6.2 Semi-geometric monotone multigrid methods
6.2.1 Conversion into a monotone multigrid method
In this paragraph, we outline the additional ingredients required to change the linear semi- geometric multigrid method into a monotone method for contact problems. The details about the characteristics of obslib++ can be found in [124]. We focus on the issues which are special to the present purpose.
Let a sequence of non-nested vector-valued finite element spaces (X`)`=0,...,L be given.
Also assume that the prolongation matrices ( ¯Π``−1)`=1,...,L with the block structure (6.1)
have been computed according to a suitable transfer concept. To solve the Signorini prob- lem, namely the variational inequality (1.19) in the finite element space XL, one needs
to distinguish between the components of the displacements in the normal direction and the tangential directions at the possible contact boundary ΓC. This may be done by an
orthogonal transformation, e. g., realized as local Householder reflections. At each node p ∈ NL∩ ΓC, the basis (ei)1≤i≤3 is rotated to a new system (epi)1≤i≤3 such that ep1= n(p);
see [129]. The Euclidean basis vectors remain unchanged at the majority of nodes, i. e., we have epi = ei for p ∈ NL\ (NL∩ ΓC), 1 ≤ i ≤ 3. This yields a locally modified basis
Λ0L= (λLpepi)p∈NL,1≤i≤3 of XL. In particular, the non-penetration condition becomes KL= {v ∈ XL| v(p) · e1 ≤ g(p), ∀ p ∈ NL∩ ΓC}
if v ∈ XL is written with respect to Λ0L as we do in the following.
The blocks from (6.1) of the prolongation matrix ¯ΠLL−1 ∈ R3nL×3nL−1 which are as- sociated with possible contact nodes are adjusted accordingly. We denote the resulting matrix, which is the representation of the transfer operator from XL−1to XL with respect
to the standard basis ΛL−1 and the modified (“rotated”) basis Λ0L, by ¯Π 0L
L−1. For the
nodes p ∈ NL and q ∈ NL−1, let the entries of the block ( ¯Π 0L
L−1)pq ∈ R3×3 be denoted by
( ¯ΠLL−1)ijpq, 1 ≤ i, j ≤ 3. We have the analogous notation for the blocks ( ¯Π``−1)pq from (6.1)
associated with nodes p ∈ N` and q ∈ N`−1.
Consequently, the definition of the coarse level bases is slightly more involved than before in Section 6.1.6, where the block entries of the prolongation matrices were diagonal
6.2 Semi-geometric monotone multigrid methods 151
at all levels. Here, the basis eΛL−1 = (eλ L−1 q,j )q∈NL−1,1≤j≤3 of VL−1 reads as e λL−1q,j := X p∈NL 3 X i=1 ( ¯Π0LL−1)ijpqλLpepi, ∀ q ∈ NL−1, 1 ≤ j ≤ 3.
Further, note that the bases of the spaces X` have not been modified at the coarse levels,
i. e., for ` < L. Therefore, to construct the spaces (V`)`=0,...,L−2, we have the recursive
relation e λ`q,j := X p∈N`+1 (Π`+1` )pqλe `+1 p,j , ∀ q ∈ N`, 1 ≤ j ≤ 3, (6.2)
for the bases eΛ` = (eλ `
q,j)q∈N`,1≤j≤3 with ` ∈ {0, . . . , L − 2}.
As indicated before, we employ a non-linear block Gauß–Seidel method, which is de- scribed in detail in [124]. The non-penetration conditions are treated only on the possi- ble contact boundary represented at the finest level, i. e., at the nodes in NL∩ ΓC. Let
ukL ∈ VL = XL be some intermediate iterate. For this approximate solution, we denote
the set of active nodes where the constraints are binding by AkL:= {p ∈ NL∩ ΓC | ukL(p) · e1 = g(p)}.
The paradigm of monotone multigrid methods is that the coarse level correction must not change the active constraints. Therefore, a linear multilevel preconditioner depending on the current iterate is employed which acts only on the subspace
VkL:= {v ∈ VL| v(p) · e1 = 0, ∀ p ∈ AkL} ⊂ VL.
For this purpose, the coarse spaces need to be constructed in a non-trivial way as, in general, the constraints in AkL cannot be represented at the coarser levels ` < L in the standard multilevel basis. This difficulty is overcome by using so-called truncated basis functions. As the approach developed in [121, 124] is of purely algebraic character, this idea can be applied to the semi-geometric framework quite straightforwardly.
To derive a multilevel hierarchy of subspaces of VkL from the semi-geometric spaces (V`)`=0,...,L−1, consider the sets
Ak` := {p ∈ N`| ∃ q ∈ Ak`+1, (Π`+1` )pq> 0}
recursively defined for ` ∈ {0, . . . , L − 1}. We remark that the scalar quantities (Π`+1` )pq
are indeed appropriate in this definition. Moreover, note that these sets are used for a recursive modification of the spaces exclusively. No special treatment of the nodes in Ak` by the smoothing operators at the coarse levels is necessary; the standard smoothers are adequate.
We obtain the truncated coarse level spaces by local modifications. The respective bases ( eΛ`)`=0,...,L−1 are changed to eΛ k ` = (eλ `,k q,j)q∈N`,1≤j≤3 with e λL−1,kq,j := eλL−1q − X p∈Ak L ( ¯Π0LL−1)1jpqλL,kp,1, ∀ q ∈ NL−1, 1 ≤ j ≤ 3, (6.3)
152 6 Numerical results
Figure 6.12. Illustration of the considered contact problem: sketch of the solu- tion (left); an unstructured mesh of the warped geometry (center) with zoom in on a corner (right). The coloring of the left image reflects the component of the displacement field in direction of the obstacle, i. e., downward.
and then, recursively for ` ∈ {0, . . . , L − 2},
e λ`,kq,j := X p∈N`+1 (Π`+1` )pqλe `+1,k p,j , ∀ q ∈ N`, 1 ≤ j ≤ 3, (6.4)
which corresponds to (6.2). This yields coarse spaces (Vk`)`=0,...,L−1 contained in VkL. Note
that generally Vk` 6⊂ V`.
The outlined truncation procedure is efficiently implemented by local algebraic modifi- cations. Concerning the prolongation matrices, we see by (6.3) and (6.4) that only entries between the levels L − 1 and L need to be modified, namely set to zero, in blocks associated with nodes in AkL. In contrast, the stiffness matrices at the levels ` < L change at the nodes in (Ak`)`=0,...,L−1. This holds true in the standard (not semi-geometric) case of [124], too.
All in all, this approach leads to a convergent non-linear iteration process which reduces to a linear subspace correction method as soon as the actual contact boundary is identified; see [122, 124]. Note that the truncated basis functions are of crucial importance for the convergence and the efficiency of the method, particularly for complicated geometries in case d = 3. As a matter of fact, monotone multigrid methods for contact show to be of optimal complexity, even for frictional problems; see also [125, 126].
6.2.2
Numerical results
In the following experiment, we study the asymptotic convergence behavior of the semi- geometric monotone multigrid method for Signorini’s problem. For this purpose, we con- sider the cube-like geometry depicted in Figure 6.12 where one of the six faces, namely the possible contact boundary, is warped. Let the rigid obstacle be the half space given by the tangential plane of the center point of the warped surface. This problem remotely reminds of the one studied analytically by Hertz [110] as early as in the year 1881. For the assessment of the performance of solvers for contact problems, it is crucial to choose a setting where at least the geometry or the obstacle are not flat. Otherwise, the correct discrete contact zone is likely to be identified in the very first step of the iteration.
6.2 Semi-geometric monotone multigrid methods 153 #elements #dof |AL| Cgr Cop KV(2,2) ρ˜V(2,2) KV(3,3) ρ˜V(3,3) KV(4,4) ρ˜V(4,4) 6,568 4,197 9 1.15 1.34 19 (4) 0.238 16 (4) 0.166 14 (3) 0.133 63,645 36,102 44 1.13 1.33 21 (7) 0.295 18 (7) 0.202 16 (7) 0.162 310,198 168,978 155 1.11 1.33 20 (7) 0.325 17 (6) 0.226 15 (6) 0.158 543,408 293,346 240 1.11 1.32 23 (8) 0.357 19 (8) 0.237 18 (7) 0.196 1,037,557 555,198 418 1.12 1.36 25 (8) 0.422 20 (8) 0.309 18 (8) 0.231 1,206,114 643,704 478 1.10 1.31 27 (9) 0.449 21 (9) 0.340 19 (8) 0.269
Table 6.12. Convergence of the semi-geometric monotone multigrid method.
The setting of this study needs to differ from the ones before as the problem is non- linear. In particular, it is not sufficient to study the convergence behavior to the trivial solution only. We prescribe non-zero Dirichlet boundary conditions for the displacements at the top quadratic surface of the domain, pointing towards the obstacle. The influence of the size of the boundary values is discussed below; the data does not need to be physically reasonable for our purposes. May the material parameters of the linear elastic body be chosen as in Section 6.1.6.
A usual estimate for the algebraic error in the context of iterative methods for variational inequalities is the energy norm of the computed correction as indicated in (1.26); see [121]. We denote by KV(ν1,ν2)the number of monotone multigrid steps, i. e., of non-linear V(ν1, ν2)- cycles specified by the construction in Section 6.2.1, to reach an estimated algebraic error less than 10−10starting from the initial iterate u0L= 0. Let ˜ρV(ν1,ν2) be the corresponding approximate asymptotic convergence rate given by (1.26) with k = KV(ν1,ν2)− 1.
Table 6.12 contains the results of the convergence study in case of Dirichlet values of size 0.08. For comparison, the size of the domain is two in all space dimensions. Moreover, the initial distance to the obstacle at the center point of the warped surface is zero. Similar to before in Section 6.1.1, we have generated several completely independent meshes of different sizes, which are then treated as given fine level problems. Then, the coarse meshes are chosen appropriately by means of the outlined strategy to construct a suitable coarse level hierarchy. In addition to the usual problem data, namely the number of elements and the number of degrees of freedom, we state the quantity |AL| := |A
KV(ν1,ν2)
L |. To describe
the convergence behavior over this range of problems, we consider the mentioned rates ˜
ρ, the total number of multigrid steps and also the number of included non-linear steps constituting the transient phase at the beginning of the iteration. The latter are given in brackets. Both has not been necessary for the linear problems presented in the previous section; there, the convergence behavior was completely characterized by the quantities ¯ρ. The presented numbers show a moderate increase of the convergence rates and iteration counts with increasing problem size. Operator and grid complexities stay in a rather small range. The monotone multigrid cycles tend to need one or two more smoothing iterations to achieve results comparable to the linear method. This is an observation which has in principle been made for the case where the truncated spaces are constructed from standard multilevel finite elements in [124], too. However, comparing the results in Table 6.12 with the ones in Table 6.10, recall that both the problem settings and the error measurements
154 6 Numerical results Active set |A L | 0.04 0.08 0.12 0.16 0 200 400 600 800 Numb er of steps 0.04 0.08 0.12 0.16 0 5 10 15 20 Convergence rates ˜ρ 0.04 0.08 0.12 0.16 0 0.1 0.2 0.3
Boundary values Boundary values Boundary values
Figure 6.13. Studying the dependence of the convergence on the magnitude of the prescribed boundary data: number of nodes in the actual contact zone (left); total number of semi-geometric monotone multigrid steps KV(3,3) marked by ×and included non-linear steps marked by ∗ (center); convergence rates ˜ρV(3,3) (right).
The values are given for the third and fifth problem of Table 6.12 in red and blue, respectively.
are different. A detailed investigation of other practical issues in the context of monotone multigrid methods for Signorini’s problem can be found in [124]. For instance, we do not study the effect of highly varying normals here.
Let us now examine the dependence of the convergence on the magnitude of the pre- scribed boundary data. For this purpose, we consider two of the above problems of different sizes and apply Dirichlet boundary values at the top surface in direction of the obstacle between 0.02 and 0.16 in eight independent runs each. Before, in Table 6.12, a medium size of 0.08 was studied. It turns out that the convergence is only weakly affected. The result of this study is illustrated in Figure 6.13 for the monotone multigrid iteration with V(3, 3)-cycles. As an illustration, we give the increasing number of nodes which are in contact, namely |AL|, in the left diagram.
7
Multigrid methods based on
parametric finite elements
In this chapter, we turn our attention to a selected technique for the application of ele- mentary multilevel ideas to problems with complicated boundaries. This is done in the context of the numerical simulation of elastic contact problems. Combining the general multilevel setting with a different perspective, namely an advanced modeling point of view, we present a (monotone) multigrid method based on a hierarchy of parametric finite element spaces. For this purpose, a full-dimensional parameterization is employed which accurately represents the computational domain.
Although the developed concept is related to several considerations made throughout this thesis, especially to the discussion in Section 4.1, we prefer to organize this as a supple- ment at the end. Indeed, the development of this particular focus does not need to be linked too closely to the previous chapters. However, we mainly do this because the purpose of the parametric finite element discretization put forward in this chapter is two-fold. On the one hand, it allows for an elegant multilevel hierarchy to be used in the mentioned multigrid algorithms. But, on the other hand, it comes with particular advantages for the modeling of contact problems. As a matter of fact, a combination of the parametric concept with other ideas, which can take advantage of an enhanced representation of the computational domain to improve some modeling aspects, is certainly advisable. This is elaborated in more detail in Section 7.1. After all, the long-term objective lies in an increased flexibility of hp-adaptive methods for contact problems.
7.1
Introduction
In the numerical simulation of elastic contact problems, the treatment of the non-penetra- tion conditions at the potential contact boundary is of particular importance for both the quality of a finite element approximation and the overall efficiency of the algorithms. A vital challenge is to achieve an accurate description of geometric features, e. g., of warped sur- faces, often incorporated in three-dimensional models from computer-aided design (CAD). Here, we investigate a new connection of different numerical methods, namely modern dis- cretization techniques for partial differential equations on complex geometries on the one side and fast multilevel solvers for constrained minimization problems on the other side.
It is fair to say that the development of hp-adaptive methods for contact problems has not yet reached a mature state; see, e. g., [54] and the references therein. Partly, this is due to the difficulties concerning the geometric representation of the computational domain. A generally accepted paradigm is, though, that high order (finite element or boundary element) methods need high order meshes [114, 140]. This is especially difficult for three-dimensional multi-body contact problems. In this case, the application of non- conforming domain decomposition techniques [173] to realize the information transfer across geometrically non-matching warped contact interfaces is a highly demanding task. For low order finite elements, this has been achieved, among others, by the author; see [71].
156 7 Multigrid methods based on parametric finite elements
The perspective we offer here is a parametric finite element method. For hp-adaptive methods it is convenient to have a parameterization describing the geometry accurately ready to hand. This is because a change of the computational domain due to locally altered polynomial degree is not desirable. Therefore, it is reasonable to uncouple the representation of the geometry on the one hand and of a scale of approximation spaces for the discrete solution on the other hand. These two purposes are usually not separated properly. But of course, one can find curved elements of other than isoparametric structure in some form or another in the literature; see, e. g., [93, 205] or the monograph [56] and the references therein. Note that, for similar reasons, an “isogeometric” concept, which uses NURBS bases for both the description of the geometry and the discrete solution of the differential equation, has been introduced in [114].
For practical computations, the development of fast and robust solvers is equally im- portant. As this issue has not yet been in the main focus of, e. g., the isogeometric analysis [114], we would like to contribute ideas from the field of multilevel methods for variational inequalities. More precisely, as indicated before, we show how to use a monotone multigrid method to efficiently solve the non-linear contact problem discretized with low order para- metric finite elements. Note that the actual treatment of higher order elements is beyond the scope of the present discussion.
To obtain multilevel parametric finite element spaces in case d = 3, we use a full- dimensional parameterization, constructed by tetrahedral transfinite interpolation [158] of CAD data, to lift standard Lagrange elements to the computational domain. Note that, similarly, a surface parameterization has been used in a wavelet Galerkin scheme for bound- ary integral equations, see [108]. Such a procedure may serve as an essential prerequisite to tackle the problems mentioned above. In particular, many of the issues arising in the generation of p-version meshes for curved boundaries [140] can be avoided in a quite ele- gant way. In this sense, although rather expensive, the use of a high order parameterization permits maximal freedom in an hp-adaptive discretization scheme. We presume that the present concept can also be combined with the ideas in [71].
All in all, the results presented in this excursus constitute at least a little progress on the way to an efficient hp-adaptive numerical simulation of contact problems in case of complex three-dimensional geometries.