Sensitivity derivatives Can obtain sensitivity derivatives of structural

Sensitivity derivatives Can obtain sensitivity derivatives of structural response at several levels • Finite difference sensitivity (section 7. 1) • Analytical sensitivity of continuum equations (Chapter 8) • Analytical sensitivities of discretized equations (Chapter 7) • Analytical sensitivities of computer program (survey paper by van Keulen, Haftka and Kim).

Finite difference derivatives • Forward divided differences • Central differences usually more accurate. Why don’t we usually use them? • Higher order formulae also available

Truncation error • Taylor series expansion • Leading to the following truncation error in forward difference approximation • Similarly for the central difference approx. • When do we expect forward to be more accurate?

Example 7. 1. 1 • u found as solution to: • Derivative at x=100

Now with poorer conditioning • New system for solving for u • Derivative at x=10, 000

Condition error • For small step sizes we are limited by the scatter in the numerical calculation of u • This scatter can be caused by: – Round-off error due to the use of finite-digit calculations. – Convergence criterion for iterative solution techniques – Automatic remeshing • General name for this error is condition error

Optimal step size forward difference approximation • Formula • Truncation error • Bound for total error • Optimal step size • Example: Calculate optimum step size for

Problems step size • Derive the truncation error for the central difference derivative. • Provide an estimate for the optimal step size when using the central difference formula for derivatives • How will you reduce the truncation error with a given step size when you cannot use the central difference formula because you can use only a positive step size (function is not defined on the left) • Generate the figure on Slide 5 for the precision used in Matlab

Effect of derivative magnitude • Large derivatives are easier to estimate than small ones. • What does it mean to say that derivative is large? • One measure is logarithmic derivative • What does it mean to have a logarithmic derivative equal to one?

Large errors in small derivatives • Example: y=10+(x-5)2. Compare the accuracy of forward difference derivatives with a step size of one at x=10 and x=6. Relate to size of logarithmic derivative

Some uses of logarithmic derivatives • The logarithmic derivative of y=xn is n • If you link design variables to a single variable, their logarithmic derivatives will add up • If you scale up all the cross-sectional areas of a truss, or all the moment of inertias of a frame, displacements and stresses will scale inversely • So the sum of the logarithmic derivatives of displacement of a truss with respect to all crosssectional areas is -1 • What are the implications on each derivative?

Problems logarithmic derivative • Rather than the size of the logarithmic derivative what is the true ratio that determines the relative accuracy of finite difference derivative calculations? • Logarithmic derivatives do not make much sense when the function changes sign. What else can you use to normalize the derivative in that case?