Computability and Adaptivity in CFD

We give a brief introduction to research on adaptive computational methods for laminar compressible and incompressible flow, and then focus on computability and adaptivity for turbulent incompressible flow, where we present a framework for adaptive finite element methods with duality- based a posteriori error control for chosen output quantities of interest. We show in concrete examples that outputs such as mean values in time of drag and lift of a bluff body in a turbulent flow are computable to a tolerance of a few percent, for a simple geometry using some hundred thousand mesh points, and for complex geometries some million mesh points

Adaptive Algorithms

Overwhelming empirical evidence in computational science and engineering proved that self-adaptive mesh-generation is a must-do in real-life problem computational partial differential equations. The mathematical understanding of corresponding algorithms concerns the overlap of two traditional mathematical disciplines, numerical analysis and approximation theory, with computational sciences. The half workshop was devoted to the mathematics of optimal convergence rates and instance optimality of the Dörfler marking or the maximum strategy in various versions of space discretisations and time-evolution problems with all kind of applications in the efficient numerical treatment of partial differential equations

Schnelle Löser für Partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch (Leipzig), and Gabriel Wittum (Frankfurt am Main), was held May 22nd–May 28th, 2011. This meeting was well attended by 54 participants with broad geographic representation from 7 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds

Output error estimation strategies for discontinuous Galerkin discretizations of unsteady convection‐dominated flows

We study practical strategies for estimating numerical errors in scalar outputs calculated from unsteady simulations of convection‐dominated flows, including those governed by the compressible Navier–Stokes equations. The discretization is a discontinuous Galerkin finite element method in space and time on static spatial meshes. Time‐integral quantities are considered for scalar outputs and these are shown to superconverge with temporal refinement. Output error estimates are calculated using the adjoint‐weighted residual method, where the unsteady adjoint solution is obtained using a discrete approach with an iterative solver. We investigate the accuracy versus computational cost trade‐off for various approximations of the fine‐space adjoint and find that exact adjoint solutions are accurate but expensive. To reduce the cost, we propose a local temporal reconstruction that takes advantage of superconvergence properties at Radau points, and a spatial reconstruction based on nearest‐neighbor elements. This inexact adjoint yields output error estimates at a computational cost of less than 2.5 times that of the forward problem for the cases tested. The calculated error estimates account for numerical error arising from both the spatial and temporal discretizations, and we present a method for identifying the percentage contributions of each discretization to the output error. Copyright © 2011 John Wiley & Sons, Ltd.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/88080/1/3224_ftp.pd

Computational modelling of iron-ore mineralisation with stratigraphic permeability anisotropy

This study develops a computational framework to model fluid transport in sedimentary basins, targeting iron ore deposit formation. It offers a simplified flow model, accounting for geological features and permeability anisotropy as driving factors. A new finite element method lessens computational effort, facilitating robust predictions and cost-effective exploration. This methodology, applicable to other mineral commodities, enhances understanding of genetic models, supporting the search for new mineral deposits amid the global energy transition

Toward Accurate, Efficient, and Robust Hybridized Discontinuous Galerkin Methods

Computational science, including computational fluid dynamics (CFD), has become an indispensible tool for scientific discovery and engineering design, yet a key remaining challenge is to simultaneously ensure accuracy, efficiency, and robustness of the calculations. This research focuses on advancing a class of high-order finite element methods and develops a set of algorithms to increase the accuracy, efficiency, and robustness of calculations involving convection and diffusion, with application to the inviscid Euler and viscous Navier-Stokes equations. In particular, it addresses high-order discontinuous Galerkin (DG) methods, especially hybridized (HDG) methods, and develops adjoint-based methods for simultaneous mesh and order adaptation to reduce the error in a scalar functional of the approximate solution to the discretized equations. Contributions are made in key aspects of these methods applied to general systems of equations, addressing the scalability and memory requirements, accuracy of HDG methods, and efficiency and robustness with new adaptation methods. First, this work generalizes existing HDG methods to systems of equations, and in so doing creates a new primal formulation by applying DG stabilization methods as the viscous stabilization for HDG. The primal formulation is shown to be even more computationally efficient than the existing methods. Second, by instead keeping existing viscous stabilization methods and developing a new convection stabilization, this work shows that additional accuracy can be obtained, even in the case of purely convective systems. Both HDG methods are compared to DG in the same computational framework and are shown to be more efficient. Finally, the set of adaptation frameworks is developed for combined mesh and order refinement suitable for both DG and HDG discretizations. The first of these frameworks uses hanging-node-based mesh adaptation and develops a novel local approach for evaluating the refinement options. The second framework intended for simplex meshes extends the mesh optimization via error sampling and synthesis (MOESS) method to incorporate order adaptation. Collectively, the results from this research address a number of key issues that currently are at the forefront of high-order CFD methods, and particularly to output-based hp-adaptation for DG and HDG methods.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/137150/1/jdahm_1.pd