An adjoint-based hp-adaptive stabilized finite-element method with shock capturing for turbulent flows

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Abstract

In this study, an adjoint-based hp-adaptation algorithm has been developed within a Petrov–Galerkin finite-element method. The developed mesh adaptation algorithm is able to perform non-conformal mesh adaptations. To account for hanging nodes in a consistent manner, the constrained approximation method has been utilized. Hierarchical basis functions have been employed to facilitate the implementation of the constrained approximation. The methodology has been demonstrated on numerous cases using the Euler and Reynolds Average Navier–Stokes (RANS) equations, equipped with negative variant of Spalart–Allmaras (SA) turbulence model. Also, a PDE-based artificial viscosity has been added to the governing equations, to stabilize the solution in the vicinity of shock waves. For accurate representation of the geometric surfaces, high-order curved boundary meshes have been generated and the interior meshes have been deformed through the solution of a modified linear elasticity equation. Fully implicit linearization has been used to advance the solution toward a steady-state. Dirichlet boundary conditions have been imposed weakly and the functional outputs have been modified according to the weak boundary conditions in order to provide a smooth adjoint solution near the boundaries. To accelerate the error reduction in presence of singularity points, an enhanced h-refinement, based on solution’s smoothness, has been used. Numerical results illustrate consistent accuracy improvement of the functional outputs for both h- and hp-adaptation, and also capability enhancement in capturing complex viscous effects such as shock-wave/turbulent boundary layer interaction.

Introduction

After several decades of development, high-order finite-element methods are now being considered for realistic and large-scale Computational Fluid Dynamics (CFD) simulations. This necessitates further studies on utilization of mesh adaptation techniques in order to reach reliable solutions at minimal computational cost. In many engineering applications, a specific scalar objective, like lift or drag coefficient, is of particular interest. The adjoint-based adaptation techniques  [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21] are able to detect regions in the mesh where the discretization errors directly affect the accuracy of the computed objective. Among adaptive high-order finite-element methods being developed for compressible flow problems, the Discontinuous-Galerkin (DG) schemes have been arguably the most utilized. However, for elements with low polynomial degrees, a Petrov–Galerkin (PG) scheme requires significantly less degrees of freedom (DOFs) and non-zero matrix entries than a DG scheme for comparable accuracy  [22], [23], [24], [25]. This advantage can be further enhanced using adaptation techniques. Although in the CFD community, a considerable amount of research has been conducted on adjoint-based adaptation for DG schemes  [8], [11], [12], [13], [18], [20], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], the methodology for PG schemes is still not rigorously established. The objective of the present work is multifold. Here, an adjoint-based adaptation algorithm has been developed for PG schemes such that it can be effectively employed in a wide range of compressible flow applications, including subsonic and transonic turbulent flows. To this end, numerous techniques must be implemented to simulate over these flow regimes. This paper describes these techniques in detail.

Since the adaptation process is navigated by an adjoint-based technique, the quality of adjoint solution is of critical importance. In particular, in presence of Dirichlet boundary conditions, special attention should be given to the boundary conditions and to the definition of the functional output. In the present work, the spatial discretization is based on a Streamline-Upwind Petrov–Galerkin (SUPG) scheme  [36], [37], [38]. To obtain a smooth adjoint solution near the no-slip walls, the boundary conditions have been imposed weakly  [39], [40], [41] and the definition of the functional outputs have been modified to be compatible with these boundary conditions. The implementation of the boundary conditions is, in essence, based on the Nitsche’s method  [42] and its particular formulation has been taken from a Symmetric Interior Penalty Galerkin (SIPG) method  [13], [24], [43], [44] that is commonly used in the DG discretizations.

A major concern in the solution of transonic and supersonic flows is the need to stabilize the numerical scheme in vicinity of the shock waves. Although several methods exist to address this problem  [45], [46], [47], [48], [49], present study seeks a method that has been previously tested in an adjoint-based hp-adaptation algorithm. To this end, a PDE-based artificial viscosity based on Refs.  [48], [49] has been added to the governing equations.

In the context of finite elements, the hp-adaptation is the most effective approach to enhance the accuracy. Such an approach utilizes p-enrichment in regions with smooth solution, and h-refinement in regions with sharp gradients. A common approach in hp-adaptation is to use only one level of refinement for p- and/or h-refinements at each adaptation cycle. In the present work, for p-enrichment, one level of enrichment is used, but, for h-refinement, depending on the smoothness of the solution, either one or two levels of refinement may be used at each adaptation cycle. This approach has been shown to be particularly beneficial in the vicinity of the singularity points.

Due to simplicity, speed, and versatility, a non-conformal mesh adaptation approach has been developed in this study. However, a major difficulty that arises during non-conformal h- and p-adaptation is the generation of hanging nodes. In PG schemes, the discrete solution is required to be continuous over the computational domain. Clearly, hanging nodes can violate this requirement across the interface between refined and unrefined elements. In a previous study  [21], a technique known as constrained approximation  [50], [51], [52], [53], [54], [55], [56], [57] was employed in which a function value at a hanging node is constrained by the function values at adjacent nodes such that a continuous solution is obtained across all element interfaces. In that study, Lagrange polynomials were used as shape functions. Although the implementation was successful up to cubic elements for two dimensional problems, the extension of the method to three dimensional problems was cumbersome. To alleviate this problem, in the present work, hierarchical polynomials have been employed. As it will be described later, this choice results in a more modular implementation.

To represent the geometries accurately, high-order curved boundary edges have been generated. It should be noted that curving the boundary edges could result in collapsed cells when high aspect ratio elements are present in the viscous boundary layers. To address this problem, the interior meshes have been deformed through a linear elasticity solver.

Finally, it should be mentioned that the present adaptive PG methodology has been developed within a general framework, denoted as FUNSAFE (Fully UNStructured Adaptive Finite Elements)  [22], [24], [25], [58], [59], [60], [61], [62], [63].

An outline of the paper is as follows. In Section  2, the governing equations for compressible turbulent flows are reviewed. In Section  3, first, the Petrov–Galerkin formulation is presented for a regular computational mesh. This includes description of the weak implementation of Dirichlet boundary conditions as well as the time integration scheme used to drive solution toward a steady state. Next, the implementation of constraint approximation as well as appropriate basis functions are discussed. In Section  4, the details of adjoint-based adaptation including error estimation formulation and decision making mechanisms for grouping elements in h-refinement and p-enrichment groups are discussed. In Section  5, the mesh curving strategy is explained. In Section  6, the numerical test cases are presented and finally, in Section  7, conclusion are made.

Section snippets

Governing equations

The governing equations consist of the compressible Reynolds Averaged Navier–Stokes (RANS) equations coupled with the negative variant of the one-equation Spalart–Allmaras (SA) turbulence model  [64]. In the conservative form, these equations can be written as Qt+Fixi=SinΩ. Here, the bold letters denote vector variables due to multiple equations, and i indexes the spatial dimension. Also, as seen in the following, () denotes a vector in nsd spatial dimensions. The vector of the

Spatial discretization

To start, the strong form of the problem is written as an initial boundary value problem: L(q)=SxΩandt[0,)Fi=FibxΓFandt[0,)q(x,t)=qDxΓDandt[0,)q(x,0)=q0(x)xΩ where, ΩRnsd is a bounded domain with Lipschitz-continuous boundary Γ, Fib are the prescribed boundary fluxes through ΓF portion of the boundary, and qD is the Dirichlet boundary condition on the ΓD portion of the boundary, and the operator L is defined as (see also Refs.  [66], [67]) L[Aq]t+[AiE]xixi(Gijxj) where [

Adjoint-based adaptation

In many simulations, a specific functional output, usually defined in an integral form, is of particular interest. Lift and drag coefficients are familiar examples in aeronautical applications. In such cases, output-, or adjoint-based methods perhaps offer the most reliable approaches for mesh adaptation, as they target the chosen functional and try to adapt the mesh such that a prescribed precision is ensured. For this purpose, the sensitivity of the functional with respect to the local

Mesh curving strategy

A common practice in CFD, and particularly in the second-order finite-volume schemes, is that the boundaries of the geometry are represented by a series of linear elements. However, to achieve higher accuracies, an increased conformity is required to account for surface curvature. Therefore, a mechanism is required to project the boundary edges to the exact geometry. However, such projection may generate collapsed elements near the boundary, especially when high aspect ratio elements are used,

Numerical results

In this section, the developed adjoint-based adaptive algorithm is applied to five numerical examples including four subsonic flows, and one transonic flow. For four examples, the NACA0012 airfoil is used and the exact geometry of the airfoil is calculated based on a modified formula from Turbulence Modeling Resource (TMR) website  [76], which is supported by NASA Langley Research Center for verifying and validating turbulence models. This formula results in a sharp trailing edge. y=±0.594689181

Conclusion

In this study, an adjoint-based adaptation algorithm was implemented within a Petrov–Galerkin finite-element method. The mesh modification mechanisms included h- and hp-adaptation which were performed in a non-conformal manner. The constrained approximation method was utilized to retain the continuity of the solution space. The resulting method can be utilized within any continuous Galerkin method and it is particularly beneficial for multidisciplinary applications. For the geometric surfaces,

Acknowledgment

This work was supported by Tennessee Higher Education Commission Center of Excellence for Applied Computational Science and Engineering.

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    Present address: Computational AeroSciences branch, NASA Langley Research Center, Hampton, VA 23681, United States.

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