Optimization with COMSOL Multiphysics COMSOL Tokyo Conference Walter Frei, PhD Applications Engineer

Transcription

1 Optimization with COMSOL Multiphysics COMSOL Tokyo Conference 2014 Walter Frei, PhD Applications Engineer

2 Product Suite COMSOL 5.0

3 Agenda An introduction to optimization A lot of concepts, and a little bit of math What do these options mean? Demo Overview of Examples

4 A quick conceptual introduction, and some terminologies - Dimensions - Material Constrained Properties Design - Operating Variables: Conditions χ -etc Black Box u(χ) K(χ)u=b(χ) - Performance Objective: f(u(χ)) - Failure criteria & -etc Constraints: g(u(χ)) Optimization

5 More formally, optimization is min χ R such that : χ p g L f ( χ) ( u( χ) ) χ 0 ( u( χ) ) = χ U 0 Objective function Simple bounds on the design variables Pointwise constraints on the design variables General Equality constraints h ( u( χ) ) 0 General Inequality constraints

6 The design variables: χ χ 2 Design Space χ 2,U Pointwise constraints p(χ) 0 χ 1,L χ 1,U χ 1 χ 2,L Upper and Lower Bounds

7 The design space must be continuous χ 2 Design Space 1 Design Space 2 χ 1 Break this up into two separate optimization problems

8 Why no equality constraints for χ? χ 2 χ 2,U p(χ) = 0 χ 1,L χ 1,U χ 1 χ 2,L An equality constraint is equivalent to a different optimization problem with one less design variable

9 Introduce a new design variable instead χ 2 χ 2,U χ A,U χ 1 = f(χ A ) χ 2 = f(χ A ) χ A χ 1,L χ 1,U χ 1 χ A,L χ 2,L

10 The design space must be continuous in the real number space χ 2 χ 1 Optimizing over a set of discrete values is an Integer Programming problem LiveLink for MATLAB & LiveLink for Excel can be used to interface to 3 rd party optimizers

11 It is helpful if the design space is convex Usually more difficult Every point can see every other design point

12 Now lets look at the objective function: f(u(χ)) or f(χ) χ 2 χ 1 χ 1 χ 2

13 We are always starting somewhere f We want to improve this χ 2 Tip: Always start optimizing from a feasible design Initial design χ 1

14 Lets first assume a smooth and differentiable objective function with a single minimum f χ f χ 2 1) Find the gradient 2) Search along the line 3) Find the minimum 4) Repeat χ 1

15 Repeat until converged 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

16 Adjoint method is used to compute derivatives χ u χ f χ ( ) u b( χ) K χ ( K( χ) u b( χ) ) ( ) K χ χ = ( ) u + K χ 0 u χ 1 b = K ( χ) χ f u = u χ = 0 ( χ) b = χ ( χ) K( χ) χ u Finite element equations Differentiate w.r.t. χ Expand Re-arrange Assuming f is differentiable w.r.t. u Computing this derivative only doubles the computational requirements, regardless of how many design variables there are

17 If we hit a constraint, follow it 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

18 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

19 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

20 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

21 What if we want to include general equality and inequality constraints? g(u(χ)) = 0 χ 1 χ 2 Highly constrained design space h(u(χ)) 0 χ 1 χ 2 Strong dependence on initial conditions

22 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

23 Demo: A bracket with a hole Fixed Load

24 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

25 Next, add a constraint on the maximum stress within the part σ < σ max But the location of the peak stress is not known, So we use a maximum coupling operator & a constraint

26 What about moving & resizing the hole?

27 Let s look at the constraints... How can we express these mathematically?

28 Add one more design variable R A

29 Let s take these constraints a few at a time With a bit of (behind the scenes) trigonometry: B = (1-0.25*A)/(1+sqrt(4.25)/2) R A R Which leads to the constraint: B-R>0

30 What about the other limits? A R>0.2 A R

31 Sometimes we can just ignore a constraint Based upon the simulations so far, its likely this constraint will never be an issue

32 The available optimization solvers Optimization Module Gradient-Free Methods Gradient-Based Methods Monte -Carlo MMA SNOPT Levenberg- Marquardt Coordinate Search Nelder- Mead BOBYQA COBYLA

33 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

34 Faster Slower Usually the Most Robust 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

35 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

36 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

37 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

38 Comparison of Algorithms Objective Function Design Variables Gradient-Free Any scalar output Anything, including geometric dimensions Gradient-Based Must be both smooth and differentiable Anything, but cannot result in remeshing of the geometry Allows Yes No Remeshing Constraints Can only constrain scalar outputs Constraints must be smooth and differentiable, but can be at each point in space Relative Performance Increases exponentially with the number of design variables Performance is not very sensitive to the number of design variables

39 So what else can you do? Parameter Estimation & Curve Fitting Shape & Dimension Topology

40 Optimization of joint positions in a truckmounted crane. Reduces force on boom lift cylinder for a range of operation conditions Uses the Multibody Dynamics Module Structural Sizing

41 Multi-study Structural Sizing Weight minimization of a mounting bracket. Multi-study constraints Maximum stress under static load Lowest eigenfrequency

42 Estimating the material properties based upon experimental data

43 Example: Optimizing a Flywheel, with Constraints Make stress distribution as uniform along the radius as possible Constrain the mass not to change Constrain the moment of inertia not to change Gradient-based approach

44 Example: Band dispersion in a microchannel Minimize the difference is transit time between inside and outside Gradient-Free optimization

45 Example: Optimizing a Horn Maximize the sound intensity along the axis of the horn The shape of the horn is described by a sum of sine waves Truncated Fourier series The Deformed Mesh functionality is used to avoid remeshing the domain Gradient-Based approach

46 Example: Optimizing a Heater Maximize the temperature at the outlet Such that the temperature at each heater stays below a target

47 Example: Optimizing a Spinning Wheel Minimize the Mass By changing: r inner r outer Constrain the maximum stress

48 Topology Optimization: Buy this book

49 Example: Optimizing a Beam Minimize the compliance Add constraint on total material SIMP method

50 Example: Tesla Microvalve Flow to the right: Minimize p Flow to the left: Maximize p Laurits Højgaard Olesen, Fridolin Okkels and Henrik Bruus, A high-level programming-language implementation of topology optimization applied to steady-state Navier Stokes flow, Int. J. Numer. Meth. Engng 2006; 65:

51 COMSOL Conference Papers on Topology Optimization Topology Optimization in Multiple Physics Problems, O. Sigmund, DTU Mechanical Engineering, Multiphysics Topology Optimization of Heat Transfer and Fluid Flow Systems, E. Dede, Toyota Research Institute of North America, Simulation of Topology Optimized Electrothermal Microgrippers, O. Sardan, D. Petersen, O. Sigmund, & P. Boggild, DTU Mechanical Engineering, Implementation of Structural Topology Optimization in COMSOL, B. Lemke, Z. Liu, & J. G. Korvink, Department of Microsystems Engineering, University of Freiburg, Topology Optimization of Dielectric Metamaterials Based on the Level Set Method Using COMSOL Multiphysics, M. Otomori & S. Nishiwaki, Kyoto University, Optimization of a Photonic Crystal Waveguide Termination, W. Frei, COMSOL,

52 When to use which one? Early in the design process, when you have a lot of time to run analyses, and don t have a very rigid idea about your final design? Already have the basic topology fixed, but can change the shape? It s about to go to production, and we realized it doesn t work!!!

53 When to use which one? Early in the design process, when you have a lot of time to run analyses, and don t have a very rigid idea about your final design? Topology Optimization Already have the basic topology fixed, but can change the shape? Shape Optimization It s about to go to production, and we realized it doesn t work!!! Dimensional Optimization

54 Some general optimization tips Scale your objective function to be close to unity Do not get frustrated, the software doesn t know that you haven t asked the right question Start as simple as possible, especially with constraints

55 Concluding remarks min χ R such that : χ p g h L f ( u( χ) ) χ χ U ( χ) 0 ( u( χ) ) = 0 ( u( χ) ) 0