Explicit and Implicit Optimization

Models based on Mathematical Programming give the user clear insight into the possible ways of structural optimization. Let us begin with the topological optimum design. Changing the layout of structural elements makes the entries of C variable. Hence, all relations where this matrix appears become non-linear. This circumstance explains why the topological formulation of the optimum design problem is still an open research area.

The search for optimum sizing of structural elements under given topology becomes much easier if we neglect elastic strains. Assuming additionally that a form of each element is given up to certain parameters, we can take the entries of k as unknowns, retaining the fixed matrices C and N. Hence, basic relations of the rigid-perfectly plastic model remain linear. Note, that is not possible for elastic structures: in the sizing problem matrix E becomes variable which destroys linearity of the constitutive relations.

Let us assume that the unknown plastic modulae k are governed by a relatively small number of design variables the : k = GTz , where G is (g x r) – matrix of configuration. In order to maintain linearity of the problem, we adopt linear cost function f (z) = cT z, where the entries of column matrix c0 є Rg are given cost coefficients. A problem to be solved reads: “given the structural layout find the optimum sizing z„ that minimizes f and assures given safety factor p„ against plastic collapse”. In fact, we know an expected load carrying capacity p„ = ,и*р0 of the optimized structure, since both the safety factor p„ and the reference load p0 are given.

Table 5. Governing relations for optimum plastic design.

X

w

s

z

1

VL,=

0

0

– NT

GT

0

> 0

VLw =

0

0

CT

0

– p.

= 0

VLS =

– N

C

0

0

0

= 0

<1

и

G

0

0

0

– c0

< 0

X> 0, z > 0, XTVL^ = 0, zTVLz = 0

Table 5 shows the internal structure of the problem of optimum plastic design. This problem is equivalent to the following pair of dual LP-problems:

min { cT z | GTz – NTs > 0, CTs = pj (25)

s, z >0

max { pTw | – N X + Cw = 0, G X< c0}

w, X >0

Duality reveals an interesting role of the cost coefficients: according to the second constraint of the kinematic energy principle (26) these coefficients bound linear combinations of plastic multipliers. If we would take G = I and g = r which means that each plastic modulus is treated as an independent design variable, then the second constraint in (26) would reduce to the inequality k< c. In those parts of optimum structure that undergo yielding this constraint must be satisfied as equality. Hence, assuming certain cost coefficients we in fact impose certain collapse mechanism on the optimum structure. It can be shown that this mechanism ensures uniform dissipation of energy over the structure.

Sizing is the case of explicitly formulated optimization problem. It is worth noting that the MP – approach allows us to uncover also certain possibilities of optimization hidden in the problems formulated from the analysis point of view. A good example of that is the problem of rigid-perfectly plastic structure brought to the state of plastic collapse by purely kinematic loading. Let us split again the degrees of freedom of the discrete structural model into the parts denoted by the indices p and w, as it was already done in the model (14), (15). Then the considered problem can be formulated in the following way: “given the structural layout and the distribution of plastic modulae k0 find the state s„, Я„, W„ of the structure brought to the plastic collapse by a given kinematic load W0 ”[9].

Table 6. Governing relations for kinematically induced plastic collapse.

к

w

p

vs

w

s

r

1

VLX =

0

0

0

– NT

0

k0

> 0

VL =

wp

0

0

0

CT

p

0

0

= 0

VL =

ww

0

0

0

CT

w

-1

0

= 0

VLs =

– N

Cp

Cw

0

0

0

= 0

VLr =

0

0

-1

0

0

w0

= 0

Ik > 0, XT VLX = 0

Table 6 shows the governing relations for this problem. They correspond to the following pair of dual LP-problems:

min { kT Я | – N Я + Cpw p + Cw w w = 0,w w = w 0} (27)

wp, ww, Я>0

max { wT r | NTs > k0, CT s = 0, CT s – r = 0} (28)

s, r

The kinematic principle (27) says us that the collapse mechanism Я„, w„ corresponds to the minimum dissipated power D = kTJl. According to the static principle (28), the stresses s„ and the reactions r„ at the plastic collapse correspond to the maximum power of reactions done on the prescribed displace­ment rates W 0. Note that if the kinematic loading would be introduced in unilateral manner, which means replacing the last constraint in (27) by Ww > W 0, then the reactions would become sign con­strained: r > 0.

A. Cyras and his co-workers looked at the problem (28) from different perspective. They treated p as unknown loading and introduced linear quality measure of load f = p . Here do є Rn is a given column matrix of weight factors. Then the following load optimization problem was formulated: “given the structural layout and the distribution of plastic modulae k0 find the ultimate load p„ that

has the highest quality index f ”. Obviously, the dual problem revealed that the entries of d0 should be treated as prescribed displacement rates.

Approaching the problem from the kinematic side seems to be more natural. The implicit optimization of reaction forces that comes out via duality is probably more interesting in the continuum formulation. It can be shown then that by prescribing displacement rates on a part of the surface of the rigid-perfectly plastic body we obtain the distribution of surface tractions optimal in a certain sense (Borkowski, 2004).