closure" of FE numerical dissipa- tion, in terms of its impact on both the solution accuracy and the matrix-conditioning. Several linear algebra solvers have been implemented and tested, including the ten- sor product (TP) approximate factorization, the line Gauss-Seidel relaxation (LGS), the conjugate gradient square (CGS), and the LGS and TP preconditioned CGS. The tests, which are conducted using rather large Courant numbers (between 100 and 150), demonstrate the LGS preconditioned CGS being the best solver and the "cycled composite LGS '' the best line relaxation solver (far superior to TP).

There are also some side-aspects of the documented research, such as the establish- ment of a single variable convection model of clear yet analyzabe nonlinearity/shock mechanism (serving for numerical verification), the theoretical justification of the positive-definiteness of the numerical dissipation for CPF (being a necessary condi- tion upon the Second Law of Thermodynamics), and the development and application of an incompressible potential flow based procedure for solution initialization in general geometry (by solving the SPD Laplacian).