Engineering Optimization – Concepts and Applications Engineering Optimization Concepts and Applications Fred van Keulen Matthijs Langelaar CLA H21.1

Engineering Optimization – Concepts and Applications Contents ● Sensitivity analysis

Engineering Optimization – Concepts and Applications Sensitivity analysis ● Sensitivity: derivative of response w.r.t. design variable: ● Note: components of s should be independent! u constant s i constant Sensitivity of system response (state variables) Total derivative Partial derivatives

Engineering Optimization – Concepts and Applications 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

Engineering Optimization – Concepts and Applications Logarithmic sensitivity ● Definition: ● Advantages: – Dimensionless, allows comparisons between parameters – Clearly indicate the relative “strength” of the influence of parameters: >1 : influential, important parameter <<1 : not very influential parameter

Engineering Optimization – Concepts and Applications Example logarithmic sensitivity ● Logarithmic sensitivity gives information on relative importance ● Always use logarithmic sensitivities when comparing sensitivity values of different variables!

Engineering Optimization – Concepts and Applications Aspects of sensitivity analysis ● Implementation effort ● Efficiency ● Accuracy and consistency Response Design variable Exact Numerical model

Engineering Optimization – Concepts and Applications Sensitivity analysis approaches ● Global finite differences Involves repetitive design evaluations ● Discrete derivatives Based on differentiation of numerical model ● Continuum derivatives Based on differentiation of governing equations ImplementationEfficiency Very easyTerrible* ModerateAs good as it gets Lots of workAs good as it gets

Engineering Optimization – Concepts and Applications Sensitivity analysis approaches (2) Schematically: Model x x x+  x f f+  f + - GFD Governing equations DiscretizationDifferentiation x Discrete Governing equations DiscretizationDifferentiation x Continuum

Engineering Optimization – Concepts and Applications Automated differentiation ● Automatic generation of code that computes sensitivities: ● Many different tools exist: ADIFOR, ADOL-F (Fortran), ADIC, ADOL-C (C/C++), … ● Convenient, but generally code is several times slower than hand-coded derivatives Analysis code Automatic differentiation Derivative code

Engineering Optimization – Concepts and Applications Finite difference derivatives ● Finite differences for sensitivity analysis (GFD): – Simple – Computationally inefficient (however …) – Accuracy depends on design perturbation ● Based on Taylor series:

Engineering Optimization – Concepts and Applications Finite difference derivatives (2) ● First order forward / backward FD: - ● Central FD:

Engineering Optimization – Concepts and Applications Finite difference derivatives (3) ● Similarly: (forward) (central) ● Forward FD error analysis: Condition errorTruncation error

Engineering Optimization – Concepts and Applications FD accuracy ● Perturbation h determines error: h Error

Engineering Optimization – Concepts and Applications Practical aspect: noise ● Numerical noise can spoil FD accuracy! ● Example of noise source: effect of remeshing Normalized stress constraint Hole radius

Engineering Optimization – Concepts and Applications Nonlinear elastic case ● Relatively cheap FD sensitivities (exception): – Solution technique: incremental-iterative approach Involves solution of many linear systems, e.g. Load Displacement – FD: start the solution process for the perturbed case from the unperturbed solution ● Much less expensive than full analysis!

Engineering Optimization – Concepts and Applications Nonlinear path-independent case ● Consider: (e.g. ) ● Solution obtained by Newton iterations: ● For FD, solve perturbed case by iterating from nominal solution:

Engineering Optimization – Concepts and Applications Nonlinear path-independent case ● Pitfall: make sure to include the finite residuals in the FD calculation! – Consider first iteration for perturbed case: – For small design perturbation, this approaches: Original residual – Interpretation: just an additional Newton iteration Original residual dominates over effect of design perturbation

Engineering Optimization – Concepts and Applications Finite residual problem: solution ● To improve FD accuracy with finite residuals: instead of solving solve i.e. subtract original residual from new residual. ● Ok for  s = 0. Original residual no longer dominates

Engineering Optimization – Concepts and Applications 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

Engineering Optimization – Concepts and Applications State variable vector sensitivity ● Then: Discrete derivatives ● Consider linear discretized equations (e.g. linear elastic FE model): and response (e.g. equivalent stress):

Engineering Optimization – Concepts and Applications State variable sensitivity ● State variable derivatives follow from differentiation of original equation: Pseudo-load vector ● Decomposed K already available (direct solver)!

Engineering Optimization – Concepts and Applications State variable sensitivity (2) ● Nonlinear case similar: Already decomposed tangent matrix (direct solver)

Engineering Optimization – Concepts and Applications Semi-analytical approach ● Semi-analytical: use FD to compute pseudo-load: ● Advantages: – Easy implementation (can be done at top level) – Efficient computation

Engineering Optimization – Concepts and Applications Already decomposed SA: nonlinear case ● Geometrically nonlinear (history-independent) setting: ● SA approach: computed using FD:

Engineering Optimization – Concepts and Applications CHEAP!! Discrete derivatives ● 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! ● Options for calculation of pseudo-load vector: a) Analytical differentiation (lots of work) b) Automated differentiation (code generator programs) c) Finite difference approach

Engineering Optimization – Concepts and Applications 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

Engineering Optimization – Concepts and Applications Eigenvalue sensitivities ● Important class of responses: eigenvalues ● Discrete sensitivity analysis:

Engineering Optimization – Concepts and Applications 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...

Engineering Optimization – Concepts and Applications Contents ● Sensitivity analysis: – Brief recap discrete / SA approach – Adjoint method – Continuum sensitivities ● Topology optimization ● Closure

Engineering Optimization – Concepts and Applications Adjoint discrete sensitivities ● Discussed direct approach: and ● One backsubstitution needed for every design variable: not attractive for many design variables ● Alternative: adjoint formulation

Engineering Optimization – Concepts and Applications Adjoint sensitivities ● Starting point: augmented response: = 0 ● To avoid computation of state vector derivatives, choose i such that vanishes!

Engineering Optimization – Concepts and Applications Adjoint sensitivities (2) ● Result: ● One backsubstitution per response: attractive in case of many design variables and few responses

Engineering Optimization – Concepts and Applications Adjoint vs. direct ● Direct method attractive when # variables # responses ● Note, adjoint method requires load vector composed of response derivatives (specific implementation) ● Difference consists of order of computations:

Engineering Optimization – Concepts and Applications Sensitivities in transient case ● Transient analysis: ● Sensitivities at time t i 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

Engineering Optimization – Concepts and Applications 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) ● Direct and adjoint versions ● SA: accuracy problems for structures under large rotations (beams, shells) Governing equations DiscretizationDifferentiation x

Engineering Optimization – Concepts and Applications Contents ● Sensitivity analysis: – Brief recap discrete / SA approach – Adjoint method – Continuum sensitivities ● Topology optimization ● Closure

Engineering Optimization – Concepts and Applications Continuum derivatives ● Example: beam bending (Euler-Bernoulli beam) Governing equations DiscretizationDifferentiation x q(x)q(x) x I(x,s) + boundary conditions Governing equation:

Engineering Optimization – Concepts and Applications Continuum derivatives (2) ● Now differentiate w.r.t. s : Sensitivity equation ● For nonlinear / complex problems, the continuum sensitivity equations are often simpler Governing equation ● Compare:

Engineering Optimization – Concepts and Applications 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) – Implementation effort, access to source code

Engineering Optimization – Concepts and Applications Sensitivity analysis summary (2) Finite difference Discrete derivatives Semi-analytical Continuum derivatives Implementation Efficiency Accuracy Adjoint mode Points of attention Perturbation size critical Efficient for nonlinear elastic & explicit transient case Inaccurate for large rotations Remedies: exact / refined version