Spectral/hp finite element models for fluids and structures.

Abstract

We consider the application of high-order spectral/hp finite element technology to the numerical solution of boundary-value problems arising in the fields of fluid and solid mechanics. For many problems in these areas, high-order finite element procedures offer many theoretical and practical computational advantages over the low-order finite element technologies that have come to dominate much of the academic research and commercial software of the last several decades. Most notably, we may avoid various forms of locking which, without suitable stabilization, often plague low-order least-squares finite element models of incompressible viscous fluids as well as weak-form Galerkin finite element models of elastic and inelastic structures. 7.2 Transient flow of an incompressible viscous fluid inside a square cavity induced by a moving cylinder at t = 0.25, 0.50 and 0.70 (from left to right respectively): (a) velocity component v x (b) velocity component v y and (c) nondimensional pressure field p.

A spectral/hp least‐squares finite element analysis of the Carreau–Yasuda fluids

“… reformulated a least‐squares functional based on a new first‐order system of the Navier–Stokes equations, where the mass conservation is improved by increasing the pressure‐velocity coupling. The iterative penalty least‐squares formulations proposed by Prabhakar and Reddy and Prabhakar, Pontaza, and Reddy and the modified unconstrained least‐squares formulation equipped with normalized volumetric flow imbalance by Payette also showed improvement in both mass conservation and velocity‐pressure coupling.…”

“…To reduce the order of the differentiability of the approximation functions in the least‐squares finite element models, the Navier–Stokes equations are re‐written as a set of equivalent first‐order equations with auxiliary variables such as vorticity, stresses, dilatation, or velocity gradient. By doing this, we do not need to use C 1 ‐continuous functions for the velocity field, which could be perceived as a practical disadvantage .…”

“…One of the deficiencies of least‐squares models applied to the solution of flows of viscous incompressible fluids is that it is weak in local mass conservation for both steady and unsteady flow problems and poor in coupling between the velocity and pressure for unsteady flow problems . Several works have been carried out to enhance the local mass conservation and the coupling of velocity and pressure.…”

Multi‐level hp‐adaptivity for cohesive fracture modeling

“…As the rest of the mesh remains unchanged, this approach allows to keep the refinement local and focused and no global re-meshing is required. The high computational performance of this approach has been demonstrated in the context of various applications [40][41][42][43][44][45].The high approximation quality of hp-finite elements seems to be a natural choice to overcome efficiently the problem of artificial, discretization-induced oscillations in cohesive fracture propagation. However, the moving crack front requests a continuously adapted change of the discretization throughout the simulation to ensure a sufficient locally refined cohesive process zone.…”

“…As the rest of the mesh remains unchanged, this approach allows to keep the refinement local and focused and no global re-meshing is required. The high computational performance of this approach has been demonstrated in the context of various applications [40][41][42][43][44][45].…”