ORF 307: Lecture 16 Linear Programming: Chapter 16 ... · Designing an \Optimal" Wall Bracket Clickherefor two-phase simplex method animation tool. Clickherefor parametric self-dual

ORF 307: Lecture 16

Linear Programming: Chapter 16:Structural Optimization

Robert J. Vanderbei

April 18, 2017

Slides last edited on May 3, 2018

http://www.princeton.edu/∼rvdb

Designing an “Optimal” Wall Bracket

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1

Structural Optimization

Forces: xij = tension in beam (aka member) {i, j}.• xij = xji.

• Compression = -Tension.

Force Balance:

Look at joint 2:

x12

[−10

]+ x23

[−0.60.8

]+ x24

[01

]= −

[b12b22

]

Notations:

~pi = position vector for joint i

~uij =~pj −~pi‖~pj −~pi‖

( Note ~uji = −~uij)

Constraints:∑j:

{i,j}∈A

~uijxij = −~bi i = 1, . . . ,m.

2

Matrix Form

Ax = −b

xT =[

x12 x13 x14 x23 x24 x34 x35 x45

]

A =

1

2

3

4

5

[10

] [01

] [.6.8

][−10

] [−.6.8

] [01

][

0−1

] [.6−.8

] [10

] [.6.8

][−.6−.8

] [0−1

] [−10

] [−.6.8

][−.6−.8

] [.6−.8

]

, b =

b11b21b12b22b13b23b14b24b15b25

.

Notes:• ‖~uij‖ = ‖~uji‖ = 1.

• ~uij = −~uji.

• Each column contains a ~uij, a ~uji, andrest are zero.

• In one dimension, exactly a node-arcincidence matrix.

3

Minimum Weight Structural Design

minimize∑{i,j}∈A

lij|xij|

subject to∑j:

{i,j}∈A

~uijxij = −~bi i = 1, 2, . . . ,m.

Not quite an LP.Our favorite trick:

xij ≤ tij−tij ≤ xij

Reformulated as an LP:

minimize∑{i,j}∈A

lijtij

subject to∑j:

{i,j}∈A

~uijxij = −~bi i = 1, 2, . . . ,m,

xij ≤ tij {i, j} ∈ A−tij ≤ xij {i, j} ∈ A

4

Another Absolute Value Trick

minimize∑{i,j}∈A

lij|xij|

subject to∑j:

{i,j}∈A

~uijxij = −~bi i = 1, 2, . . . ,m.

Not quite an LP.Use a common trick:

xij = x+ij − x−ij, x+

ij, x−ij ≥ 0

|xij| = x+ij + x−ij

Reformulated as an LP:

minimize∑{i,j}∈A

(lijx+ij + lijx

−ij)

subject to∑j:

{i,j}∈A

(~uijx+ij −~uijx

−ij) = −~bi i = 1, 2, . . . ,m

x+ij, x

−ij ≥ 0 {i, j} ∈ A.

5

Redundant Equations

Recall network flows:

number of redundant equations = number of connected components.

Row combinations:~yTi ~uij +~y

Tj~uji

Sum of “x”-component rows:

[1 0

] [ ~u(x)ij

~u(y)ij

]+[1 0

] [ ~u(x)ji

~u(y)ji

]= 0

Sum of “y”-component rows, “z”-component rows, etc. is similar.

6

Are There Others?

Yes. Put

~yi = R~pi, R =

[0 −11 0

], RT =

[0 1−1 0

]= −R.

Compute:

~yTi ~uij +~yTj~uji = ~pTi R

T~uij +~pTj R

T~uji

= (~pi −~pj)TRT~uij

= −(~pj −~pi)TRT (~pj −~pi)

‖~pj −~pi‖= 0

Last equality follows from:[ξ1 ξ2

] [ 0 1−1 0

] [ξ1ξ2

]= ξ1ξ2 − ξ1ξ2 = 0 for all ξ1, ξ2

7

Skew Symmetric Matrices

Definition.RT = −R

For d = 1: no nonzero ones.

For d = 2: [0 −11 0

]

For d = 3: 0 −1 01 0 00 0 0

, 0 0 −10 0 01 0 0

, 0 0 00 0 −10 1 0

Structure is stable if the redundancies just identified represent the only redundancies.

8

Conservation Laws

Suppose a combination of rows of A vanishes.Then the same combination of elements of b must vanish.

Force Balance: ∑i

b(x)i = 0 and

∑i

b(y)i = 0

What is meaning of the other redundancies?∑i

(R~pi)T~bi = 0

Answer...

9

Torque Balance

Consider two-dimensional case:

R =

[0 −11 0

].

Physically, this matrix rotates vectors 90◦ counterclockwise.

Let ~vi =~pi/‖~pi‖ be a unit vector pointing in the direction of ~pi:

~pi = ‖~pi‖~vi.

Then,

(R~pi)T~bi = ‖~pi‖(R~vi)T~bi

= (length of moment arm)(component of force perp to moment arm)

In three dimensions, three independent torques: roll, pitch, yaw.

They correspond to the three basis matrices given before.

Note: torque balance is invariant under parallel translation of axis.

10

Trusses (analog of a Tree in Network Flows)

Definition.

• Stable (analog of connected)

• Has md− d(d + 1)/2 beams, d is dimension (analog of acyclic)

Anchors

No force balance equation at anchored joints.Earth provides counterbalancing force.

If enough (d(d + 1)/2) independent constraints are dropped (due to anchoring), then noforce balance or torque balance limitations remain.

11

The Michell Bracket (1904)

Constraints: 5,396Variables: 193,310Time: 270 seconds

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12

The Michell Bracket (1904)

Constraints: 14,996Variables: 536,972Time: 1994 seconds

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