Crédito BCP

As already discussed in Section 2.2.6, automatic mesh generators, such as [77, 80] for example, may result in meshes which depend in a discontinuous manner on the shape parameters F , even the number of elements may not be constant. As a compromise between the smoothness properties of parametric meshes and the geometric flexibility of automatically generated meshes, we proposed to use a hybrid approach of defining a base

mesh by means of an automatic mesh generator and then use deformed versions of it as a parametric mesh in a neighbourhood of the base shape parameters. The definition of the deformations as well as a discussion of how these have to be taken into account in the evaluation of the derivatives with respect to the shape parameters are the subject of this subsection.

The stationary Lamé equations (linear elasticity) are a mathematical model for elastic deformations of solid bodies under internal or external forces, see e.g. [19]. The pure Dirichlet problem, find u ∈ H1_g(Ω)2such that

aL(u, v) = 0 ∀u ∈ H10(Ω)2 (2.6.21) aL(u, v) := Z Ω λ (∇ · u) (∇ · v) + 2 µ 2

∑

can be used to propagate a deformation g of the boundary of a domain into the interior, resulting in a deformation vector field u of minimal potential energy. This model contains two material constants λ > 0 and µ > 0 (Lamé constants) which form a continuous ana- logue to spring constants in a discrete network of springs. In this present work they are chosen to take the values for steel,

µ = 7.7 · 104, λ = 1.15 · 105. (2.6.23)

Finite element discretisation of (2.6.21) on the reference mesh allows a node-wise specifi- cation of a deformation g_hof the boundary, i.e. the displacement vectors for each bound- ary node. In order to define the displacement of the interior nodes an equation system of the form " K_i,i K_i,b 0 I # " u_i u_b # = " 0 g # (2.6.24) has to be solved, where the subscript ∗idenotes a block corresponding to interior nodes

while ∗bdenotes a block corresponding to boundary nodes. The vector s of node positions

in the deformed mesh is defined as the positions in the base mesh x₀ plus the displace- ments due to the deformation,

s = x₀+ u. (2.6.25)

This basic linear elasticity approach has one major disadvantage: regions of the mesh with relatively small elements (e.g. to resolve boundary layers) are treated the same way

as those with large elements, even though deformations in these regions of small ele- ments have a much stronger influence on the mesh quality than in the coarser regions of the mesh. In [87] a modification to the discretised Lamé equations was proposed in order to address this issue. From a modelling point of view this modification adapts the param- eters λ and µ such that the material is more rigid in regions of small elements, thus the deformations in those areas will tend to be smaller than in regions of large elements. This can easily be implemented by choosing λ and µ to be element-wise constant and mul- tiplying the base values (2.6.23) by |Ak|α, where α ∈ R is a parameter and |Ak| denotes

the surface area of element k. The tests in [87] suggest that α = −1 forms a reasonable compromise, balancing the deformations between areas of small elements and areas of larger elements. Conveniently this parameter choice is equivalent to dropping the term |A_k| completely from the assembly routines14_{, thus it even simplifies the computations.} For these reasons the choice α = −1 is used in this present work as well. Note that even with these modifications a multigrid preconditioned CG solver performs optimally for the symmetric reformulation of (2.6.24). Thus, the costs for solving these problems are comperatively small in the context of the Navier-Stokes solver.

To illustrate the resulting mesh deformations, Figure 2.15 shows a close-up of two superimposed versions of the mesh around an interior obstacle in a channel (Example

2.2from Section2.7.1.2): one before and one after the mesh deformation. For moderate deformations of the boundary the mesh quality in terms of the size of the interior angles of the elments is usually maintained. However, strong deformations may result in degen- erating meshes and thus re-definition of the base mesh may be required as discussed in Section2.2.6.

Remark 5. Note that simply deforming the boundary of the mesh only is not appropriate, because this would drastically restrict the amount by which the shape parameters can change before remeshing becomes necessary. For example a mesh cell can degenerate if the boundary is moved inward. Further, the quality of the boundary cells would strongly depend on the shape parameters, and the changes in this quality might even affect the quality of the discrete approximation of the performance functional I more strongly than the actual change of the boundary geometry. On the other hand, if the interior mesh nodes are moved in an appropriate manner as well, then these effects are far less pronounced. To illustrate these issues we ask the reader to look again at Figure2.15. Note that some of the smaller elements at the top of the obstacle would have almost collapsed if the interior nodes of the mesh where not moved as well. Yet, the fully deformed mesh is of similar quality as the initial mesh.

14_{The surface area |A}

-0.4 -0.2 0 0.2 0.4 0 0.2 0.4 0.6 0.8 1 initial deformed

Figure 2.15: Mesh deformation by means of linear elasticity

An implication of equations (2.6.25) and (2.6.24) is that for the deformed mesh the node positions of the interior nodes are linearly dependent on the boundary displacements g,

s_i= x_0,i− K_i,i−1K_i,bg and s_b= x_0,b+ g.

This dependency has to be taken into account in the computation of the derivatives DI/DF . This is easily done applying the adjoint equation approach, yielding

DI Dg= DI Ds_b− K T i,bKi,i−T DI Ds_i, (2.6.26)

where in analogy to (2.6.24) the vector of node positions s is split into interior and bound- ary parts. The derivatives with respect to the node positions in the mesh DI/Ds are computed using the discrete adjoint method with the approaches discussed in Subsec-

tion 2.6.2.1. The vector of boundary displacements g itself is a function of the shape

parametersF . Thus the computation of DI/DF is completed by DI DF = DI Dg  ∂ g ∂F  .

Remark 6. Note that this deformation approach has advantages in the context of multi- grid solution techniques as well. Successful application of multigrid requires a hierarchy of meshes, with the coarsest meshes in the hierarchy consisting of a relatively small num- ber of elements or mesh cells only. Such hierarchical meshes are most easily created by refining an initial coarse mesh. However, for domains with curved boundaries some means of adjusting the refined mesh to fit the domain is required, for which the mesh deformation approach can be easily applied.