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 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 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 for 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=x n 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 cross- sectional 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?