Abstract

As already demonstrated by different authors, the Taylor-Galerkin (TG) scheme, in the context of the Finite Element Method (FEM), is particularly suitable for the solution of supersonic flows. The TG scheme, using hexahedral finite elements with analytical evaluation of element matrices, is applied in this work. Tools to avoid locking and a shock capturing technique for the solution of supersonic viscous and non-viscous compressible flows are also employed. However, TG scheme usually presents instabilities in subsonic flows. Even in cases in which the free stream Mach number corresponds to supersonic flows, there are always flow regions, specifically near the walls of the immersed obstacles, where the speed is lower and the local Mach number corresponds to a subsonic flow. The Characteristic Based Split (CBS) scheme was developed in order to obtain a single method to improve the behavior with respect to TG method in subsonic and supersonic regimes. In the last two decades some works have shown advantages in convergence rates of the CBS method when compared to the TG algorithm. However, simulation time increases in the CBS method since split operations, typical of this algorithm, imply in additional element loops. In this paper a hybrid algorithm called Modified-Taylor-Galerkin scheme (MTG) is proposed. This algorithm presents advantages with respect to TG and CBS schemes in terms of convergence properties and computational processing time. In order to get an efficient algorithm, the element matrices are analytically integrated. This is performed with two different approaches. In the first approach the inverse matrix and the determinant of the Jacobian matrix at element level are evaluated with a reduced integration form, using the point located in the center of the element for mass, convective, diffusive and stabilization element matrices; all these matrices are integrated analytically. In the second approach, mass and convective matrices are calculated by a complete integration scheme (including the inverse matrix and the determinant of the Jacobian matrix at element level in the analytical expression to be integrated) and the diffusive and stabilization matrices are calculated with a reduced integration form, using the point located at the center of the element to calculate the inverse matrix and the determinant of the Jacobian matrix at element level. Finally, this work incorporates the Spalart-Allmaras (S-A) turbulence model using a conservative version of the transport equation, as proposed by the authors of the original S-A model in a later paper. Algorithms are tested to determine convergence rate improvements in both laminar and turbulent cases and for different Mach numbers (supersonic, transonic and subsonic flows). 

Abstract

We examine a residual and matrix-free Jacobian formulation of compressible and nearly incompressible (v → 0.5) displacement-only linear isotropic elasticity with high-order hexahedral finite elements. A matrix-free p-multigrid method is combined with algebraic multigrid on the assembled sparse coarse grid matrix to provide an effective preconditioner. The software is verified with the method of manufactured solutions. We explore convergence to a predetermined L₂ error of 10^-4, 10^-5 and 10^-6 for the compressible case and 10^-4, 10^-5 for the nearly-incompressible cases, as the Poisson's ratio approaches 0.5, based upon grid resolution and polynomial order. We compare our results against results obtained from C3D20H mixed/hybrid element available in the commercial finite element software ABAQUS that is quadratic in displacement and linear in pressure. We determine, for the same problem size, that our matrix-free approach for displacement-only implementation is faster and more efficient for quadratic elements compared to the C3D20H element from ABAQUS that is specially designed to handle nearly-incompressible and incompressible elasticity problems. However, as we approach the near incompressibility limit, the number of Conjugate Gradient iterations required to achieve the desired solution increases significantly.