Anisotropic mesh adaptation for continuous finite element discretization through mesh optimization via error sampling and synthesis

Highlights

• DOF efficiency of Continuous FEM realized in adaptive context.

• Optimal mesh anisotropy demonstrated for CG.

• Nodal error models exploit continuity of basis functions.

• Algorithm uses edge–based local solve patches.

Abstract

Anisotropic output-based mesh adaptivity is a powerful technique for controlling the output error of finite element simulations, particularly when used in conjunction with higher-order discretization. The Mesh Optimization via Error Sampling and Synthesis (MOESS) algorithm makes use of the continuous mesh model which encodes local mesh sizing and anisotropy in a Riemannian metric field, and was developed for Discontinuous Galerkin (DG) discretization. In this paper, we outline an extension of the MOESS algorithm for discretizations which are defined by basis functions that are continuous across element boundaries. Error models are defined in terms of local error estimates defined at vertices, and local solutions are computed on local patches defined in terms of the edges of the mesh. The resulting algorithm displays reduced error for the same number of degrees of freedom compared to the MOESS algorithm for DG discretization, as illustrated by some numerical examples for $L^2$ projection and linear advection diffusion.

Introduction

Modern computational architectures have given rise to a drastic increase in the complexity of numerical simulations performed on a regular basis. As the complexity of these simulations has increased, the requirement to control errors within these simulations has increased commensurately. Output-based estimation enables the quantifiable control of error in measures more useful to scientific and engineering applications than typical residual norms. For instance the drag or heat flux on a boundary, or the average temperature or species consumption within a volume.

The Dual-Weighted Residual (DWR) framework developed for the Galerkin Finite Element Method (FEM) by Becker and Rannacher [1] provides a localizable estimate of the error in these quantities. The DWR estimate has been analyzed for other discretization methods by Carson et al. [2] and extended to other localization methods by Richter and Wick [3]. Another approach to output error estimation within FEM is the implicit estimation approach. Patera and Peraire utilize the coercivity of PDEs to bound the output with respect to the solution on a ‘truth’ mesh [4]. This technique was extended by Sauer-Budge et al. to bounding the computed output with respect to the true output [5], [6].

An initial approach to anisotropic adaptive algorithms for simplices used the Hessian of the solution, which controls linear interpolation error [7], [8]. This approach was codified in the work of Loseille and Alauzet [9], [10] who introduce the concept of mesh-metric duality and provide interpolation error estimates. In the case of a system of equations, the Hessian of a scalar derived quantity is used, for instance the Mach number for a fluid dynamics simulation. Alauzet and Loseille give a comprehensive review of the development of anisotropic adaptivity [11].

Various approaches for output-based anisotropic adaptation have been proposed. Venditti and Darmofal constructed a Hessian of a scalar component of the solution [12]. Fidkowski and Darmofal [13] and Leicht and Hartmann [14] extended this approach to higher order discretizations and Formaggia et al. used the Hessian of the dual [15], [16]. Loseille et al. applied mesh-metric duality to goal-based adaptation for the Euler equation by weighting the interpolation error of the fluxes by the dual [17]. Though successful these methods exhibit shortcomings, with the anisotropy detection being restricted to either primal or dual features alone, or only single components of multi-variable systems. Fidkowski and Darmofal provide a review of output error estimation and mesh adaptation in the context of computation fluid dynamics [18].

Driving adaptive decisions using local solves has been done by Houston et al. [19] and Ceze and Fidkowski [20] for quad-based meshes utilizing hanging nodes. Richter [21] and Leicht and Hartmann [22] used the quad/hex structure to pose discrete optimization problems. Yano and Darmofal used a local sampling process on simplex meshes to develop the MOESS algorithm [23]. MOESS consists of building surrogate models for the effect of local mesh perturbations on the global error via local sampling, then optimizing these models before re-meshing. The MOESS algorithm has proven particularly successful for aerodynamic applications where the physics are largely advection-dominated [24], [25]. Fidkowski proposed an alternative to the local sampling procedure of Yano and Darmofal, using projection of the higher-order adjoint between locally divided meshes to simulate local sampling [26].

The local sampling process of MOESS was developed particularly for the Discontinuous Galerkin (DG) discretization, though it can be naturally extended to any discretization with an elementally localizable estimate and discontinuous basis functions. One shortcoming of the DG discretization is the large number of degrees of freedom that come from duplication on element boundaries. This contrasts with the classical Continuous Galerkin (CG) discretization where basis functions are shared by adjacent elements. These shared degrees of freedom though efficient, prevent the application of the MOESS algorithm to this class of discretizations. MOESS has been extended to the Embedded Discontinuous Galerkin (EDG) discretization where Fidkowski distributed the error associated with the continuous features to the surrounding elements [27], thus the modeling process remains fundamentally elemental in nature. A comparison of MOESS algorithms for continuous and discontinuous discretizations, as applied to aerospace applications was shown by Carson et al. [28]

In this paper, we present a new MOESS class algorithm that works with discretizations that share degrees of freedom amongst elements, thereby enabling the design of anisotropic output adapted meshes for CG discretizations. For simplicity of exposition we consider here only the classic unstabilized CG discretization, though there is no fundamental difference to the algorithm when adding the stabilization necessary for more complex partial differential equations.

The outline of the paper is as follows: Section 2 introduces the metric optimization framework along with the notion of mesh-metric duality and the necessary machinery for manipulation of a metric field. Section 3 describes the MOESS algorithm for discontinuous-type discretizations. Section 4 outlines a new MOESS algorithm for continuous-type discretizations utilizing vertex-based local error models. Section 5 demonstrates the new algorithm in comparison to the original discontinuous-type algorithm for $L^2$ error control in two dimensions. Section 6 outlines the vertex localized Dual-Weighted Residual output error estimate. Section 7 compares the algorithm to the original MOESS algorithm for a three dimensional linear advection diffusion PDE.