Quiz 3 Flashcards
Why are convex optimisers better than evolutionary optimisers?
Evolutionary optimisers are very slow and convex optimisers are easier and quicker
How does convex optimisation work and what is the standard form?
- Involves minimising (or maximising) convex functions over convex sets
- Standard form:
minimise f(x)
subject to
gi(x) <= 0, i = 1, …, m
hi(x) = 0, i = 1, …, p
x = set of optimisation variable
gi = inequality constraints
hi = equality constraints
What are 5 categories of convex optimisation in order for slowest to quickest?
Cone Problem (CP)
Semidefinite Program (SDP)
Second-Order cone Problem (SOCP)
Quadratic Program (QP)
Linear Program (LP)
What does a linear program function comprise of?
- a linear objective function to minimise
or maximise - problem variables (continuous)
- a series of linear constraints, either
equalities or inequalities
What makes a a problem solution status unbounded or infeasible?
Unbounded: If there is the objective function can be minimised or maximize without limit
Infeasible: if no solution satisfying all the constraints can be obtained
What are integer or mixed integer problems
Integer programming (IP) or Mixed Integer Linear Programming (MILP) can be used if respectively all or some of the variables must take on integer values
Integer problems are not strictly speaking convex and hence are more challenging to solve as a consequence
What are the steps to model a given LP problem
- Identify the problem variables
- Define the objective function
- Identify the problem constraints
- Rearrange in standard form (or in a form
that can be solved using a standard LP
tool)
What is Maxwell’s Theorem and what does it tell use about the volume of material required to carry tensile and compressive forces?
Maxwell’s theorem states that for any given truss problem, the sum of the products of the tensile forces and the corresponding lengths of the bars minus the sum of the products of the compressive forces and the corresponding lengths of the bars is equal to a constant that is related to the external forces/reactions.
It also assumes that the tensile and compressive strength are equal and if the volume of material required to carry tensile forces 𝑉+ increases the volume of material required to carry compressive forces 𝑉− must also increase.
How does the constant c related to external forces/reactions?
The sum of the products of the x coordinate and force in the x direction at each node/support + the sum of the products of the y coordinate and force in the y direction at each node/support equals c.
How do we find a minimum volume structure?
When V+ or V- equals 0.
What is the more efficient way of connecting tension sand compression elements in a system?
Orthogonally so that there is a 90 degree angle between the two components
Why are beams the least efficient structural form?
Lacks of structural depth
Prismatic sections used
For a single load case, what is the LP formulation for minimising volume?
How is this adapted more multiple loads?
min V = L^T * a
Where L^T = transpose of the length matrix
a = area matrix
such that B * q = f
B is the angle matrix
q = internal force matrix
f = applied nodal forces
-Stress * a <= q <= Stress *a
The b, q and f matrices are put in a matrix that is n x n where n is the number of nodes
What are some key points of numerical layout optimisations?
- Size optimization formulation for ground structures with bars that can have zero cross-sectional area
- Statically determinate solution for single load case problems, minimum volume and stiffest elastic layout
- Assumes truss bars yield plastically for multiple load cases
- Can be applied to both 2D and 3D problems
How can numerical layout optimisation be made more practical@
a) First rationalise solution via geometry optimization (moving/merging joints to improve the solution)
* Add nodal positions as optimisation variables
* Resulting problem is non-linear, but relatively small-scale as it only involves a
small subset of the original nodes and bars
b) Then simplify solution either manually or automatically (e.g. reducing numbers of joints or members)
* Minimize number of members or joints subject to given volume increase
* Smooth ‘Heaviside’ representation of 0-1 (off-on) variables
* Advantage: short run time
c) Use enriched formulations to account for e.g. local buckling and/or global instability