White paper multi objopt

Multi-Objective Optimization:
Design of Control Arm Using HyperWorks

Altair Engineering,

November 2010

Altair Engineering: United States, Brazil, Canada, China, France, Germany, India, Italy, Japan, Korea, Sweden, United Kingdom

Multi-Objective Optimization in the Design Cycle

Design optimization seeks a design that minimizes the
objective function while satisfying design constraints. Real-
world design problems are usually characterized by the
presence of many conflicting objectives. Therefore, it is
natural to look at the engineering design problems as Multi-
Objective Optimization (MOO) problems. For example, we
may want to maximize the range and payload mass while
trying to minimize the manufacturing costs of the airplane.

Challenge:
Engineers constantly face
challenges making
informed design decisions
in the presence of
conflicting design
objectives

Figure 1. Multiple Objectives: A design dilemma

So, in practice, the designer often needs to consider several
design criteria or objective functions simultaneously. In
most cases, trade-offs exist among the objectives, where
Solution: improvement in one objective cannot be achieved without
Using Multi-Objective deteriorating another. As a consequence, it is rare for a
Optimization (MOO) multi-objective optimization problem to produce a single
methods, engineers can optimal solution; instead, a set of equally valid solutions
logically choose an will be proposed.
acceptable trade-off
solution to satisfy In this article, we will cover the various MOO methods
conflicting design available in Altair HyperStudy; solver-neutral design
objectives exploration, optimization and stochastic study tool. Further,
we demonstrate the use of MOO methods on the design of
a control arm with conflicting design objectives.

1

Introduction to Multi-Objective Optimization

Design variables are a set of independent parameters that
can be changed to improve system performance. The
objective function measures the quality of a design.
Constraints are design requirements, and they must be
satisfied by the optimal design. Both the objective and the
constraints are functions of design variables. So, the
optimization problem reads: Find the design variable values
that minimize (maximize) the objective function(s) and
satisfy the design constraints.
Multi-objective problem
has several objective A multi-objective optimization (MOO) problem is
functions as opposed to a formulated as follows:
single objective function
minimize f(x)={f1(x),f2(x),…,fn(x)}
subject to gj(x) < 0

where x is the design variables, f1(x), …, fn(x) are multiple
objective functions and gj(x) are the constraints.

When dealing with multiple objectives (f1, f2, …) the
natural “better than”/“worse than” relationships between
alternative solutions no longer hold, and the relationship
among individuals is described as a “domination”
relationship, leading to Pareto-Optimal set instead of a
single optimal design. Let’s introduce these two key
definitions.

Domination

 For any two solutions x1 and x2
x1 is said to dominate x2 if these conditions hold:
o x1 is not worse than x2 in all objectives
o x1 is strictly better than x2 in at least one
objective
 If one of the above conditions does not hold x1 does
not dominate x2

2

Pareto-Optimal set

 Non-dominated set
The set of all solutions that are not dominated by
Multi-objective any other solution in the design space.
optimization methods
search for a set of non-  Global Pareto optimal set
dominated designs instead There exists no other solution in the entire design
of a single optimal design. space that dominates any member of the set.
Navigation along the Pareto front allows engineers
to do trade-off analyses and pick the best design for
various scenarios.

From these definitions, in MOO, the approach is to search
the design space for a set of Pareto optimal solutions, from
which the designer can choose the final design. Pareto
optimality is defined as a set where every element is a
possible solution for which no other solutions can be better
in all design objectives. A solution in the Pareto optimal set
cannot be declared as “better” than others in the set without
using other information (such as preference) to rank the
competing objectives. For the two-dimensional case, the
Pareto front is a curve that clearly illustrates the tradeoff
between the two objectives (see figure 2).

Figure 2. Pareto Front and non-domination illustration

3

Handling Multi-Objective Optimization Problems with
HyperStudy

HyperStudy provides a number of algorithms (SQP,
ARSM, MFD, GA, SORA) to cover a wide range of
optimization problems: constrained optimization, gradient-
based methods vs. heuristic approaches, continuous and
discrete variables, adaptive response surface based method,
reliability-based design optimization.

Multi-objective optimization problems cannot be solved
using the usual design optimization methods for a scalar
objective function, and specific algorithms are needed
HyperStudy has three instead. HyperStudy’s offering for MOO includes:
multi-objective
optimization methods:  Weighted sum approach
Weighted sum, MOGA and  MOGA (Multi-Objective Genetic Algorithm)
GMMO  GMMO (Gradient based Method for Multi-
Objective Optimization)

Weighted Sum is an approach for treating a MOO problem
as a single objective optimization problem by summing up
the weighted individual objectives. HyperStudy provides a
user-friendly GUI to perform this task:

min f(x)={f1(x),f2(x),…,fn(x)} à min Σ ωi.fi

Since the problem is no longer a MOO problem, any
standard optimizer of HyperStudy (GA, ARSM, MFD,
SQP) can be used to solve it.

Multi-objective genetic algorithm, unlike the weighted
sum approach, produces a set of Pareto-optimal solutions
(non-dominated solutions). The MOGA implementation in
HyperStudy is based on the standard feature of GA and
includes:

 Non-dominated classification strategy
 Crowding distance evaluation to help create a good
distribution of the points on the Pareto front.
 Storage of all the global non-dominated points
 Flexible termination criterion for better run control

4

Gradient-based method for multi-objective optimization
also produces a set of Pareto-optimal solutions (non-
dominated solutions). The GMMO implementation in
HyperStudy includes:

 A proprietary method that extends a typical
gradient-based algorithm to multi-objective
formulation
 A method constructed for the orderly and efficient
exploration of the Pareto-front
 A method where the user can control the number of
analyses to be performed

Multi-Objective Optimization of an Arm

This example illustrates the tradeoff challenges faced by
the designer of a typical mechanical component. The arm
shown below is clamped at one end and is under an axial
loading on the other end (see figure 4).

The model has been meshed and modeled in HyperMesh.
Problem: Linear static analysis is performed using RADIOSS, the
Minimize the mass of the HyperWorks finite element solver,.
arm and the maximum arm
displacement while
respecting the constraint
on the maximum allowable
stress.

Figure 3. Arm model

Figure 4. Boundary and loading conditions

5

The problem is stated as follows: minimize volume and
minimize the maximum arm displacement, but respect an
upper limit on the stress level throughout the part.

The current (nominal) design gives a volume of 1.7667E06
mm3, a maximum displacement of 1.41 mm and a
maximum stress of 195.29 MPa.

The designer can change the following properties of the
arm: six shape variables for the overall length and height of
Problem Formulation: the part, three shape variables for the three radii (see figure
5 below).

Step 1:
Create shape variables
using HyperMoprh.

Figure 5. Shapes defined on the model

The first step is to create the shape variables to represent
shape changes. We use HyperMesh’s HyperMorph module
to set up these parametric mesh-based shape changes
(morphing). In the morphing process, domains are created
around the FE model features. Handles are defined on the
edges of these domains. By moving a handle, nodes in the
associated domain also will move. Each of these
movements then can be saved as a shape variable and can
be exported directly to HyperStudy.

In HyperStudy, the study is set up by identifying these
variables (nine shape variables), setting up the simulations
to run (RADIOSS), and extracting the responses (maximum
displacement, volume, maximum stress).

6

Figure 6. Table of all the design variables

After defining the problem, a nominal run is executed
within HyperStudy. Once the user makes sure that nominal
run is executed correctly, the next task is to pick a study
method (i.e. DOE, optimization, stochastics). In this case,
an optimization based on an approximation is chosen.
Step 2:
Run a screening DOE
study to understand the
physics of the problem and
possibly to reduce the
problem size which
decreases the runtime.

Figure 7. Study flow and solver execution in HyperStudy

The study starts with an attempt to reduce the number of
variables. A fractional Factorial (16 runs) DOE (design of
experiment), using the nine variables is realized. The main
effects plots (figure 8) show that three design variables (the
three radii are of smaller impact on the responses than the
other six variables and thus can be screened out.

7

Step 3:
Since MOO problems take Figure 8. Main Effects
longer time to run, it is
recommended to replace Only six design variables are then kept to perform:
the exact simulations with
a good approximate model.
 A full factorial DOE (64 runs) and an Hammersley
DOE (86 runs), to be used as input matrix for the
approximation
 A complementary Latin HyperCube DOE (20 runs)
Use approximation DOE
to be used for approximation validation
study to create a good
approximate model.
Moving Least Square Method (MLSM) approximations are
built and checked for all responses using 64+86 runs as the
input matrix and 20 runs as the validation matrix.

A multi-objective optimization is performed using MOGA.
The target is to minimize both the volume and the
maximum displacement, while satisfying a constraint on
the maximum stress in the part.

Figure 9. MOO setup in HyperStudy

8

After running 25 iterations of MOGA, with a total of 1,878
analyses on the meta-model, the results of the last iteration
as shown below are obtained (figure 10). Note that,
contrary to standard optimization, an MOO will produce,
for a given iteration, a set of designs (a Pareto front). The
front displayed below is the 25th iteration and is made of
116 non-dominated points.

Step 4:
Run a Multi-Objective
Optimization method to
obtain a Pareto front.

Figure 10. Pareto front obtained at iteration 25 of a MOGA
optimization

In this plot, X-axis is the volume and Y-axis is the
maximum displacement: The tradeoff is based on selecting
the best compromise between the two competing
objectives.

The black lines on the Figure 11 show where the nominal
point (initial design) is situated.

Figure 11. Initial design (at the intersection of the black
lines) vs. Pareto Front

9

Notice also that MOGA produces a front where all points
are admissible (maximum stress < 300 MPa).

For example in the graph below, we can see that, of the 116
non-dominated points of iteration, 25 are all giving a
max_stress response under 297.97 MPa (x-axis)

Using the Pareto fronts
obtained from MOO
methods, engineers can
make informed decisions
on problems that involve
conflicting objectives.

Figure 12. Pareto front for the space (volume ;
max_stress)

Conclusions

Handling multiple objectives is a constant dilemma for
designers. Often, the “most important” objective is kept and
other objectives are transformed into constraints. But this
does not reflect the tradeoff among all possible designs.
The implementation of MOO methods in HyperStudy
allows for extended exploration of the solution and
tradeoffs.

This study illustrates the use of HyperWorks to search for an
optimal arm design such that the volume (mass) is minimized
and the maximum displacement is minimized, while meeting
stress constraints. In this study, HyperStudy automatically
updates the design using HyperMorph for shape variables, runs
RADIOSS as the FE solver and uses a specific algorithm for
multi-objective optimization. With this process, a set of solutions
is proposed to the designer to select the best compromise.

10

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