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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
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