Engineering Optimization

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# Engineering Optimization - PowerPoint PPT Presentation

Concepts and Applications. Engineering Optimization. Fred van Keulen Matthijs Langelaar CLA H21.1 A.vanKeulen@tudelft.nl. Contents. Sensitivity analysis. Sensitivity of system response (state variables). u constant. s i constant. Sensitivity analysis.

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## Engineering Optimization

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Presentation Transcript
Concepts and ApplicationsEngineering Optimization
• Fred van Keulen
• Matthijs Langelaar
• CLA H21.1
• A.vanKeulen@tudelft.nl
Contents
• Sensitivity analysis

Sensitivity of system response (state variables)

u constant

si constant

Sensitivity analysis
• Sensitivity: derivative of response w.r.t. design variable:

Total derivative

Partial derivatives

• Note: components of s should be independent!
Sensitivity analysis (2)
• What for?
• Avoid curse of dimensionality by using higher-order optimization algorithms (gradient-based, Newton, …)
• Examine sensitivity / robustness of optimized design solutions (parameter sensitivity)
• When?
• Attractive when sensitivity information can be obtained relatively cheaply
Logarithmic sensitivity
• Definition:
• Dimensionless, allows comparisons between parameters
• Clearly indicate the relative “strength” of the influence of parameters:>1: influential, important parameter<<1: not very influential parameter
Example logarithmic sensitivity
• Logarithmic sensitivity gives information on relative importance
• Always use logarithmic sensitivities when comparing sensitivity values of different variables!

Response

Design variable

Aspects of sensitivity analysis
• Implementation effort
• Efficiency
• Accuracy and consistency

Exact

Numericalmodel

Sensitivity analysis approaches

Implementation Efficiency

Very easy Terrible*

Moderate As good

as it gets

Lots of work As good

as it gets

• Global finite differencesInvolves repetitive design evaluations
• Discrete derivativesBased on differentiation of numerical model
• Continuum derivativesBased on differentiation of governing equations

x

Governingequations

Discrete

Discretization

Differentiation

x

Governingequations

Continuum

Differentiation

Discretization

Sensitivity analysis approaches (2)

Schematically:

f

x

Model

x

GFD

-

+

f+Df

x+Dx

Model

Automated differentiation
• Automatic generation of code that computes sensitivities:

Derivativecode

Automaticdifferentiation

Analysiscode

• Convenient, but generally code is several times slower than hand-coded derivatives

Based on Taylor series:

Finite difference derivatives
• Finite differences for sensitivity analysis (GFD):
• Simple
• Computationally inefficient (however …)
• Accuracy depends on design perturbation

Central FD:

-

Finite difference derivatives (2)
• First order forward / backward FD:

(forward)

(central)

• Forward FD error analysis:

Condition error

Truncation error

Finite difference derivatives (3)
• Similarly:
FD accuracy
• Perturbation h determines error:

Error

h

Practical aspect: noise
• Numerical noise can spoil FD accuracy!
• Example of noise source: effect of remeshing

Normalized stress constraint

Nonlinear elastic case
• Relatively cheap FD sensitivities (exception):
• Solution technique: incremental-iterative approachInvolves solution of many linear systems, e.g.

• FD: start the solution process for the perturbed case from the unperturbed solution

Displacement

• Much less expensive than full analysis!

Solution obtained by Newton iterations:

• For FD, solve perturbed case by iterating from nominal solution:
Nonlinear path-independent case

(e.g. )

• Consider:

For small design perturbation, this approaches:

Originalresidual

Nonlinear path-independent case
• Pitfall: make sure to include the finite residuals in the FD calculation!
• Consider first iteration for perturbed case:
• Interpretation: just an additional Newton iterationOriginal residual dominates over effect of design perturbation
Finite residual problem: solution
• To improve FD accuracy with finite residuals:instead of solvingsolve

i.e. subtract original residual from new residual.

• Ok for Ds = 0. Original residual no longer dominates
Finite difference summary
• Easy to implement, black box approach
• Inefficient, except for nonlinear path-independent and explicitly solved transient case
• Choice of proper relative design perturbation critical
• No adjoint formulation possible: unattractive in cases with many design variables and few responses

Then:

State variable vector sensitivity

Discrete derivatives
• Consider linear discretized equations (e.g. linear elastic FE model):and response (e.g. equivalent stress):

State variable sensitivity
• State variable derivatives follow from differentiation of original equation:

Semi-analytical approach
• Semi-analytical: use FD to compute pseudo-load:
• Easy implementation (can be done at top level)
• Efficient computation

• SA approach: computed using FD:
SA: nonlinear case
• Geometrically nonlinear (history-independent) setting:

Options for calculation of pseudo-load vector:

a) Analytical differentiation (lots of work)

b) Automated differentiation (code generator programs)

c) Finite difference approach

Discrete derivatives

CHEAP!!

• Note, computation of discrete derivatives
• Only involves a linear equation, also in nonlinear case
• Allows re-use of the decomposed system matrix

 Sensitivity analysis much cheaper than analysis itself!

SA accuracy problem
• Accuracy of semi-analytical (SA) sensitivities w.r.t. shape variables reduces for cases with substantial rotations (slender structures)
• Problem increases with mesh refinement!?!

SEE APPENDIX

Eigenvalue sensitivities
• Important class of responses: eigenvalues
• Discrete sensitivity analysis:
Eigenvalue sensitivities (2)
• Result:
• Note, no need to compute eigenvector sensitivities v’! If needed, one can use Nelson’s method(but rather expensive)
• Difficulties: eigenvalue multiplicity, mode switching ...
Contents
• Sensitivity analysis:
• Brief recap discrete / SA approach
• Continuum sensitivities
• Topology optimization
• Closure

and

• Discussed direct approach:
• One backsubstitution needed for every design variable: not attractive for many design variables

= 0

• To avoid computation of state vector derivatives, choose li such that vanishes!
• Starting point: augmented response:
• Result:
• One backsubstitution per response: attractive in case of many design variables and few responses

Difference consists of order of computations:

• Direct method attractive when #variables < #responses,adjoint method attractive when #variables > #responses
• Note, adjoint method requires load vector composed of response derivatives (specific implementation)
Sensitivities in transient case
• Transient analysis:
• Sensitivities at time ti depend on sensitivities at previous instants
• Direct method: forward time integration of sensitivities
• Adjoint method: backward time integration of sensitivities (unattractive, storage problem)
• FD often preferred for explicitly solved transient problems

x

Governingequations

Discretization

Differentiation

Discrete derivative summary
• Generally efficient and easy to implement, particularly semi-analytical case (combination with FD)
• Reuse of decomposed stiffness matrix (with direct solver – with iterative solver, reuse of preconditioner)
• SA: accuracy problems for structures under large rotations (beams, shells)
Contents
• Sensitivity analysis:
• Brief recap discrete / SA approach
• Continuum sensitivities
• Topology optimization
• Closure

x

Governingequations

Differentiation

Discretization

q(x)

I(x,s)

x

Governing equation:

+ boundary conditions

Continuum derivatives
• Example: beam bending(Euler-Bernoulli beam)

Compare:

Governing equation

Sensitivity equation

Continuum derivatives (2)
• Now differentiate w.r.t. s:
• For nonlinear / complex problems, the continuum sensitivity equations are often simpler
Sensitivity analysis summary
• Sensitivities important in optimization:
• Efficient higher-order optimization algorithms
• Evaluation of robustness of results
• Choice of sensitivity analysis method depends on:
• Number of design variables vs. number of responses (adjoint vs. direct)
• Type of model (cheap / expensive, linear / nonlinear / transient)

Perturbation size critical

• Efficient for nonlinear elastic & explicit transient case
• Inaccurate for large rotations
• Remedies: exact / refined version
Sensitivity analysis summary (2)

Points of attention

Implementation