Topology Optimization Topology optimization is concerned with material distribution and how the members within a structure are connected. It treats the “equivalent density” of each element as a design variable. The solver calculates an equivalent density for each element, where 1 is equivalent to 100% material, while 0 is equivalent to no material in the element. The solver then seeks to assign elements that have a low stress value a lower equivalent density before analyzing the effect on the remaining structure. In this way extraneous elements tend towards a density of 0, with the optimum design tending towards 1. As a designer, you will need to exercise your judgment. For example, you may decide that you will omit material from all (finite) elements whose density is less than 0.3 (or 30%). Using an iso-plot of element densities helps to visualize the “remaining” structure as elements with a density below this threshold can be masked leaving behind the optimum design. Then you will need to take this geometry back to your CAD modeler, smooth it out (that is, use geometrically regular edges or surfaces, etc.) and re-evaluate the design for stresses, displacements, frequencies etc..

1. OPTIMIZATION IN DESIGN
1

2. OptiStruct For Optimization -Agenda
Intro to
OptiStruct&
Theoretical
Background
Model
Definition &
Optimization
Setup
Concept
Design
Session 1 Session 2 Session 3 Session 4 Session 5 Session 6
Fine Tuning
Design
Run Options
and Output
Management
Advanced
Optimization
Responses
Optimization
with
Parallelization
Advanced
Optimization
Solutions
Additive
Manufacturing
Session 7 Session 8 Session 9
2

3. Introduction to OptiStruct & Theoretical Background
Sessions
• Optimization
• Design Process
• Structural Optimization
• Sensitivities
• Gradient-Based
Methodology
• Example
• Terminology
• Interpreting the results
• Techniques
• Workflow
• Design interpretation
Sensitivities &
Gradient-Based
Methodology
Terminology
and
Interpreting
the Results
Techniques,
workflow and
design
interpretation
Optimization
Basics
00‘ 10‘ 15‘ 20‘ 30‘
3

4. © Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.
OptiStruct for Optimization, v2021.2
Introduction to OptiStruct & Theoretical Background
Sessions
• Optimization
• Design Process
• Structural Optimization
• Sensitivities
• Gradient-Based
Methodology
• Example
• Terminology
• Interpreting the results
• Techniques
• Workflow
• Design interpretation
Sensitivities &
Gradient-Based
Methodology
Terminology
and
Interpreting
the Results
Techniques,
workflow and
design
interpretation
Optimization
Basics
00‘ 10‘ 15‘ 20‘ 30‘
4

5. Optimization
Definitions
• World English Dictionary
• “To find the best compromise among several often-conflicting requirements, as in engineering design.”
• Webster-Merriam Dictionary
• “A mathematical technique for finding a maximum or minimum value of a function of several variables subject to a
set of constraints, as linear programming or systems analysis.”
• The Wikipedia definition for mathematical optimization
• “It is the selection of a best element (with regard to some criteria) from some set of available alternatives.
• In the simplest case, an optimization problem consists of maximizing or minimizing a real function by
systematically choosing input values from within an allowed set and computing the value of the function.”
5

6. Design Process
Overview
• Classical Design Process Integrating Manual Optimization
• Creation of design
• Analysis of design(s)
• Evaluation of analysis results
• Summation of limiting factors (cost, requirements, time)
• Definition of updates for a new design
• Return to analysis
• Design Process withOptiStruct
• Creation of FE model
• Definition of design variables, objective and constraints
• Automated computational evaluation of the design space
• Evaluation of analysisresults
• Definition of updates for a new improved design
• Return to analysis
6

7. Structural Optimization
Development Time
• Fact
• Most of the product cost is determined at
the concept design stage.
• Problem
• Concept design offers minimum knowledge,
but maximum design freedom.
• Need
• OptiStruct provides effective concept design
tools to minimize downstream “re-design”
costs and time-to-market.
7

8. Optimization
Problem Statement
• Design Variables – What should I operate on to achieve my target?
xi
L ≤ xi ≤ xi
U i = 1, 2, 3,…, N
• Responses – What characteristics are relevant to my problem?
j j = 1, 2, 3, …, M
• Constraints – What performance targets must be met?
gj(x) ≤ 0 j = 1, 2, 3, …, M
• Objective – If all constraints are satisfied, what should OptiStruct minimize/maximize?
min f(x) also min [max f(x)]
• Example: explicit y(x) = x² – 2x or implicit y³ – y²x + yx = 0
Note: The functions f(x), gi(x), can be linear, non-linear, implicit or explicit and are continuous.
9

9. Optimization
Algorithms
• This optimization problem has to be solved, i.e. the optimum solution has to be found.
• Two major optimization algorithm groups exist to solve optimization problems
• Mathematical Programming Methods (OptiStruct uses only these methods)
• Usually require sensitivity information as they rely on gradients
• Solve the optimization problem in a “steepest descent” fashion using mathematical logic
• Few function evaluations – good if function evaluation is time consuming, such as a FE simulation
• Convergence to a localminimum
• Example: Sequential Quadratic Programming or Method of feasible directions
• Evolutionary algorithms
• Do not require sensitivity information
• Often mimic natural behavior to improvedesign
• Require many function evaluations – good if function evaluation is fast
• More likely to find the global optimum
• Example: Genetic Algorithm or Particle Swarm optimization
9

10. Introduction to OptiStruct & Theoretical Background
Sessions
• Optimization
• Design Process
• Structural Optimization
• Sensitivities
• Gradient-Based
Methodology
• Example
• Terminology
• Interpreting the results
• Techniques
• Workflow
• Design interpretation
Sensitivities &
Gradient-Based
Methodology
Terminology
and
Interpreting
the Results
Techniques,
workflow and
design
interpretation
Optimization
Basics
00‘ 10‘ 15‘ 20‘ 30‘
10

11. Optimization
Sensitivities
Sensitivities are calculated if the optimization algorithm requires gradient information
• It is the derivative of a response with respect to a design variable.
They are calculated for each defined response and each design variable
• The simplest way to calculate them is global finite difference
• Each Design Variable is perturbed and the function is evaluated
• This is very slow, as a FE model has to be solved each time
• In OptiStruct, analytical sensitivities are calculated, which is muchfaster.
11

12. Gradient-Based Optimization
Workflow
1. Start from ax0 point
2. Evaluate the function f(xi) and the gradient of the
function  f(xi) at thexi
3. Determine the next point using the negative gradient
direction
xi+1 = xi -  f(xi)
4. Repeat the step 2 to 3 until the function converged
to the minimum
x0
x1
x3
x2
12

13. Simple Beam
Example
• A cantilever beam is modeled with 1D beam elements and loaded with force F = 2400 N
• The width and height of cross-section are optimized to minimize weight
• Ensure that normal and shear stresses do not exceed yield
• The height h should not be larger than twice the width b
13

14. Simple Beam
Example
• Design Variables – cross-section of the beam
width bL < b < bU 20 < b < 40
height hL < h < hU 30 < h < 90
• Responses
normal stress , shear stress , mass
• Constraints
 (b, h)  max, where max = 160
 (b, h)  max, where max = 60
h  2 b
• Objective
weight min mass (b, h)
14

15. beam width b
15
beam
height
h
max = 160
max = 60
h  2 b
30 ≤ h ≤90
20 ≤ b ≤40
Feasible
Domain
Infeasible
Domain
m=11
m=9
m=7
Simple Beam
Example
Mathematical Design Space

16. Introduction to OptiStruct & Theoretical Background
Sessions
• Optimization
• Design Process
• Structural Optimization
• Sensitivities
• Gradient-Based
Methodology
• Example
• Terminology
• Interpreting the results
• Techniques
• Workflow
• Design interpretation
Sensitivities &
Gradient-Based
Methodology
Terminology
and
Interpreting
the Results
Techniques,
workflow and
design
interpretation
Optimization
Basics
00‘ 10‘ 15‘ 20‘ 30‘
16

17. Optimization
Terminology
• Design Variables
• System parameters that are varied to optimize system
performance.
• Beam width b and beam height h
• Design Space
• Selected parts which are designable during optimization
process.
• For example, material in the design space of a topology
optimization.
20 < b < 40 and 30 < h < 90
17

18. Optimization
Terminology
• Response
• Measurement of system performance:  (b, h),  (b, h), mass (b, h)
• Constraint Functions
• Bounds on response functions of the system that need to be satisfied for the design to be acceptable
 (b, h)  160
 (b, h)  60
h  2 b
• Objective Function
• Any response function of the system to be optimized.
• The response is a function of the design variables.
• Examples are Mass, Stress, Displacement, Moment of Inertia, Frequency, Center of Gravity, Buckling factor, etc.
min mass (b, h)
18

19. Optimization
Terminology
• Feasible Design
• One that satisfies all the constraints.
• Infeasible Design
• One that violates one or more
constraint functions.
• Optimum Design
• Set of design variables along with the
minimized (or maximized) objective
function and satisfy all the constraints.
19

20. Interpreting the Results
Process Concerns
• Objective
• Did we reach our objective?
• How much did the objective improve?
• Design Variables
• Values of variables for the improved design
• Constraints
• Did we violate any constraints?
• Two ways of determining each of these in OptiStruct
• .out file from optimization run
• .mvw and _hist.mvwfiles from optimization run
20

21. Interpreting the Results
Common Issues
• Local vs. global extreme (minimum/maximum)
• Problem may be over constrained
• Review the objective, constraints and design variables to allow more design
freedom
• Efficiency of Optimization
• Relation between constraints and design variables with respect to their numbers
• Unconstrained OptimizationProblem
• Optimization problem setup is not appropriate
• Issues related to FEAmodeling
• Stress constraints on nodes connected to rigids
21

22. Introduction to OptiStruct & Theoretical Background
Sessions
• Optimization
• Design Process
• Structural Optimization
• Sensitivities
• Gradient-Based
Methodology
• Example
• Terminology
• Interpreting the results
• Techniques
• Workflow
• Design interpretation
Sensitivities &
Gradient-Based
Methodology
Terminology
and
Interpreting
the Results
Techniques,
workflow and
design
interpretation
Optimization
Basics
00‘ 10‘ 15‘ 20‘ 30‘
22

23. Optimization
Concept LevelTechniques
• Topology
• Given a design envelope, topology optimization finds the optimum
material placement within that space according to the constraints and
objective.
• Free Size
• Given a shell structure, free size optimization finds the optimum
thickness on an element-by-element basis that meets the constraints
and objective.
• Topography
• Given a shell structure, topography optimization creates a bead
pattern from the elements that meets the constraints and objective.
Topology
Free Size
Topography
23

24. Optimization
Fine Tuning-LevelTechniques
• Parameter/Size
• Given a structure, size optimization finds the optimum component
thickness that meets the constraints and objective.
• Shape
• Given a structure and a number of user-defined shapes, shape
optimization finds the optimum fractional summation of those
shapes that meets the constraints and objective.
• Free Shape
• Given a structure with features on its boundaries, free shape
modifies the boundary nodes to find a more optimal structure that
meets the constraints andobjectives.
Parameter/
Size
Shape
Free Shape
24

25. Optimization Process
Workflow
Topology
Optimization
Results
Interpretation
Topology
Optimization
Design Space
Creation
Size and Shape
Optimization Design
Fine-tuning
Optimized
Torsion Link
Topology
Optimization
Analysis &
Optimization
Setup
25

26. Design Interpretation
OSSmooth
OSSmooth is a semi-automated design interpretation software, facilitating the recovery of a modified
geometry resulting from a structural optimization, for further use in the design process and FEA
reanalysis.
• OSSmooth can be used in three different ways:
• OSSmooth for geometry
• FEA topology reanalysis
• FEA topography reanalysis
• The tool has two incarnations:
• Standalone version that comes with the OptiStruct installation.
• Dependent version that is embedded in HyperWorks.
Only this version can handle the reanalysis as it is using HyperWorks features.
• OSSmooth will be covered in detail in the followingchapters.
26

27. © Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.
OptiStruct for Optimization, v2021.2
28
Summary
Optimization is faster than traditional design
processes and leads to betterdesigns.
• An optimization problem contains thefollowing:
• Objective function (what is minimized or maximized, e.g. mass)
• Constraints (what does the design have to fulfill, e.g. stresses)
• Design variables (what can be changed in the model/structure, e.g.
thickness)
• Optimization algorithms solve the optimization problem.
• There are two groups of optimization algorithms:
• Mathematical programming methods, e.g. SQP
• Evolutionary algorithms, e.g. Geneticalgorithm
• There are the following optimization techniques:
• Topology, Free-size and Topography (concept level)
• Parameter/Size, Shape and Free-Shape (fine-tuning level)

28. © Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.
OptiStruct for Optimization, v2021.2
29
Questions & Answers
1. Name at least four of the six optimization techniques
supported by OptiStruct
_
2. What is the difference between a feasible design
and an infeasible design for optimization analyses?
a) There are not differences for optimization analyses, only
for structural analyses.
b) The feasible design satisfies all the constraint functions,
the infeasible design violates one or more constraint
functions.
c) The feasible design satisfies one or more constraint
functions, the infeasible design violates all the constraint
functions.
d) The feasible design satisfies all the constraint functions,
the infeasible design violates all the constraint functions.

29. © Altair Engineering, Inc. Proprietary and Confidential. All rights reserved.
OptiStruct for Optimization, v2021.2
30
Questions & Answers
3. What is the function of an optimization constraint?
a) To describe which part(s) are designable during the
optimization process.
b) To restrict a response to a lower and/or upper bound.
c) To set a parameter within the model to be monitored.
d) To select one response which should be optimized as a
function of other responses.
4. What are some common issues encountered during
optimization?
a) Infeasible design by over constraint.
b) Local maxima/minima instead of global maxima/minima.
c) Unconstrained problem.
d) All of the above.

30. THANK YOU