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Appendix a Some Elements of Convex Analysis Appendix A Some Elements of Convex Analysis A.1 Convex Sets Let C be a set of linear space V . This set is convex if Definition A.1 ∀x ∈ C, ∀y ∈ C, ∀θ ∈ ]0, 1[, point θx + (1 − θ)y ∈ C. Segment [0, 1] of space V = R is convex. The interior of a circle is a convex set of V = R2. On the contrary the exterior of a circle is not convex. Convex sets are useful in mechanics to describe numerous and various properties. For instance, the possible positions of a soccer ball above the football field is a convex set: it is C = {x = (xi), i = 1, 3 |xi ∈ R, x3 ≥ 0} . A.2 Convex Functions Let R = R ∪ {+∞} where the regular addition is completed by the rules ∀a ∈ R , a + (+∞) =+∞, +∞ + (+∞) =+∞. Multiplication by positive numbers is completed by ∀a ∈ R , a > 0, a × (+∞) =+∞. In this context, it is forbidden to multiply either by 0 or by negative numbers. Let f an application from linear space V into R. This function is convex if © Springer-Verlag Berlin Heidelberg 2017 259 M. Frémond, Collisions Engineering: Theory and Applications, Springer Series in Solid and Structural Mechanics 6, DOI 10.1007/978-3-662-52696-5 260 Appendix A: Some Elements of Convex Analysis Definition A.2 ∀x ∈ V, ∀y ∈ V, ∀θ ∈ ]0, 1[, f (θx + (1 − θ)y) ≤ θf (x) + (1 − θ)f (y). Remark A.1 Function f is actually multiplied by positive numbers because θ ∈ ]0, 1[. A.2.1 Examples of Convex Functions It is easy to prove that functions fp where V = R 1 x ∈ R → f (x) = |x|p ,with p ∈ R , p ≥ 1, p p are convex. Let function I from V = R into R is defined by I(x) = 0, if x ∈ [0, 1] , I(x) =+∞, if x ∈/ [0, 1] . This function is convex. It is called the indicator function of segment [0, 1] (Fig. A.2). More generally, we denote indicator function of set C ⊂ V , function IC defined by IC(x) = 0 , if x ∈ C, IC(x) =+∞, if x ∈/ C. It is easy to connect the convex function and convex set notions. It is shown that Theorem A.1 A convex C ⊂ V is convex if and only if its indicator function IC is convex. Indicator functions may seem a little bit strange. In the sequel it is to be seen in the examples that they are productive tools in mechanics for dealing with internal constraints relating mechanical quantities. Three other indicator functions are useful for V = R. They are the indicator functions I+, I− and I0 of the sets of the non negative and non positive numbers and of the origin: I+(x) = 0 , if x ≥ 0, I+(x) =+∞, if x < 0, I−(x) = 0 , if x ≤ 0, I−(x) =+∞, if x > 0, Appendix A: Some Elements of Convex Analysis 261 and I0(0) = 0, I0(x) =+∞, if x = 0. The positive part function, pp(x), x ∈ R, is defined by x, if x ≥ 0, pp(x) = sup {x, 0} = (A.1) 0, if x ≤ 0, and the negative part function, np(x), x ∈ R, 0, if x ≥ 0, np(x) = (A.2) −x, if x ≤ 0, are convex functions as well as their difference the absolute value function, abs(x) = |x|, |x|=abs(x) = pp(x) + np(x). (A.3) A.3 Linear Spaces in Duality Definition A.3 Two linear spaces V and V ∗are in duality if there exist a bilinear form ·, · defined on V × V ∗ such that ∈ , = , ∗ ∈ ∗ , , ∗ = ; for any x V x 0 there exists y V such that x y 0 for any y∗ ∈ V ∗ , y∗ = 0 , there exists x ∈ V , such that x, y∗ = 0. A.3.1 Examples of Linear Spaces in Duality Spaces V = R, V ∗ = R are in duality with the bilinear form which is the usual product x, y = x · y; (A.4) V = Rn, V ∗ = Rn are in duality with the bilinear form which is the usual scalar product i=n x, y = x · y = xiyi, (A.5) i=1 262 Appendix A: Some Elements of Convex Analysis n where x = (xi) and y = (yi) are vectors of R , the coordinates of which are xi and yi; V = S, V ∗ = S where S is the linear space of symmetric matrices 3 × 3, are in duality with the bilinear form i,j=3 ∗ e ∈ V, s ∈ V , e, s = e : s = ei,jsi,j i,j=1 = ei,jsi,j = e11s11 + e22s22 + e33s33 + 2e12s12 + 2e13s13 + 2e23s23, (A.6) where we use the Einstein summation rule. Be careful, linear spaces V = S and V ∗ = S are also in duality with the bilinear form e, s = e11s11 + e22s22 + e33s33 + e12s12 + e13s13 + e23s23, which is different from the preceding one. Let us give two more examples. Linear space of the velocities U in a domain Ω of R3 V = U(x) U ∈ L2(Ω) , is in duality with the linear space of the forces f applied to the points of the domain V ∗ = f(x) f ∈ L2(Ω) , with bilinear form U, f = U(x) · f(x)dΩ, Ω which is the power of the force applied to the domain. The linear space of the strain rates 2 V = D = Dij(x) Dij = Dji, Dij ∈ L (Ω) , is in duality with the stresses linear space ∗ 2 V = σ = σij(x) σij = σji,σij ∈ L (Ω) , with bilinear form D,σ = σ(x) : D(x)dΩ = σij(x)Dij(x)dΩ, Ω Ω which is the power of the stresses. This bilinear form is used in the definition of the work of the interior forces, see Chaps.7 and 8. Appendix A: Some Elements of Convex Analysis 263 A.4 Subgradients and Subdifferential Set of Convex Functions Convex function fp given above, is differentiable for p > 1 but it is not for p = 1 where it is equal to the absolute value function x → |x|, which has no derivative at the origin. In the same way, indicator function I is not differentiable because its value is +∞ at some points and it has no derivative at points x = 0 and x = 1. Let us recall that for smooth convex functions of one variable, the derivative is an increasing function and this property yields df df (y) − (z) (y − z) ≥ 0. (A.7) dx dx Thus it seems that we have to loose all the calculus properties related to derivatives. Fortunately, this is not the case. Indeed, it is possible to define generalized derivatives and keep a large amount of properties related to derivatives. Consider function f shown in Fig. A.1. It is convex but it is not differentiable because it has no derivative at point A. At a point where the function has a derivative, the curve is everywhere above the tangent. At point A, there exist several lines which have this properly. The slopes of these lines are the subgradients which generalize the derivative: Definition A.4 Let convex function f defined on V in duality with V ∗. A subgradient of f at point x ∈ V is an element x∗ ∈ V ∗ which satisfies ∀z ∈ V , z − x, x∗ + f (x) ≤ f (z). (A.8) The set of the x∗which satisfy (A.8) is the subdifferential set f at point x, denoted ∂f (x). Remark A.2 The subgradient depends on f but depends also on the bilinear form ·, · . Fig. A.1 Convex function f has not a derivative at point A. It has generalized derivatives: the slopes of the lines which pass at point A and are under the curve representing function f . These slopes are the sub-gradients which constitute the subdifferential set 264 Appendix A: Some Elements of Convex Analysis A.4.1 Two Properties of the Subdifferential Set The subdifferential set keeps usual properties of the derivative: a function is not differentiable where its value is +∞ and a differentiable convex function satisfies relationship (A.7). For a convex function first property becomes: Theorem A.2 Let convex function f =+∞. If this function is subdifferentiable at point x, i.e., if ∂f (x) =∅, then it is finite at that point: f (x)<+∞. Proof Let us assume f is subdifferentiable at point x:lety∗ ∈ ∂f (x). Let us rea- son ab absurdo and assume that f is not finite at that point: f (x) =+∞. Let us write relationship (A.8) at a point z where f (z)<+∞. Such a point exists because f =+∞. Then we have +∞ = z − x, y∗ + f (x) ≤ f (z). We deduce that +∞ = f (z). Which is contradictory with the assumption. Let us note that if the only point z where f (z)<+∞ is x itself, then f (x)<+∞ for f not to be identical to +∞. This theorem applies in numerous constitutive laws where we have the relationship B ∈ ∂I(β): because ∂I(β) is not empty, we have I(β) < +∞ which implies that I(β) = 0 and 0 ≤ β ≤ 1, see Fig. A.2. Thus relationship B ∈ ∂I(β) implies that quantity β which may be a phase volume fraction is actually in between 0 and 1. Now let us prove that relationship (A.7) is satisfied in some sense by the subdif- ferential set : Theorem A.3 Let f a function convex. We have ∀y∗ ∈ ∂f (y), ∀z∗ ∈ ∂f (z), y − z, y∗ − z∗ ≥ 0. Proof It is sufficient to write relationship (A.8) at points x and y. Remark A.3 It is said that the subdifferentiation operator is a monotone operator. A.4.2 Examples of Subdifferential Sets Let us begin by the subdifferential set of indicator function I of interval [0, 1].Itis easy to see that the subdifferential set is (Figs.A.2 and A.3) ∂I(0) = R− ; ∂I(1) = R+ ; if x ∈ ]0, 1[ ,∂I(x) = {0} ; if x ∈/ [0, 1] ,∂I(x) =∅.
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