Truss Design and Convex Optimization - dspace.mit.edu · Outline Physical Laws of a Truss System The Truss Design Problem Second-Order Cone Optimization Truss Design and Second-Order

Truss Design and ConvexOptimization

Robert M. Freund

April 23-25, 2002

Outline�Physical Laws of a Truss System

�The Truss Design Problem

�Second-Order Cone Optimization

�Truss Design and Second-Order Cone Optimization

�Some Computational Results

�Extensions of the Truss Design Problem

c�2002 Massachusetts Institute of Technology. All rights reserved. 15.094 1

Outline�Semi-Definite Optimization

�Truss Design and Semi-Definite Optimization

�Truss Design and Linear Optimization


Physical Laws ofa Truss System

A Truss Structure

A truss is a structure in � � � or � � � dimensions,formed by � nodes and� bars joining these nodes.

2

3

4

5

6

1

Example of a truss in � � � dimensions,with � � � nodes and � � �� bars.

Examples of trusses include bridges, cranes, and the Eiffel Tower.



Data Description of a Truss

The data used to describe a truss is:

� a set of nodes (given in physical space)

� a set of bars joining pairs of nodes with associateddata for each bar:

– the length �� of bar �– the Young’s modulus �� of bar �

– the volume �� of bar �

� an external force vector � on the nodes




2

3

4

5

6

1f

f

F13

12

3

A truss problem.




Static and Free Nodes

Nodes can be:

� static nodes (fixed in place), or

� free nodes (movable when the truss is stressed).

2

3

4

5

6

1f

f

F13

12

3

A truss problem.




Degrees of Freedom

The allowable movements of the nodes defines thedegrees of freedom (dof) of the truss.

We say that the truss has degrees of freedom.

� ��

2

3

4

5

6

1f

f

F13

12

3

A truss problem.



Displacements and Forces

Movements of nodes in the truss will be repre-sented by a vector � of displacements � � �� .

The external force on the truss is given by avector � � �� .

Displacements of the nodes in the truss causeinternal forces of compression and/orexpansion to appear in the bars in the truss.

Let �� denote the internal force of bar �. Thevector � � �� is the vector of forces of thebars.

2

3

4

5

6

1f

f

F13

12

3

A truss problem.



Example of a Truss Problem

2

3

4

5

6

1f

f

F13

12

3

A truss problem.

There is a single external force appliedto node � in the direction indicated.

This will result in internal forces � alongthe bars in the truss and will simultane-ously cause small displacements � inall of the nodes.

Nodes � and � are fixed, and the othernodes are free.




2

3

4

5

6

1f

f

F13

12

3

A truss problem.

We can label the 13 different bars by the nodes theylink. The bars will be: ��, ��, , ��.




Expansion and Compression

2

3

4

5

6

1f

f

F13

12

3

A truss problem.

The internal force �� of bar � can be positive or negative.



Forces on the Bars

Expansion

Lk∆ > 0

A bar under expansion.

If �� , bar � has been expanded and its internalforce counteracts the expansion with compression.



Forces on the Bars

Compression

Lk

∆ < 0

A bar under compression.

If �� , bar � has been compressed and its internalforce counteracts the compression with expansion.



Physical Laws

Force Balance Equations...

In a static truss the internal forces will balance the external forces in everydegree of freedom. This is a law of conservation of forces.

2

3

4

5

6

1f

f

F13

12

3

A truss problem.

Consider the balance of forces on node 3:

� ��

� ��



Physical Laws

...Force Balance Equations...

For the entire truss we write linear equations thatrepresent the balance of forces in each degree offreedom.

In matrix notation this is:

��

� is an �� matrix.

Each column of �, denoted as �, is the projection ofthe bar onto the degrees of freedom of the nodes thatbar � meets.c�2002 Massachusetts Institute of Technology. All rights reserved. 15.094 15


Physical Laws

...Force Balance Equations...

2

3

4

5

6

1f

f

F13

12

3

A truss problem.



Physical Laws

...Force Balance Equations

In our example, we have:

�

��

��

� ��

� � � � � �

��

� � ��

� � � � � �

� ��

� � � � ��

� � �

� � � � ��

� � � � ��

� � �

� � � ��

� � ��

� �

� � � ��

� � � � ��

� �

� � � � ��

� � � � � � ��

� ��

� � � � ��

� � � � � � ��

� � ��

��

��

��

� ��

��



Physical Laws

Constitutive Relationship Between Forces and Distortion of a Bar

If bar � has length �� and cross-sectional area ��,Young’s modulus ��, and its length is changed by ��,then the internal force �� is given by:

��

��

��

��

� ��

We write:

��

��



Physical Laws

Distortion and Displacements ...

Displacements in the nodes will cause the lengths of the bars to change.

Suppose that �� , with all other barsmeasured proportionately.

Consider a displacement of: � � ��

2

2

3

4

5

6

1

F3

Example of a small displacement in a node.



Physical Laws

...Distortion and Displacements ...

� � ��

2

2

3

4

5

6

1

F3

Example of a small displacement in a node.



Physical Laws


Changes in the lengths of the bars under nodal displacement.Bar � New Length Linearized Length Linearized Change��

��

12

��

13 1 1 0 014 1 1 0 016

��

�� 0 0

23

��

�� 0 0

24

��

25

��

��

34 1 1 0 035 1 1 0 036

��

�� 0 0

45

��

�� 0 046 1 1 0 056 1 1 0 0



Physical Laws


If � is small, then the distortion of bar � is nicelyapproximated by the linear expression:

��

We approximate the internal force on bar � due to afeasible displacement � as:

��

��

��



Physical Laws

...Distortion and Displacements

Define:

� ��

��

��

�

. . .

� ��

��

� � ��

��

��

�

. . .

� ��

��

��

��

can be written as: � � �� which is �� c�2002 Massachusetts Institute of Technology. All rights reserved. 15.094 23


Physical Laws

Equilibrium Conditions...

We write the physical laws of the truss system as:

�� (conservation of forces)

�� (forces, distortions, and displacements)



Physical Laws

...Equilibrium Conditions� � �� (conservation of forces)


Combining these yields: � � ��

� ��

� ��

where: � � ��

The matrix � is called the stiffness matrix of the truss.

Note that � is an SPSD matrix.



Physical Laws

Solving the Equations� � �� (conservation of forces)


Form: � � ��

Solve for �: ��

Then compute: � � ��

This last expression is simply: ��



Physical Laws

Compliance

The compliance of the truss is the work (or energy)performed by the truss.

The compliance is the sum of the forces timesdisplacements:

� � � � ��

The compliance will generally be a positive quantity.



Physical Laws

A View from Optimization...

Consider the following optimization problem:

�� minimize ��

��

��

� ��

��

��

s.t.

��

The constraints can be written:

��

��



Physical Laws

...A View from Optimization...


��

��

��

��

��

s.t.

��

This is a convex quadratic problem. The optimality conditions are:

��

��

��

�� for � � ��



Physical Laws


Optimality conditions:

��

��

��

�� for � � � � � � ��

Re-write as:

� � ��

�� c�2002 Massachusetts Institute of Technology. All rights reserved. 15.094 30


Physical Laws


Optimality conditions of ��:

� � ��

��

These are the original equilibrium conditions of the truss.

The truss system is nature’s solution to an optimization problem.



Physical Laws

...A View from Optimization


��

��

��

��

��

s.t.

��

The optimal objective function value is:

��

��

��

��

��

��

��

The optimal objective value of �� is the compliance of thetruss system.


The TrussDesign Problem

We now consider the volumes �� of the bars � to be design parameters that we wish to determine.

� � ��

� ��

�

��

�

�

��

�

��

��

where ��

��

��

��



��

��

��

� � �� is a weighted sum of rank-one matrices(weighted by the volumes ��).

� � �� is a weighted sum of outer-productmatrices.



In designing the truss, we choose the volumes � of the barssubject to linear constraints:

� � � � � �

Typically, these constraints include upper and lower bounds onthe volumes of certain bars, as well as an overall cost or volumeconstraint:

��

The criteria in truss design is to choose the volumes � of the barsso as to minimize the compliance of the truss, namely:

� ��

Such a truss will be the most resistant to the external force � .c�2002 Massachusetts Institute of Technology. All rights reserved. 15.094 35


Single-load Truss Design Problem

��

��

� � �

� �

� � � � ��

��

��

��

��

��

��

� � �

� � �

� �

� � � � �� c�2002 Massachusetts Institute of Technology. All rights reserved. 15.094 36



��

��

��

��

��

��

��

� � �

��

� � ��

The decision variables are � and �.

The real decision variables are � only.

Once � is chosen, � will be determined by the solutionto the system of equations: ��




� ��

��

��

��

��

� � �

� � �

��

� � �

� � ��

Note that as written, the truss design problem TDP isnot a convex problem.



A Convex Version of TDP

Recall problem ��:


��

��

��

��

��

s.t.

��

Convert � � � � � � � to be decision variables and addconstraints

��

� � � �




Write this as:

(TDP�� minimize��

��

��

��

��

��

s.t.

��

��

� � ��





��

��

��

��

��

s.t.

��

(TDP� minimize��

��

��

��

��

��

s.t.

��

��

� � �

� � ��




Two modifications:

1. bars with zero volume (�� ) are no longercounted

2. � instead of ��




(TDP� minimize��

��

��

��

��

s.t.

��

� � �

� �

� � ��




Re-write as:


��

��

s.t.

��

��

�� for � � ��

� � ��





��

��

s.t.

��

��

�� for � � ��

� � ��




Further re-write as:


��

��

s.t. � � ��

��

�� for � � ��

� � ��





��

��

s.t. � � ��

��

��

�� for � � ��

� � ��




We have a linear objective function and almost all constraints arelinear.

It is pretty easy to show that the constraints:��

��

describe a convex region.

This is a convex optimization problem.


Second-OrderCone Optimization

Definition of SOCP

A second-order cone optimization problem (SOCP) isan optimization problem of the form:

��

��

��

� � ��

� � � ��

In this problem, the norm �� is the standard Euclidean norm:

��

The norm constraints in SOCP are called“second-order cone” constraints.



Linear Optimization is an SOCP��

��

��

� � ��

� � � ��

SOCP is a convex problem, because

��

is a convex function.

Linear optimization is a special case of SOCP: just set

�� , �� , �� , and �� to be the �th unit vector,

� � �� .



Quadratic Constraints and SOCP

��

��

��

Suppose we have the constraint: ��

Factor � ��

��

�

��

�

Square both sides and collect terms.



Quadratic Objective Function and SOCP

��

��

��

Suppose we have a convex quadratic objective:

��

Create a new variable �� and write:

��

��

...

��

This can be further re-written as an SOCP.


Truss Design and Second-Order Cone Optimization

CTDP

Recall:(TDP�� minimize��

��

��

s.t. � � ��

��

�� for � � ��

� � ��



CTDP

Make the simple change of variables:

��

��

��

��

��

Then �� and � � �� , and so:

��

� � ��



CTDP

The constraint

��

becomes:

��

Rearrange this and take square roots:��

��

� � ��

and re-write as: ��

��

��



CTDP


��

��

s.t. � � ��

��

��

�� for � � ��

� � ��



CTDP

can be re-written as:

(CTDP): minimize��

��

s.t. � � ��

��

��

��

��

��



CTDP

(CTDP): minimize��

��

s.t. � � ��

��

��

��

��

��



CTDP

CTDP is a second-order cone problem.The LHS is the norm of a 2-dimensional vector:

��!�� !��

��

��

��

and the RHS is the linear expression:

��

Proposition Suppose that �� is a feasible or optimal solution of CTDP.Let:

� � � � and � � � ��

Then �� is the corresponding feasible or optimal solution of TDP�.



Other Formulations of TDP

We have seen TDP�, which is a convex problem.

We have also seen CTDP, which is a second-order cone problem.

We can also formulate the TDP as a “semidefinite optimization”(SDO for short) problem.

In the special case when the constraints on the volume variables

� � � � � are:

� �

��

TDP can actually be solved by linear optimization.


SomeComputational Results

Three Truss Design Problems

Bridge Design Problem

0 2 4 6 8 10 12 14 16

0

1

2

3

4

5

6

7

The forces and nodes for the basic bridge design model.



Three Truss Design Problems

Possible Bars for the Bridge

0 2 4 6 8 10 12 14 16

0

1

2

3

4

5

6

7

The set of possible bars for the basic bridge design problem.



Hanging Sign Design Problem

0 5 10 15 20 25 30

0

2

4

6

8

10

12

14

16

18

20

The forces and nodes for the hanging sign design model.



Crane Design Problem

0 2 4 6 8 10 12 14 16 18 20

0

5

10

15

20

25

30

35

40

The forces and nodes for the crane design model.



Solution of the Basic Bridge Design Problem

0 2 4 6 8 10 12 14 16

0

1

2

3

4

5

6

7

Optimal solution to the basic bridge design problem.



Solution of the Hanging Sign Design Problem

0 5 10 15 20 25 30

0

2

4

6

8

10

12

14

16

18

20

Optimal solution to the hanging sign design problem.



Solution of the Crane Design Problem

0 2 4 6 8 10 12 14 16 18 20

0

5

10

15

20

25

30

35

40

Optimal solution to the crane design problem.



Solution of the Enhanced Bridge Design Problem

Lower Bounds on “Road-Surface” Bars

0 2 4 6 8 10 12 14 16

0

1

2

3

4

5

6

7

Optimal solution of the bridge design problem,with lower bounds on the “road surface” bar volumes.



Details of Problems Solved

Dimensions of the Truss Design Problems

Dimensions of the truss design problemsName Size of Nodes Arcs Degrees of Maximum

Grid � � Freedom � VolumeBridge-16�7 16�7 136 1,547 268 1,600Bridge-20�10 20�10 231 2,878 458 2,000Bridge-30�10 30�10 341 4,368 678 4,000

Sign-10�20 10�20 231 2,878 444 1,000Sign-20�30 20�30 651 9,058 1,280 2,000Sign-30�40 30�40 1,271 18,438 2,512 3,000

Crane-10�20 10�20 96 818 188 2,000Crane-20�40 20�40 291 3,188 578 5,000Crane-30�40 30�40 401 4,678 798 10,000




Variables and Inequalities

Number of variables and inequalities in the truss design problemsLP Model SOCP Model

Problem Variables Inequalities Variables Inequalities

�� LBsBridge-16�7 4,641 1,564Bridge-20�10 8,634 2,899Bridge-30�10 13,104 4,399

Sign-10�20 5,756 5,756 8,634 2,879Sign-20�30 18,116 18,116 27,174 9,059Sign-30�40 36,876 36,876 55,314 18,439

Crane-10�20 1,636 1,636 2,454 819Crane-20�40 6,376 6,376 9,564 3,189Crane-30�40 9,356 9,356 14,034 4,679

� The linear optimization model cannot be used for the modified bridge design model.




Variables and Equations

Number of variables and equations in the truss design problemsLP Model SOCP Model SDO Model

Problem Variables Equations Variables Constraints Matrix Dimension

�� LBs � ��

Bridge-16�7 4,641 1,832 1,685Bridge-20�10 8,634 3,357 3,111Bridge-30�10 13,104 5,077 4,711

Sign-10�20 5,756 444 8,634 3,323 3,111Sign-20�30 18,116 1,280 27,174 10,339 9,711Sign-30�40 36,876 2,512 55,314 20,951 19,711

Crane-10�20 1,636 188 2,454 1,007 916Crane-20�40 6,376 578 9,564 3,767 3,481Crane-30�40 9,356 798 14,034 5,477 5,081





Compliance of the Optimized Truss Design

Compliance of the optimized truss designfor the truss design problems

Problem LP SOCP SDPBridge-16�7 27.43365 27.15510Bridge-20�10 52.58314 52.10850Bridge-30�10 132.46038

Sign-10�20 0.77279 0.77293 0.77279Sign-20�30 2.52003 2.52056

Sign-30�40 4.08573 4.08681

Crane-10�20 28.67827 28.67840 28.67753Crane-20�40 286.24854 286.24954 286.19340Crane-30�40 596.73177 596.73324 596.63740

� The linear optimization model cannot be used for the for the modified bridge design model.

� The data to run the SDO model could not be prepared for this instance due to memoryrestrictions.




Iterations of Interior-Point Method

LOQO was used tosolve the LP and SOCPmodels.

�� was used tosolve the SDO models.

Number of iterations of the interior-pointalgorithm to solve the truss design problems

Problem LP SOCP SDOBridge-16�7 38 36Bridge-20�10 55 43Bridge-30�10 47

Sign-10�20 18 61 37Sign-20�30 21 47

Sign-30�40 24 53

Crane-10�20 17 52 34Crane-20�40 31 158 58Crane-30�40 31 144 60






Running Times (in seconds)

Running time (in seconds) of the interior-pointalgorithm to solve the truss design problems

Problem LP SOCP SDOBridge-16�7 93.90 257.01Bridge-20�10 460.96 1,088.09Bridge-30�10 904.86

Sign-10�20 3.33 513.37 1,081.54Sign-20�30 32.08 4,254.08

Sign-30�40 111.03 26,552.74

Crane-10�20 0.52 41.12 446.98Crane-20�40 6.46 1,299.12 33,926.20Crane-30�40 10.67 1,599.85 95,131.49




Semi-DefiniteOptimization

Semi-definite Notation

If � is a � � � matrix, then � is a symmetric positivesemi-definite (SPSD) matrix if � is symmetric:

��" � �"� for any ��

and

� �� for any � � ��

If � is a � � � matrix, then � is a symmetric positivedefinite (SPD) matrix if � is symmetric and

� �� for any � � ��




Let �� denote the set of symmetric � � � matrices.

Let �� denote the set of symmetric positivesemi-definite (SPSD) � � � matrices.

Let �� denote the set of symmetric positive definite(SPD) � � � matrices.




Let � and! be any symmetric matrices.

We write “� � �” to denote that � is symmetric andpositive semi-definite.

We write “� �” to denote that � is symmetric andpositive definite.



�� is a Convex Set

Remark 1 �� is a convex set in ��.

Proof : Suppose that ��! � ��. Pick any scalars

"�# � � for which "� # � �. For any � � ��, wehave:

� "� � #!� � "� �� #� !� � � �

whereby "� � #! � ��. This shows that �� is aconvex set. q.e.d.



Definition of SDO

� �� $

�� % �

��$��

�$ � �

The matrices %�� are symmetric matrices.



Interpretation of SDO

� �� $

�� % �

��$��

�$ � �

The objective is to minimize the linear function:��

��$�

of the� scalar variables $ � $�� $�.




� �� $

�� % �

��$��

�$ � �

$ � $�� $� must satisfy�$ � �.




��

��

��

��

$ � $�� $� must satisfy the condition that thematrix �, defined by:

� �� % �

��$��

must be positive semi-definite. That is,

� �� % �

��$�� c�2002 Massachusetts Institute of Technology. All rights reserved. 15.094 82


Illustration of SDO�� !��

��

��

� ! �

� � "�

��

��

� � "

� " ��

��

�� #

� � �

# � �

� � # � �

�� "��

��

�� !��

�� #�� ! � �� "��

� � �� #�� "�� " � ��

� � �

�� "��



SDO is Convex

� �� $

�� % �

��$��

�$ � �

Remark 2 . SDO is a convex minimization problem.



Further Remarks on SDO

Semi-definite optimization is a unifying model that includes:

linear optimization;

quadratic optimization; second-order cone optimization;

certain other convex optimization problems.

Semi-definite optimization has applications that span convexoptimization, discrete optimization, and control theory.

Semi-definite optimization may very well become the canonicalway that optimizers will think about optimization in the nextdecade.


Truss Design and Semi-Definite Optimization

STDP�� $ &

��

�& �

�

��

��

��

� �

��

� � �

& � ��

Notice that STDP is a semi-definite optimization problem.



TDP Yields Equivalent Solutions to STDP�� !

��

��

��

��

� � � ��

�! � �

�

��

��

��

�

� � � � � �

� � � �

� � � � �� ! � � � ��

Proposition 3 . Suppose that � � is a feasible solution of TDP.Let:

! � � �� Then � ! is a feasible solution of STDP, and ! � � ��.



STDP Yields Equivalent Solutions to TDP�� !

��

��

��

��

� � � ��

�! � �

�

��

��

��

�

� � � � � �

� � � �

� � � � �� ! � � � ��

Proposition 4 . Suppose that �� & is a feasiblesolution of STDP. Then there exists a vector � forwhich �� is feasible for TDP, and in fact:

� � � & c�2002 Massachusetts Institute of Technology. All rights reserved. 15.094 88


Solving TDP via STDP�� !

��

��

��

��

� � � ��

�! � �

�

��

��

��

�

� � � � � �

� � � �

� � � � �� ! � � � ��

Solve STDP for � ! and then solve the following linearequation system for �:�

��

��

��

� � �� c�2002 Massachusetts Institute of Technology. All rights reserved. 15.094 89

Truss Design andLinear Optimization

Special Case of Truss Design

��

��

��

��

��

� � �

��

� �

� � � � ��

The only constraint on the volumes of the bars � is a volumeconstraint limiting the total volume of the bars to not exceed thegiven value � .

This is a special case of TDP, but it is not too unusual.



Special Case of Truss Design

��

��

��

��

��

� � �

�� %

� � ��

��

��

��

� ��

��

�� %

� � ��



A Dual of TDP

��

��

��

� �

� � � � ��

�� " � � #

��

��"��

��# � � � � � � �

� � � �



Weak Duality Property�� " � � #

��

��"��

��# � � � � � � �

��

� �

� � � � ��

Proposition 5 . Suppose that �� is feasible for TDPand that �� ' is feasible for DTDP. Then:

� � � �� ( ' c�2002 Massachusetts Institute of Technology. All rights reserved. 15.094 93


A Pair of Linear Models

Consider the following pair of primal and dual linearoptimization models:

��

��

��

��

��

��

��

��

��

� � � �



Linear Models and Original Models

Dual Problems

�$��

��

��

��

�

�$�� %��

��

��

� � ��

��

�� %�� ! � % &

��

��!� � ��

��& � � � ��

�� % � � ��

� � ��



Linear Model Solves the Original Problem

Proposition 6 . Suppose that � �� is an optimal solution ofLP and that �� is an optimal solution of LD, and consider thefollowing assignment of variables:

$ �

��

��

� � �

��

��

��

�" � ��

��

�# � ��

� �

Then �� solves TDP and ��" �# solves DTDP.



Comparison of the Models�$��

��

��

��

�

�$�� %��

��

��

� � ��

��

�� %�� ! � % &

��

��!� � ��

��& � � � ��

�� % � � ��

� � ��


Extensions of theTruss Design Problem

Multiple Loads

Conservative Model

We consider a finite set of external loads on the truss:

� � �� '� � )*�

A conservative strategy would be to minimize themaximum compliance:

�� "��'�� " �"�

��

� � � �� " � �"



Multiple Loads

Averaging Model

We consider a finite set of external loads on the truss:

� � �� %$ �

Let &� denote the relative frequency or importance associatedwith the truss structure undergoing the external force ��.

&� � � � &� � � �

��&� � ��

��

��&��

� ��

� � �

� � ��



Self-Weights

We have assumed that the truss structure itself is not affected by its own weight.

Add an external force corresponding to the gravitational force associated withbar �, and linearly proportional to �� for � � �� .

Let �� ()� denote the vector that projects the gravitational force of bar �

onto the appropriate nodes.

��

� �

��

�

��

� �

��

��

� � ��



Reinforcement

We are given an existing truss structure.

We must determine how to strengthen it.

We add lower bounds on the volumes of the existing bars equal tothe current volumes of the bars:

��

��

� � �

� �

� � � � � � � � � �

� � � � ��



Robustness of Solutions

The optimal solution of the TDP might yield a trussdesign that is not rigid for a different external forcethan the one used.

To obtain a robust solution, we must consider multipleexternal loads that will contain the possible loads thatthe truss will be subject to.



Buckling Constraints

We have ignored the fact that if a bar is under greatcompression it might actually collapse instead ofcounteracting that external force with its internal force

For this we have to add lower bounds on the allowableinternal forces in terms of the design variables �� andthe geometry of the bars.

These constraints cause the resulting problem to loseits convex structure.c�2002 Massachusetts Institute of Technology. All rights reserved. 15.094 103

Documents

Truss Design and Convex Optimization - dspace.mit.edu · Outline Physical Laws of a Truss System The Truss Design Problem Second-Order Cone Optimization Truss Design and Second-Order