An abstract inf-sup problem inspired by limit analysis in perfect plasticity and related applications

This paper is concerned with analysis of the abstract duality problem

where $P\subset X$ is a closed, convex set with $0_{X}\in P$, $X,Y$ are Banach spaces, $L$ is a non-trivial continuous linear functional in $Y$, and $a\colon X\times Y\rightarrow\mathbb{R}$ is a bilinear form continuous with respect to both arguments. Henceforth the problem in the right hand side of (1.1) is called primal, while the one in the left hand side is called dual. It is easy to check that $0\leq\lambda^{*}\leq\zeta^{*}\leq+\infty$. In general, necessary and sufficient conditions for $\lambda^{*}=\zeta^{*}$ are unknown (therefore, (1.1) uses the symbol $\stackrel{{\scriptstyle?}}{{=}}$). One of our main goals is to identify cases where (1.1) holds as the equality.

Problem (1.1) and similar problems appear in various applications, from mechanics to economics [12, 9, 3]. In finite dimensions, minimax and maximin variants of these problems are known in game theory [19] and linear, cone or convex programming [11, 5, 21, 18].

In classical elastic-perfect plasticity, (1.1) is known as the limit analysis problem. In this case, $\lambda^{*}$ is the factor that determines the critical load $\lambda^{*}L$ ($L$ is a linear functional associated with external loads), subject to the constraint set $P$ of plastically admissible stresses; see for example [17, 9, 32, 8, 10, 29, 30, 28, 16]. For the load $\lambda L$ with $\lambda>\zeta^{*}$, no solution of the primal and dual problems exists; the body is unable to sustain the loading and collapses. Also, we note the similarity between (1.1) and the shakedown analysis problem (see [33] and the references therein).

Although the limit analysis problem has been studied for several decades, it is still unsolved in the general setting and presents a challenging problem from the theoretical and numerical points of view. There are several reasons that stimulate further analysis of the problem. First, we notice that the equality $\lambda^{*}=\zeta^{*}$ can be analyzed in a rather general framework introduced in [12] or by using particular results from [9, 32, 16]. However, these results do not cover any interesting cases. Second, additional and hidden constraints appear in the primal and dual problems (that follow from their inf- and sup-definitions). They often make the numerical analysis difficult. The third reason is related to the choice of the function spaces $X$ and $Y$. This question becomes especially important if the primal problem is related to minimization of a functional with linear growth at infinity and a certain problem relaxation must be done to find a minimizer (see e.g. [17, 32, 29, 10]). Then we arrive, for example, at a formulation in which the $BD$- or $BV$- spaces of functions of bounded deformation and bounded variation, respectively, are appropriate for the problem setting [32]. Nevertheless, standard Sobolev spaces seem to be sufficient or even more appropriate for analysis of numerical errors [27, 28, 16]. Finally, reliable estimates of $\lambda^{*}$ and $\zeta^{*}$ are often required because they define safety factors of structures. Lower bounds of $\lambda^{*}$ and upper bounds of $\zeta^{*}$ can be found by analytical approaches for specific geometries [8] or, more generally, by finite element methods; see [30] and the references therein. Computable majorants of $\zeta^{*}$ can be found in recent papers [28, 16].

In order to investigate the abstract problem (1.1), we use the ideas applied in [14, 15, 28, 16] for analysis of limit load problems. This extension is not always straightforward and requires innovative techniques. In particular, we derive conditions for the equality $\lambda^{*}=\zeta^{*}$ to hold, the existence of a solution to the dual problem in (1.1), a regularization method for solving (1.1) with related convergence results, and a computable majorant of $\zeta^{*}$, which can be used for a posteriori analysis of numerical results.

One of the key assumptions in the results presented is the so-called inf-sup condition on convex cones which was introduced in [16]. This condition generalizes the Babuška-Brezzi condition defined on function spaces [1, 4]. Conditions of this type are important for analysis of saddle point problems generated by various mixed finite element approximations [3].

Generalization and abstraction of results is a basic procedure that allows results and insights in a particular application to be applied to broad classes of problems. In our case, we show that the results presented here are useful in problems of gradient-enhanced plasticity and in delamination problems. We choose the strain gradient model studied in [25, 24, 6, 26] and use (1.1) for the description of a global yield surface and for limit load analysis. In related work, limit analysis has been considered for a model in which size-dependence is through the gradient of a scalar function of plastic strain, see [13, Section 7] or [23]. One can expect further applications of the problem (1.1), at least within nonlinear mechanics.

The rest of the paper is organized as follows. In Section 2, we introduce the primal and dual problems, discuss them in more detail, and present criteria ensuring their solvability and the principal duality relation $\lambda^{*}=\zeta^{*}$. One of the criteria is based on the inf-sup condition on convex cones. The proof of this new result is carried out in Section 3 and its extensions are studied in Section 4. Section 5 is devoted to a regularization of the problem (1.1). The regularized problem provides a lower and sufficiently sharp bound of $\lambda^{*}$, reduces the constraints in the dual problem, and thus it is convenient for numerical solution. In Section 6, a computable majorant of the quantity $\zeta^{*}$ is derived. Section 7 contains particular examples of the abstract problem (1.1), including classical and strain-gradient plasticity and a delamination problem.

First, we recapitulate the basic assumptions used in the problem (1.1):

• (A1)

  $X,Y$ are two Banach spaces equipped with the norms $\|.\|_{X}$ and $\|.\|_{Y}$, respectively. The corresponding dual spaces are denoted by $X^{*}$ and $Y^{*}$;

• (A2)

  $a\colon X\times Y\rightarrow\mathbb{R}$ is a continuous bilinear form;

• (A3)

  $L\colon Y\rightarrow\mathbb{R}$ is a non-trivial continuous linear functional (i.e., $L\neq 0$ in $Y^{*}$);

• (A4)

  $P\subset X$ is a nonempty, closed and convex set with $0_{X}\in P$.

The primal problem in (1.1) reads

where

The functional $\mathcal{J}$ is convex, proper and 1-positively homogeneous. In addition, the effective domain $\mathrm{dom}\,\mathcal{J}$ is a convex cone; see Section 6 for more details. We shall assume that all cones considered in the text have a vertex at zero, so henceforth do not emphasize this property. We say that the problem (2.1) has a solution if the functional $\mathcal{J}$ has a minimizer in the feasible set $\mathrm{dom}\,\mathcal{J}\cap\{y\in Y\ |\;L(y)=1\}$. Using the positive homogeneity of $\mathcal{J}$, we obtain the following useful and equivalent definition of $\zeta^{*}$:

To rewrite the dual problem in (1.1) we define the functional

and the related set

Then, we have

We shall say that the problem (2.6) has a solution if $\lambda^{*}<+\infty$ and there exists $\bar{x}\in P\cap\Lambda_{\lambda^{*}}$.

Now, we present three different results ensuring the equality $\lambda^{*}=\zeta^{*}$ and the existence of primal or dual solutions. The first result follows from [12, Proposition VI.2.3 and Remark VI.2.3].

It is worth noting that for the validity of the theorem it suffices to assume that the set $P_{C}$ is only closed and convex in $X$ and satisfies $(D4)$. On the other hand, we have

This fact (independence of the scaling parameter) explains why we assume that $P_{C}$ is a convex cone. In addition, we shall see in Section 6 that the cones $P_{C}$ and $\mathrm{dom}\,\mathcal{J}$ are closely related.

We also note that any closed linear subspace of $X$ is a special case of the cone $P_{C}$. Then, we arrive at the standard inf-sup condition on function spaces. This case will be considered in Theorem 4.2 and in Section 7.

The next lemma shows that the function $\varphi$ is closely related to the problems (2.6) and (2.1).

Next, consider the following problem: given $\lambda\geq 0$,

Proof of Theorem 2.3. The proof is based on Lemma 3.3. We show that the problem (3.10) has a solution for any $\lambda\geq 0$ under the assumptions of Theorem 2.3. From Lemma 3.1, we know that the function $\Phi_{\lambda}$ is convex and Lipschitz continuous in $X$ for any $\lambda\in\mathbb{R}_{+}$. Using the assumptions (D3) and (D4) we prove that $\Phi_{\lambda}$ is also coercive in $P$ for any $\lambda\geq 0$. Indeed, for any $x\in P$ and $\lambda\in\mathbb{R}_{+}$, we have:

where $c_{*}>0$ is the inf-sup constant from (2.7) and $\rho_{A}$ is a positive constant characterizing the boundedness of $A$. Since $X$ is a reflexive Banach space, the properties of $\Phi_{\lambda}$ guarantee that (3.10) has a solution for any $\lambda\geq 0$. ∎

Now we present three different generalizations (or extensions) of Theorem 2.3. Since their proofs are quite analogous, we only sketch them.

First, the assumption (D4) cannot hold if the subspace

contains an element $x_{0}\in P_{C}$ such that $x_{0}\neq 0$. In Section 7, we will show that this case may arise in some applications. To weaken the assumption (D4) we introduce the quotient space $X/H$ (with the norm $\|\cdot\|_{X/H}$) whose elements are the equivalence classes induced by the equivalence relation:

Let $P/H$ denote the set of equivalent classes generated by the set $P$. It is easy to verify that $P/H$ is a closed, convex, and nonempty set in $X/H$. Similarly, one can introduce the sets $P_{A}/H$ and $P_{C}/H$, where $P_{A}$ and $P_{C}$ are defined in accordance with the assumption (D3). These sets have properties analogous to properties of $P_{A}$ and $P_{C}$. In particular, for any $x\in P/H$ there exists $x_{A}\in P_{A}/H$ and $x_{C}\in P_{C}/H$ such that $x=x_{A}+x_{C}$. We note that if $X$ is a Hilbert space then $X/H$ can be identified with the orthogonal complement $H^{\perp}$ of $H$ in $X$ and $P/H$ with the projection of $P$ onto $H^{\perp}$.

Sketch of the proof: It suffices to show that (3.10) has a solution in $P$ for any $\lambda\geq 0$ under the assumptions of this theorem. Let $\lambda\geq 0$ be fixed. Using (4.1), we see that the function $\Phi_{\lambda}$ defined in (3.1) satisfies

Therefore, (3.10) has a solution in $P$ if and only if $\Phi_{\lambda}$ has a minimum in $P/H$. From (4.2), one can prove the coercivity of $\Phi_{\lambda}$ in $P/H$ analogously as in the proof of Theorem 2.3. Therefore, $\Phi_{\lambda}$ has a minimum in $P/H$ and thus the result of Theorem 4.1 holds. ∎

Second, it turns out that the assumption (D2) of Theorem 2.3 can be extended to some reflexive Banach spaces associated with a bounded Lipschitz domain $\Omega\subset\mathbb{R}^{d}$, $d=2,3$.

Sketch of the proof: It suffices to modify formulae (3.4)–(3.9). To this end, we set

Then (3.5) holds and there exists a unique element $y_{x}\in Y$ such that

or equivalently

Consequently,

and

where $1/q=1-1/p$. The rest of the proof is analogous to that of Section 3. ∎

The third extension illustrates that Theorem 2.3 remains valid even if the space $Y$ is replaced by its conic subset.

Sketch of the proof: The following two changes in the proof of Theorem 2.3 are necessary:

1. 1.

   We define

   $$\Phi_{\lambda}(x):=\sup_{\begin{subarray}{c}y\in Y_{C}\\ y\neq 0_{Y}\end{subarray}}\frac{-a(x,y)+\lambda L(y)}{\|y\|_{Y}}$$

   and consider the minimization problem

   $$\mbox{find }y_{x}\in Y_{C}:\quad\mathcal{L}(x,y_{x})\leq\mathcal{L}(x,y)\quad\forall y\in Y_{C}$$

   with $\mathcal{L}$ defined by (3.4). Since $Y_{C}$ is a closed convex cone, the corresponding necessary and sufficient condition characterising $y_{x}$ reads

   $$(y_{x},y)_{Y}\geq\lambda L(y)-a(x,y)\quad\forall y\in Y_{C},\quad(y_{x},y_{x})_{Y}=\lambda L(y_{x})-a(x,y_{x}).$$

   We obtain

   $$\|y_{x}\|_{Y}=\sup_{\begin{subarray}{c}y\in Y_{C}\\ y\neq 0_{Y}\end{subarray}}\frac{(y_{x},y)_{Y}}{\|y\|_{Y}}\geq\sup_{\begin{subarray}{c}y\in Y_{C}\\ y\neq 0_{Y}\end{subarray}}\frac{-a(x,y)+\lambda L(y)}{\|y\|_{Y}}\geq\frac{-a(x,y_{x})+\lambda L(y_{x})}{\|y_{x}\|_{Y}}=\|y_{x}\|_{Y},$$

   so that $\|y_{x}\|_{Y}=\Phi_{\lambda}(x)$.

2. 2.

   To prove Lemma 3.3, we modify (2.6) as follows:

   $$\lambda^{*}=\sup\{\lambda\in\mathbb{R}_{+}\ |\;\;P\cap\Lambda_{\lambda}\neq\emptyset\},\quad\Lambda_{\lambda}:=\{x\in X\ |\;\;a(x,y)\geq\lambda L(y)\;\;\forall y\in Y_{C}\}.$$

Then the proof of Theorem 2.3 is applicable without any substantial changes.

Regularization methods are often used for solving nonsmooth, constrained, or ill-posed problems. As an example, we mention proximal point methods [22] which can be used for solving the problems (2.6) and (2.1).

Here we consider another regularization method which has been subsequently developed in [31, 7, 14, 15] and used also in [28, 16]. In these recent papers, this method has been called either the “indirect incremental method” or the “penalization method”. Below, we generalize, results of [14, 15] and show that some of these can be established in a simpler way. Within this section it is assumed that the conditions (A1)–(A4) from Section 2 hold.

To regularize the functional $\mathcal{J}$ defined by (2.2) we introduce the functional

where $\alpha>0$ is a given parameter. It is easy to see that $\mathcal{J}_{\alpha}$ is convex and finite-valued in $Y$ (unlike the functional $\mathcal{J}$) and $\mathcal{J}_{\alpha_{1}}\leq\mathcal{J}_{\alpha_{2}}\leq\mathcal{J}$ for any $\alpha_{1},\alpha_{2}>0$, $\alpha_{1}\leq\alpha_{2}$.

The regularization of the primal problem (2.1) with respect to the parameter $\alpha$ defines the function $\psi:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ :

In view of (5.1) and [12, Proposition VI 2.3], it holds

Thus, the main duality relation holds without any gap, unlike the original primal-dual problem (1.1). The properties of the function $\psi$ are set out in the following theorem.