4.1.1. What to do with this estimate?

The right-hand side is a spatial integral, the integrand of which should be used to decide where to refine the mesh. The iso-distribution of the error can be approximated by refining according to a tolerance, as in Becker and Rannacher (1996). In Giles and Suli (2002a), it is proposed to use this right-hand side as a correction that importantly improves the quality (in particular the convergence order) of the approximation of j by setting

However, by substituting w^∗ by we introduce an error in which results in being the main error term when we use j^corrected. In Venditti and Darmofal (2002, 2003), it is proposed to keep the corrector and to adapt the mesh to this higher order error term, that is,

[4.4]

or, equivalently,

[4.5]

In order to evaluate numerically these terms, Venditti and Darmofal (2002, 2003) chose to approach these approximation errors, that is, and w_h − w, by interpolation errors:

[4.6]

relying on the differences between the linear representation and a quadratic representation reconstructed on a finer mesh.

4.1.2. Adjoint-L¹ approach

In the approach described in this chapter, metric analysis and goal-oriented analysis are complementary. Indeed, a metric-based method specifies the object of our search through an accurate description of the ideal optimal mesh, while a goal-oriented method specifies precisely the purpose of the search, specifying which error will be reduced. It is then very motivating to seek for a combination of both methods, with the hope of obtaining a metric-based specification of the best mesh for reducing the error committed on a target functional. A few works address this purpose. In Venditti and Darmofal (2003), an anisotropic step relying on the Hessian of the Mach number is introduced into the a posteriori estimate. In Rogé and Martin (2008), an ad hoc formula gives a better impact to the anisotropic component. This chapter presents a different approach to the combination of Hessian and estimates.

The first key point is to work in a continuous (non-discrete) formulation by following the continuous interpolation analysis described in Chapters 3 and 4 of Volume 1. But feature-based metric methods minimize an interpolation error, the deviation between an exact sensor and its linear interpolation on the mesh. This assumes the knowledge of the solution; this is an a priori standpoint. In contrast, goal-oriented methods are generally envisaged from an a posteriori standpoint; we refer to Becker and Rannacher (1996), Verfürth (1996), Apel (1999), Giles and Suli (2002a), Picasso (2003), Formaggia et al. (2004). With that option, the error committed is element-wise known on the mesh under study. Mesh refinement scheme based on a such a posteriori estimations relies on an equi-distribution principle and is thus intrinsically isotropic. Fortunately, goal-oriented methods do not need to be systematically associated with an a posteriori analysis. According to Babuška and Strouboulis (2001), a priori analysis can bring many useful information. Check these information such as anisotropy (see Formaggia and Perotto 2001). This chapter will demonstrate that the goal-oriented error can be globally expressed by an a priori analysis containing anisotropy information. For exploiting this information, a third key point is to work with a numerical scheme that allows expressing the difference Ψ_h(w) − Ψ(w) in term of interpolation errors. This can be done in a straightforward way by considering finite-element variational formulations.

4.1.3. Outline

This chapter begins with the introduction of a theoretical abstract framework in section 4.2. Within this framework, a first a priori goal-oriented error estimate (equation [4.10]) is derived. Its application to the compressible Euler equations is then studied in section 4.3 for a class of specific Galerkin-equivalent numerical schemes. From this study, a generic anisotropic error estimate (equation [4.13]) is expressed. The estimate is then minimized globally on the abstract space of continuous meshes (section 4.4). We give in section 4.5 some details on the main modifications of the adaptive loop as compared to classical Hessian-based mesh adaptation. The practical optimal metric field minimizing the goal-oriented error estimate is then exhibited. 3D detailed examples conclude this chapter by providing a numerical validation of the theory (section 4.6).

4.2.1. Assumptions and definitions

Let V be a Hilbert space of functions. We write the state equation under a variational statement:

[4.7]

where the operator (, ) is the Hilbertian product, and w is the solution of this equation. Symbol (Ψ(w),φ) holds for a functional that is linear with respect to test function φ but a priori nonlinear with respect to w. The continuous adjoint w^∗ is solution of

[4.8]

where g is the jacobian of a given functional j. Let V_h be a subspace of of finite dimension N, and we write the discrete state equation as follows:

Then, we can write

[4.9]

For the a priori analysis, we assume that the solutions w and w^∗ are sufficiently regular to be interpolated:

and that we have an interpolation operator:

4.2.2. A priori estimation

We start from a functional defined as: j(w) =(g, w), where g is a function of V. Our objective is to estimate the following approximation error on the functional:

as a function of continuous solutions, continuous residuals and discrete residuals. The error δj is split as follows:

δj is now composed of an interpolation error and an implicit error, Π_hw − w_h. Let us introduce the following discrete adjoint system:

We derive the following extension of δj with the choice ψ_h = Π_hw − w_h:

According to [4.9], we have

which gives, by using a Taylor extension,

where the remainder R₁ is

Thus, we get the following expression of δj:

We now apply a second Taylor extension to get

with remainder term

This implies

In contrast to an a posteriori analysis, this analysis starts with a discrete adjoint . However, our purpose is to derive a continuous description of the main error term. Thus, we get rid of the discrete solutions in the dominating terms. To this end, we re-write δj as follows:

where

The latter expression of δj can be even more simplified because of the continuous adjoint of equation [4.8], leading to

[4.10]

At least formally, the R_i and the D_k are higher order terms, and the first term in the right-hand side of [4.10] is the dominating one. In order to exhibit, from [4.10], a formulation specifying the optimal mesh, we need to make the physical and numerical models more precise.

4.3.1. Variational analysis

Equation [1.5] from Volume 1 will play the role of equation [4.7] in the abstract analysis of the previous section. For the discretisation, we consider a discrete domain Ω_h and a discrete boundary Γ_h, which are not necessarily identical to the continuous ones. The MUSCL scheme introduced in Chapter 1 of Volume 1 can be written as a variational Galerkin-type scheme complemented by a higher order term, which can be assimilated to a numerical viscosity term D_h. This gives, ∀ϕ_h ∈ V_h,

[4.11]

The numerical diffusion term is of higher order as soon as it is applied to the interpolation of a smooth enough field W on a sufficiently regular mesh (see Mer 1998 for a detailed analysis):

As a result, the dissipation term will be neglected in the same way we neglect the remainders R_i and D_k of relation [4.10]. Even in the case of a flow with shocks, we get inspired by the Hessian-based study of previous chapters and choose to neglect the error term from artificial dissipation.

4.3.2. Approximation error estimation

Returning to the output functional j(W) = (g, W) and according to estimate [4.10], the main term of the a priori error estimation of δj becomes

where W^∗ is the continuous adjoint state. Using the exact solution W in equations [1.5] and [4.11] while neglecting the dissipation D_h leads to

By integrating by parts the previous estimate, we get

where fluxes are given by

By definition, F_h is the linear interpolate of F, that is,Π_hF = F_h, thus we have

[4.12]

We observe that this estimate of δj is expressed in terms of interpolation errors for the fluxes and in terms of the continuous functions W and W^∗. The integrands in [4.12] contain positive and negative parts, which could compensate for some particular meshes. In our strategy, we prefer to avoid these parasitic effects in our estimate³. To this end, all integrands are bounded by their absolute values:

[4.13]

4.5.1. Adjoint solver

As compared to Feature-based mesh adaptation, an extra step to add to the solver is the resolution of the linear system providing the adjoint state: where g_h is the approximated jacobian of j(W_h) with respect to the conservative variables and is the adjoint state. The state is solved with the linearized implicit pseuso-unsteady method using a spatially first-order accurate Jacobian A_h. It is described in Chapter 1 of Volume 1. We define as the adjoint of A_h. In the Wolf code, the linear systems for (as for W_h) is solve with the iterative GMRES method coupled with an incomplete BILU(0) preconditioner (Saad 2003). Once is computed, its vertex gradient

is recovered by using a L² projection from the neighboring element-wise constant gradients (Alauzet and Loseille 2009a).

4.5.2. Optimal goal-oriented discrete metric

The continuous optimal metric discussed in section 4.4 is composed of a volume tensor field M_go defined in Ω and a surface one defined on Γ_h. For the discrete case, we compute the following:

– for each vertex x of H, the pseudo-Hessian matrix arising from the volume contribution of each component of the Euler fluxes:

where

with denoting the j^th component of the vector ;

– for each vertex x of ∂H, the pseudo-Hessian matrix arising from the surface contribution:

where n =(n₁, n₂, n₃) is the outward normal of Γ.

H_R stands for the operator that recovers numerically the second derivatives of an initial piecewise linear by element solution field. The recovery method is based on the Green formula method, described in section 5.4.3 of Volume 1. The standard L¹ norm normalization is then applied independently on each metric tensor field:

[4.17]

Constants C and depend on the desired complexity N.

A natural synthesis between the volumic pseudo-Hessian H and the surfacic pseudo-Hessian is to define the final optimal metric as

Of course, a special decimation⁴ needs to be applied to match smoothly the boundary metric and the volumic one. In practice, numerical experiments (see Loseille 2008) show that the resulting functional prediction is not notably improved by accounting the boundary metric and that measured convergence order slightly degrades due to the extra vertices. A particular case is the case of a symmetric flow. Computing the complete flow leads to having no special boundary term on the symmetry plan, while computing the half flow adds a couple vertices near the symmetry plan. This tends to show that at least with slip conditions the boundary metric is not necessary⁵.

In the presented experiment, the surfacic terms of the adaptation criterion are neglected.

REMARK 4.1 (Non-convergence of goal-oriented formulation).– We want to emphasize that the goal-oriented formulation is designed to favor the best convergence of the prediction of the scalar functional, but it is not designed to produce the flow field convergence. This point is illustrated in next section.

4.5.3. Controlled mesh regeneration

We use Yams (Frey 2001) for the adaptation of the surface and an anisotropic extension of Gamhic (George 1999) for the volume mesh. When the surface is not adapted, we use Mmg3d (Dobrzynski and Frey 2008).

4.6.1. High-fidelity pressure prediction of an aircraft

We consider the flow around a supersonic business jet (SSBJ). The code Wolf is used. The geometry provided by Dassault-Aviation is depicted in Figure 6.3 (left) of Volume 1. Flight conditions are Mach 1.6 with an angle of attack of 3^◦. As for a body flying at a supersonic speed, each geometric singularity generates a shock wave having a cone shape; a multitude of conic shock waves are emitted by the aircraft geometry. They generally coalesce around the aircraft while propagating to the ground. The goal, here, is to compute accurately the pressure signature only on a plane located 100 m below the aircraft. The scope of this test case is to evaluate the ability of the adjoint to prescribe refinements only in areas that impact the observation region. The functional is given by

[4.18]

In order to exemplify how adjoint-based mesh adaptation gives an optimal distribution of the degrees of freedom to evaluate the functional, this adaptation is compared to the Hessian-based mesh adaptation (the adaptation is done on the local Mach number and the interpolation error is controlled in L² norm) presented in section 6.4 of Volume 1.

We apply the convergent steady fixed point Algorithm 4.1. The adaptive loop is divided into five outer steps. The complexity sequence is [10,000, 20,000, 40,000, 80,000, 160,000]. Each step is composed of six sub-iterations at constant complexity for a total of 30 adaptations. Hessian-based and adjoint-based final adapted meshes are composed of almost 800,000 vertices. They are represented in Figure 4.1 where several cuts in the final adapted meshes for the adjoint-based (top) and Hessian-based (bottom) adaptations are displayed.

For the Hessian-based adaptation, the mesh is adapted in the whole computational domain along the Mach cones and in the wake (see Figure 4.1, right). Such an anisotropic adapted mesh provides an accurate solution everywhere in the domain. But, if the aim is to only compute an accurate pressure signature on surface γ, then we clearly notice that a large amount of degrees of freedom is wasted in the upper part of the domain and in the wake where accuracy is not needed.

With regard to the adjoint-based adaptation, the mesh is mainly adapted below the aircraft in order to capture accurately all the shock waves that impact the observation plane (Figure 4.1, left). On the contrary, areas that do not impact the functional are ignored with the goal-oriented approach: the region over the aircraft, the wake of the SSBJ and in the region just behind the aircraft, only the lower half of the conic shock waves are refined and the angular amplitude of the refined part keeps on decreasing along with the distance to the aircraft.

Another point of interest, which is more technical, is that the Hessian-based adaptation in L² norm prescribes a mesh size that depends on the shock intensity. A stronger shock is then more refined than a weaker one. In this simulation, shocks directly below the aircraft have a lower intensity than lateral or upper shocks emitted by the wings (see Figure 4.1, bottom). Consequently, the adapted meshes with the Hessian-based method are less accurate in regions that directly affect the observation plane. On the contrary, with the adjoint-based strategy, shocks below the aircraft are not uniformly refined. The shock waves are all the more refined as they are influent on the functional independently of their amplitudes. This has a drastic consequence on the accuracy of the observed functional (Figure 4.2).

As a conclusion, it is clear in the present computation that the Hessian-based adaptation gives a non-optimal result for the accuracy of the output functional. It gives an inappropriate distribution of the degrees of freedom. It is also demonstrated how the adjoint defines an optimal distribution of the degrees of freedom for the specific target. However, it is important to note that the mesh obtained with the Hessian-based strategy is somewhat optimal to evaluate globally the local Mach number field. It is also worth mentioning that the present method is completely automatic and gives an optimal result.

4.7. Conclusion

This chapter has described a method providing the anisotropic adapted mesh minimizing a first error term estimate in the approximation of a functional depending on the solution of a flow problem. This method is based on a formal a priori estimate of the functional approximation error and its minimization in an abstract continuous framework. The estimate is rather pessimistic since it does not take into account possibly compensating errors. In Chapter 5 of this volume, we discuss a more efficient estimate. With the present estimate, the goal-oriented anisotropic adaptation has been applied to the compressible Euler equations. The method has the following features:

– it produces an optimal anisotropic metric uniquely specified as the optimum of a functional and explicitly given by variational calculus from the continuous state and the adjoint state. The coupled system of the metric and the two states is the object of the discretization. This should be put in contrast with the usual process of starting from a (discrete) mesh and then improving it;

– to apply it, there is no need to choose in a more or less arbitrary way any local refinement “criterion” and no need to fix any parameter except the total number of vertices which represents the error threshold;

– mesh convergence is performed in a natural way by increasing the metric complexity at each stage of the mesh adaptation process.

The goal-oriented method is illustrated on a challenging 3D sonic boom problem. Other calculations are presented in Loseille (2008); Loseille et al. (2010a,b) and Alauzet and Loseille (2010). Numerical experiments show that the goal-oriented method enjoys at best level the advantages of Hessian-based anisotropic methods and of goal-oriented methods. As compared to the Hessian-based method, the anisotropic stretching of the meshes is not lost but even more strengthened and better distributed along shocks. Compared to traditional goal-oriented methods, the anisotropic goal-oriented method behaves like a goal-oriented method, but also naturally takes the anisotropy related to the functional into account.

4.8. Notes

This chapter revisits the study of Loseille et al. (2010b).

A paradox. O. Pironneau raised some paradox in the proposed goal-oriented adaptation algorithm. The question was: How can you adequately adapt with a criterion expressed in terms of the state and adjoint state if you do not adapt for a better state and a better adjoint?

This question is related to a particular step in the design of the mesh adaptation algorithm. Indeed, the first step in our theory is to apply a continuous metric analysis, which produces a continuous optimality system involving continuous state, continuous adjoint and continuous stationarity with respect to the metric.

In a second step, we have to discretize the whole optimality system. For this, an option could be to use a sequence of uniformly refined meshes. State and adjoint would be well refined. This option would finally produce (with a prohibitive computational cost) the optimal metric.

Once we have a good approximation of the optimal metric, we can generate unit adapted meshes based on it. By construction, these meshes are adequate for the evaluation of the functional through the evaluation of the state. In other words, the state computed with an unit mesh of the metric is a good mesh for the functional evaluation. At the same time, with this adapted mesh, the state is not well computed since only its features influencing the functional are supposed to be well computed. Adapting for a better state would then be not optimal.

Adapting for a better adjoint is a somewhat second-order approach, since it does not influence directly the quality of the approximation of the functional, but solely how close is the metric from the perfectly optimal one: formally, an ε-large deviation of the (unconstrained) quasi-optimum provokes an ε²-large deviation of the (smooth) functional value.

Lastly, an examination of [4.4] and [4.5] show the adjoint itself does not need to be accurately computed, as far as RHS of [4.4] is accurately computed. Now this RHS is equal to the RHS of [4.5], which is minimized in the present method. □