A contribution to the finite element analysis of high-speed compressible flows and aerodynamic shape optimization

This work covers a contribution to two most interesting research fields in aerodynamics,
the finite element analysis of high-speed compressible flows (Part I)
and aerodynamic shape optimization (Part II).
The first part of this study aims at the development of a new stabilization
formulation based on the Finite Increment Calculus (FIC) scheme for the Euler
and Navier-Stokes equations in the context of the Galerkin finite element method
(FEM). The FIC method is based on expressing the balance of fluxes in a spacetime
domain of finite size. It is tried to prevent the creation of instabilities
normally presented in the numerical solutions due to the high convective term
and sharp gradients.
In order to overcome the typical instabilities happening in the numerical
solution of the high-speed compressible ows, two stabilization terms, called
streamline term and transverse term, are added through the FIC formulation in
space-time domain to the original conservative equations of mass, momentum
and energy. Generally, the streamline term holding the direction of the velocity
is responsible for stabilizing the spurious solutions produced from the convective
term while the transverse term smooths the solution in the high gradient zones.
An explicit fourth order Runge-Kutta scheme is implemented to advance the
solution in time.
In order to investigate the capability of the proposed formulation, some
numerical test examples corresponding to subsonic, transonic and supersonic
regimes for inviscid and viscous ows are presented. The behavior of the proposed
stabilization technique in providing appropriate solutions has been studied
especially near the zones where the solution has some complexities such
as shock waves, boundary layer, stagnation point, etc. Although the derived
methodology delivers precise results with a nearly coarse mesh, the mesh refinement
technique is coupled in the solution to create a suitable mesh particularly
in the high gradient zones.
The comparison of the numerical results obtained from the FIC formulation
with the reference ones demonstrates the robustness of the proposed method for
stabilization of the Euler and Navier-Stokes equations. It is observed that the
usual oscillations occur in the Galerkin FEM, especially near the high gradient
zones, are cured by implementing the proposed stabilization terms. Furthermore,
allowing the adaptation framework to modify the mesh, the quality of
the results improves significantly.
The second part of this work proposes a procedure for aerodynamic shape optimization
combining Genetic Algorithm (GA) and mesh refinement technique.
In particular, it is investigated the effect of mesh refinement on the computational
cost and solution accuracy during the process of aerodynamic shape
optimization. Therefore, an adaptive remeshing technique is joined to the CFD
solver for the analysis of each design candidate to guarantee the production of
more realistic solutions during the optimum design process in the presence of
shock waves.
In this study, some practical transonic airfoil design problems using adaptive mesh techniques coupled to Multi-Objective Genetic Algorithms (MOGAs)
and Euler ow analyzer are addressed. The methodology is implemented to
solve three practical design problems; the first test case considers a reconstruction
design optimization that minimizes the pressure error between a predefined
pressure curve and candidate pressure distribution. The second test considers
the total drag minimization by designing airfoil shape operating at transonic
speeds. For the final test case, a multi-objective design optimization is conducted
to maximize both the lift to drag ratio (L/D) and lift coeficient (Cl). The
solutions obtained with and without adaptive mesh refinement are compared in
terms of solution accuracy and computational cost. These design problems under
transonic speeds need to be solved with a fine mesh, particularly near the
object, to capture the shock waves that will cost high computational time and
require solution accuracy.
By comparison of the the numerical results obtained with both optimization
problems, the obtainment of direct benefits in the reduction of the total computational
cost through a better convergence to the final solution is evaluated.