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
How can you save material on a floor system?
Avoid long spans between columns
* Reduce span L, reduce M = wL2/8; = 5wL4/384EI (for simply
supported beam)
- Vary beam sizes across the floorplate
- i.e. avoid standardisation to maximise utilisation of material
- Consider using non-prismatic sections
- Although convenient, prismatic sections (e.g. standard I-beam)
may only be fully utilised at one location - Consider changing the layout of beams in the floorplate
- Which of the following is NOT a feature of linear programming (LP)?
a) It can solve problems involving many thousands of variables and constraints
b) It can solve many problems more quickly than meta-heuristic methods
c) It can solve both convex and non-convex problems
d) It can solve problems involving both equality and inequality constraint
c) It can solve both convex and non-convex problems
- To convert the linear programming constraint 4x - 2y + 7z ≥ 12 into standard form requires:
a) No action – the constraint is already in standard form
b) The constraint to be rewritten as a “≤” constraint, with terms on both sides of the expression then multiplied by -1 to convert it
c) The constraint to be rewritten as an “=” constraint, with terms on both sides of the expression then multiplied by -1 to convert it
d) The constraint to be rewritten as “≤” and “≥” constraints, with terms on both sides of the “≥” expression then multiplied by -1 to convert it
b) The constraint to be rewritten as a “≤” constraint, with terms on both sides of the expression then multiplied by -1 to convert it
How to calculate the volume of a half wheel truss?
Volume of rim = Length * area
Where length equal 2piRadius / 2
- Which of the following statements about the solutions obtained using the linear programming (LP) based truss
layout formulation is correct?
a) The generated structures will always be in equilibrium with the applied loads
b) A unique layout will always be obtained
c) The generated structures will always be stable
d) The stresses in the generated structures will only exceed the specified yield stress at the supports
a) The generated structures will always be in equilibrium with the applied loads
The equilibrium constraint (Bq = f) ensures that “a” is true; none of the other statements are correct
Consider a three-load case truss layout optimisation problem comprising 10 nodes and 84 bars. How many variables will be present in the plastic linear programming (LP) formulation?
Number of bars = 84
3 loads cases
1 area
Total number of variables = 84 * (3+1) = 336
Consider a four-load case truss layout optimisation problem comprising 15 nodes and 200 bars. Assuming that three of these nodes have fixed pin supports, how many equality constraints will be present in the plastic linear programming (LP) formulation? Assume that the problem is two-dimensional
Equality constraints are associated to the equilibrium relation B * q = f
Number of nodes = 15
3 nodes are fixed = -3
x and y direction = 2
4 load cases = 4
Number of equilibrium constraints = (15-3) * 2 * 4 = 96
Which of the following statements relating to the basic linear programming (LP) grillage layout optimisation formulation covered in the lectures is correct?
a) It is always assumed that the cross-sectional areas of the beams are fixed between endpoints
b) The problem variables are the internal moments, the shear forces and the cross-sectional areas of the beams
c) The depths of the beams in the grillage can vary according to the magnitude of the moment
d) The cross-sectional areas of most beams in the ground structure will normally be zero
d) The cross-sectional areas of most beams in the ground structure will normally be zero
This is because a layout optimisation problem can be considered as a size optimisation problem, though with many of the beams carrying zero moment and having zero cross-sectional area.
- Consider a grillage layout optimisation problem discretised using 4 nodes. Assuming that each node will be connected to every other node by a potential beam, how many beams (including overlapping beams) will be present in the ‘ground structure’?
6
For a fully connected ground structure containing n nodes, the total number of connections m = n (n – 1) /2 =
4 (4 - 1) / 2 = 6
