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

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concepts and applications
Concepts and ApplicationsEngineering Optimization
  • Fred van Keulen
  • Matthijs Langelaar
  • CLA H21.1
  • Sensitivity analysis
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
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
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
example logarithmic sensitivity
Example logarithmic sensitivity
  • Logarithmic sensitivity gives information on relative importance
  • Always use logarithmic sensitivities when comparing sensitivity values of different variables!
aspects of sensitivity analysis


Design variable

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



sensitivity analysis approaches
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
sensitivity analysis approaches 2











Sensitivity analysis approaches (2)












automated differentiation
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
finite difference derivatives

Based on Taylor series:

Finite difference derivatives
  • Finite differences for sensitivity analysis (GFD):
    • Simple
    • Computationally inefficient (however …)
    • Accuracy depends on design perturbation
finite difference derivatives 2

Central FD:


Finite difference derivatives (2)
  • First order forward / backward FD:
finite difference derivatives 3



  • Forward FD error analysis:

Condition error

Truncation error

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



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

Normalized stress constraint

Hole radius

nonlinear elastic case
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


  • Much less expensive than full analysis!
nonlinear path independent case

Solution obtained by Newton iterations:

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

(e.g. )

  • Consider:
nonlinear path independent case1

For small design perturbation, this approaches:


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
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
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
discrete derivatives


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

Decomposed Kalready available (direct solver)!

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

Pseudo-load vector

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


  • SA approach: computed using FD:
SA: nonlinear case
  • Geometrically nonlinear (history-independent) setting:
discrete derivatives1

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


  • 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
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!?!


eigenvalue sensitivities
Eigenvalue sensitivities
  • Important class of responses: eigenvalues
  • Discrete sensitivity analysis:
eigenvalue sensitivities 2
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 ...
  • Sensitivity analysis:
    • Brief recap discrete / SA approach
    • Adjoint method
    • Continuum sensitivities
  • Topology optimization
  • Closure
adjoint discrete sensitivities


Adjoint discrete sensitivities
  • Discussed direct approach:
  • One backsubstitution needed for every design variable: not attractive for many design variables
  • Alternative: adjoint formulation
adjoint sensitivities

= 0

  • To avoid computation of state vector derivatives, choose li such that vanishes!
Adjoint sensitivities
  • Starting point: augmented response:
adjoint sensitivities 2
Adjoint sensitivities (2)
  • Result:
  • One backsubstitution per response: attractive in case of many design variables and few responses
adjoint vs direct

Difference consists of order of computations:

Adjoint vs. direct
  • 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
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
discrete derivative summary





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)
  • Sensitivity analysis:
    • Brief recap discrete / SA approach
    • Adjoint method
    • Continuum sensitivities
  • Topology optimization
  • Closure
continuum derivatives








Governing equation:

+ boundary conditions

Continuum derivatives
  • Example: beam bending(Euler-Bernoulli beam)
continuum derivatives 2


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
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
sensitivity analysis summary 2

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


Adjoint mode



Finite difference

Discrete derivativesSemi-analytical

Continuum derivatives