Dual View of Mechanics

It seems that the first new insight brought by the MP-approach to Structural Mechanics was discovering the equivalence of kinematic and static formulations. In Mathematical Programming this property is known as duality: under certain premises each problem of constrained extremum has its dual and the values of cost functions for such problems attained at the solutions coincide. We already quoted the dual LP-problems (1) and (2). If x„ is the solution of (1) and if the constrained minimum f ‘(xt) = cTx„ is finite, then there exists a solution y„ of the dual problem (2) and the optimum values of the cost functions coincide: f "(yt) = bTy„ = f ’(xt) .

Let us expand slightly the QP-problem (4) by introducing additional variables y є Rn :

Here bx є Rm, by є Rn, A є Rmxm, A є Rnxn, A є Rnxm. Moreover, A is positive definite

X y XX yy yX XX [8]

and Ayy is negative definite. The dual of (6) reads

Duality has simple geometrical interpretation. Solving a pair of dual problems (6), (7) is equivalent to finding the saddle point

L(x„ ,y„) = min m ax L(x, y)

x>0 y>0

of the Lagrange function

Such point can be reached in two ways. One can first establish the parabola that contains all maxima with respect to y – variables and then find the minimum on this curve. This sequence corresponds to the problem (6). Alternatively, one can begin with finding the parabola that corresponds to all minima with respect to the x – variables and then look for the maximum of this concave function. This way leads to the dual problem (7).

Point (x„ ,y „) is the saddle point of L if it satisfies Kuhn-Tucker conditions (KT-conditions): VLx > 0, VLy < 0, x > 0, y > 0 (10)

xTVLx = 0, yTVLy = 0 (11)

Here VLx є Rn and VLy є Rm are gradients of L with respect to x and y. If all variables were free, then the KT-conditions would reduce to the common stationarity conditions:

VLx = 0, VLy = 0 (12)

or, explicitly, to the set of linear algebraic equations

Axxx + Axyy + bx = 0 (13)

Ayxx + Ayyy + by = 0

Note two features that distinguish this set: a) its matrix of coefficients is symmetric; b) the sub­matrices situated along the diagonal have special properties – Axx is positive definite, Ayy is negative definite

It is easy to check by inspecting the Table 1 that the set of equations governing linear static analysis of elastic structures follows exactly the template (13). The goal of such analysis is to find displacements1 w, stresses s and reactions r of elastic structure caused by a given static load p0 and

by a given kinematic load w0 . The structure is represented by a common discrete model, where E is the matrix of elasticity, C is the matrix of compatibility, subscript p refers to the degrees of freedom with prescribed external forces and subscript w refers to the degrees of freedom with prescribed displacements.

It is seen from the Table 1 that displacements play the role of x – variables, whereas stresses and reactions correspond to y – variables in the model (13). The first two rows of the Table 1 contain the equilibrium equations. The third row comes from substituting strains q = Cw = Cpwp + Cwww into the constitutive equation q = E^S. The last row merely says that ww = w0. A matrix with zero entries can be treated either as positive semi-definite or as negative semi-definite. The inverse of matrix of elasticity is strictly positive definite. Hence, — E-1 is strictly negative definite.

Note that there are no inequalities in the Table 1 and that all variables are free with respect to sign. Hence, we don’t need to take the KT-conditions into account. 1

What do we gain by using the MP-based approach in elastic analysis? First, having filled the Table 1, we can derive easily the dual energy principles (compare the templates (6) and (7)):

a) kinematic principle –

m і n

s, w

b) static principle –

max {- 2 sTE_1 s + w^ | CT s = Po, CW s – r = 0}

s, r 2

Second, since the QP-problems (14), (15) contain no inequality constraints or non-negative variables, each of them can be reduced to a set of equations. This leads us very naturally to the fundamental computational tools of elastic analysis: the Stiffness (Force) Method and the Flexibility (Displace­ment) Method.

Third, the existence and uniqueness of solution for any given loading p0,w0 follows immediately from the convexity of problems (14), (15). Moreover, a generalization of the model (14), (15) to unilateral contact is straightforward. The replacement of ww = w0 in (14) by less restrictive

condition ww > w0 induces sign constraint on r in the dual problem:

min { S1^1 s – wTP0 1 Cp wp + Cwww – E~1s = 0 ww ^ w0}

s, w 2

Seemingly minor, this modification has dramatic consequences: a) the linearity of the problem is lost due to the KT-condition rT(w0 — ww) = 0 ; b) the energy principles (16), (17) can not be replaced by the sets of equations. Moreover, for certain loads p0,w0 the constraints of the problems (16), (17) might become contradictory. Thus the existence of solution is not warranted any more.