Abstract
Chapter 13 develops dual variational formulations for the two dimensional equations of the nonlinear elastic Kirchhoff-Love plate model. We obtain a convex dual variational formulation which allows non positive definite membrane forces. In the third section, similar to the triality criterion introduced in [36], we obtain sufficient conditions of optimality for the present case. Again the results are based on fundamental tools of Convex Analysis and the Legendre Transform, which can easily be analytically expressed for the model in question.
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Keywords
- Dual Application
- Kirchhoff Love Plate Model
- Dual Variational Formulation
- Positive Definite
- Duality Principle
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1 Introduction
In the present work we develop dual variational formulations for the Kirchhoff–Love thin plate model. Earlier results establish the complementary energy under the hypothesis of positive definiteness of the membrane force tensor at a critical point (please see [30–32, 36, 65] for details). In more recent works Gao has applied his triality theory to models in elasticity (see [33–35] for details) obtaining duality principles which allow the optimal stress tensor to be negative definite. Here for the present case we have obtained a dual variational formulation which allows the global optimal point in question to be not only positive definite (for related results see Botelho [11, 13]) but also not necessarily negative definite. The approach developed also includes sufficient conditions of optimality for the primal problem. Moreover, a numerical example concerning the main duality principle application is presented in the last section.
It is worth mentioning that the standard tools of convex analysis used in this text may be found in [13, 25, 54], for example. Another relating result may be found in [14].
At this point we start to describe the primal formulation.
Let \(\varOmega \subset \mathbb{R}^{2}\) be an open, bounded, connected set which represents the middle surface of a plate of thickness h. The boundary of Ω, which is assumed to be regular, is denoted by ∂ Ω. The vectorial basis related to the Cartesian system {x 1, x 2, x 3} is denoted by (a α , a 3), where α = 1, 2 (in general Greek indices stand for 1 or 2) and where a 3 is the vector normal to Ω, whereas a 1 and a 2 are orthogonal vectors parallel to Ω. Also, n is the outward normal to the plate surface.
The displacements will be denoted by
The Kirchhoff–Love relations are
Here − h∕2 ≤ x 3 ≤ h∕2 so that we have u = (u α , w) ∈ U where
It is worth emphasizing that the boundary conditions here specified refer to a clamped plate.
We define the operator \(\varLambda: U \rightarrow Y \times Y\), where \(Y = Y ^{{\ast}} = L^{2}(\varOmega; \mathbb{R}^{2\times 2})\), by
The constitutive relations are given by
where \(\{H_{\alpha \beta \lambda \mu }\}\) and \(\{h_{\alpha \beta \lambda \mu } = \frac{h^{2}} {12} H_{\alpha \beta \lambda \mu }\}\) are symmetric positive definite fourth-order tensors. From now on, we denote \(\{\overline{H}_{\alpha \beta \lambda \mu }\} =\{ H_{\alpha \beta \lambda \mu }\}^{-1}\) and \(\{\overline{h}_{\alpha \beta \lambda \mu }\} =\{ h_{\alpha \beta \lambda \mu }\}^{-1}\).
Furthermore {N α β } denote the membrane force tensor and {M α β } the moment one. The plate stored energy, represented by \((G\circ \varLambda ): U \rightarrow \mathbb{R}\), is expressed by
and the external work, represented by \(F: U \rightarrow \mathbb{R}\), is given by
where \(P,P_{1},P_{2} \in L^{2}(\varOmega )\) are external loads in the directions a 3, a 1, and a 2, respectively. The potential energy, denoted by \(J: U \rightarrow \mathbb{R}\), is expressed by
It is important to emphasize that the existence of a minimizer (here denoted by u 0) related to J(u) has been proven in Ciarlet [22]. Some inequalities of Sobolev type are necessary to prove the above result. In particular, we assume the boundary ∂ Ω of Ω is regular enough so that the standard Gauss–Green formulas of integration by parts and the well-known Sobolev imbedding and trace theorems hold. Details about such results may be found in [1].
Finally, we also emphasize from now on, as their meaning is clear, we may denote L 2(Ω) and \(L^{2}(\varOmega; \mathbb{R}^{2\times 2})\) simply by L 2 and the respective norms by \(\|\cdot \|_{2}.\) Moreover derivatives are always understood in the distributional sense, θ denotes the zero vector in appropriate Banach spaces, and the following and relating notations are used:
and
2 The Main Duality Principle
In this section, we develop a duality principle presented in similar form in [13, 14]. The novelty here is its suitability for the Kirchhoff–Love plate model.
Theorem 13.2.1.
Let \(\varOmega \subset \mathbb{R}^{2}\) be an open, bounded, connected set with a regular boundary denoted by Γ. Suppose \((G\circ \varLambda ): U \rightarrow \mathbb{R}\) is defined by
and let \((F \circ \varLambda _{2}): U \rightarrow \mathbb{R}\) be expressed by
where
where \(u = (u_{\alpha },w) \in U = W_{0}^{1,2}(\varOmega; \mathbb{R}^{2}) \times W_{0}^{2,2}(\varOmega ).\)
Also, define \(F_{1}: U \rightarrow \mathbb{R}\) by
where \(\tilde{P} = (P,P_{\alpha }),\) and let \(J: U \rightarrow \mathbb{R}\) be expressed by
Under such hypotheses, we have
where, when the meaning is clear, denoting \(Y = Y ^{{\ast}} = L^{2}(\varOmega, \mathbb{R}^{2\times 2}) \equiv L^{2}\), \(Y _{1} = Y _{1}^{{\ast}} = L^{2}(\varOmega, \mathbb{R}^{2}) \equiv L^{2},\) \(v^{{\ast}} = (v_{1}^{{\ast}},v_{2}^{{\ast}},v_{3}^{{\ast}})\) , and \(v_{1}^{{\ast}} =\{ N_{\alpha \beta }\},\) \(v_{2}^{{\ast}} =\{ Q_{\alpha }\}\) , and \(v_{3}^{{\ast}} =\{ M_{\alpha \beta }\},\) we have
that is, recalling that \(\tilde{P} = (P,P_{\alpha }),\) we may write
Here
and
Also,
and
if N K is positive definite, where
Moreover,
and
Here we recall that \(z_{12}^{{\ast}} = z_{21}^{{\ast}} = 0\mbox{ in }\varOmega.\)
Finally, if there is a point \((u_{0},v_{0}^{{\ast}},z_{0}^{{\ast}}) \in U \times A^{{\ast}}\times Y _{0}^{{\ast}}\) such that
where K > 0 is such that
we have that
Proof.
Observe that
\(\forall u \in U,v^{{\ast}}\in A^{{\ast}},z^{{\ast}}\in Y _{0}^{{\ast}}.\)
Thus,
\(\forall u \in U,v^{{\ast}}\in A^{{\ast}},z^{{\ast}}\in Y _{0}^{{\ast}},\) so that, taking the supremum in z ∗ at both sides of last inequality, we obtain
\(\forall u \in U,v^{{\ast}}\in A^{{\ast}}.\) Hence,
Finally, suppose that \((u_{0},v_{0}^{{\ast}},z_{0}^{{\ast}}) \in U \times A^{{\ast}}\times Y _{0}^{{\ast}}\) is such that
From the variation in v ∗ we obtain
that is,
and
From the variation in z ∗ we get
that is, from this and (13.16) we get
so that
From the variation in u, we get
so that \(v_{0}^{{\ast}}\in A^{{\ast}}\). Therefore, from (13.17), (13.20), and (13.21) we have
To complete the proof just observe that from the condition indicated in (13.10), the extremal relations (13.18) and (13.19) refer to a global optimization (in z ∗, for a fixed v 0 ∗), so that the infimum indicated in the dual formulation is attained for the z 0 ∗ in question.
From this and (13.22), the proof is complete.
3 Another Duality Principle
In this section we present another result, which is summarized by the following theorem.
Theorem 13.3.1.
Considering the introduction statements, let \((G\circ \varLambda ): U \rightarrow \mathbb{R}\) be expressed by
where
where \(u = (u_{\alpha },w) \in U = W_{0}^{1,2}(\varOmega; \mathbb{R}^{2}) \times W_{0}^{2,2}(\varOmega ).\)
As above, define \(F: U \rightarrow \mathbb{R}\) by
and \(J: U \rightarrow \mathbb{R}\) by
Under such assumptions, we have
where
so that
Moreover,
and
if
where \(\hat{w} \in W_{0}^{2,2}(\varOmega )\) is the solution of equation
Also,
and
Finally, if there exists u 0 ∈ U such that δJ(u 0 ) = θ and \(N_{0} =\{ N_{\alpha \beta }(u_{0})\} \in A_{1}\) , where \(N_{\alpha \beta }(u_{0}) = H_{\alpha \beta \lambda \mu }\gamma _{\lambda \mu }(u_{0}),\) then
Proof.
Clearly
Hence,
Now suppose there exists u 0 ∈ U such that δ J(u 0) = θ and \(N_{0} =\{ N_{\alpha \beta }(u_{0})\} \in A_{1}\).
First, note that from δ J(u 0) = θ, the following extremal equation is satisfied:
that is, N 0 ∈ A 2, so that \(N_{0} \in A_{1} \cap A_{2} = A^{{\ast}}.\)
Thus, from N 0 ∈ A 1, we obtain
where \(\hat{w} \in W_{0}^{2,2}(\varOmega )\) is the solution of equation
Replacing such a relation in (13.29), we obtain
Hence, also from the equation δ J(u 0) = θ and (13.30), we may get \(\hat{w} = w_{0},\) so that from this and (13.29), we obtain
Finally, considering that
we get
so that
From this and (13.28) and also from the fact that N 0 ∈ A ∗, the proof is complete.
Remark 13.3.2.
From the last duality principle, we may write
where
\(\hat{A} = A_{1} \cap A_{2} \cap A_{3} \cap A_{4},\)
and A 1 and A 2 as above specified, that is,
where
and
Finally, we could suggest as a possible approximate dual formulation the problem of maximizing \(-J_{K}^{{\ast}}(M,N,u)\) on \(A_{1} \cap A_{2} \cap A_{3},\) where K > 0 and
A study about the system behavior as K → +∞ is planned for a future work. Anyway, big values for K > 0 allow the gap function \(\frac{1} {2}\int _{\varOmega }N_{\alpha \beta }w_{,\alpha }w_{,\beta }\;dx\) to be nonpositive at a possible optimal point inside the region of convexity of J K ∗.
4 An Algorithm for Obtaining Numerical Results
In this section we develop an algorithm which we prove, under certain mild hypotheses; it is convergent up to a subsequence (the result stated in the next lines must be seen as an existence one and, of course, it is not the full proof of convergence from a numerical analysis point of view). Such a result is summarized by the following theorem.
Theorem 13.4.1.
Consider the system of equations relating the boundary value form of the Kirchhoff–Love plate model, namely
where
Define, as above,
and \(J: U \rightarrow \mathbb{R}\) by
Assume \(\{\|P_{\alpha }\|_{2}\}\) are small enough so that (from [22] pages 285–287) if either
or
then
Let \(\{u_{n} = ((u_{n})_{\alpha },w_{n})\} \subset U\) be the sequence obtained through the following algorithm:
-
1.
Set n = 1.
-
2.
Choose \((z_{1}^{{\ast}})_{1},(z_{2}^{{\ast}})_{1} \in L^{2}(\varOmega ).\)
-
3.
Compute u n by
$$\displaystyle\begin{array}{rcl} u_{n}& =& \mbox{ argmin}_{u\in U}\left \{G(\varLambda u) + \frac{K} {2} \int _{\varOmega }(w_{x})^{2}\;dx + \frac{K} {2} \int _{\varOmega }(w_{y})^{2}\;dx\right. {}\\ & & \left.-\langle w_{x},(z_{1}^{{\ast}})_{ n}\rangle _{L^{2}} -\langle w_{y},(z_{2}^{{\ast}})_{ n}\rangle _{L^{2}}\right. {}\\ & & \left.+ \frac{1} {2K}\int _{\varOmega }(z_{1}^{{\ast}})_{ n}^{2}\;dx + \frac{1} {2K}\int _{\varOmega }(z_{2}^{{\ast}})_{ n}^{2}\;dx - F(u)\right \}, {}\\ \end{array}$$which means to solve the equation
$$\displaystyle{ \left \{\begin{array}{lll} M_{\alpha \beta,\alpha \beta } + (N_{\alpha \beta }w_{,\alpha })_{,\beta } + P + Kw_{,\alpha \alpha } - (z_{n}^{{\ast}})_{\alpha,\alpha } = 0,&\mbox{ in }\varOmega \\ \\ N_{\alpha \beta,\beta } + P_{\alpha } = 0 &\mbox{ in }\varOmega \\ \\ u_{\alpha } = w = \frac{\partial w} {\partial \mathbf{n}} = 0 &\mbox{ on }\partial \varOmega \end{array} \right. }$$(13.41) -
4.
Compute \(z_{n+1}^{{\ast}} = ((z_{1}^{{\ast}})_{n+1},(z_{2}^{{\ast}})_{n+1})\) by
$$\displaystyle\begin{array}{rcl} z_{n+1}^{{\ast}}& =& \mbox{ argmin}_{ z^{{\ast}}\in L^{2}\times L^{2}}\left \{G(\varLambda u_{n}) + \frac{K} {2} \int _{\varOmega }(w_{n})_{x}^{2}\;dx + \frac{K} {2} \int _{\varOmega }(w_{n})_{y}^{2}\;dx\right. {}\\ & & \left.-\langle (w_{n})_{x},z_{1}^{{\ast}}\rangle _{ L^{2}} -\langle (w_{n})_{y},z_{2}^{{\ast}}\rangle _{ L^{2}}\right. {}\\ & & \left.+ \frac{1} {2K}\int _{\varOmega }(z_{1}^{{\ast}})^{2}\;dx + \frac{1} {2K}\int _{\varOmega }(z_{2}^{{\ast}})^{2}\;dx - F(u_{ n})\right \}, {}\\ \end{array}$$that is,
$$\displaystyle{(z_{1}^{{\ast}})_{ n+1} = K(w_{n})_{x},}$$and
$$\displaystyle{(z_{2}^{{\ast}})_{ n+1} = K(w_{n})_{y}.}$$ -
5.
Set n → n + 1 and go to step 3 till the satisfaction of a suitable approximate convergence criterion.
Assume \(\{u_{n} = ((u_{n})_{\alpha },w_{n})\} \subset U\) is such that for a sufficiently big K > 0 we have
Under such assumptions, the sequence {u n } is uniquely defined (depending only on \((z^{{\ast}})_{1}\) ), and such that, up to a subsequence not relabeled, for some \(u_{0} = ((u_{0})_{\alpha },w_{0}) \in U\) , we have
and
where
is a solution for the system of equations indicated in (13.35).
Proof.
Since \(J: U \rightarrow \mathbb{R}\) is defined by
we have
\(\forall u \in U,\;z^{{\ast}}\in L^{2}(\varOmega ) \times L^{2}(\varOmega ).\)
From the hypotheses, {u n } is inside the region of strict convexity of the functional in U (for z ∗ fixed) in question, so that it is uniquely defined for each z n ∗ (through the general results in [11] we may infer the region of convexity of the functional
corresponds to the satisfaction of constraints
Denoting
we may easily verify that {α n } is a real nonincreasing sequence bounded below by \(\inf _{u\in U}\{J(u)\};\) therefore, there exists \(\alpha \in \mathbb{R}\) such that
From the hypotheses
\(\mbox{ if either }\|u_{\alpha }\|_{W_{0}^{1,2}(\varOmega )} \rightarrow \infty \) or \(\|w\|_{W_{0}^{2,2}(\varOmega )} \rightarrow \infty.\)
From this, (13.44), (13.47), and (13.48) we may infer there exists C > 0 such that
and
Thus, from the Rellich–Kondrachov theorem, up to a subsequence not relabeled, there exists \(u_{0} = ((u_{0})_{\alpha },w_{0}) \in U\) such that
and
so that, considering the algorithm in question,
where
From these last results, the Sobolev imbedding theorem and relating results (more specifically, Korn’s inequality and its consequences; details may be found in [21]), we have that there exist K 1, K 2 > 0 such that
and
On the other hand, u n ∈ U such that
is also such that
and
in the sense of distributions (theoretical details about similar results may be found in [25]).
Fix \(\phi \in C_{c}^{\infty }(\varOmega ).\) In the next lines, we will prove that
as \(n \rightarrow \infty,\;\forall \alpha,\beta \in \{ 1,2\}\) (here the repeated indices do not sum).
Observe that, since
from the Hölder inequality, we obtain
Moreover, from the generalized Hölder inequality, we get
where also up to here the repeated indices do not sum.
Thus (13.52) has been proven, so that we may infer that
Since \(\phi \in C_{c}^{\infty }(\varOmega )\) is arbitrary, we obtain
in the distributional sense.
Similarly,
for α ∈ { 1, 2}, also in the distributional sense.
From the convergence in question, we also get in a weak sense
The proof is complete.
Remark 13.4.2.
We emphasize that for each \(n \in \mathbb{N}\), from the condition indicated in (13.42), {u n } is obtained through the minimization of a convex functional. Therefore the numerical procedure translates into the solution of a sequence of convex optimization problems.
5 Numerical Results
In this section we present some numerical results. Let \(\varOmega = [0,1] \times [0,1]\) and consider the problem of minimizing \(J: U \rightarrow \mathbb{R}\) where
{N α β } denote the membrane force tensor and {M α β } the moment one, so that from the constitutive relations,
Also, \(F: U \rightarrow \mathbb{R}\) is given by
Here
\(\varGamma _{0} =\{ [0,y] \cup [x,0],\;0 \leq x,y \leq 1\},\) and \(P,P_{1},P_{2} \in L^{2}\) denote the external loads in the directions a 3, a 1, and a 2, respectively.
We consider the particular case where all entries of {H α β λ μ } and {h α β λ μ } are zero, except for H 1111 = H 2222 = H 1212 = 105 and \(h_{1111} = h_{2222} = h_{1212} = 10^{4}\). Moreover P = 1000, P 1 = −100, and P 2 = −100 (units refer to the international system). In a first moment, define the trial functions \(w:\varOmega \rightarrow \mathbb{R}\), \(u_{1}:\varOmega \rightarrow \mathbb{R}\), and \(u_{2}:\varOmega \rightarrow \mathbb{R}\) by
respectively.
The coefficients {a 1, a 2, a 3, a 4} will be found through the extremal points of J.
We have obtained only one real critical point, namely,
so that the candidate to optimal point is \(((u_{0})_{1},(u_{0})_{2},w_{0})\) where
With such values for the coefficients, it is clear that N 11(u 0) and N 22(u 0) are negative in Ω so that \(\{N_{\alpha \beta }(u_{0})\}\) is not positive definite. Even so, as we shall see in the next lines, the optimality criterion of the second duality principle developed may be applied. Let
We have that
Now observe that
Therefore s 11 > 0, s 22 > 0, and \(s_{11}s_{22} - s_{12}^{2} = 8.31155 {\ast} 10^{12} > 0\) so that W(a 1, a 2) is a positive definite quadratic form.
Hence, from the second duality principle, we may conclude that \(((u_{0})_{1},(u_{0})_{2},w_{0})\) is indeed the optimal solution (approximate global minimizer for J).
Refining the results through finite differences using the algorithm of last section, we obtain again the field of displacements w 0(x, y) (please see Fig. 13.1).
Vertical axis: w 0(x, y)-field of displacements
6 Conclusion
In this chapter, we develop duality principles for the Kirchhoff–Love plate model. The results are obtained through the basic tools of convex analysis and include sufficient conditions of optimality. It is worth mentioning that earlier results require the membrane force tensor to be positive definite in a critical point in order to guarantee global optimality, whereas from the new results here presented, we are able to guarantee global optimality for a critical point such that N 11(u 0) < 0 and \(N_{22}(u_{0}) < 0,\mbox{ in }\varOmega\). Finally, the methods developed here may be applied to many other nonlinear models of plates and shells. Applications to related areas (specially to the shell models found in [23]) are planned for future works.
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Botelho, F. (2014). Duality Applied to a Plate Model. In: Functional Analysis and Applied Optimization in Banach Spaces. Springer, Cham. https://doi.org/10.1007/978-3-319-06074-3_13
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