Optimization with COMSOL MultiphysicsCOMSOL Tokyo Conference 2014

Walter Frei, PhD Applications Engineer

Product Suite COMSOL 5.0

Agenda An introduction to optimization

A lot of concepts, and a little bit of math What do these options mean?

Demo

Overview of Examples

A quick conceptual introduction, and some terminologies

- Dimensions - Material Properties- Operating Conditions- etc K()u=b()

- Performance- Failure criteria- etc

Constrained Design Variables:

Objective: f(u())&

Constraints: g(u())

Black Boxu()

Optimization

More formally, optimization is

( )

( )( )( ) 0uh

0ug0p

u

=

)()(:such that

)(min

UL

f Objective functionSimple bounds on the design variables

Pointwise constraints on the design variables

General Equality constraints

General Inequality constraints

The design variables:

1

2

2,L

2,U

1,L 1,U

Design Space

Upper and Lower Bounds

p() 0

Pointwise constraints

The design space must be continuous

1

2

Break this up into two separate optimization problems

Design Space 1 Design

Space 2

Why no equality constraints for ?

1

2

2,L

2,U

1,L 1,U

An equality constraint is equivalent to a different optimization problem with one less design variable

p() = 0

Introduce a new design variable instead

1

2

2,L

2,U

1,L 1,UA

A,L

A,U

1= f(A)

2 = f(A)

The design space must be continuous in the real number space

1

2

Optimizing over a set of discrete values is an Integer Programming problem

LiveLink for MATLAB & LiveLink for Excelcan be used to interface to 3rd party optimizers

It is helpful if the design space is convex

Usually more difficult Every point can see every other design point

Now lets look at the objective function:f(u()) or f()

1

2

1 2

1

2

f

We are always starting somewhere

Initial design

We want to improve this Tip: Always start

optimizing from a feasible design

Lets first assume a smooth and differentiable objective function with a single minimum

1

2 f

1) Find the gradient

2) Search along the line

3) Find the minimum

4) Repeat

f

Start from a point, find the direction of steepest descent (the gradient) and search in that direction for a minimum

Repeat

Once the gradient is zero, or the boundary of the design space is reached, stop

Repeat until converged

Adjoint method is used to compute derivatives

( ) ( )( ) ( )( )

( ) ( ) ( )

( ) ( ) ( )

u

u

uK

bK

u

b

uKu

K

0buK

0buK

=

=

=

+

=

=

ff

1

Computing this derivative only doubles the computational requirements, regardless of how many design variables there are

Finite element equations

Differentiate w.r.t.

Expand

Re-arrange

Assuming f is differentiable w.r.t. u

If we hit a constraint, follow itStart from a point, find the direction of steepest descent (the gradient) and search in that direction for a minimum

Repeat

Once the gradient is zero, or the boundary of the design space is reached, stop

What if we have multiple minima?

Depends on where you start!

You are never guaranteed of finding the global minimum, but you can find a local minimum

Approximate the shape by evaluating the objective function repeatedly

For example, Nelder-Mead evaluates n+1 points when optimizing n design variables

What if the objective function is not smooth or differentiable?

Approximate the shape by evaluating the objective function repeatedly

For example, Nelder-Mead evaluates n+1 points when optimizing n design variables

What if the objective function is not smooth or differentiable?

What if we want to include general equality and inequality constraints?

1 2

g(u()) = 0

1 2

h(u()) 0

Strong dependence on initial conditionsHighly constrained design space

Summary Design variables must be continuous and real-valued Design space:

Simple (Cartesian) bounds Pointwise inequality constraints If you want to set up an equality constraint, get rid of one design variable Convex design space is better

Objective function If it is smooth and differentiable,

can use the Adjoint method and the gradient-based optimization technique If it is non-smooth or non-differentiable

Use the gradient-free approach General Equality and Inequality Constraints

Equality constraints can severely complicate the optimization problem Inequality constraints can make the design space non-continuous

Demo: A bracket with a hole

Load

Fixed

First, minimize the mass by changing the hole radius

R

The radius must be greater than zero, and not so large as to cut the bracket in half

Next, add a constraint on the maximum stress within the part

But the location of the peak stress is not known,So we use a maximum coupling operator & a constraint

< max

What about moving & resizing the hole?

Lets look at the constraints...

How can we express these mathematically?

Add one more design variable

A

R

Lets take these constraints a few at a time

R

RA

B = (1-0.25*A)/(1+sqrt(4.25)/2)

With a bit of (behind the scenes) trigonometry:

Which leads to the constraint: B-R>0

What about the other limits?

A

AR>0.2

R

Sometimes we can just ignore a constraint

Based upon the simulations so far, its likely this constraint will never be an issue

The available optimization solversOptimization

Module

Gradient-Free Methods

Gradient-Based Methods

Monte-Carlo

Coordinate Search

MMA Levenberg-Marquardt

Nelder-Mead BOBYQA

SNOPT

COBYLA

When to use gradient-free methods? Non-differentiable objective function, and/or constraints

Few design variables Optimization time increases exponentially with number of variables Aim for less than 10 design variables

Whenever re-meshing will occur Re-meshing results in a non-smooth objective function

The gradient-free solvers COBYLA

Similar to BOBYQA, but uses a linear approximation Can consider constraints

BOBYQA Constructs a quadratic approximant to the objective function Probably the fastest, but needs a reasonably smooth objective

function Nelder-Mead

Construct a simplex, and improve the worst point Probably the best if the objective function is relatively noisy Can consider constraints

Coordinate Search Search along one design variable at a time Estimate the gradients along that line, move on to next variable,

repeat Monte-Carlo

Random choices of design variables are evaluated Only a very dense statistical sampling can find the global optimum

Faster

Slower

Usually the

Most Robust

When to use gradient-based methods? Differentiable objective function, and/or constraints

Many design variables Optimization speed does not depend strongly on number of variables 100,000+ design variables are not unreasonable

Topology Optimization

The gradient-based solvers SNOPT

Sequential Quadratic Programming algorithm

MMA Linear convergence rate near the optimum Popular in the Topology Optimization community

Levenberg-Marquardt Only for unconstrained least squares minimization problems Very fast

Scaling and Tolerances Specify scales for all control variables

In Optimization study step for global parameters In Optimization interface features for fields

All solvers work with rescaled variables Solver tolerances are relative to these

Keep objectives and constraints close to 1 Solvers may use scaled gradient for termination

Comparison of Algorithms

Gradient-Free Gradient-Based

ObjectiveFunction

Any scalar output Must be both smooth and differentiable

Design Variables Anything, including geometric dimensions

Anything, but cannot result in remeshing of the geometry

Allows Remeshing

Yes No

Constraints Can only constrain scalar outputs Constraints must be smooth and differentiable, but can be at each point in space

RelativePerformance

Increases exponentially with the number of design variables

Performance is not very sensitive to the number of design variables

So what else can you do?

Parameter Estimation & Curve Fitting

Shape & Dimension

Topology

Structural Sizing Optimization of joint

positions in a truck-mounted crane.

Reduces force on boom lift cylinder for a range of operation conditions

Uses the MultibodyDynamics Module

Multi-study Structural Sizing Weight minimization of

a mounting bracket. Multi-study constraints

Maximum stress under static load

Lowest eigenfrequency

Estimating the material properties based upon experimental data

http://www.comsol.com/model/transient-optimization-fitting-material-properties-of-a-wall-10905

Examp