Truss Topology Optimization - An Alternative View

Mathias Stolpe

February 19, 2007

Abstract

In these notes we consider a single load minimum compliance trusstopology design problem stated in both design and displacement vari-ables. This problem is reformulated as a linear program stated indisplacement variables only. The reformulated problem is related tominimum volume problems with stress constraints. These problemsare shown to be equivalent up to a scaling.

Introduction and problem statement

We have so far considered minimum volume problems with stress constraints.This problem can be reformulated as the linear program

minimizegIRn,hIRn

nj=1

lj(gj hj )

subject to R(g h) = f,g 0, h 0,

(1)

where g IRn and h IRn correspond to bar forces in tension and compres-sion, respectively. The linear programming dual of (1) is given by

maximizeyIRd

fT y

subject to 1 rTj y

lj 1, j = 1, . . . , n,

y free.

(2)

In these notes we consider the minimum compliance problem with a volumeconstraint

minimizeaIRn,uIRd

fTu (Compliance)

subject to K(a)u = f, (Elastic equilibrium)n

j=1

ljaj = V, (Volume)

a 0,

(3)

where V > 0 is a given bound on the volume. Minimizing the compliancefTu is equivalent to maximizing the global stiffness of the structure underthe external load f IRd. The stiffness matrix K(a) IRdd is given by

K(a) =n

j=1

ajKj =n

j=1

ajE

ljrjr

Tj .

The objectives of these notes are to reformulate the nonlinear problem (3)as a linear program and relate the reformulated problem to the minimumvolume problem (1) and the dual problem (2).

Reformulation of the minimum compliance problem

The potential energy for a given design a IRn is given by

(a, u) =12uTK(a)u fTu.

For a given vector a the displacement vector u should be a minimizer of thepotential energy, i.e.

u argminu

(a, u).

The first order optimality conditions for this unconstrained optimizationproblem are given by

u(a, u) = K(a)u f = 0,which are equivalent to the equilibrium equations already stated in (3).These conditions are both necessary and sufficient since the potential energyis a convex function in u. Inserting a u satisfying K(a)u f = 0 into(a, u) we obtain

12(u)TK(a)u fTu = 1

2fTu fTu = 1

2fTu.

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Hence, the minimum compliance problem (3) can be reformulated as themax-min problem

maxa0,lT a=V

minu

{12uTK(a)u fTu

}. (4)

The max and min in (4) may be interchanged

minu

maxa0,lT a=V

{12uTK(a)u fTu

}. (5)

The inner maximization problem in (5) is a linear program in a (u is fixedin the inner problem). For a 0 and lTa = V we have

nj=1

ajuTKju V max

j=1,...,n

{1ljuTKju

}.

These inequalities are satisfied with equality if exactly one aj > 0. Hence,the problem (5) becomes

minu

maxj=1,...,n

{V

2ljuTKju fTu

}.

This is an unconstrained but non-smooth problem. A common trick inoptimization is to replace the max term by an additional continuous variableand reformulate the problem as a smooth constrained problem

minimizeIR,uIRd

fTu

subject toV

2ljuTKju , j = 1, . . . , n.

(6)

Up to a scaling problem (6) is equivalent to

minimizeuIRd

fTu

subject toV

2ljuTKju 1, j = 1, . . . , n.

(7)

Problem (7) is a constrained nonlinear problem with quadratic, and hencesmooth, inequality constraints. So far we have not used that we are design-ing a truss structure. For the next phase of the reformulation we use the

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mathematical properties of the element stiffness matrices Kj . Since Kj is arank-1 matrix the quadratic terms

1ljuTKju = uT (

E

l2jrjr

Tj )u =

(ErTj u

lj

)2.

The constrainsV

2ljuTKju 1

can thus be rewritten(ErTj u

lj

)2 1 iff 1

ErTj u

lj 1.

The constrained problem (7) and hence the minimum compliance problem(3) can thus be reformulated as the linear program

maximizeuIRd

fTu

subject to 1

V E

2rTj u

lj 1, j = 1, . . . , n,

y free.

(8)

This implies that (3) is equivalent (up to a scaling) to the dual problem (2)!Hence, the problems (1) and (3) are also equivalent up to a scaling!

Exercises

1. Show that the potential energy

(a, u) =12uTK(a)u fTu.

is a convex function in u.

2. Show that the compliance fTu is constant for all u satisfying the equi-librium equations K(a)u = f .

3. Show the precise correspondence between the nonlinear problem (3)and the linear problem (1).

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