A Comparison of Hybridized and Standard DG Methods for Target-Based hp-Adaptive Simulation of Compressible Flow

We present a comparison between hybridized and non-hybridized discontinuous Galerkin methods in the context of target-based hp-adaptation for compressible flow problems. The aim is to provide a critical assessment of the computational efficiency of hybridized DG methods. Hybridization of finite element discretizations has the main advantage, that the resulting set of algebraic equations has globally coupled degrees of freedom only on the skeleton of the computational mesh. Consequently, solving for these degrees of freedom involves the solution of a potentially much smaller system. This not only reduces storage requirements, but also allows for a faster solution with iterative solvers. Using a discrete-adjoint approach, sensitivities with respect to output functionals are computed to drive the adaptation. From the error distribution given by the adjoint-based error estimator, h- or p-refinement is chosen based on the smoothness of the solution which can be quantified by properly-chosen smoothness indicators. Numerical results are shown for subsonic, transonic, and supersonic flow around the NACA0012 airfoil. hp-adaptation proves to be superior to pure h-adaptation if discontinuous or singular flow features are involved. In all cases, a higher polynomial degree turns out to be beneficial. We show that for polynomial degree of approximation p=2 and higher, and for a broad range of test cases, HDG performs better than DG in terms of runtime and memory requirements

Adjoint-Based Error Estimation and Mesh Adaptation for Hybridized Discontinuous Galerkin Methods

We present a robust and efficient target-based mesh adaptation methodology, building on hybridized discontinuous Galerkin schemes for (nonlinear) convection-diffusion problems, including the compressible Euler and Navier-Stokes equations. Hybridization of finite element discretizations has the main advantage, that the resulting set of algebraic equations has globally coupled degrees of freedom only on the skeleton of the computational mesh. Consequently, solving for these degrees of freedom involves the solution of a potentially much smaller system. This not only reduces storage requirements, but also allows for a faster solution with iterative solvers. The mesh adaptation is driven by an error estimate obtained via a discrete adjoint approach. Furthermore, the computed target functional can be corrected with this error estimate to obtain an even more accurate value. The aim of this paper is twofold: Firstly, to show the superiority of adjoint-based mesh adaptation over uniform and residual-based mesh refinement, and secondly to investigate the efficiency of the global error estimate

Unsteady Output-Based Adaptation Using Continuous-in-Time Adjoints

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/143081/1/6.2017-0529.pd

Error Estimation and Adaptation in Hybridized Discontinous Galerkin Methods

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/140705/1/6.2014-0078.pd

Verification of Unstructured Grid Adaptation Components

Adaptive unstructured grid techniques have made limited impact on production analysis workflows where the control of discretization error is critical to obtaining reliable simulation results. Recent progress has matured a number of independent implementations of flow solvers, error estimation methods, and anisotropic grid adaptation mechanics. Known differences and previously unknown differences in grid adaptation components and their integrated processes are identified here for study. Unstructured grid adaptation tools are verified using analytic functions and the Code Comparison Principle. Three analytic functions with different smoothness properties are adapted to show the impact of smoothness on implementation differences. A scalar advection-diffusion problem with an analytic solution that models a boundary layer is adapted to test individual grid adaptation components. Laminar flow over a delta wing and turbulent flow over an ONERA M6 wing are verified with multiple, independent grid adaptation procedures to show consistent convergence to fine-grid forces and a moment. The scalar problems illustrate known differences in a grid adaptation component implementation and a previously unknown interaction between components. The wing adaptation cases in the current study document a clear improvement to existing grid adaptation procedures. The stage is set for the infusion of verified grid adaptation into production fluid flow simulations

Output Error Control Using r-Adaptation

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/143062/1/6.2017-4111.pd

Output-based Adaptive Meshing Using Triangular Cut Cells

This report presents a mesh adaptation method for higher-order (p > 1) discontinuous Galerkin (DG) discretizations of the two-dimensional, compressible Navier-Stokes equations. The method uses a mesh of triangular elements that are not required to conform to the boundary. This triangular, cut-cell approach permits anisotropic adaptation without the difficulty of constructing meshes that conform to potentially complex geometries. A quadrature technique is presented for accurately integrating on general cut cells. In addition, an output-based error estimator and adaptive method are presented, with emphasis on appropriately accounting for high-order solution spaces in optimizing local mesh anisotropy. Accuracy on cut-cell meshes is demonstrated by comparing solutions to those on standard boundary-conforming meshes. Adaptation results show that, for all test cases considered, p = 2 and p = 3 discretizations meet desired error tolerances using fewer degrees of freedom than p = 1. Furthermore, an initial-mesh dependence study demonstrates that, for sufficiently low error tolerances, the final adapted mesh is relatively insensitive to the starting mesh

Improving High-Order Finite Element Approximation Through Geometrical Warping

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/140695/1/1.J055071.pd

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

This work covers a contribution to two most interesting research elds in aerodynamics, the fi nite element analysis of high-speed compressible flows (Part I) and aerodynamic shape optimization (Part II). The fi rst 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 nite element method (FEM). The FIC method is based on expressing the balance of fluxes in a spacetime domain of nite 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 flows, 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 flows 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 signi cantly. The second part of this thesis proposes a procedure for aerodynamic shape optimization combining Genetic Algorithm (GA) and mesh re nement technique. In particular, it is investigated the e ect of mesh re nement 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 adap- tive mesh techniques coupled to Multi-Objective Genetic Algorithms (MOGAs) and Euler flow analyzer are addressed. The methodology is implemented to solve three practical design problems; the fi rst test case considers a reconstruction design optimization that minimizes the pressure error between a prede ned 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 coe cient (Cl). The solutions obtained with and without adaptive mesh re nement are compared in terms of solution accuracy and computational cost. These design problems under transonic speeds need to be solved with a ne 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 bene ts in the reduction of the total computational cost through a better convergence to the final solution is evaluated. Indeed, the improvement of the solution quality when an adaptive remeshing technique is coupled with the optimum design strategy can be judged.El presente trabajo pretende contribuir a dos de los campos de investigaci on m as interesantes en la aerodin amica, el an alisis num erico de flujos compresibles a alta velocidad (Parte I) y la optimizaci on de la forma aerodin amica (Parte II). La primera parte de este estudio se centra en la soluci on num erica de las ecuaciones de Navier-Stokes, que modelan el comportamiento de flujos compresibles a alta velocidad. La discretizaci on espacial se lleva a cabo mediante el m etodo de elementos nitos (FEM) y se pone especial enfasis en el desarrollo de una nueva formulaci on estabilizada basada en la t ecnica de c alculo de Incremento fi nitos (FIC). En este ultima, los t erminos de estabilizaci on convectiva se obtienen de manera natural de las ecuaciones de gobierno a trav es de postulados de conservaci on y equilibrio de flujos en un dominio espacio-tiempo de tamaño nito. Ello lleva a la obtenci on de dos t erminos de estabilizaci on que funcionan de manera complementaria. Uno act ua en direcci on de las lineas de corriente proporcionando la estabilizaci on necesaria para contrarestrar las inestabilidades propias de la forma discreta de Galerkin y el otro t ermino, de tipo shock capturing, act ua de manera transversal a las l neas de corriente y permite mejorar la soluci on num erica alrededor de discontinuidades y otro tipos de fen omenos localizados en el campo de soluci on de problema. La forma discreta de las ecuaciones de gobierno se completa mediante un esquema de integraci on temporal expl icito de tipo de Runge-Kutta de 4to orden. El esquema de soluci on b asico propuesto se complementa con una t ecnica de re namiento adaptativo de malla que permite mejorar autom aticamente la soluci on num erica en zonas localizadas del dominio en que, dadas las caracter sticas del flujo, se necesita una mayor resoluci on espacial. Con el prop osito de investigar el comportamiento de la formulaci on num erica se estudian diferentes casos de an alisis que implican flujos viscosos y no viscosos en r egimen subs onico, trans onico y supers onico y se estudia con especial detalle el funcionamiento de la t ecnica de estabilizaci on propuesta. Los resultados obtenidos demuestran una exactitud satisfactoria y una buena correlaci on con resultados presentes en la literatura, incluso cuando se trabaja con discretizaciones espaciales relativamente gruesas. Adicionalmente, los estudios num ericos realizados demuestran que el empleo del esquema adaptativo de malla es e ficaz para incrementar la exactitud de la soluci on numerica manteniendo un bajo coste computacional. En la segunda parte de este estudio se propone un m etodo para la optimizaci on de formas aerodin amicas que combina algoritmos gen eticos multiobjetivo (MOGAs) y remallado adaptativo con el objetivo de asegurar, con un coste computacional m nimo, la calidad de la soluci on numerica empleada en el proceso de b usqueda de un determinado diseño objetivo, particularmente cuando el flujo presenta discontinuidades y gradientes muy localizados, ti picos de flujos a alta velocidad. La metodolog a se aplica a resolver tres problemas pr acticos de diseño de per les aerodin amicos en flujo trans onico que implican la optimizaci on de la distribuci on de presiones, minimizaci on de la resistencia de onda y maximizaci on conjunta de la sustentaci on y la relaci on sustentaci on/resistencia. Para cada uno de ellos se estudia el efecto del re namiento en la calidad de la soluci on num erica as como tambi en en el coste computacional y la convergencia del problema. Los estudios realizados demuestran la e cacia de la metodolog a propuesta