Approximations in structural optimization

Local and Local-Global ApproximationsLocal algebraic approximationsVariants on Taylor seriesLocal-Global approximationsVariants on fudge factor

In todays lecture we will be covering the use of derivatives for constructing an approximation to structural response. First we will cover the use of a first order Taylor series, where we can gain some flexibility by use of intervening variables, such as the reciprocal of the variables.

Next we will cover the use of derivative in order to localize a global approximation to the function. For example, we may have fitted a global approximation based on a sample of points, and we want to take advantage of the derivatives. This is called global-local approximation.1Local algebraic approximationsLinear Taylor series

Intervening variables

Transformed approximationMost common: yi=1/xi

The simplest local algebraic approximation is a linear Taylor series approximation shown in the first equation. Instead we can define an intermediate variable y, and use a Taylor series approximation in y. That would make sense if we can gain accuracy.

In structural response, reciprocal variables y=1/x often make sense for sizing variables. This is because there is diminishing returns to increasing sizes. You can reduce the stress by 50% by doubling a thickness, for example. However, tripling the variable would reduce the stress only by an additional 16.6% to one third.2Beam exampleTip displacement

Intervening variables yi=1/Ii

For statically determinate structures, using reciprocal sizing variables often makes the approximation exact. This is illustrated in the tip displacement of a cantilever beam where the design variables are the moments of inertia.3Reciprocal approximationIt is often useful to write the reciprocal approximation in terms of the original variables x instead of the reciprocals y

For the case of the reciprocal approximation, it is useful to write the approximation in terms of the original variables instead of the reciprocal variables. It is easy to derive the expression given in the equation in the slide.4Conservative-convex approximationAt times we benefit from conservative approximations

All second derivatives of gC are non-negativeConvex linearization obtained by applying the approximation to both objective and constraints

It is common to combine the linear and reciprocal approximations to form a convex approximation that is also more conservative than either. If you take the difference of the linear and reciprocal approximation you see that the sign of the difference depends on the sign of he derivative times or divided by the variable. For variables with a positive sign the linear is more conservative (because larger g means more critical) and for negative sign the reciprocal is more conservative.

By using the sign to decide for each variable whether to use linear or reciprocal, we get an approximation that is more conservative than either. It also has the nice property that all the second derivatives are positive, and hence it is a convex approximation.

If we use this approximation for both objective function and constraints, we get a convex problem.5Three-bar truss example

We will compare the approximation for a three-bar truss with a large vertical load and small horizontal load. Specifically we will look at the stress in member C that can change sign depending on the values of the cross sectional areas. It is assumed that member A and member C have the same cross sectional area.6Stress constraint on member CStress in terms of areas

Stress constraint

Using non-dimensional variablesWhat assumption on stress?

The stress in member C is given by the top equation. Note that the first term represents the compressive stresses due to the horizontal loads and the second term is the tensile stresses due to the vertical load. The stress constraint is given by the second equation, and it has the implicit assumption that the stress in member C is tensile. This will not hold if member B is very large leaving very little tensile stress in members A and C, but there is no reason to have so much area invested in member B.

To non-dimensionalize the variables, it makes sense to define a reference area equal to the load divided by the stress allowable. This is the area needed to carry safely p in a single member. We define x1 and x2 as the areas divided by the reference area, and get the last equation for the constraint in terms of the non-dimensional variables. 7Results around (1,1).x1x2ggLgRgC0.750.750.36350.27830.36350.38501.000.750.42270.34260.44930.44931.250.750.42050.40700.50080.51370.751.00-0.0856-0.0417-0.0631-0.04171.251.000.06190.08700.07410.08710.751.25-0.3786-0.3617-0.3191-0.29771.001.25-0.2440-0.2974-0.2334-0.23341.251.25-0.1819-0.2330-0.1819-0.1690

The table compares the approximations for departures of 25% from nominal values of both variables equal to 1. We see that the reciprocal approximation is exact when both variables are scaled by the same factor (0.75,0.75) and (1.25,1.25). This is due to the fact that the stress is scaled by the reciprocal of the factor when both areas are scaled.

For these cases the reciprocal approximation is better. On the other hand, when they go in opposite directions, the linear approximation is more accurate. The conservative approximation is always the most conservative, but often the least accurate. Note that it is not guaranteed to be conservative as here. It is only guaranteed to be at least as conservative as the more conservative of the two others..8Problems local approximationsWhat are intervening variables? There are also cases when we use intervening function in order to improve the accuracy of a Taylor series approximation. Can you give an example? AnswersWhat is conservative about the conservative approximation? Why is that a plus? Why is it useful that it is convex? Answers

9Local Approximations pros and consDerivative based local approximations have several advantagesDerivatives are often computationally inexpensiveDerivatives are needed anyhow for optimization algorithmsThese approximations allow rigorous convergence proofsThere are some disadvantages tooThey can have very small region of acceptable accuracyThey do not work well with noisy functions

Derivative based local approximations are useful mostly because as we have seen, it is possible to get derivatives of structural response cheaply. In addition, for gradient based optimization algorithms we need the derivatives anyhow for the optimization. Finally, derivatives are often needed for convergence proofs, or to test optimality conditions such as the Karush-Kuhn-Tucker conditions.

On the other hand, local approximations often have very small region of acceptable accuracy. They also do not work well for noisy functions.10Global approximationsCan be based on more approximate mathematical modelCan be based on same mathematical model with coarser discretizationCan be based on fitting a meta-model (surrogate, response surface) to a number of simulationsPro and cons complement those of local approximations: Wider range, noise tolerance, but more expensive, and less amenable to math proofs

Global approximations may be based on a less expensive mathematical model. That may mean simpler physics, such as 2D approximation instead of 3D, or it may mean coarser discretization. Global approximation may also be based on fitting a meta-model (aka surrogate or response surface) to a number of simulations.

The pro and cons of global approximations are the inverse of those of local approximations. They do not easily take advantage of derivatives, but they have wider regions of applicability and noise tolerance.11Combining local and global approximationsCan use derivatives to combine the two models

The combined approximation matches the value and slope at x0.

It is quite common to use a fudge factor to reconcile analytical model with experimental results. This means that you multiply the analytical model by a constant (fudge factor) to obtain a prediction of the experimental results.

The local-global approximation suggested here is for f factor beta that is a linear function of the design variables, so that the approximation matches both the function value and the derivative at the point x0.12ExampleApproximating the sine function as a quadratic polynomial

For an example we take the sine function. The approximation in the form of a quadratic polynomial was selected to match the value of the function at three points x=0,0.5, 1.

Now assume that we want to refine the approximation in the vicinity of x=0.1. At that point we find that the simple model is about 15% higher than the sine function, and furthermore, its derivative is about 7% higher. This means that the percentage differences between the two will reduce as x is increased.

By using the formulas on the previous slide we obtain a correction that applies a 14% percent correction at x=0.1, and increases beta, thus decreasing the correction, so that near x=0.3 the correction will change direction, and it will eventually be worse than the simple model.13Overall comparison.

The figure shows that indeed near x=0.1 the correction is greatly improved. We also see that the error in the correction is not as bad as a pure linear approximation. However, we also see that it is not good for large departure from x=0.1. By the time we get to x=0.3, we are better off with the simple model than with the corrected one.14Without linear.

It is a bit easier to see the trend without the linear approximation.15Problems local-globalGiven the function y=sinx, compare the linear, reciprocal, and global local approximation about x0=p/3, where the global approximation is yS=2x/p

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