Mueller 21/22 Flashcards
(30 cards)
Give a brief outline of the SIMP method
Introduces material density variability from 0 to 1, where 0 is no material and 1 is a solid. Calculates derivative of the objective function (e.g. weight) w.r.t. density. Then reduces density of elements with lowest derivative.
What is an Adjoint Solution?
One which directly expresses the sensitivities of a single cost function w.r.t. many design variables. (Think of the car which showed where to push in/push out for better aerodynamic performance.)
State the Optimality conditions for optimization
If F'(x) = 0 and F''(x) >0 and if these are satisfied by x=x* F(x)>F(x*) for all other x
Then x* is a minimum.
State the steps for the Bisection Method
For an interval a
List the key properties of the Bisection Method
- User define interval which must contain the min
- Will find any min, not not necessarily the global
- Convergence is slow, depends of width of the initial and desired brackets
- N = [log(xb-xa)+log(eta)]/log(2)
- Gradient free
Describe the Secant Method and list the key steps
A gradient based optimization technique which uses linearly interpolates between a bracket (although extrapolation can be used under certain conditions), using values of x and F’(x).
- Set x1 = a and x2 = b (Bracket)
- Compute F’1 = F’(x1) and F’2
- Set k=2
- Use linear interpolation formula to find x’k+1
- Compute F’k+1
- Set k=k+1
- Repeat until convergence
State the conditions for extrapolation for the Secant Method
If x1x2 & F’1>F’2>0
State the key properties of the Secant Method
- Needs gradients
- Only need first derivatives
- May converge to a max
- Faster than Bisection Method
- Flexibility in choosing xk
- Can be generalised to multi-variate problems.
State the three methods of choosing xk in the Secant method and briefly describe them
Chronological - xk becomes xk-1, xk+1 becomes xk. Simple and quick.
Smallest gradient - The two F’(x) with the smallest absolute values are used as the bracketing points, as they are, in theory, closer to the min. Faster
Bracketed - The above may find a max, by choosing two F’(x) with opposite signs for the brackets, we make sure we find a min. Slower but no risk of finding max.
List the key differences in origin of the Bisection, Secant, and Newton Methods
Bisection - Literal calculation of function values over points and converging to the smallest values
Secant - Uses Linear Interpolation to find the zero of the gradient to find the min
Newton - Uses Taylor expansion to approximate the zero of the function using gradients.
What is the process for safeguarding the Bisection Method? Explain.
From the initial point, calculate the negative gradient and follow until it starts increases in steps s. This will select an interval with a min and if the function is unimodal, there will only be one min.
What is the process for safeguarding Newton’s Method? Explain.
If F’‘(x) < then we will tend to a max.
If F’‘(x) = 0 then we will divide by zero.
If the above conditions are true, then use deltax = -F’(k)
To ensure the next step, s = alphadeltax is within the interval.
If deltax < 0, alpha = min{1, (a-xk)/deltax}
If deltax > 0, alpha = min{1, (b-xk)/deltax}
What is the process for safeguarding the Secant Method? Explain.
Use a bracketed interval. This ensures a local max is not found.
State a Taylor Expansion for two variables
F(x+dx, y+dy) = F+p^tg+0.5p^tHp
where H is the hessian
g = gradient matrix = [Fx Fy]^t
p = step vector = [deltax, deltay]^t
State the multivariate optimality conditions
For F(x+dx, y+dy) = F+p^tg+0.5p^tHp
H is positive definite
p^t*g<0
State the three Wolfe Conditions and what they mean.
p^tgk<= -eta0mag(p)*mag(gk)
This is stronger than the p^t*g<0 condition as it ensures a minimum angle of from the contour is achieved
F(xk+spk) - F(xk) <=eta1sp^t*gk
This ensures the step size is not too big
{F(xk+spk) - F(xk)}>=(1-eta2)sp^t*gk
This ensures the step size isn’t too small.
What is an Augmented Lagrangian?
It is a penalty function which, using approximation methods, approximates the first-order constrained optimality criteria by using Lagrange multipliers. It then adds a small penalty to correct for the error in the approximation of the Lagrange multipliers.
How do projected gradient methods work?
By removing the component of the gradient that is perpendicular to the constraint, however, they need the gradient of the constraint.
What are interior point methods good for?
Inequality problems, as they can explore the feasible are well.
What are projected gradient methods good for?
Constraint problems with linear constraints
What are the principles in the derivation of SQP?
Perform a TE to approximate the objective function to obtain a quadratic model, then include an approximated 2nd derivative of the constraints within the Hessian. This improves stability and convergence. So even though the constraint approximation is still linear, the hessian includes the second derivative, so it can handle non-linear constraints better.
When should you use SQP?
When you’ve got non-linear constraints and you already have a design which is close to the minimum. SQP is expensive when far from the min, so if you’re establishing a first design, it might not be the best method.
Describe the finite difference method and when to use it with regard to calculation of derivatives
The finite difference method calculated the function at two locations:
For forward difference, its at x and x+del_x
For central difference it’s x+del_x and x-del_x
It then subtracts the first from the second and divides by the interval between the first and second.
FD is FO accurate, CD is SO accurate. Good for use with 50 or so variables but calcs scale with number of desvar so not appropriate for 100 +, can get v expensive. Also suffers from precision error of pc, and too small a change in x will blow the solution up.
What is the principle behind complex variable derivative calculation?
By implementing an imaginary perturbation, a Taylor Expansion eliminations the subtraction that occurs in FD methods. This eliminates the cancellation error that occurs when delta approaches zero, so we get a more accurate calculation. It’s slightly more expensive but has increased precision.
