C. Tiago

Comparison of high-order continuous and hybridizable discontinuous Galerkin methods for incompressible fluid flow problems

Abstract The computational efficiency and the stability of Continuous Galerkin (CG) methods, with Taylor–Hood approximations, and Hybridizable Discontinuous Galerkin (HDG) methods are compared for the solution of the incompressible Stokes and Navier–Stokes equations at low Reynolds numbers using direct solvers. A thorough comparison in terms of CPU time and accuracy for both discretization methods is made, under the same platform, for steady state problems, with triangular and quadrilateral elements of degree k = 2 − 9 . Various results are presented such as error vs. CPU time of the direct solver, error vs. ratio of CPU times of HDG to CG, etc. CG can outperform HDG when the CPU time, for a given degree and mesh, is considered. However, for high degree of approximation, HDG is computationally more efficient than CG, for a given level of accuracy, as HDG produces lesser error than CG for a given mesh and degree. Finally, stability of HDG and CG is studied using a manufactured solution that produces a sharp boundary layer, confirming that HDG provides smooth converged solutions for Reynolds numbers higher than CG, in the presence of sharp fronts.

Geometrically exact analysis of shells by a meshless approach

There is a growing interest in the geometrically exact analysis of structures. The innate elegance of this king of formulations arises from the exact representation of the rotations. In this case, the rotation vector is parameterized by the Euler-Rodrigues formula. The internal power arises from the first Piola- Kirchhoff stress tensor and the deformation gradient. A consistent plane stress condition is imposed in a hyperelastic material to derive the appropriate (symmetric) constitutive operator.

Coupling of Continuous and Hybridizable Discontinuous Galerkin Methods: Application to Conjugate Heat Transfer Problem

A coupling strategy between hybridizable discontinuous Galerkin (HDG) and continuous Galerkin (CG) methods is proposed in the framework of second-order elliptic operators. The coupled formulation is implemented and its convergence properties are established numerically by using manufactured solutions. Afterwards, the solution of the coupled Navier–Stokes/convection–diffusion problem, using Boussinesq approximation, is formulated within the HDG framework and analysed using numerical experiments. Results of Rayleigh–Bénard convection flow are also presented and validated with literature data. Finally, the proposed coupled formulation between HDG and CG for heat equation is combined with the coupled Navier–Stokes/convection diffusion equations to formulate a new CG–HDG model for solving conjugate heat transfer problems. Benchmark examples are solved using the proposed model and validated with literature values. The proposed CG–HDG model is also compared with a CG–CG model, where all the equations are discretized using the CG method, and it is concluded that CG–HDG model can have a superior computational efficiency when compared to CG–CG model.