In this chapter the application of the discrete adjoint method to shape optimisation for fluid dynamics problems has been discussed. The method has been successfully applied for two discretisation techniques, namely finite elements and finite volumes. Since shape optimisation invariably requires repetitive evaluation of the performance criterion and its derivatives, particular emphasis has been given to the need for efficient solution tech- niques for the linear systems arising in both the forward and the adjoint problems. It has been demonstrated that efficient solution strategies for the forward problem lead to efficient solution methods for the adjoint problem in a natural way, requiring only a few modifications. Various aspects arising in the application of the discrete adjoint method have been discussed in Section2.6, including the computation of partial derivatives of the discretisation with respect to the mesh. The utility of the presented approaches has been demonstrated by successfully applying them to two shape optimisation example prob- lems.
A more detailed discussion of conclusions regarding this chapter, including a discus- sion of opportunities for future research, will be given in Chapter 4 together with the conclusions for the second major part of this thesis. This second part deals with an en- tirely different application of the discrete adjoint method: adaptive mesh design.
Adaptive mesh design
In this chapter we consider adaptive mesh design as a topic of its own, not limited to problems from fluid dynamics. Thus, the focus is on PDEs in general, using a model problem and the finite element discretisation to develop a (hopefully) general approach. The discussion is meant to be independent of the previous chapter, apart from Section
3.4.2.2, where some results from Chapter 2regarding the application of the discrete ad-
joint method in the finite element discretisation are used.
3.1
Introduction
The use of a posteriori error estimation in order to guide local mesh refinement is now common in the finite element (FE) solution of PDEs [3, 13, 74, 107]. Such techniques not only provide a reliable indication as to the overall accuracy of a computed solution, but also provide a reliable indication as to which regions of the computational domain contribute most (and least) greatly to the overall error in a given solution. Recently, this approach has been augmented by the development of a posteriori error estimation tech- niques for quantities of interest which are derived from the solution of the PDE. Typical examples are described in [11,37]. This development is significant since it is frequently the case that quantities such as drag, lift, local fluxes, etc., are of more interest to the user than the overall solution. Hence the numerical solution procedure should seek to approximate these chosen quantities as efficiently as possible.
For the purposes of developing efficient adaptive finite element software reliable error estimation, whilst necessary, is not sufficient. Some mechanism is also required for using this information in order to construct an improved trial space from which to seek a new solution. One possibility is to locally enrich the polynomial order of the FE trial space (p- refinement) in regions of the domain where the error is largest and the solution is judged to be sufficiently smooth [2, 9]. Alternatively, isotropic local refinement (h-refinement) of the FE mesh may be used [70,83] and recently significant research has focused on the efficient combination of these two [4, 78]. In this work our focus is only on techniques for improving the trial space based upon adapting the FE mesh, rather than enriching the polynomial degree, which we take to be piecewise linear throughout this chapter.
For many problems isotropic local refinement of the current mesh provides a per- fectly satisfactory mechanism for the adaptive procedure. However, this is not always the case. A significant number of practical problems have solutions which possess features that are highly anisotropic (shocks or boundary layers for example). In such situations, unless one starts from a mesh designed using prior knowledge concerning the location and orientation of such features, regular mesh refinement is generally far from optimal. Numerous authors have considered mechanisms for addressing this problem through the development of techniques for anisotropic refinement. Examples include the approach of [32], which makes use of a solution-dependent metric based upon an approximation to the Hessian matrix, or the technique described in [81], which is suitable for tensor product meshes. See also the work of Kunert (e.g. [53, 54]) on a posteriori error estimation for anisotropic meshes or the experimental results of [7].
This chapter investigates an alternative approach to the problem of automatically adapting a mesh, or meshes, based upon a posteriori error estimates for problems whose solutions exhibit anisotropic behaviour. This is based upon not only computing an error estimate, but also calculating the sensitivity of this estimate to the positions of the nodes in the current mesh, which can be done efficiently with the discrete adjoint method.
This sensitivity information may then be used to improve the quality (in the sense of reducing the estimated error) of the existing mesh without increasing the dimension of the FE trial space. The ultimate goal would be to adjust the mesh in order to position the nodes in locally optimal locations, by employing techniques from mathematical optimi- sation for example. However, this is still an extremely demanding and computationally expensive task, even when the sensitivity information can be computed inexpensively. Furthermore, it is essential to ensure that the error estimate itself remains reliable on the meshes produced and so a number of constraints are proposed to reflect this requirement. An approach has been developed therefore that seeks to balance the goal of reducing the
estimated error for a given mesh connectivity with the need to maintain control over both the computational cost and the quality of the error estimate.
The developed approach falls into the class of r-refinement methods (e.g. [10,12,48,
56, 99]), which generally covers adaptivity by node redistribution (hence r-refinement). However, the use of the sensitivity of a posteriori error estimates distinguishes it from previous approaches, which either construct the mesh movement as the solution of a non- linear PDE (e.g. [10, 48, 56]) or by solving sequences of local optimisation problems only (e.g. [12,99] solve small optimisation problems for each node in a Gauß-Seidel like fashion). We refer to [10] for a review of r-refinement approaches.
The new approach has been implemented and tested for the solution of a class of singularly-perturbed reaction-diffusion equations. These problems were selected because a priorianalysis is available to guide the design of anisotropic meshes, for example using so-called Shishkin meshes (see e.g. [8] and the survey article [71]), thus allowing com- parisons to be made against our new, more general a posteriori approach. The extension to wider classes of problems is also discussed in the concluding remarks in Section4.2.
Note that the material in this chapter is in large parts a reproduction of our previous work [76].