Structural Optimization

Structural Engineering Civil Engineering Department Cairo University Faculty of Engineering

Graduate Course STR651

STR651 High-Rise Building

Topic # 1 (Part 2) Structural Optimization

Prepared by:

Dr. Hazem Elanwar

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Outline  Motivational Example  Introduction to topology optimization  Introduction to optimization  Topology optimization

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Example  For the shown truss:  Find the top lateral deflection  Assume all member to be IPE100  E=2100 t/cm2

10t

+10𝑡

−10 2 𝑡

5m

 Solution • IPE-100  Area=10.3 cm2 • dtop= No  N1 dL



10t

+10𝑡

E A

−20 2 𝑡 −10𝑡 5m +30𝑡

5m 3

Example  For the shown truss:

1t

 Find the top lateral deflection  Assume all member to be IPE100  E=2100 t/cm2

+1𝑡

−1 2 𝑡

5m

 Solution • IPE-100  Area=10.3 cm2 • dtop= No  N1 dL

+1𝑡

• dtop=1/EI*(10x1x500+10 2 x1 2 x500 2 +10x1x500+30x2x500+20 2x1 2x500 2 + 10x1x500)=4.04cm

−1 2 𝑡



E A

−1 𝑡 5m

+2𝑡

5m 4

Example  For the shown truss:

10t

 Find the top lateral deflection  Assume all member to be IPE100  E=2100 t/cm2

5m

 Solution • IPE-100  Area=10.3 cm2 • dtop= No  N1 dL



10t

E A

• dtop=1/EA*(…)=4.5cm

5m

• Then the first system is more efficient than the second system • How many systems should be studied? 5m 5

Example  For the shown truss:

10t

 Find the top lateral deflection  Assume all member to be IPE100  E=2100 t/cm2

5m

 Solution The shorter the load path:

10t • IPE-1001 The Area=10.3 cm2 more optimum design N1 dL • dtop= No 2- The smaller the deflection



E A

• dtop=1/EA*(…)=4.5cm

5m

• Then the first system is more efficient than the second system • How many systems should be studied? 5m 6

Example  How many topologies need to be checked?

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Intro. to Topology Optimization

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Introduction  For reference check the following seminar:  William F. Baker: "On the Harmony of Theory and Practice in the Design of Tall Buildings"

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Introduction: William F. Baker Seminar Lecture  Optimizing the structural components:  Reduces the project cost  Protects the environment • 37% of embodied energy

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Introduction: William F. Baker Seminar Lecture  The lateral system optimization is based on Maxwell theorem on load paths.

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Introduction: William F. Baker Seminar Lecture  The lateral system optimization is based on Maxwell’s theorem on load paths.  It states that: the sum of compression and tension load paths is always constant

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Introduction: William F. Baker Seminar Lecture  Maxwell’s Theorem on Load Path: Example  By selecting any topology of the truss, the difference between the tension and compression path shall be “P.B”.

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Introduction: William F. Baker Seminar Lecture  Maxwell’s Theorem on Load Path: Example  It is logic to assume a truss geometry that follows the moment diagram of a cantilever beam.

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Introduction: William F. Baker Seminar Lecture  Maxwell’s Theorem on Load Path: Example  However, Warren shape truss is more economic and yields less deflection!

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Introduction: William F. Baker Seminar Lecture  Choice of topology significantly impacts the cost of the structure:  Truss A: requires 27% more volume of material to satisfy safety limits.  Truss A: requires 60% more volume of material to give the same deflection as Truss B

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Introduction: William F. Baker Seminar Lecture  Maxwell’s Theorem:

 Since the sum of the tension path and the compression path is constant, then:  The longer the tension path the longer the compression path and vice versa  The efficiency is paid twice in the tension and compression path  By knowing the value of P.r, you need only to calculate the tension or comp. path to know the other  If you have only tension or compression path then this is the optimum structure 17

Introduction: William F. Baker Seminar Lecture  Maxwell’s Theorem:

 Since the difference between the tension path and the compression path is constant, then:  The longer the tension path the bench longer theTherefore, compressiona path andmark (lower vice versa bound solution) is essential to guide the twice designer’s  In efficiency is paid in the decisions tension and compression path  By knowing the value of P.r, you need only to calculate the tension or comp. path to know the other  If you have only tension or compression path then this is the optimum structure 18

Introduction: William F. Baker Seminar Lecture  In order to reach the bench mark design:  Use analytical approaches: i.e. Michell’s optimal trusses.  Use optimization techniques: i.e. topology optimization

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Introduction  Topology optimization might provide a solution for selecting the most optimum geometry, shown a comparison between Michell’s solution and topology optimization software (TopOpt).

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Introduction  Topology optimization might provide a solution for selecting the most optimum geometry, shown a comparison between Michell’s solution and topology optimization software (TopOpt).

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Introduction  Topology optimization might provide a solution for selecting the most optimum system

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Introduction to Optimization

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Optimization  Optimization can be classified into 3 main categories  Size  Shape  Topology

Topology Optimization: Theory, Methods, and Applications, M.P. Bendsoe, O.Sigmund, Springer, 2004

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Optimization  Optimization can be either:  Gradient Based  Non-Gradient (Random) Based

 Gradient based optimization  (+) Utilize the gradient and Hessian matrix  (+) Computationally efficient  (+) Feasible for large scale problems  (-) The function must be continuous and differentiable  (-) The output depends on the initial guess  (-) Can be trapped in Local minimum solution  Examples: Interior point method, Golden search method, etc.

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Optimization  Optimization can be either:  Gradient Based  Non-Gradient (Random) Based

 Non- Gradient based optimization  (-) Requires long computational time  (-) Infeasible for large scale problems  (+) The function doesn’t need to be continuous nor differentiable  (+) The output doesn’t depends on the initial guess  (+) More efficient when dealing with Local minimum  (+) Can handle problems even if the function is not accessible  Examples: Genetic Algorithm, Ant Colony, etc.

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Optimization  Optimization problem is divided into:  Objective (cost) function  Design variables  Constraints

 Objective function  The value that need to be maximized or minimized  i.e. minimize the volume of member A

 Design variables  The parameter that can be changed to satisfy the objective function  i.e. the distance “X”

 Constraints  The set of limitations that must be satisfied  i.e. The upper and lower bounds for “X”  i.e. Linear and non-linear constraints on stress limits, buckling, etc.

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Optimization: Example  Solve the following problem:

Prof. Glaucio Paulino Lecture Notes

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Optimization: Example  Solve the following problem: 1. Graphical

Prof. Glaucio Paulino Lecture Notes

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Optimization: Example  Solve the following problem: 1. Graphical

Prof. Glaucio Paulino Lecture Notes

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Optimization: Example  Solve the following problem: 1. Graphical

Prof. Glaucio Paulino Lecture Notes

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Optimization: Example  Solve the following problem: 1. Graphical

Prof. Glaucio Paulino Lecture Notes

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Optimization: Example  Solve the following problem: 1. Graphical

Prof. Glaucio Paulino Lecture Notes

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Optimization: Example  Solve the following problem: 1. Graphical 2. Linear programing (i.e. MATLAB) Prof. Glaucio Paulino Lecture Notes

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Optimization: Example  Solve the following problem: 1. Graphical 2. Linear programing (i.e. MATLAB) 3. Genetic Algorithm (i.e. MATLAB) Prof. Glaucio Paulino Lecture Notes

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Optimization: Example  Solve the following problem: 1. 2. 3. 4.

Graphical Linear programing (i.e. MATLAB) Genetic Algorithm (i.e. MATLAB) Interior Point Method (i.e. fmincon)

Prof. Glaucio Paulino Lecture Notes

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Optimization: Example  Tools such as GA and IPM are very useful in case of having a group of complicated linear and non-linear constraints

Prof. Glaucio Paulino Lecture Notes

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Optimization: Example#2  Find the dimensions of the box with largest volume if the total surface area is 64 cm2:  Obj Fnc: max(X*Y*Z)  Variables: X, Y, & Z  Constraints: 2XY+2YZ+2XZ=64 XY+YZ+XZ=32

 Solve using Lagrange Multiplier     

L=XYZ+λ(XY+YZ+XZ-32) L,X=dL/dX=YZ+ λ(Y+Z)=0  (*X) -XYZ=λX(Y+Z) L,Y=dL/dY=XZ+ λ(X+Z)=0  (*Y) -XYZ=λY(X+Z) L,Z=dL/dZ=XY+ λ(X+Y)=0  (*Z) -XYZ=λZ(X+Y) L,λ=dL/dλ=XY+YZ+XZ-32=0

…(1) …(2) …(3) …(4)

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Optimization: Example#2  Find the dimensions of the box with largest volume if the total surface area is 64 cm2:  From eq(1) & eq(2)  λX(Y+Z)=λY(X+Z)  λ(XZ-YZ)=0 • Either λ=0 (refused, it means the constraint is not applied) • Or (XZ-YZ)=0  X=Y

 Repeat the same procedure eq(2) & eq(3) to get X=Y=Z  From eq(4) X2+X2+X2=32  X=Y=Z=3.266

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Topology Optimization

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Introduction  Topology optimization:  Given: • • • •

Feasible domain (area or volume) Boundary conditions (B.C.) Load conditions (L.C.) Required openings or holes

 Variable: The density of each element in the domain should take a value either 0 or 1.  Objective function: There are several objective functions that can be utilized. For example min. compliance problem.

Sigmund et.al, 2011 “ Efficient topology optimization in MATLAB using 88 lines of code.

• Min.: L(u)

 Constraints • aE(u,v)=L(v) For Future Reference: Topology Optimization, Theory, Methods and Applications (Bendsoe and Sigmund)

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Objective Function  A simple objective function is the minimum compliance (maximum global stiffness)  Assumptions:  Eijkl(x) is variable over the domain  Virtual work at equilibrium (u) and small displacement (v) applies: • a . ( u ,v )

  Eijkl ( x)   ij (u)   ij (v) d  

dui du j  )  Linearize strain:  ij (u)  0.5  ( dx j dxi





 Load linear form: L(u)= L(u )  P  u d   t  u d  



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Solution algorithm  The optimization problem is as follows

 Its Lagrange formula is as follows

 The derivative of Lagrange equation w.r.t the design variable r is:

 With switching conditions 43

Solution algorithm  Lagrange multiplier (λ) is either 0 or >0,  if λ > 0 it means that the term in the bracket must be =0, which implies: • r=0 or 1 without intermediate value (black or white mesh)

 If λ=0 it means that the term in the bracket must NOT be 0, which implies that • r is intermediate value between 0 and 1 (grey mesh)

 The problem start from the intermediate values and we want to reach 0 or 1. In this case we can assume λ+ & λ- =0, Lagrange equation becomes as follows:  Define new parameter

Beta :

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Solution algorithm  The problem is formulated such that the volume ratio to total domain volume must be a pre specified constant (i.e. 0.5).  At first Iteration the volume of Lagrange multiplier “Λ” is assumed using the bisection method Λ mid between Λ min=0 & Λ max=10^6.  Then the volume constant is checked, if the value is greater than the required number (i.e. 0.5), then:  Λ min= Λ mid & Λ max= Λ max, in another word increase “Λ”  From b equation if “Λ” increase b decreases  In this case the function must direct selective r(s)  r(s)-move to decrease overall volume

 Now we can follow the set of equations for iteration procedure

ζ=move (i.e. 0.2) 45

Solution algorithm  Then the volume constant is checked, if the value is greater than the required number (i.e. 0.5), then:  Λ min= Λ mid & Λ max= Λ max, in another word increase “Λ”  From b equation if “Λ” increase b decreases  In this case the function must direct selective r(s)  r(s)-move to decrease overall volume

 Now we can follow the set of equations for iteration procedure

 If beta is small there is a higher possibility to have r r-move. Then the overall volume is decreased aiming to find optimum solution  On the other hand, if volume
Solution algorithm

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TopOpt 88 Code

 nelx: no. of meshes in xdirection.  nely: no. of meshes in ydirection.  volfrac= volume ratio to the overall domain.  Numbering system of nodes follow the figure, it starts from the top left corner moving down, then to the left direction.  F=to add forces by node number and value.  Fixeddofs= to add support nodes and direction. 48

TopOpt 88 Code

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TopOpt 88 Code

 nelx: no. of meshes in xdirection.  nely: no. of meshes in ydirection.  volfrac= volume ratio to the overall domain.  Numbering system of nodes follow the figure, it starts from the top left corner moving down, then to the left direction.  F=to add forces by node number and value.  Fixeddofs= to add support nodes and direction. 50

TopOpt 88 Code

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TopOpt Example  This is a very simple example to describe the input of TopOpt.m program

1

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Assignment

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Assignment: Special Problems  SP#1:  Use fmincon to solve the following optimization problem:

 SP#2:  Use TopOpt.m software to find the optimum topography of the shown system. Comment on the results.  Given: • • • • • •

Nelx=200 Nely=160 VolFrac=0.55 Penal=3 Rmin=1.5 Ft=1

1

1

Row=80 Columns 99,101

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