Abstract
In this chapter, we shall present a few among many applications of mixed methods to plate problems. In the first section, we shall describe a mixed method for the linear thin plates theory and in the second, a dual hybrid method. In the last section, we shall report some recent results on the discretisation of the Mindlin-Reissner formulation for moderately thick plates.
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In this chapter, we shall present a few among many applications of mixed methods to plate problems. In the first section, we shall describe a mixed method for the linear thin plates theory and in the second, a dual hybrid method. In the last section, we shall report some recent results on the discretisation of the Mindlin-Reissner formulation for moderately thick plates.
10.1 A Mixed Fourth-Order Problem
10.1.1 The ψ − ω Biharmonic Problem
Let us now see, as a new example of application of the abstract results of Chaps. 4 and 5, some simple cases of fourth-order problems. We shall start with formulation (1.3.65) which we may now rewrite in the form (4.2.6) by setting
We shall denote by (ω, ψ) instead of (u, p) the solution of the problem in order to be consistent with the usual physical notations. It is easy to see that we are now in the situation of Sect. 3.6: the bilinear form a(ω, ϕ) is not coercive on V (nor is it on KerB but only on H : = L 2(Ω)). A loss of accuracy is therefore to be expected. Another pitfall is that we cannot use the abstract existence results of Chap. 4 for the continuous problem and that we must deduce the existence of a solution through another channel. In the present case, we know that the solution of our mixed problem: find \(\psi \in H_{0}^{1}(\varOmega )\) and ω ∈ H 1(Ω) such that
should be a solution of a biharmonic problem
From a regularity result on the biharmonic problem, we know, for instance, that if Ω is a convex polygon [234, 281, 362], for f ∈ H − 1(Ω), the solution of (10.1.5) belongs to H 3(Ω) so that \(\omega = -\bigtriangleup \psi\) belongs to H 1(Ω). It is then direct to verify that we have thus obtained a solution of (10.1.4). This is an example of an “ill-posed” mixed problem. It should be remarked that the discussion of existence made above does not apply when the right-hand side of the first equation of (10.1.4) is not equal to zero.
To get a discrete problem, we take, following the notations of Chap. 2,
The case k = 1 requires a more special analysis [197, 226, 344]. We then have that the constant S(h), appearing in (5.2.40), can now be bounded by S(h) ≤ c h − 1 so that a direct application of Proposition 5.2.6 gives
Indeed, the inf-sup condition is quite straightforward. The operator B is nothing here but the Laplace operator from H 1(Ω) to H − 1(Ω), which is obviously surjective. To check the discrete condition, we use the criterion of Proposition 5.4.3:given ω ∈ H 1(Ω), we want to build ω h ∈ V h such that
We recall, however, that we have chosen Q h ⊂ V h so that (10.1.8) will, a fortiori, hold if we take μ h ∈ V h . However, (10.1.8) is then nothing but a discrete Neumann problem for which a solution exists and can be chosen (it is defined up to an additive constant) so that
It must be noted that the condition Q h ⊂ V h is essential to the above result. In practice, this is not a restriction as (10.1.6) is a natural and efficient choice. Result (10.1.7) is far from optimal and may suggest at first sight that the method is not worth being used. It can however be sharpened in two ways. First it is possible to raise the estimate on \(\vert \omega -\omega _{h}\vert _{0}\) by half an order [197, 345] by a quite intricate analysis using L ∞-error estimates. The second way is a more direct variant of the duality method of Sect. 5.5.5 and shows that the expected accuracy can be obtained for ψ ∈ H 3(Ω), that is,
and under a supplementary regularity assumption
We refer the reader to [107, 189, 342, 345] and [192] for this analysis.
On the other hand, the particular structure of problem (10.1.4) allows the use of sophisticated but effective techniques for the numerical solution [150, 225, 227], so that this method and its variants have a considerable practical interest. In fact, it provides a correct setting for the widely used ψ − ω approximations in numerical fluid dynamics. We refer to [222] for more informations on this subject. Still in the case of fourth-order problems, we could also consider instead formulation (1.3.70) which is more related to plate bending problems. We now set
and we define, following (1.3.70) for \(\underline{\underline{\sigma }}\) and \(\underline{\underline{\tau }}\) in V ,
In order to consider a weaker form of the saddle point problem (1.3.70), we introduce
This enables us to look for \(w \in H_{0}^{1}(\varOmega )\) instead of \(H_{0}^{2}(\varOmega )\), the second boundary condition being implied by this variational formulation as a natural condition. This is again an “ill-posed” mixed problem: we must obtain existence of a solution through a regularity result on the standard problem. Two approaches have been followed in the approximation of this mixed problem. One of them consists in taking (see [300])
With respect to (10.1.14), it is, however, possible to use a second approach and to work not in \(V = ({H}^{1}(\varOmega ))_{s}^{2\times 2}\) but in the weaker space
Discretisations of this space can be built through composite elements. We refer to [262] and [27] for the analysis of this case.
In the first case, the results are the same as for the ψ − ω approximation discussed above. We get, by Proposition 5.2.6, an error estimate which is O(h k − 1). Duality methods (see [192]) would enable us to lift the estimate on ψ at the right level. For the second case, we can have optimal error estimates (see the above references).
10.1.2 Eigenvalues of the Biharmonic Problem
We now briefly consider the possibility of computing eigenvalues of the biharmonic problem using the elements introduced above. If we refer to Sect. 1.2.1 of Chap. 6, we are considering a (0, g) situation. This means that, fortunately for us, we do not need a coercivity condition. Our eigenvalue problem can indeed be written as: find \(\psi \in H_{0}^{1}(\varOmega )\) and ω ∈ H 1(Ω) such that
In the notation of Sect. 6.5.5, we have V = H 1(Ω) and Q = H 0 1(Ω). We take H Q = L 2(Ω) and we assume that Ω is a convex polygon. We then have
so that with obvious notation
For any given polygon, V H 0 and Q H 0 will be slightly more regular, according to the maximum angle (see e.g. [233]).
For every given regular sequence \(\{\mathcal{T}_{h}\}\) of triangulations of Ω and for every integer k ≥ 2, we can take as in [152, 224, 298]:
Notice that \(Q_{h}^{k} = V _{h}^{k} \cap H_{0}^{1}(\varOmega )\). We can now define Π h w in V h as the solution of:
Clearly, (6.5.54)–(6.5.56) hold. Similarly, (6.5.53) holds by taking p I (here ψ I) as the usual interpolant. On the other hand, to check (6.5.52), we have to assume quasi-uniformity of the decomposition and then proceed, as we did for Dirichlet’s problem in (7.1.43), using an inverse inequality to obtain: for v h ∈ KerB h and \(q \in {H}^{3}(\varOmega ) \cap H_{0}^{2}(\varOmega )\),
This shows the utility of the requirement k ≥ 2. However, a more sophisticated proof, following the arguments of Scholz [344], shows that (1.2.50) also holds for k = 1.
We thus have checked all the hypotheses of Theorem 6.5.3 and our eigenvalue problem is properly posed.
10.2 Dual Hybrid Methods for Plate Bending Problems
We now consider as a final example an application of our general theory to hybrid methods. We go back again to Example 1.3.8 and set, for the sake of simplicity, ν = 0 and \({\mathit{Et}}^{3}/12 = 1\). The consideration of the true values would not change the mathematical structure of the problem, but would result in more lengthy formulae. The condition \(D_{2}^{{\ast}}(\underline{\underline{\tau }}) = f\) in (1.3.74) is, in general, difficult to enforce directly. Hence, following [321], we may think of working with stresses satisfying \(D_{2}^{{\ast}}(\underline{\underline{\tau }}) = f\) inside each element of a given decomposition. This will imply that we have to enforce some continuity of the stresses by means of a Lagrangian multiplier; moreover, it will be convenient to assume f ∈ L 2(Ω). In order to make the exposition clearer, we need some Green’s formulae. We have indeed, on any triangle K of a triangulation \(\mathcal{T}_{h}\) of Ω,
for all \(\underline{\underline{\tau }} \in ({H}^{2}(T))_{s}^{2\times 2}\) and v ∈ H 2(T), where
It is essential, in the definition of K n , to consider the derivative ∂ ∕ ∂ t in the distributional sense, that is, to take into account the jumps of \((\underline{\underline{\tau }} \cdot \underline{ n}) \cdot \underline{ t}\) at the corners of K (the so-called corner forces).
It is easy to check that the condition \(D_{2}^{{\ast}}(\underline{\underline{\tau }}) = f\) in Ω is equivalent to
Setting
the problem can now be written as
If now \({\underline{\underline{\sigma }}}^{f}\) is a given element of \(V _{f}(\mathcal{T}_{h})\), that is, a particular solution of \(D_{2}^{{\ast}}(\underline{\underline{\sigma }}) = f\) in each K, we have
where obviously \({\underline{\underline{\sigma }}}^{0} +{ \underline{\underline{\sigma }}}^{f} := \underline{\underline{\sigma }}\). Problem (10.2.8) has now the form (4.2.6), where \(V = V _{0}(\mathcal{T}_{h})\), Q = H 0 2, \(a(\underline{\underline{\sigma }},\underline{\underline{\tau }}) = (\underline{\underline{\sigma }},\underline{\underline{\tau }})\), and \(b(\underline{\underline{\tau }},v)\) is given by (10.2.5). The right-hand side is obviously \(-({\underline{\underline{\sigma }}}^{f},\underline{\underline{\tau }})\) for the first equation and \(-b({\underline{\underline{\sigma }}}^{f},v)\) for the second equation. It is natural to use in V the L 2-norm, and in Q the norm \(\Vert v\Vert _{Q} =\Vert \underline{\underline{D}}_{2}v\Vert _{V } =\Vert \underline{\underline{D}}_{2}v\Vert _{0}\). It is clear that condition (4.2.12), that is, the ellipticity of \(a(\cdot ,\cdot )\), is trivially satisfied in the whole V (and not only in KerB) with α = 1. A different value for E, t, ν would obviously yield a different value for α, but the V -ellipticity will still be true. It is clear that KerB t cannot be empty; indeed, any v with support in a single K will satisfy \(b(\underline{\underline{\tau }},v) = 0\) for all \(\underline{\underline{\tau }}\), and hence is a zero energy mode. However, it is not difficult to see that ImB is closed.
Proposition 10.2.1.
The image of B is a closed subset of \({Q}^{{\prime}} := {H}^{-2}(\varOmega )\) .
Proof.
We have to show that if a sequence \(\chi _{n} := B\underline{\underline{\tau }}_{n}\) converges to χ in H − 2, then \(\chi = B\underline{\underline{\tau }}\) for some \(\underline{\underline{\tau }} \in V _{0}(\mathcal{T}_{h}) =: V\). We first note that
which is quite obvious from (10.2.5) and (10.2.6). Now let \(\phi \in H_{0}^{2}(\varOmega )\) be such that △ 2 ϕ = χ and let \(\underline{\underline{\tau }} := \underline{\underline{D}}_{2}\phi\) (so that \(D_{2}^{{\ast}}\underline{\underline{\tau }} =\chi\)). For every ϕ ∈ H 0 2, we have
Now, since \(\chi _{n} = B\underline{\underline{\tau }}_{n} \rightarrow \chi\) in H − 2, we have
that is, \((\underline{\underline{\tau }}_{n} -\underline{\underline{\tau }},\underline{\underline{D}}_{2}\phi ) \rightarrow 0\) for all ϕ ∈ H 0 2(Ω). This easily implies \(D_{2}^{{\ast}}\underline{\underline{\tau }} = 0\) in each T, so that \(\underline{\underline{\tau }} \in V _{0}(\mathcal{T}_{h})\). Hence, \(\langle \chi ,\phi \rangle = (\underline{\underline{\tau }},\underline{\underline{D}}_{2}\phi ) = b(\underline{\underline{\tau }},\phi ) =\langle B\underline{\underline{\tau }},\phi \rangle\), that is, χ ∈ ImB. □
Proposition 10.2.2.
We have \(\mathrm{Ker}{B}^{t} =\prod _{K}\limits H_{0}^{2}(K)\) .
Proof.
It is obvious from (10.2.5) that if \(\phi \vert _{K} \in H_{0}^{2}(K)\) for all K, then \(b(\underline{\underline{\tau }},\phi ) = 0\) \(\forall \underline{\underline{\tau }}\) and hence ϕ ∈ KerB t. Therefore, we only need to prove that \(\text{Ker}{B}^{t} \subset \prod _{K}\limits H_{0}^{2}(K)\). For this, let ϕ ∈ KerB t, that is,
We want to show that \(\phi \in \prod _{K}(H_{0}^{2}(K))\), that is,
Let ψ be defined in each K by
clearly, \((\underline{\underline{\tau }},\underline{\underline{D}}_{2}\psi ) = 0\) for all \(\underline{\underline{\tau }}\) in \(V _{0}(\mathcal{T}_{0})\) so that from (10.2.12),
However, \(D_{2}^{{\ast}}\underline{\underline{D}}_{2}(\psi -\phi ) = {\bigtriangleup }^{2}(\psi -\phi ) = 0\) in each K, so that we can take \(\underline{\underline{\tau }} = \underline{\underline{D}}_{2}(\psi -\phi )\) in (10.2.15) and obtain \(\underline{\underline{D}}_{2}(\psi -\phi ) \equiv 0\). Since both ψ and ϕ are in H 0 2(Ω), this implies ψ = ϕ so that from (10.2.14), we get (10.2.13). □
Proposition 10.2.3.
We have
where \(\bar{\phi }\) is the function in H 0 2 (Ω) such that
Proof.
By definition, we have
Now from Proposition 10.2.2 and the definition of \(\Vert \chi \Vert _{Q} :=\Vert \underline{\underline{D}}_{2}\chi \Vert _{0}\), we have
It is now an easy matter to check that, for each K,
for \(\bar{\phi }\) defined in (10.2.17) and (10.2.18). Hence, (10.2.21) and (10.2.20) prove (10.2.16). □
We are now able to prove the inf-sup condition
because \(\phi -\bar{\phi }\) is the projection (in Q) of ϕ onto KerB t so that \(\bar{\phi }\) and \(\phi -\bar{\phi }\) are orthogonal in Q.
Remark 10.2.1.
A way of getting rid of KerB t (which is infinite dimensional) is to consider as a space of Lagrange multipliers the space
This is what has been done in [114, 127]. The drawback in the choice (10.2.23) is that the actual transversal displacement w does not belong to \(\tilde{Q}\) so that, as a solution, we have the unique function \(\bar{w}\) in \(\tilde{Q}\) that coincides with w (with its first derivatives) at the inter-element boundaries (as in (10.2.17) and (10.2.18)). □
Let us continue our analysis of problem (10.2.8). We already noted that (4.2.12) is satisfied in our case. Hence, we have to check that the right-hand side of the second equation in (10.2.8) (that is \(-b({\underline{\underline{\sigma }}}^{f},v)\)) is in ImB; this means that we have to find a particular solution of (10.2.8), which is obvious by taking \({\underline{\underline{\sigma }}}^{f} := \underline{\underline{D}}_{2}w -{\underline{\underline{\sigma }}}^{f}\).
We can now go to the discretisation of (10.2.8); for this, we have to choose subspaces \(V _{h} \subset V _{0}(\mathcal{T}_{h})\) and Q h ⊂ Q. For instance, for any triple (m, r, s) of integers, we may choose
Note that V h is made of tensor-valued polynomials of degree ≤ m which are completely discontinuous from one element to another and verify \(D_{2}^{{\ast}}\underline{\underline{\tau }}\,=\,0\) in each T. On the other hand, Q h is clearly infinite dimensional (which is quite unusual); however, this does not show up in the computations, where only the values of ϕ and ∂ ϕ ∕ ∂ n on \(\mathcal{E}_{h}\) are considered. To get coercivity, we now have to choose (m, r, s) in such a way that \(\mathrm{Ker}B_{h}^{t} \subset \mathrm{Ker}{B}^{t}\). This means, in our case, that we have to show
The proof of (10.2.26) (or, rather, the finding of sufficient conditions on m for having (10.2.26)) will be easier with the following characterisation of \(V _{h}^{m}\).
Lemma 10.2.1.
We have
where \(\underline{\underline{S}}\) is defined, for \(\underline{q} = (\alpha ,\beta )\),
Proof.
The inclusion \(\underline{\underline{S}}\,[{(\mathcal{L}_{m+1}^{0}(\mathcal{T}_{h}))}^{2}] \subseteq V _{h}^{m}\) is trivial; the opposite inclusion is an exercise (see [127] for more details). □
We now notice that if \(\underline{\underline{\tau }} = \underline{\underline{S}}(\underline{q})\), then
where \(\underline{t}\) is the tangent to ∂ T. We also notice that
We may now use (10.2.27)–(10.2.30) in (10.2.26) which becomes
Now, (10.2.31) is implied by
(but not vice-versa). Now let k be the degree of \(\mathrm{g\underline{ra}d}\ \phi\) on ∂ T, that is,
The following technical lemma is proved in [127].
Lemma 10.2.2.
If ϕ ∈ H 1 (K) and \(\phi \vert _{e_{i}} \in P_{k}(e_{i})\) (i = 1,2,3), and if
then
where, on each e i , we define \(\ell_{k}^{i}\) as the kth Legendre polynomial (normalised with value 1 in the second endpoint in the anticlockwise order).
Formula (10.2.35), for k odd, directly implies that ϕ is constant on ∂ K. We therefore have a first result.
Proposition 10.2.4.
If \(m + 1 = k =\max (r - 1,s)\) and k is odd, then (10.2.32) holds.
If m + 1 is even, we can apply Lemma 10.2.2 to both ∂ ϕ ∕ ∂ x and ∂ ϕ ∕ ∂ y and get
on each e i . If now r − 1≠s, there must exist a combination of ∂ ϕ ∕ ∂ x and ∂ ϕ ∕ ∂ y on each e i (to get ∂ ϕ ∕ ∂ n) which has degree lower than k. This easily implies that both ∂ ϕ ∕ ∂ x and ∂ ϕ ∕ ∂ y are constants on ∂ K. We therefore have the following result:
Proposition 10.2.5.
If \(m + 1 = k =\max (r - 1,s)\) and r − 1≠s, then (10.2.32) holds.
We are finally left with the last and worst case in which \(r - 1 = s\) is even. We have several escapes. First, brutally, we may take \(m + 1 = k + 1\). It is easy to see that, then, (10.2.32) always holds. As a second possibility, we may take \(m + 1 = k\) and enrich \({(\mathcal{L}_{m+1}^{0}(\mathcal{T}_{h}))}^{2}\) into \((\mathcal{L}_{m+1}^{0}(\mathcal{T}_{h}))_{\mathit{enr}}^{2}\) by adding, in each K, a pair of functions \(\underline{q}\) in \({(P_{m+1})}^{2}\) such that \(\partial q_{j}/\partial t\vert _{e_{i}} =\ell_{ k}^{i}\) (j = 1, 2 and = 1, 2, 3). Again, it is easy to check that (10.2.32) is satisfied if we take the enriched space \((\mathcal{L}_{m+1}^{0}(\mathcal{T}_{h}))_{\mathit{enr}}^{2}\) instead of the original one. Then, of course, we must consider \(V _{h,\mathit{enr}}^{m} = \underline{\underline{S}}\,[(\mathcal{L}_{m+1}^{0}(\mathcal{T}_{h}))_{\mathit{enr}}^{2}]\) instead of V h . Finally, we might give up (10.2.32) and go directly to (10.2.31). It is easy to check that in (10.2.36), the values of c, c 1, γ, and γ 1 must remain constants from one K to another due to the continuity of \(\mathrm{g\underline{ra}d}\ \phi \vert _{e}\) across the edges. Hence, since \(\phi \in H_{0}^{2}(\varOmega )\), we must have \(c = c_{1} =\gamma =\gamma _{1} = 0\) and, actually, (10.2.31) holds for \(m + 1 = k =\max (r - 1,s)\) in any case, that is, also for \(r - 1 = s =\) even. However, we shall see in a moment that (10.2.32) has other basic advantages over (10.2.31) that we are not very willing to give up. We summarise the results in the following theorem.
Theorem 10.2.1.
The condition \(\mathrm{Ker}B_{h}^{t} \subset \mathrm{Ker}{B}^{t}\) holds whenever
Moreover, (10.2.32) holds when (10.2.37) is satisfied, unless \(r - 1 = s =\) even. In that case, (10.2.32) is satisfied by taking m + 1 > k or by using an enriched \(V _{h,\mathit{enr}}^{k-1}\) (between \(V _{h}^{k-1}\) and V h k ) as described above.
The condition \(\mathrm{Ker}B_{h}^{t} = \mathrm{Ker}{B}^{t}\) implies, by Proposition 5.5.2, the existence of an operator Π h from \(V _{0}(\mathcal{T}_{h})\) to V h m such that
However, in view of the use of Proposition 5.4.3, we would also like to show that there exists a Π h which satisfies (10.2.38) and
with c independent of h. Since V h m is finite dimensional, (10.2.39) will always hold, but the constant might depend on h. Now, if (10.2.32) holds, we see that Π h can be defined element by element. Now, the dimension of \(V _{h}^{m}\vert _{K}\) depends only on m, but not on h. A continuous dependence argument on the shape of the element can now prove (10.2.39) without major difficulty (but, to be honest, not quickly); we refer to [127] for a detailed proof of (10.2.39). Once we have (10.2.38) and (10.2.39), we apply Proposition 5.4.3 to prove the discrete inf-sup condition. Then, Theorem 5.2.5 immediately gives
where \(\tilde{w}_{h}\) is the (unique) element in Q h r, s that satisfies \({\bigtriangleup }^{2}\tilde{w}_{h} = f\) in each K and belongs to the set of discrete solutions.
Theorem 10.2.2.
If \(m + 1 \geq \max (r - 1,s)\) (and m + 1 > s for \(r - 1 = s\) is even), we have
with \(t =\min (m + 1,r - 1,s)\) .
Proof.
The proof is obvious from (10.2.40) and the standard approximation results. □
We end this section with a few computational remarks. First, we notice that our discretisation of (10.2.8) has obviously the matrix structure
where A, corresponding to the approximation of the identity in V h m, is obviously block diagonal because V h m is made of discontinuous tensors. Hence, one usually makes an a priori inversion of A, to end with the matrix BA − 1 B t which operates on the unknown w h and is symmetric and positive definite. However, the computation of the right-hand side is, in general, a weak point in the use of dual hybrid methods, unless f is very special (zero, Dirac mass, constant, etc.) and allows the use of a simple \({\underline{\underline{\sigma }}}^{f}\). A few computational tricks for dealing with more general cases can be found in [127, 289, 290]. Here, we recall from [115] a simple method that works for low-order approximations (more precisely, when t in Theorem 10.2.2 is ≤ 2). We first define the operator R : = orthogonal projection onto V h . We then remark that the discretisations (10.2.24) and (10.2.25) of (10.2.8) may be written as
Solving a priori in \(\underline{\underline{\sigma }}_{h}^{0}\) from the first equation and substituting into the second equation, we obtain
Now, the left-hand side of (10.2.44) corresponds to the matrix BA − 1 B t acting on the unknown w h . The right-hand side is actually computable because both \((f,\phi ) - ({\underline{\underline{\sigma }}}^{f},\underline{\underline{D}}_{2}\phi )\) and \((R{\underline{\underline{\sigma }}}^{f},\underline{\underline{D}}_{2}\phi )\) depend (looking carefully) only on the values of ϕ and its gradient at the inter-element boundaries. However, the computation, in general, is not easy. Therefore, in some cases, it can be convenient to use a rough approximation of it, for instance
where the V j are the vertices of K. It can be shown (see [115]) that this involves an additional error of order O(h 2) (essentially because V h contains all piecewise linear stress functions and therefore \(\Vert {\underline{\underline{\sigma }}}^{f} - R{\underline{\underline{\sigma }}}^{f}\Vert _{0} \leq c{h}^{2}\)) and hence this procedure is recommended whenever t ≤ 2 in (10.2.41).
Finally, we provide a few remarks on the choice of the degrees of freedom in V h m and Q h r, s. As we have seen, the unknown \(\underline{\underline{\sigma }}_{h}^{0}\) is usually eliminated a priori at the element level due to the complete discontinuity of \(V _{h}^{m}\). As a consequence, the choice of the degrees of freedom in \(V _{h}^{m}\) is of little relevance. In general, it is more convenient to start from \({(\mathcal{L}_{m+1}^{0}(\mathcal{T}_{h}))}^{2}\) and to derive V h through (10.2.27).
When m is “large” (say m ≥ 4, to fix the ideas), however, the resulting matrix A can be severely ill-conditioned unless the degrees of freedom in V h m are chosen in a suitable way. We refer to [289, 290] for a discussion of this point. On the other hand, the degrees of freedom in Q h r, s are the ones that count in the final stiffness matrix, and, besides, they have to take into account the C − 1 continuity requirements. We sketch in Fig. 10.1 some commonly used choices for different values of r and s.
Some common choices for the space Q h r, s
Remark 10.2.2.
It is impossible to say what is, in general, the best choice for r and s. Numerical evidence shows obviously that the accuracy/number of degrees of freedom ratio is improved for large r and s, at least when the solution is smooth. However, it is clear that the simplest (and most widely used) choice \(r = 3,\,s = 1\) allows a much easier implementation. Similar considerations also hold with the choice of m, in particular in the case of an even \(r - 1 = s\), for instance for \(r = 3,\,s = 2\). The use of the enriched V h, enr 1 implies a smaller matrix to be inverted on each element than with the “brutal” choice V h 2 (11 ×11 instead of 17 ×17), but the latter may allow some simplification in writing the program. □
Remark 10.2.3.
We have used, so far, homogeneous Dirichlet boundary conditions corresponding to a clamped plate. Nothing changes when considering non-homogeneous Dirichlet conditions. If, instead, a part of the plate is simply supported (w = given; M nn = 0) or free (M nn = 0; K n = 0), then we have two possibilities for dealing with them. Let us discuss a simple case: let ∂ Ω = Γ D ∪Γ N and assume that \(w = \partial w/\partial n = 0\) on Γ D and \(M_{n} = K_{n} = 0\) on Γ N . One possibility is to choose Q h r, s so that its elements vanish only on Γ D , and to let V h m unchanged. In this case, the conditions \(M_{n} = K_{n} = 0\) on Γ N will be satisfied only in a weak sense. A second possibility is to choose V h m in such a way that its elements satisfy, a priori, the boundary condition \(M_{n} = K_{n} = 0\) on Γ N . However, care must be taken in this case to enrich conveniently the stress field in the boundary elements so that the inf-sup condition still holds. Otherwise, a loss in the order of convergence is likely to occur. □
Remark 10.2.4.
One may think to use other discretisations of the dual hybrid formulations than the ones discussed here (see, for instance, the previous remarks). In any case, the inf-sup condition should be checked. Although this is not evident from our discussion (because we wanted to deal with many cases at the same time), nevertheless, it is true that to check the inf-sup condition in hybrid methods is basically an easy task. What is really needed is the following: for any element K, the only displacement modes with zero energy on K, that is, the only modes ϕ such that
must be the rigid modes (that is, \(\mathrm{g\underline{ra}d}\ \phi =\) constant on T). If this condition is violated, one can expect trouble (minor or major, depending on the cases). □
10.3 Mixed Methods for Linear Thin Plates
We consider the variational formulation of a problem discussed in Chap. 1 which we recall here for the convenience of the reader. We had
where
In order to use a more compact notation, we set
and write \(L(\underline{\underline{\tau }},\phi )\) as
Assume that we are given a triangulation \(\mathcal{T}_{h}\) of Ω and that we are willing to discretise the stress field \(\underline{\underline{\sigma }}\) by means of piecewise polynomials for which the normal bending moment
is continuous from one element to another. We recall the following Green’s formulae,
valid for all \(\underline{\underline{\tau }}\) and ϕ smooth in K; we recall again that, here, \(\underline{t}\) is the unit tangent (anticlockwise) vector and
If \(M_{\mathit{nn}}(\underline{\underline{\tau }})\) is continuous and ϕ is smooth, we can write
A little functional analysis shows that every integral in (10.3.15) makes sense (at least as a suitable duality pairing), provided \(\underline{\underline{\tau }}\) and ϕ are, respectively, in the following spaces:
Remark 10.3.1 (For mathematicians).
We have to choose p > 2 in (10.3.17) because for ϕ ∈ H 1(K) we have \(\partial \phi /\partial t \in {H}^{-1/2}(\partial K)\) whereas \(M_{\mathit{nt}}(\underline{\underline{\tau }})\) is in \(\prod _{e_{i}}{H}^{1/2}(e_{i})\) but not in H 1 ∕ 2(∂ K). On the other hand, for ϕ ∈ W 1, p, we have \(\partial \phi /\partial t \in {W}^{-1/p,p}(\partial K)\). Since \(M_{\mathit{nt}}(\underline{\underline{\tau }})\) is in H s(∂ K) for all s < 1 ∕ 2 and since \({W}^{-1/p,p}(\partial K) \subset {H}^{-1}(\partial K)\) for s > 1 ∕ p, the boundary integral which appears in (10.3.15) can now be interpreted as a duality pairing between H − s(∂ K) and H s(∂ K) for \(1/p < s < 1/2\) (which is possible since p > 2). □
The Euler equations of (10.3.15) can now be written as:
which has the form (5.1.9) if we set
Unfortunately, problem (10.3.18) and (10.3.19), as it stands, does not satisfy any of the conditions given in Chap. 4 in order to have a well posed problem. However, we know that the original problem (1.2.4) has a solution w. If \(\underline{\underline{\sigma }} = {C}^{-1}(\underline{\underline{D}}_{2}w)\) is in H 1(Ω), that is if the solution w of (1.2.4) is smooth enough, it is easy to check that the pair \((\underline{\underline{\sigma }},w)\) solves (10.3.18) and (10.3.19). Hence, we only have to prove the uniqueness of the solution of (10.3.18) and (10.3.19).
Proposition 10.3.1.
Problem (10.3.18) and (10.3.19) has a unique solution.
Proof.
It is obvious that
Let us now check a weaker inf-sup condition. For every ϕ in Q, we define \(\underline{\underline{\tau }}(\phi )\) by
It is immediate to check that \(M_{\mathit{nt}}(\underline{\underline{\tau }})\) is continuous across the inter-element boundaries, so that
and therefore
It is also easy to check, using (10.3.23) and the Poincaré’s inequality (1.2.14), that
hence, we have from (10.3.25) and (10.3.26) that
Now using (10.3.22) and (10.3.27), we have the desired uniqueness by standard arguments. □
We are now ready to discretise our problem. Following [132] and [261], for any integer k ≥ 0, we set
with the notation of Chap. 2. Note that the space V h in (10.3.28) is made of tensors whose normal bending moment is continuous across the inter-element boundaries. The degrees of freedom for Q h will be the usual ones (see Sect. 2.2). As degrees of freedom for V h , we may choose, for instance, the following ones:
The possibility of choosing (10.3.30) and (10.3.31) as degrees of freedom in V h is shown by the following lemma and by a standard dimensional count.
Lemma 10.3.1.
Let \(\underline{\underline{\tau }} \in (P_{k-1}(T))_{s}^{2\times 2}\) be such that
Then, \(\underline{\underline{\tau }} \equiv 0\) .
Proof.
We only give a hint of the proof. From (10.3.32), we get \(M_{\mathit{nn}}(\underline{\underline{\tau }})\,=\,0\). We first show that \(D_{2}^{{\ast}}(\underline{\underline{\tau }}) =\,0\). This is trivial for k ≤ 1; for k > 1, take \(\underline{\underline{p}} = \underline{\underline{D}}_{2}b\) with \(b = b_{3}D_{2}^{{\ast}}\underline{\underline{\tau }}\) in (10.3.33) to get \(\int _{K}b_{3}{(D_{2}^{{\ast}}(\underline{\underline{\tau }}))}^{2}\,\mathit{dx} = 0\) and hence \(D_{2}^{{\ast}}(\underline{\underline{\tau }}) = 0\). Now use the formula (see Sect. 10.2)
for ϕ ∈ P k + 1(T); thus, we get
and easily obtain that \(\mathcal{K}_{n}(\underline{\underline{\tau }}) = 0\). It is now simple to show that \(\underline{\underline{\tau }} = \underline{\underline{S}}(\underline{q})\) (see (10.2.27) for the definition of \(\underline{\underline{S}}\)) for some \(\underline{q} \in {(P_{k+1}(K))}^{2}\) with \(\underline{q} = 0\) on ∂ K. Therefore, q 1 (for instance) has the form b 3 z with z ∈ P k − 2(K). Let us now choose, in (10.3.33), p 11 such that ∂ p 11 ∕ ∂ y and \(p_{12} = p_{22} = 0\). We then get
so that z = 0 and q 1 = 0. Similarly, one proves that q 2 = 0. □
We are now able to define the operator Π h . We set, for \(\underline{\underline{\tau }} \in V\),
Lemma 10.3.2.
Let Π h be defined by (10.3.37) and (10.3.38). Then, we have
and
Proof.
Formula (10.3.39) is easy to check. Let us prove (10.3.40). From (10.3.12) and (10.3.21), we have
and from (10.3.41), (10.3.37), and (10.3.38), we get (10.3.40). □
Lemma 10.3.3.
If \(\underline{\underline{\tau }}_{h} \in V _{h}\) is such that
then
Proof.
We have, from (10.3.13) and (10.3.21),
Integrating \(\int _{\partial K}M_{\mathit{nt}} \frac{\partial \phi } {\partial t}\mathit{ds}\) by parts and recalling the definition of \(\mathcal{K}_{n}\) in (10.2.3), we then have
Note that (10.3.45) holds for any \(\underline{\underline{\tau }}_{h}\) and ϕ piecewise smooth. If now (10.3.42) holds, we first have \(D_{2}^{{\ast}}(\underline{\underline{\tau }}_{h}) = 0\) by choosing \(\phi \vert _{K} = b_{3}D_{2}^{{\ast}}(\underline{\underline{\tau }}_{h})\) (for k ≥ 2, otherwise the property is trivial). Hence, we are left with
Since \(\mathcal{K}_{n}\) is made of Dirac measures at the vertices and of polynomials of degree less or equal to k − 1 on each edge, it is easy to see that (10.3.46) implies \(\mathcal{K}_{n}(\underline{\underline{\tau }}_{h}) = 0\). Therefore, we have proved that if \(\underline{\underline{\tau }}_{h} \in V _{h}\) satisfies (10.3.42), then \(D_{2}^{{\ast}}(\underline{\underline{\tau }}_{h}) = 0\) and \(\mathcal{K}_{n}(\underline{\underline{\tau }}_{h}) = 0\). We now insert those two equations into (10.3.45) and we get (10.3.43). □
This last property was denoted, in Chap. 5, as Z h (0) ⊂ Z(0). We have seen that, together with the existence of the operator Π h , this property is so important that it can provide optimal error estimates even in desperate situations (no ellipticity, no inf-sup condition) like ours.
Actually, we first remark that (10.3.27) and Lemma 10.3.2 provide, through Proposition 5.4.3, the following inf-sup type condition:
On the other hand, since Q h and V h are finite dimensional, (10.3.22) and (10.3.47) ensure that the discrete problem has a unique solution. We are now ready for error estimates.
Proposition 10.3.2.
If \((\underline{\underline{\sigma }},w)\) is the solution of (10.3.18) and (10.3.19) and \((\underline{\underline{\sigma }}_{h},w_{h})\) is the discrete solution of (10.3.18) and (10.3.19) , then, through (10.3.28) and (10.3.29) , we have
□
The proof is immediate from the standard theory of Chap. 5.
From (10.3.48) and standard approximation results, we then have
Proposition 10.3.3.
With the notation of Proposition 10.3.2 , we have
Proof.
Let ϕ h ∈ Q h to be chosen. From (10.3.47), we have for some \(\underline{\underline{\tau }}_{h} \in V _{h}\)
It is now elementary to see that ϕ h can be chosen in such a way that
With such a choice, we have
so that from (10.3.51), (10.3.54) and (10.3.49) we get (10.3.50). □
Remark 10.3.2.
Result (10.3.50) is not optimal as far as the regularity of w is involved. Actually, it says
while an (s + 1)-norm on w should be enough for optimality. Furthermore, a more sophisticated analysis [44, 192] shows that
for k ≥ 1 and
In particular, the approach of [44] has a special interest because, by a suitable use of mesh-dependent norms in V h and Q h , they can show that the discretised problem (in the new norms) satisfy the abstract assumptions (5.2.33) and (5.2.34) so that optimal error estimates (in the new norms) can be directly obtained by Theorem 5.2.5. Their approach also works for other fourth-order mixed methods, like those analysed in Sects. 10.1 and 10.2. □
Remark 10.3.3.
For the actual solution of the discretised problem, the most convenient method is to disconnect the continuity of \(\underline{\underline{\sigma }}_{h} \cdot \underline{ n}\) and to enforce it back via Lagrange multipliers λ h . Then, one eliminates \(\underline{\underline{\sigma }}_{h}\) at the element level and one solves a symmetric and positive definite system for the unknowns λ h and w h . The procedure is identical to the one described in Sect. 7.2 and we refer to it for a detailed description. As far as the error estimates for the Lagrange multipliers λ h are concerned, recent results have been obtained in [158]. □
Remark 10.3.4.
It is interesting to analyse the relationship between the mixed methods described here and some nonconforming methods for fourth-order problems. For instance, the following result is proved in [23]. Let us consider the space built by means of the Morley element \(\mathcal{L}_{2}^{2,\mathit{NC}}\) described in Example 2.2.6 and let us define
For every \(\phi _{h} \in \mathcal{L}_{2}^{2,\mathit{NC}}\), let ϕ h I be the piecewise linear interpolant of ϕ h (that is \(\phi _{h}^{I} \in \mathcal{L}_{1}^{1}\) and \(\phi _{h}^{I} =\phi _{h}\) at the vertices). Consider now the modified Morley problem: find \(\psi _{h} \in \mathcal{L}_{2}^{2,NC}\) such that
Then, we have
where \((\underline{\underline{\sigma }}_{h},w_{h})\) is the discrete solution of the mixed problem (10.3.18) and (10.3.19) through (10.3.28) and (10.3.29) for k = 0. We note explicitly that, in the case of variable coefficients, the equivalence is more complicated. Also note that \(\partial \psi _{h}/\partial n\vert _{e} =\lambda _{h}\vert _{e}\) for all \(e \in \mathcal{E}_{h}\), where λ h is the Lagrange multiplier introduced in the previous remark. Notice that we have, from [23],
which improves (10.3.50) and (10.3.57) since it requires only H 3-regularity on w. This is particularly striking since the cost for computing ψ h is cheaper (or equal, using λ h ) than the cost for computing \((\underline{\underline{\sigma }}_{h},w_{h})\). □
10.4 Moderately Thick Plates
10.4.1 Generalities
We end this chapter with a hint on the theory for the so-called “Mindlin–Reissner plates”. The corresponding model stands somehow in between the standard three-dimensional linear elasticity and the two-dimensional Kirchhoff theory for thin plates. Let us recall it briefly. Assume that we are given a three-dimensional elastic body that, in absence of forces, occupies the region \(\varOmega \times \left ]-t,t\right [\), where \(\varOmega \subset {\mathbb{R}}^{2}\) is a bounded smooth domain and t > 0 is “small” (but not “too small”) with respect to diam(Ω). This is what we call a “moderately thick” plate. We shall assume, for the sake of simplicity, that the plate is clamped along the entire boundary \(\partial \varOmega \times \left ]-t,t\right [\) and that a vertical load \(\underline{f} = (0,0,f_{3})\) is imposed.
Here below, we present the “Mindlin-Reissner” model following the classical engineering “derivation”. Such derivation is questionable, from the mathematical point of view, at some points, but it has the clear merit of being short and simple. From the mathematical point of view, the derivation of [35] is much more convincing, but it is surely longer and more complicated. As the aim of this book is mainly concentrated on the mathematical properties of models and on their discretisations rather than on the modelling aspects, we decided to stick to the simpler choice.
The Mindlin model assumes that the “in plane” displacements u 1 and u 2 have the form
and that the “transversal” displacement u 3 has the form
The corresponding strain field therefore takes the form:
and assuming a linear elastic material, the stress field is
If we now write the total potential energy
in terms of θ and w through (10.4.1)–(10.4.4), we obtain (after some calculations)
where the symmetric bilinear form a is identified by
where
and k is a correction factor which is often used to account for the “nonconformity” of (10.4.4). Indeed, from (10.4.1)–(10.4.4), we deduce that σ 13 and σ 23 are constants in z, whereas the physical problem has \(\sigma _{13} =\sigma _{23} = 0\) on the upper and lower face of the plate: Ω ×{ t} and Ω ×{ − t}; hence, (10.4.4) is often corrected by assuming that σ 13 and σ 23 behave parabolically in z, vanishing for z = ± t and assuming the value (10.4.4) for z = 0. For a mathematically more convincing justification of the classical 5 ∕ 6 factor, we refer again to [35]. Actually, for the sake of simplicity, we shall assume, from now on, that
In fact, as far as we do not expect the true value (10.4.8) to go to zero or to + ∞, assuming λ = 1 will just change the numerical value of the constants appearing in the stability estimates or in the a priori error estimates, but it will not change the behaviour in function of the thickness t or the mesh-size h.
10.4.2 The Mathematical Formulation
The assumed boundary conditions lead to the kinematic constraints
Hence, we define the spaces
with the norm
When convenient, the generic element of V will be denoted \(\underline{v} = (\underline{\eta },\zeta )\) with \(\underline{\eta }= (\eta _{1},\eta _{2}) \in \Theta \) and ζ ∈ Z. We finally recall the Korn inequality
where, from now on in this section, the symmetric bilinear form a will be the one given in (10.4.7).
It is easy to check that, for any fixed t > 0, functional (10.4.5) has a unique minimiser \((\underline{\theta },w)\) on V which satisfies
In particular, we have
for any \(\underline{v} = (\underline{\eta },\zeta ) \in V\). Note that for fixed t, (10.4.15) always guarantees that (10.4.13), (10.4.14) is a nice linear elliptic problem so that, for instance, any reasonable conforming approximation of V will have optimal order of convergence.
The troubles start when we take a small t; then, the constant in (10.4.15) deteriorates and so does the constant in front of the optimal error bound. In practice, it is well known that if we use “any reasonable conforming approximation of V ”, we will get pretty bad answers for small t. Here, we shall make an analysis of the nature of the trouble. We shall also give some sufficient conditions on the discretisation so that it stays good for t smaller and smaller. The one-dimensional case was treated in [15], but the two-dimensional case, as we shall see, is more complicated.
The first thing that we have to do is to construct a sequence of physical problems \(\mathcal{P}_{t}\) (for t > 0 and, say t < T 0) that fulfil the following requirements:
-
(1)
Each \(\mathcal{P}_{t}\) is of type (10.4.13) and (10.4.14) and so has a unique solution \(\underline{\theta }(t)\), w(t);
-
(2)
There exists two constants c 1, c 2 with \(0 < c_{1} < c_{2}\) such that
$$\displaystyle{ c_{1} \leq \Vert \underline{\theta }(t)\Vert _{1} +\Vert w(t)\Vert _{1} \leq c_{2}\qquad \quad \forall \,t \in ]0,T_{0}[. }$$(10.4.16)
A possible answer is to fix Ω, E, and ν, and to choose, for each t > 0, the load f 3(x, y, z) of the form
with g(x, y) fixed (once and for all) independent of t. It is clear that (10.4.17) implies
where as usual (f, w) denotes the L 2(Ω) inner product or (with an abuse of notation) whenever f is assumed to be only in H − 1(Ω), the duality pairing between H − 1(Ω) and \(H_{0}^{1}(\varOmega )\). Hence, dividing (10.4.6) by t 3, each problem \(\mathcal{P}_{t}\) will amount to minimise, in V ,
Proposition 10.4.1.
Let \(\underline{\theta }(t)\) , w(t) be the minimiser of (10.4.19) in V . Then, (10.4.16) holds with c 1 and c 2 independent of t.
Proof.
We obviously have
Using (10.4.12) and a little algebra, we deduce from (10.4.20) that
which implies the boundedness of \(\Vert \underline{\theta }\Vert _{1} +\Vert w\Vert _{1}\) from above. Then, one observes that the minimum of Π t over all V is surely smaller than the minimum of Π t over \(V _{0} =\{ (\underline{\eta },\zeta )\,\vert \,\underline{\eta } = \mathrm{g\underline{ra}d}\ \zeta \}\) (which is clearly independent of t and negative). Hence,
for some positive c independent of t, which immediately gives
which implies that \(\|w\|_{0}\) (and hence \(\Vert \underline{\theta }\Vert _{1} +\Vert w\Vert _{1}\)) is bounded from below by a positive constant. This completes the proof. □
According to Proposition 10.4.1, we have now a sequence of problems, indexed by the thickness t, whose solutions are bounded uniformly (in t) and also bounded uniformly away from zero.
For the convenience of the reader, we repeat explicitly the general problem of our sequence.
Given a bounded domain \(\varOmega \subset {\mathbb{R}}^{2}\) with diameter T : = diam(Ω) and an element f ∈ L 2(Ω), for every thickness t ∈ ]0, T[, we consider the problem: find \((\underline{\theta }(t),w(t))\) in \(V := {(H_{0}^{1}(\varOmega ))}^{2} \times H_{0}^{1}(\varOmega )\) such that
where Π t is given by (10.4.19).
The sequence (10.4.24) is what we need to analyse the performance of numerical methods. Indeed, we expect a “good and reliable” numerical method to perform uniformly well on all the problems of our sequence, regardless of the possible smallness of t. We therefore look for error bounds (in terms of powers of the mesh-size h) which hold uniformly in t.
10.4.3 Mixed Formulation of the Mindlin-Reissner Model
It will be convenient, in order to carry on the analysis, to introduce the auxiliary variable
which is related to the shear stresses but does not go to zero with t (and could be considered as a sort of normalised shear stress). We can now write the Euler equations for Π t in the form
This is now taking the form of the abstract problems studied in Chap. 4, especially in Sect. 4.3. In particular, we can define the bilinear forms
where a is defined in (10.4.7), and
corresponding to the operator
and finally the functional
With this notation, Eqs. (10.4.26) and (10.4.27) can be written as
As we have already seen on several other examples, it is convenient, from many aspects, to consider (10.4.32) and (10.4.33) as a perturbation of the “limit problem” that we have for t = 0, namely
It is easy to check that the kernel \(K := \mathrm{Ker}\mathbb{B}\) is given by
It is then clear that the Korn inequality (10.4.12) implies that the bilinear form \(\mathcal{A}\), defined in (10.4.28), is elliptic in the kernel K of \(\mathbb{B}\):
with α 0 depending only on the Korn constant α Korn appearing in (10.4.12).
On the other hand, we note that we did not decide yet what the space Q should be, and hence where \(\underline{\delta }\) is allowed to vary in (10.4.33) or in (10.4.35). Recalling the general theory of Chap. 4, we observe that the space Q should be defined in such a way that the operator \(\mathbb{B}\), associated with the bilinear form \(\mathcal{B}\), is surjective from V to Q ′ (or, at least, that its image is a closed subspace of Q ′). It is therefore clear that the next, crucial, step has to be the characterisation of the image of \(\mathbb{B}\), that is \(\mathbb{B}(V )\) with V given in (10.4.10).
In what follows, we are going to use the notation introduced in Chap. 2 for the two-dimensional operators
Note as well that (for the same reason) we are using here (x, y, z) instead of \((x_{1},x_{2},x_{3})\).
Proposition 10.4.2.
The mapping \(\mathbb{B}\) is surjective from V onto the space \(\Gamma = H_{0}(\mathrm{curl},\varOmega )\) defined by
(where \(\underline{t}\) is the unit tangent to ∂Ω) and admits a continuous lifting.
Proof.
We shall show that there exists a β RM > 0 such that: for every \(\underline{\chi }\in H_{0}(\text{rot};\varOmega )\) there exists \((\underline{\eta },\zeta ) \in V\) verifying
and
For this, we first choose \(\underline{v} \in {(H_{0}^{1})}^{2}\) such that
this is obviously possible because
Then, we set
so that from (10.4.43) and (10.4.44) we have
Now choose ζ as the unique solution in \(H_{0}^{1}(\varOmega )\) of
we have, using (10.4.48) and (10.4.49),
We now have
which easily implies (10.4.41). On the other hand, (10.4.42) follows from (10.4.48) and (10.4.50). □
Proposition 10.4.2 tells us how to choose Q in order to have that \(\mathbb{B}\) is surjective from V to Q ′. Actually, we have little choice: Q ′ must be equal to the space Γ = H 0(curl, Ω) defined in (10.4.39). As we are dealing with Hilbert space, this implies that Q has to be the dual space of Γ:
On the other hand, a little functional analysis allows us to characterise Γ ′ as follows:
with the norm
Then, the Closed Range Theorem (see Sect. 4.2.2) tells us that Proposition 10.4.2 can be written in the form of an inf-sup condition:
Hence, to start with, we can make precise the limit problem (10.4.34) and (10.4.35) as follows
From (10.4.37) and (10.4.55), using Theorem 4.2.3, we then have the following result on the limit problem (10.4.34) and (10.4.35) in the form (10.4.56).
Proposition 10.4.3.
Let \(\mathcal{A}\) and \(\mathcal{B}\) be defined as (10.4.28) and (10.4.29) , respectively. Then, for every f ∈ L 2 (Ω), the limit problem (10.4.56) has a unique solution \((\underline{\theta }_{0},w_{0},\underline{\gamma _{0}})\) and we have
□
Remark 10.4.1.
Actually, the abstract theory of Chap. 4 tells us that we could take any framework that is much more general than the one used for problem (10.4.56). For instance, we could have allowed a general \(\mathbb{F} \in {V }^{{\prime}}\) (not necessarily of the form (10.4.31)) in the right-hand side of the first equation. Besides, we did not need to assume f ∈ L 2(Ω), as f ∈ H − 1(Ω) would clearly have been sufficient. Moreover, a right-hand side in Q ′ = Γ would also be allowed (instead of zero) in the second equation. We decided, however, to present the result in the framework of our original plate problem. □
Remark 10.4.2.
It is not difficult to check that the unique solution of (10.4.56) is related to the solution of the Kirchhoff model: find \(w_{K} \in H_{0}^{2}(\varOmega )\) such that
by the relations
□
Remark 10.4.3.
In the case of beam problems, the space Γ ′ is replaced by L 2, which makes things much easier. □
Remark 10.4.4.
We now remark that, with our choice, we have \({Q}^{{\prime}}\,\equiv \,H_{0}(\mathrm{curl};\varOmega )\hookrightarrow {({L}^{2}(\varOmega ))}^{2}\). As Q ′ is clearly dense in (L 2(Ω))2, we also have (identifying, as usual, \({({L}^{2}(\varOmega ))}^{2}\) with its dual space) \({({L}^{2}(\varOmega ))}^{2}\hookrightarrow Q\). This implies that the perturbation introduced, for positive t, in the full problem (10.4.32) and (10.4.33) has to be regarded as a singular perturbation of the limit problem (10.4.34) and (10.4.35). Hence, it has to be dealt with using the instruments of Sect. 4.3.2. □
In view of the previous remark, we introduce the space
and set the mathematical framework for the Mindlin-Reissner problem (10.4.32) and (10.4.33) as follows
Having chosen W as well as Q, we can now prove the following result.
Proposition 10.4.4.
Let the spaces V , Q, and W be defined as in (10.4.10)–(10.4.60), respectively, and let the bilinear forms \(\mathcal{A}\) and \(\mathcal{B}\) and the operator (10.4.30) be defined in (10.4.28), (10.4.29) and (10.4.30), respectively. Then, there exists an \(\tilde{\alpha }> 0\) such that
Proof.
The result is essentially trivial. Indeed, using (10.4.11), the triangle inequality, and the Poincaré inequality (1.2.14), we have first
where C 1 and C 2 depend only on the Poincaré constant. Then, we can use the Korn inequality (10.4.12) and the definition of \(\mathcal{A}\) and \(\mathbb{B}\) to obtain
and the result follows. □
We can now apply Theorem 4.3.4 (with g = 0) and obtain the following result.
Theorem 10.4.1.
With the same assumptions as in Proposition 10.4.4 , for every f ∈ V ′ and for every t ∈]0,1[, problem (10.4.61) has a unique solution \((\underline{\theta }(t),w(t),\underline{\gamma }(t))\) . Moreover, there exists a constant c, depending only on Ω, such that
□
We can now study the behaviour of the solutions of problem (10.4.61) when t → 0.
Proposition 10.4.5.
With the same assumptions as in Theorem 10.4.1 , we have
where \((\underline{\theta }_{0}\), \(w_{0}\), \(\underline{\gamma }_{0})\) is the solution of the limit problem (10.4.56).
Proof.
The weak convergence (a priori, up to a subsequence) in (10.4.64) just follows from (10.4.16) and (10.4.57). A passage to the limit in (10.4.61) gives (10.4.56). □
Remark 10.4.5.
Additional results in this direction can be found in [171]. □
We can now apply the results of Proposition 4.3.5 and of Remarks 4.3.12 and 4.3.14 to estimate the convergence rate as a function of t 2 which plays here the role of λ. This leads us to a convergence rate in \(\sqrt{\lambda } = t\). In order to improve this bound and also to enable us later to get sharper error estimates, we now introduce a decomposition principle for (10.4.26) and (10.4.27).
10.4.4 A Decomposition Principle and the Stokes Connection
We shall first prove the following decomposition principle for vector-valued functions in \({\Gamma }^{{\prime}} = H_{0}(\mathrm{curl};\varOmega )\).
Proposition 10.4.6.
Every element \(\underline{\gamma }\in {\Gamma }^{{\prime}}\) can be written in a unique way as
with ψ ∈ H 0 1 (Ω), \(p \in {L}^{2}(\varOmega )/\mathbb{R}\) , and \(\mathrm{c}\underline{\mathrm{ur}}\text{l}p ={\Bigl \{ -\partial p/\partial y,\partial p/\partial x\Bigr \}}\) . Moreover, we may use
as a norm on Γ ′ .
Proof.
Set \(\xi := \mathrm{div}\underline{\gamma } \in {H}^{-1}(\varOmega )\). We define ψ to be the unique solution of \(-\bigtriangleup \psi =\xi\), \(\psi \in H_{0}^{1}(\varOmega )\) and we set \(\underline{\alpha }=\underline{\gamma } -\mathrm{g\underline{ra}d}\ \psi\). One has \(\mathrm{div}\underline{\alpha } = 0\) so that \(\underline{\alpha }= \mathrm{c}\underline{\mathrm{ur}}\text{l}p\) and p is determined up to a constant in L 2(Ω). Condition (10.4.66) is then immediate. □
Remark 10.4.6.
The decomposition introduced in Proposition 10.4.6 also holds for \({({L}^{2}(\varOmega ))}^{2}\) and H(curl; Ω). The difference between these spaces lies in the regularity of the p component. Indeed, taking \(\underline{\gamma }= \mathrm{g\underline{ra}d}\ \psi + \mathrm{c}\underline{\mathrm{ur}}\text{l}p\) with ψ ∈ H 0 1(Ω), we have
□
It is now a simple exercise to transform problem (10.4.61) in terms of the new unknowns \(\underline{\theta }(t)\), w(t), ψ(t), and p(t). We have indeed the following basic theorem, which is of considerable help in understanding the nature of the Mindlin-Reissner equations.
Theorem 10.4.2.
Any solution of (10.4.61) is a solution of the following problem (and conversely) through the change of variables (10.4.65) : find \((\underline{\theta }(t),w(t),\psi (t),p(t))\) in \(\underline{\Theta } \times Z \times H_{0}^{1}(\varOmega ) \times {L}^{2}(\varOmega )/\mathbb{R}\) such that
Proof.
The proof is immediate: it is enough to make the substitution (10.4.65), and observe that both (10.4.61) and (10.4.70)–(10.4.72) have a unique solution. □
Remark 10.4.7.
It must be noted that (10.4.71) implies \(\partial p/\partial n\vert _{\partial \varOmega } = 0\) and p ∈ H 2(Ω) so that \(\underline{\gamma }= \mathrm{g\underline{ra}d}\ \psi + \mathrm{c}\underline{\mathrm{ur}}\text{l}p\) is indeed an element of \(\Gamma = H_{0}(\mathrm{curl};\varOmega )\). Note also that ψ(t) is actually independent of t. □
Remark 10.4.8.
It is important to note that although (10.4.70)–(10.4.72) seems, at first sight, a system of four equations, it actually decomposes immediately into equation (10.4.70) (which allows to compute ψ directly from f), plus equations (10.4.71) (which allow to compute \(\underline{\theta }(t)\) and p(t) once we know ψ) plus equation (10.4.72) (which allows to compute w(t) once we know \(\underline{\theta }(t)\) and ψ). We have thus reduced, through Theorem 10.4.2, our original problem into the following sequence
-
A Dirichlet problem (10.4.70) that is independent of t,
-
A “Stokes-like” problem (10.4.71),
-
A Dirichlet problem (10.4.72).
□
The decomposition provided by Theorem 10.4.2 shows us that it is the p component of \(\underline{\gamma }\) which depends on t. Before coming back to the quantification of this dependency, we rapidly develop the analogy between (10.4.71) and a Stokes problem. Let us set \(\underline{{\eta }}^{\perp } =\{ -\eta _{2},\eta _{1}\}\) . We can write (10.4.71) in the form
The limit problem (t = 0) is thus a standard Stokes problem and we shall be able to rely on results of Chap. 8 to build approximations. We shall not analyse here the case t≠0 in too much detail. However, it is important to see the behaviour of p as t → 0.
Proposition 10.4.7.
Let \(\underline{\theta }(t)\) , w(t), p(t), and ψ be the solution of (10.4.70)–(10.4.72) . We then have
where the constant c is independent of t. □
We refer to [122] for the proof of this result which is based essentially on the regularity properties of the Dirichlet problem and the Stokes problem.
An important point is that (10.4.74) does not improve too much for a more regular f (even in a smooth domain). It is not possible to bound \(\Vert p(t)\Vert _{2}\) uniformly in t. The reason is that the normal derivative of p(t) vanishes although this is not the case for the solution p(0) of the limit problem. We thus have a boundary layer effect which has been studied in [29]. This analysis shows that an analogue of (10.4.74) exists for \(\Vert \underline{\theta }\Vert _{\frac{5} {2} }\) and \(\Vert p\Vert _{\frac{3} {2} }\) but not for more regular spaces.
Remark 10.4.9.
We can now try to apply Remarks 4.3.12 and 4.3.14 to our problem. Denoting \(W_{+} :=\{ p\;\vert \;p \in {H}^{2}(\varOmega )/\mathbb{R},\ \partial p/\partial n\vert _{\partial \varOmega } = 0\}\), it is clear that we have
Whenever the solution p 0 of the limit problem is regular enough (this is the case for smooth data and a smooth domain), we shall have
No improvement is possible because of the fact that ∂ p(0) ∕ ∂ n≠0. We can thus apply Remark 4.3.14 to get for \(\theta < \frac{3} {4}\)
where \(\Vert p_{0}\Vert _{\theta }\) is the norm of p 0 in \([{L}^{2}(\varOmega ),W_{+}]_{\theta }\) . We can summarise (10.4.77) by saying that we have an \(O({t}^{3/2-\varepsilon })\) convergence. This requires, however, a smooth domain. In the case where ∂ Ω is only Lipschitz continuous, the best we can get is O(t). □
10.4.5 Discretisation of the Problem
We now turn our attention to the discretisation of our problem (10.4.26) and (10.4.27). Let us thus assume that we are given finite-dimensional subspaces Θ h and Z h of Θ and Z and use \(V _{h} = \Theta _{h} \times Z_{h}\) as a subspace of V . We also discretise the space W = (L 2(Ω))2 by Γ h and we consider the discretised problem: find \((\underline{{\theta }}^{h},w_{h},\underline{\gamma }_{h})\) such that
This could also be written with the notation of Sect. 10.4.3, that is, in particular, making use of the bilinear form \(\mathcal{A}\) and \(\mathcal{B}\) defined in (10.4.28) and (10.4.29). The discrete problem (10.4.78) becomes: find \(((\underline{\theta }_{h},w_{h}),\underline{\gamma }_{h}) \in V _{h} \times Q_{h}\) such that
In what follows, we shall use either the form (10.4.78) or the form (10.4.79), according to the notational convenience.
Remark 10.4.10.
Note that from the second equation of (10.4.78), we do not have in general \(\underline{\gamma }_{h} =\lambda {t}^{-2}(\mathrm{g\underline{ra}d}\ w_{h} -\underline{\theta }_{h})\) unless we take Θ h , Z h , and Γ H such that \(\mathrm{g\underline{ra}d}\ Z_{h} - \Theta _{h} \subseteq \Gamma _{h}\). This, as we shall see, could be a problem regarding the actual implementation of the method. Indeed, in the common engineering practice, one prefers to solve the discrete problems in terms of \(\underline{\theta }_{h}\) and w h alone. In this case, the use of the mixed formulation (and the introduction of the variable \(\underline{\gamma }_{h}\)) should be regarded as a mathematical artefact used in order to have a better understanding of the mathematical structure of the discretised problem. We will come back several times to this important point. □
It is easy to check that, now, the discrete kernel \(K_{h} := \text{Ker}\mathbb{B}_{h}\) is given by
and we consider the problem of having, for our discrete problem, the ellipticity in the discrete kernel;
For the continuous case, the Korn inequality (10.4.12) implied that the bilinear form \(\mathcal{A}\) is elliptic in the kernel K (see (10.4.37)). As the variable ζ does not appear in the actual expression of \(\mathcal{A}((\underline{\eta },\zeta ),(\underline{\eta },\zeta ))\), we deduce that the only possibility in order to have the ellipticity in K h is that the following property holds
and a simple necessary condition for it is that
This can easily be satisfied assuming for instance that
As we shall see, the above condition (10.4.84) is not difficult to enforce when choosing the finite element spaces and the vast majority of the good and reliable methods will satisfy it. On the other hand, the discrete inf-sup condition
is a major difficulty, and most methods will be designed in order to get around it. For this, the first methods that we are going to consider are those based on the decomposition principle given in Proposition 10.4.6 and on the re-formulation of the problem given in Theorem 10.4.2.
Remark 10.4.11.
It will often be convenient to look as well at the limit problem: find \((\underline{\theta }_{0h},w_{0h},\underline{\gamma }_{0h}) \in \Theta _{h} \times Z_{h} \times \Gamma _{h}\) such that
that could also be expressed in the form (10.4.79) with t = 0. It also comes from the results of Sects. 4.3.2 and 5.5.3 that to get a good approximation of (10.4.61) by (10.4.79) (that is, with convergence properties independent of t), it is necessary for (10.4.86) to be a good approximation of (10.4.34) and (10.4.35). □
We shall first consider the most “naive” case.
Example 10.4.1 (The direct approach).
Let us suppose that we are given Θ h ⊂ Θ and Z h ⊂ Z, and let us choose
This choice implies that
so that the ellipticity in K h (10.4.81) evidently holds. It is important to note that the choice (10.4.87) is very easy to use on the computer, as it actually corresponds to minimising the energy functional Π t given by (10.4.19) on \(V _{h} = \Theta _{h} \times Z_{h}\) and that you do not even see \(\underline{\gamma }_{h}\) (nor Γ h ). The choice (10.4.87) is then one of the most widely used choices for Γ h although, in general, one does not realise it.
However, in the limit t → 0, one is lead to minimise
on \(\text{Ker}\mathbb{B}_{h}\). Now, a quick glance to KerB h will make us understand that we have a long way to go. Consider \(\underline{\eta }_{h}^{\perp } =\{ -\eta _{2h},\eta _{1h}\}\), that is, a rotation of π ∕ 2 of \(\underline{\eta }_{h}\). It is clear that if \((\underline{\eta }_{h},\zeta _{h})\) belongs to KerB h , we then have, by (10.4.88),
Therefore, with choice (10.4.87), we are minimising Π t in (10.4.89) on a subset of functions \(\underline{\eta }_{h}\) satisfying (10.4.90). However, we have already seen in Chap. 8, for the linear Stokes problem, that it is not recommended to work with velocity fields which are exactly incompressible (because there are too few of them in general). A direct application of (10.4.87) is likely to lead to bad results (e.g. locking) unless a very special choice of Θ h and Z h has been made. □
In what follows, we shall mainly concentrate on two groups of finite element approaches: the Methods based on the decomposition principle, and the Methods based on a nonconforming approximation of the original minimisation problem (10.4.24).
10.4.5.1 Methods Based on the Decomposition Principle
The first group of methods that we present is directly guided by the decomposition principle of Propositions 10.4.6 and 10.4.2 in which a Stokes-like problem explicitly appears. For the sake of simplicity, we shall describe one possible method in this group, based on the MINI element for Stokes. However, it will be clear that starting from every finite element stable approximation for the Stokes problem using continuous pressures, one can derive a Reissner-Mindlin method belonging to the present group.
The basic idea is to give up a direct approximation of \(\underline{\gamma }\) and to approximate instead each component of its decomposition into \(\mathrm{g\underline{ra}d}\ \psi _{h} + \mathrm{c}\underline{\mathrm{ur}}\text{l}p_{h}\). Moreover, as (10.4.71) shows us that θ h and p h are analogous to a velocity field and a pressure field in a Stokes problem, we shall try to use some results of Chap. 8 to build a suitable approximation.
We assume that Ω is a convex polygon and that we are given a sequence \(\{\mathcal{T}_{h}\}\) of partitions of Ω into triangles. Let Θ h be built by employing the MINI element of Chap. 8 , that is, in the notations of Chap. 2,
These are spaces of piecewise linear polynomials enriched by a bubble function in the case of Θ h . We also introduce
This space is then a strict subspace of piecewise constant vector functions constructed by discretising the ingredients of the decomposition principle of Proposition 10.4.6 and Remark 10.4.6.
It is straightforward to check that \(\text{Ker}\mathbb{B}_{h}\) is made of the pairs \((\underline{\eta }_{h},\zeta _{h})\) in Θ h ×Z h such that
Now, condition (10.4.94) is especially nice as it implies
and hence, (10.4.82) holds and we have the ellipticity in the kernel (10.4.81). We still have to check the inf-sup condition (10.4.85) and we can do it using Proposition 5.4.3: given \((\underline{\eta },\zeta )\), we must then be able to build \((\underline{\eta }_{h},\zeta _{h}) = \Pi _{h}(\underline{\eta },\zeta )\) such that
with
Using the structure \(\underline{\delta }= \mathrm{g\underline{ra}d}\phi _{h} + \mathrm{c}\underline{\mathrm{ur}}\text{l}q_{h}\), condition (10.4.96) becomes:
In order to construct the operator Π h , we use the result already obtained in Chap. 8 to deal with the inf-sup condition for the MINI element. In particular, we proved that there exists an operator Π S , from \(\Theta = {(H_{0}^{1}(\varOmega ))}^{2}\) into Θ h , such that
with \(\|\Pi _{S}(\underline{\eta })\|_{1} \leq C\,\|\underline{\eta }\|_{1}\) and C independent of h. With the same arguments, we can obviously prove that there exists an operator Π R from \(\Theta = {(H_{0}^{1}(\varOmega ))}^{2}\) into Θ h such that
with
with C independent of h. Condition (10.4.100), taking into account the fact that \((\text{c}\underline{\mathrm{ur}}\text{l}q_{h},\mathrm{g\underline{ra}d}\ \zeta _{h}) = (\mathrm{c}\underline{\mathrm{ur}}\text{l}q_{h},\mathrm{g\underline{ra}d}\ \zeta ) \equiv 0\) (by Green’s formula), tells us that the second equation of (10.4.98) is satisfied if we take \(\underline{\eta }_{h} = \Pi _{r}(\underline{\eta })\). We now observe that the first equation of (10.4.98) reduces to
and this is a discrete Dirichlet problem for the Laplace operator for which we have easily \(\Vert \zeta _{h}\Vert _{1} \leq c\;(\Vert \zeta \Vert _{1} +\|\underline{\eta }\| _{1})\), yielding the second part of (10.4.97).
Remark 10.4.12.
It should be clear from our construction that the crucial step is to have an operator Π R satisfying (10.4.100) and (10.4.101). This, always changing as we did the div operator into rot, essentially means that we could take, instead of the MINI element, any other finite element pair that is stable for the Stokes problem and which uses continuous pressures. □
Having proved the inf-sup condition (10.4.85), we can therefore apply to the limit problem (10.4.86) the basic results of Chap. 5. We can summarise this in the following proposition.
Proposition 10.4.8.
Problem (10.4.86) with the choice (10.4.91) and (10.4.92) has a unique solution. Moreover, if \((\underline{\theta }_{0},w_{0},\underline{\gamma }_{0})\) is the solution of (10.4.34) and (10.4.35) , we have
□
Remark 10.4.13.
The result of Proposition 10.4.2 can be applied to the discrete problem in the present case. Indeed, we built, a priori, Γ h in order to obtain a decomposition principle. Problem (10.4.78) can be written in the form: find \((\underline{\theta }_{h}(t),w_{h}(t),\psi _{h}(t),p_{h}(t))\mbox{ in }\underline{\Theta }_{h} \times Z_{h} \times Z_{h} \times \mathcal{L}_{1}^{1}/\mathbb{R}\) such that
These problems can be solved sequentially and (10.4.105) is a Stokes-like problem using the MINI element of Chap. 8. This approximation has been introduced and studied for t≠0 in [122]. Using this decomposition and Proposition 10.4.8, recalling that
and bringing in the regularity result of Proposition 10.4.7, we have, for t = 0, the following estimate:
From a numerical point of view, (10.4.104)–(10.4.106) can lead to an efficient method, provided one has a Stokes solver available. □
Remark 10.4.14.
An easy duality argument would also show that we have the estimate
□
To end the discussion on this group of methods, we rapidly show how the results of Sect. 5.5 can be applied to the case t≠0. We consider the error estimate (5.5.52) from Remark 5.5.5, where we denote V = (H 0 1(Ω))2 ×H 0 1(Ω), Q = Γ ′ and W = (L 2(Ω))2. The parameter λ is, of course, t 2 in the present case. It is easily verified that all conditions are satisfied and that we have, taking into account regularity properties of Remark 10.4.6,
Using the decomposition principle and the estimate (10.4.74), we can recover the following result of [122].
Theorem 10.4.3.
For every t ∈]0,T[, problem (10.4.104)–(10.4.106) with the choices (10.4.91) and (10.4.92) has a unique solution \((\underline{\theta }_{h}(t),w_{h}(t),\psi _{h}(t),p_{h}(t))\) . If moreover \((\underline{\theta }(t),w(t),\psi (t),p(t))\) is the solution of (10.4.70)–(10.4.72) , then we have
with c independent of h and t.
We therefore have an O(h) convergence uniform in t. This result cannot be (much) improved because of the boundary layer effect already described.
10.4.5.2 Nonconforming Approximations of the Minimum Problem
The previous class of methods is, although interesting, rather remote from the actual engineering practice in which one tries to stick as closely as possible to the original formulation. In particular, as already pointed out in Remark 10.4.10, what is preferred in the engineering practice is to work only in terms of the original unknowns \(\underline{\theta }\) and w, and, possibly, having their degrees of freedom at the same nodes (in particular if one wants to extend the methods to shell problems).
As we have seen, however, in Example 10.4.1, working directly on the minimisation problem (10.4.24) would require approximations \(\underline{\theta }_{h}(t)\) and w h (t) that, in the limit for t → 0, satisfy \(\underline{\theta }_{h}(0) = \mathrm{g\underline{ra}d}\ w_{h}(0)\), and if we want to use a conforming approximation Θ h ⊂ Θ this would require \(w_{h} \in Z_{h}\) to belong to H 0 2(Ω), which is not so easy to obtain, in particular for low degree elements.
The most common escape to the troubles that we are facing is to use some kind of numerical integration (or a nonconforming approximation) for the term \({t}^{-2}\Vert \mathrm{g\underline{ra}d}\ w -\underline{{\theta }\Vert }^{2}\) which appears in (10.4.19), thus weakening condition (10.4.90). A way of formalising it is the following. We assume that we are given a linear operator r which maps \(\Theta _{h} \times Z_{h}\) into (for instance) L 2(Ω). To see an example, consider for instance the possible, but not necessarily recommended, choices:
or
at the barycentre of K. Then, one minimises, instead of Π t (as in (10.4.24)), the functional
on Θ h ×Z h . This can be regarded as obtained from the problem find \((\underline{\theta }_{h},w_{h},\underline{\gamma }_{h}) \in \Theta _{h} \times Z_{h} \times \Gamma _{h}\) such that
whenever its second equation is equivalent to
This will always be the case for choices of Γ h that verify
In this case, the limit problem (for t = 0) will be: find \((\underline{\theta }_{h},w_{h},\underline{\gamma }_{h}) \in \Theta _{h} \times Z_{h} \times \Gamma _{h}\) such that
With the notation (10.4.28) for \(\mathcal{A}\) and setting
we can write the problem (10.4.115) as
The kernel of the operator \(\tilde{\mathbb{B}}_{h}\) associated with \(\tilde{\mathcal{B}}\) will then be
which, assuming that (10.4.117) is satisfied, can also be written as
All this should be connected to the ellipticity in the kernel, or, better, to the following (more powerful) property, strongly related to (5.5.46)
where T is always the diameter of Ω as in (10.4.19).
We have for this the following result.
Proposition 10.4.9.
Let \(\mathcal{A}\) and \(\tilde{\mathcal{B}}_{h}\) be defined as in (10.4.28) and (10.4.119) for an r that satisfies (10.4.117) . If moreover we have
then (10.4.123) holds.
Proof.
The proof is almost immediate using the Korn inequality (10.4.12). It is sufficient to combine the two inequalities
and
Condition (10.4.124) might look cumbersome. We have, however, a simple sufficient condition for that.
Proposition 10.4.10.
Assume that \(r(\underline{\eta },\zeta )\) has the form
where R is a mapping from Θ h to Γ h such that
for some constant C R independent of h. Then, (10.4.124) holds.
The proof is an easy exercise.
We can now use Theorem 5.5.5 and obtain the following abstract error bound.
Theorem 10.4.4.
Assume that R is an operator from Θ h to Γ h satisfying (10.4.126) , and assume that the bilinear form \(\tilde{\mathcal{B}}\) is defined through (10.4.125) and (10.4.119). For every t ∈]0,T[, let \((\underline{\theta }(t),w(t),\underline{\gamma }(t)\) be the solution of Problem (10.4.61) and let \((\underline{\theta }_{h}(t),w_{h}(t),\underline{\gamma }_{h}(t)\) be the solution of (10.4.120). Then, for every \((\underline{\theta }_{I}(t),w_{I}(t),\underline{\gamma }_{I}(t)\) in \(\Theta _{h} \times Z_{h} \times \Gamma _{h}\) such that
we have
where C is a constant independent of t and h.
Proof.
The proof is elementary: using (10.4.126) and Proposition 10.4.10, we obtain (10.4.124). Using Proposition 10.4.9, we obtain (10.4.123), which is the crucial assumption needed to apply Theorem 5.5.5. □
Remark 10.4.15.
In many cases, the last term in the right-hand side of (10.4.128) can be better estimated by
which, in a sense, separates the errors \(\|\underline{\gamma }(t) -\underline{\gamma }_{I}(t)\|_{-1}\) and \(\|R_{h} -\mathit{Identity}\|\). It has to be pointed out that, in most cases, the difference \(R_{h}\underline{\eta }_{I}-\underline{\eta }\) will be orthogonal to all (vector-valued) polynomial of a certain degree ℓ so that
where π ℓ is the projection operator on polynomials of degree ℓ. □
As we did for the previous class of methods (the ones based on the decomposition), we will not present here a list of all methods of this type available on the market. We will instead present a single method, as an example, in order to show the general guidelines that rule their construction.
We assume again that Ω is a convex polygon and then we are given a sequence \(\{\mathcal{T}_{h}\}\) of partitions of Ω into triangles. We set, with the notation of Chap. 2,
Note that this is the rotated \(\mathcal{B}\mathcal{D}\mathcal{F}\mathcal{M}_{2}\), following Remark 2.3.2. Together with the spaces (10.4.131), we consider the operator Π h from, say, (H 1(Ω))2 into Γ h defined in each triangle K by
where \(\mathcal{R}\mathcal{T}_{0}(K)\) is the lowest order Raviart-Thomas space (see Chap. 2).
We can now define the operator r. Following the structure (10.4.125), we set
The kernel of \(\mathbb{B}_{h}\) as defined in (10.4.121) is now easily characterised as the set of \((\underline{\eta }_{h},\zeta _{h})\) such that
Since \(\Vert \Pi _{h}\underline{\eta }_{h}\Vert _{0} \leq c\Vert \underline{\eta }_{h}\Vert _{1}\) for some constant c independent of h, we can apply Proposition 10.4.10 and then Proposition (10.4.9) to get
that is, more precisely, condition (10.4.137). In order to apply Theorem 10.4.4, we just need to check that condition (10.4.127) holds for suitable \(\underline{\theta }_{I},w_{I},\underline{\gamma _{I}}\) having optimal approximation properties. For the construction of \(\underline{\theta }_{I},w_{I},\underline{\gamma _{I}}\), we can use the following lemma.
Lemma 10.4.1.
Assume that
Set \(\underline{\gamma }_{I} := \Pi _{h}\underline{\gamma }\) and assume that we can find \(\underline{{\theta }}^{i}\) and w i verifying
Then, from (10.4.138) and (10.4.139) , one obviously has
for some ζ h ∈ Z h . Then setting
one has (10.4.127) as well as
Note that, in other words, inequality (10.4.142) tells us that we can “arrange” (10.4.127) without losing accuracy. The proof is simple: first we check that
giving us (10.4.127). Inequality (10.4.142) then follows immediately from
□
Then, we just have to construct \(\underline{{\theta }}^{i}\) and w i satisfying (10.4.139). The construction of \(\underline{{\theta }}^{i}\) is easy. Indeed, denoting Π C R the B-compatible operator for the Couzeix-Raviart element for the Stokes problem and by π 1 the projection onto \(\mathcal{L}_{1}^{0}\), one has from Example 8.6.1
and similarly, from the properties of the \(\mathcal{B}\mathcal{D}\mathcal{F}\mathcal{M}\) element,
We deduce that
This says that the choice
will satisfy the first condition of (10.4.139). On the other hand, taking
and
we easily have that
and
implying the second condition of (10.4.139).
We can therefore use Theorem 10.4.2 and standard interpolation estimates (together with Remark 10.4.15) to obtain the following result.
Theorem 10.4.5.
Consider the discretised problem (10.4.120) with the choices (10.4.131)–(10.4.135) . Then, for every t ∈]0,T[, it has a unique solution \((\underline{\theta }_{h}(t),w_{h}(t),\underline{\gamma }_{h}(t)\) . Let moreover \((\underline{\theta }(t),w(t),\underline{\gamma }(t)\) be the solution of Problem (10.4.61) . Then we have
where C is a constant independent of t and h.
As we already noted, this estimate is overoptimistic because it ignores the boundary layer effects. From the results of [29], an O(h 3 ∕ 2) convergence rate should be expected.
Remark 10.4.16.
Similar estimates have been obtained in [117] for the presently discussed element and related ones, including elements defined on quadrilaterals. More refined estimates can be found in [126]. A recent review of different Mindlin-Reissner approximations, including the Linked interpolation techniques (that have not been considered here), can be found in [190]. □
Remark 10.4.17.
The choice of second-order accuracy has been made only for the sake of simplicity. Higher-order elements are possible and we shall indicate at the end of this chapter a general framework within which they could be built. On the contrary, lower-order elements are more difficult to get; see for instance [54] for the convergence analysis of a similar method, which is only O(h) accurate [55, 258]. We also refer to [28, 54, 122, 126, 181, 182, 323] for other examples. □
Remark 10.4.18.
It is possible to use a duality argument to get an O(h 3) estimate for \(\Vert \underline{\theta } -\underline{\theta }_{h}\Vert _{0}\) and \(\Vert w - w_{h}\Vert _{0}\). See [126]. □
Now to end this lengthy section, we are in a position to present general guidelines for the discretisation of Mindlin–Reissner problems.
We must emphasise again that the decomposition principle makes apparent a direct link with the Stokes problem. Indeed, all examples for which a satisfactory analysis could be achieved contained an already proven Stokes element. If we distinguish the case of continuous pressure approximation and the case of discontinuous pressure element, we get two types of strategies.
10.4.6 Continuous Pressure Approximations
-
Suppose one knows Θ h ×Q h to be a good approximation of the Stokes problem with \(Q_{h} \subset {H}^{1}(\varOmega )\).
-
Choose Z h an approximation of \(H_{0}^{1}(\varOmega )\) of the same order of accuracy.
-
Write \(\Gamma _{h} = \mathrm{g\underline{ra}d}\ Z_{h} + \mathrm{c}\underline{\mathrm{ur}}\text{l}Q_{h}\).
In this context, the definition of Γ h does not lead, in general, to a standard space and the decomposition principle of Theorem 10.4.2 and Remark 10.4.5 is the only way to handle things from a computational point of view. It may, however, happen, for a clever choice of Z h and Q h , that Γ h turns out to be a standard polynomial space. Such a situation has been encountered in [28] where, using for \(\Theta _{h} \times Q_{h}\) the MINI element, but taking Z h to be \(\mathcal{L}_{1}^{1,\mathit{NC}}\), that is, a nonconforming P 1 approximation of H 0 1(Ω), Γ h comes to be the whole space \({(\mathcal{L}_{0}^{0})}^{2}\) and not a proper subspace. For an extension of the Arnold-Falk element to higher degree, see [14, 26].
10.4.7 Discontinuous Pressure Elements
This second class of approximations to the Stokes problem has been the basis for the “reduced integration” method of the last subsection. Here, we shall try to outline the principal features of this strategy in order to provide a guide for possible extensions, some of which can be found in [117].
-
1.
Here again, our starting point is an approximation of the Stokes problem Θ h ×Q h , Q h being a space of discontinuous polynomial functions. This approximation should, of course, satisfy the inf-sup condition.
-
2.
We need to match this with an approximation Γ h of H 0(curl, Ω). More precisely, we need a couple of spaces \((\Gamma _{h},Q_{h})\) (where Q h is the same as before) and a uniformly bounded linear operator Π h → Γ h such that we have the commuting diagram:
(10.4.155)where \(\Theta = {({H}^{1}(\varOmega ))}^{2} \cap H_{0}\;(\text{curl},\varOmega )\) and P h is the L 2-projection operator.
-
3.
We finally need a space \(Z_{h} \subset H_{0}^{1}(\varOmega )\) such that
$$\displaystyle{ \mathrm{g\underline{ra}d}\ Z_{h} =\{\underline{\delta } _{h} \in \Gamma _{h},\;\text{curl }\underline{\delta }_{h} = 0\}. }$$(10.4.156)
Ingredients (1), (2), (3) will produce a plate element for which one can essentially repeat the proof of Theorem 10.4.5 and obtain optimal error estimates for \(\underline{\theta }\) and w.
References
D. N. Arnold, F. Brezzi, and L. D. Marini. Locking-free Reissner-Mindlin elements without reduced integration. Comput. Methods Appl. Mech. Engrg, 196:3660–3671, 2007.
D.N. Arnold. Discretization by finite elements of a model parameter dependent problem. Numer. Math., 37:405–421, 1981.
D.N. Arnold and F. Brezzi. Mixed and non-conforming finite element methods: implementation, post-processing and error estimates. Math. Modelling Numer. Anal., 19:7–35, 1985.
D.N. Arnold, F. Brezzi, and L. D. Marini. A family of discontinuous Galerkin finite elements for the Reissner-Mindlin plate. J. Sci. Comput, 22/23:25–45, 2005.
D.N. Arnold, J. Douglas, and C.P. Gupta. A family of higher order mixed finite element methods for plane elasticity. Numer. Math., 45:1–22, 1984.
D.N. Arnold and J.S. Falk. A uniformly accurate finite element method for the Mindlin-Reissner plate. SIAM. J. Num. Anal., 26:1276–1290, 1989.
D.N. Arnold and J.S. Falk. The boundary layer for the Reissner-Mindlin plate model. SIAM J. Math. Anal., 21:281–312, 1990.
D.N. Arnold and A.L. Madureira. Asymptotic estimates of hierarchical modeling. Math. Models Methods Appl. Sci, 13:1325–1350, 2003.
I. Babuška, J.E. Osborn, and J. Pitkaranta. Analysis of mixed methods using mesh-dependent norms. Math. Comp., 35:1039–1079, 1980.
K.J. Bathe and F. Brezzi. The convergence of a four-node plate bending element based on Mindlin-Reissner plate theory and a mixed interpolation. In J.R. Whiteman, editor, Proceedings of the Conference on Mathematics of Finite Elements and applications V , pages 491–503. Academic Press, New-York, 1985.
K.J. Bathe and E. Dvorkin. A continuum mechanics based four-node shell element for general non-linear analysis. J. Eng. Comp., 1:77–78, 1984.
J.H. Bramble and R.S. Falk. Two mixed finite element methods for the simply supported plate problem. R.A.I.R.O. Anal. Numer., 17:337–384, 1983.
F. Brezzi. Sur la méthode des éléments finis hybrides pour le problème biharmonique. Numer. Math., 24:103–131, 1975.
F. Brezzi. Hybrid approximations of nonlinear plate-bending problems. In Hybrid and mixed finite element methods (Atlanta, Ga., 1981), pages 267–280. Wiley, Chichester, 1983.
F. Brezzi, K.J. Bathe, and M. Fortin. Mixed-interpolated elements for Reissner-Mindlin plates. Int. J. Num. Meth. Eng., 28:1787–1801, 1989.
F. Brezzi and M. Fortin. Mixed and hybrid finite element methods. Springer-Verlag, New York, 1991.
F. Brezzi, M. Fortin, and R. Stenberg. Error analysis of mixed-interpolated elements for Reissner-Mindlin plates. Math. Models Methods Appl. Sci., 1(2):125–151, 1991.
F. Brezzi and L.D. Marini. On the numerical solution of plate bending problems by hybrid methods. R.A.I.R.O. Anal. Numer., pages 5–50, 1975.
F. Brezzi and P.A. Raviart. Mixed finite element methods for 4th order elliptic equations. In J. Miller, editor, Topics in Numerical Analysis III. Academic Press, New-York, 1978.
P.G. Ciarlet and R. Glowinski. Dual iterative techniques for solving a finite element approximation of the biharmonic equation. Comp. Meth. Appl. Mech. Eng., 5:227–295, 1975.
P.G. Ciarlet and P.A. Raviart. A mixed finite element method for the biharmonic equation. In C. de Boor, editor, Mathematical Aspects of Finite Element in Partial Differential Equations. Academic Press, New York, 1974.
L. Comodi. The Hellan-Hermann-Johnson method: error estimates for the Lagrange multipliers and post processing. Math. Comp., 52:17–30, 1989.
P. Destuynder. Une théorie asymptotique des plaques minces en élasticité linéaire. Masson, Paris, 1986.
R. Duran and E. Liberman. On mixed finite element methods for the Reissner-Mindlin plate model. Math. Comp, 58(198):561–573, 1992.
R. G. Duran and E. Liberman. On the convergence of a triangular mixed finite element method for Reissner-Mindlin plates. Math. Models Methods Appl. Sci, 6:339–352, 1996.
R.S. Falk. Approximation of the biharmonic equation by a mixed finite element method. SIAM J. Numer. Anal., 15, 556–567 (1978)
R.S. Falk. Finite elements for the Reissner-Mindlin plate. In D. Boffi and L. Gastaldi, editors, Mixed Finite Elements,Compatibility Conditions and Applications. Springer-Verlag, Berlin, 2008.
R.S. Falk and J.E. Osborn. Error estimates for mixed methods. RAIRO Anal. Numér., 14(3):249–277, 1980.
G.J. Fix, M.D. Gunzburger, and R.A. Nicolaides. Theory and applications of mixed finite element methods. In C.V. Coffman and G.J. Fix, editors, Constructive Approaches to Mathematical Models, pages 375–393. Academic Press, New York, 1979.
V. Girault and P.A. Raviart. Finite Element Approximation of Navier-Stokes Equations, volume 749 of Lectures Notes in Math. Springer-Verlag, Berlin, 1979.
R. Glowinski. Approximations externes par éléments finis de Lagrange d’ordre un et deux, du problème de Dirichlet pour l’opérateur biharmonique. Méthodes itératives de résolution des problèmes approchés. In J. Miller, editor, Topics in Numerical Analysis, pages 123–171, New York, 1973. Academic Press.
R. Glowinski. Approximations externes par éléments finis d’ordre 1 et 2 du problème de Dirichlet pour l’opérateur biharmonique,methode itérative de résolution des problèmes approchés. In J. Miller, editor, Topics in Numerical Analysis. Academic Press, New York, 1973.
R. Glowinski. Numerical Methods for Nonlinear Variational Problems. Springer-Verlag, Berlin, 1984.
R. Glowinski and O. Pironneau. Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem. SIAM Review, 17:167–212, 1979.
P. Grisvard. Elliptic Problems in Non-Smooth Domains. Pitman, London, 1985.
P. Grisvard. Singularities in Boundary Value Problems. Masson, Paris, 1992.
T.J.R. Hughes and T.E. Tezduyar. Finite elements based upon Mindlin plate theory with particular reference to the four-node bilinear isoparametric element. Jour. of App. Mech., 48:587–596, 1981.
C. Johnson. On the convergence of a mixed finite element method for plate bending problems. Numer. Math., 21:43–62, 1973.
C. Johnson and B. Mercier. Some equilibrium finite element methods for two-dimensional elasticity problems. Numer. Math., 30:103–116, 1978.
J.L. Lions and E. Magenes. Problèmes aux limites non-homogènes et applications. Dunod, Paris, 1968.
L.D. Marini. Implementation of hybrid finite element methods and associated numerical problems, part 1. Technical Report 36, IAN-CNR, Pavia, 1976.
L.D. Marini. Implementation of hybrid finite element methods and associated numerical problems, part 2. Technical Report 182, IAN-CNR, Pavia, 1978.
B. Mercier. Numerical solution of the biharmonic problem by mixed finite elements of class C 0. Boll. U.M.I., 10:133–149, 1974.
T. Miyoshi. A finite element method for the solution of fourth order partial differential equations. Kumamoto J. Sci. Math., 9:87–116, 1973.
T.H.H. Pian and P. Tong. Basis of finite element methods for solid continua. Int. J. Num. Meth. Eng., 1:3–28, 1969.
J. Pitkäranta. Analysis of some low-order finite element schemes for Mindlin-Reissner and Kirchhoff plates. Numer. Math., 53:237–254, 1988.
R. Scholz. Approximation von Sattelpunkten mit Finiten Elementen. Bonner Mathematischen Schriften, 89:54–66, 1976.
R. Scholz. A mixed method for fourth order problems using linear finite elements. R.A.I.R.O. Anal. Numer., 12:85–90, 1978.
R. Scholz. A remark on the rate of convergence for a mixed finite element method for second order problems. Numer. Funct. Anal. Optim., 4:269–277, 1982.
R. Temam. Navier-Stokes Equations. North-Holland, Amsterdam, 1977.
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Boffi, D., Brezzi, F., Fortin, M. (2013). Complements on Plate Problems. In: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol 44. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36519-5_10
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