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Derivatives

Derivatives. In modern structural analysis we calculate response using fairly complex equations. We often need to solve many thousands of simultaneous equations coming from finite element models. Derivatives of the solution with respect to problem parameters are useful

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Derivatives

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  1. Derivatives • In modern structural analysis we calculate response using fairly complex equations. • We often need to solve many thousands of simultaneous equations coming from finite element models. • Derivatives of the solution with respect to problem parameters are useful • For gradient based optimization. • For estimating uncertainty in solution due to uncertainty in input. • Calculating derivatives is usually more difficult than calculating the response at similar level of accuracy.

  2. Where do we differentiate • Structural analysis usually begins with differential equations. • These are discretized into algebraic equations (e.g. by finite elements). • Then the algebraic equations are coded and solve in a computer program. • We can differentiate the differential equations, the algebraic equations, or the computer program.

  3. Continuum derivatives • These are obtained by differentiating the differential or integral equations. • Example: Beam column differential equations, and derivative with respect to moment of inertia • Obtain a differential equation for derivative of structural response with a “pseudo load”

  4. Algebraic derivatives • Take the equations for laminate strains as an example: • In Section 7.2, we want to calculate the derivatives of the laminate strains with respect to the thickness of the ith ply • If the loads are fixed, we get =- • Calculate the derivatives of the strain using a pseudo load.

  5. Computational differentiation • Given a computer program, there are software packages that will augment it with code that will calculate the derivatives when it is run. • Example: • a=b+c*d; e=a^2+5 • Requesting the derivative with respect to b will produce • a=b+c*d; aprime=1; e=a^2+5; eprime=2a*aprime

  6. Derivative equations are linear • When you differentiate a nonlinear equation or set of equations with respect to a parameters, you need to solve a linear equation or set for the derivative. • Example • Differentiate with respect to a • How do we check?

  7. Finite differencing • Taylor series expansion • Forward differencing • Central differencing • How do we estimate an optimum step size? • Example: Sunrises are given in the paper to the minutes. How many days should h be?

  8. Complex differentiation • If your function can support complex arguments can use • Can take any value of h without worrying about round-off error.

  9. Strength design with thickness design variables • A first cut at strength design of complex wing and fuselage structures is usually done with fixed stacking sequence, where the thicknesses of the plies are the design variables. • This is attractive because standard gradient based techniques can be used. • Section 7.2 shows how you can differentiate all the equations leading from the loads to the stresses in each ply in order to get analytical derivatives for the optimization. • If the laminate is thick, one can often obtain a reasonable design by rounding the thicknesses to an integer number of plies.

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