Actually Mr. Rien is not correct, Dirac equation has no relevance, for problems of Metallurgy we are not relativistic
. But ok I want to show you by a concrete example a mix of three things. Computers, numerical simulation and hard theorems in a simple problem of designing a composite material
So here is the problem. I am given the following:
(a) All sorts of materials of varying density, lead, copper, Kothari chaap paan ka masala, etc, and I am also told that the densities of all the supplied materials lie between two fixed given
numbers a,b that is 0<a< density <b.
(b) Of course I am given the shape of the object
, oh I am a retard, so I may want to think of the shape as a lowly washer, some of you aim for the moon, so think of an an aircraft wing.
(c) I am also prescribed the mass M
of the object.
So (a), (b) and (c) are the constraints on me. All natural engineering constraints. Now under these constraints I have to combine my materials to build the object so it has the maximum rigidity
This means the following. I build the object taking Kothari chaap paan ka masala, lead, tin, and take a huge effing hammer and hit it. I hear a fundamental tone like a gong. I record the lowest frequency in the vibration. I now change the material percentages still keeping my constraints, now hit the guy, and my aim is such that I want at the end of the day to produce that body of course satisfying my constraints (a), (b) and (c) that has the lowest possible fundamental vibrational frequency
. This is in engineering parlance called maximizing rigidity. So natural questions arise.
1. Can I even build such a minimizer or optimal composite satisfying the constraints (a) to (c) and maximizing rigidity? Answer is yes
and the math. theorem ( Theorem 13, paper 1)says you just need the heaviest density material and the lightest density material, you can and should throw out all the material in intermediate density. Also all the lighter density material ( black zone in the pictures in paper 1) must be placed at the boundary or edges of the object. See the linked paper 1 where this quite amazing fact is rigorously proved as a mathematical theorem.
2. OK now that we know that, that is there is a solution to the optimization problem, we can proceed further. Let us say we want to design a washer, do we arrange the two materials ( lowest and highest density materials) symmetrically respecting the symmetry of the washer. Answer No
. This was first seen by computer simulation using Fortran and the thought was that perhaps there was some error in the code. But soon after a rigorous mathematical proof as a theorem ( theorem 6, paper 1) showed that indeed that is true, see paper 1. See the simulations and pictures in the linked paper for a washer called an annulus. So symmetry breaking occurs in this problem, materials have to be arranged non-symetrically to construct symmetric objects so constraints (a) to (c) are satisfied and yet maximal rigidity is established.
3. Now what about the fine structure of the junction between the two materials, cracks, dislocations etc. This too has been studied and the fine structure depends on the dimension. For 2-dimension objects like plates or layered composites the junction ( free boundary as you cannot control the interface, the optimizer will select it for you) can be a crack that looks like a cross etc. and two other configurations. Unfortunately the authors of that paper, paper 2 on interfaces never drew pictures in their paper, but it is clear they have completely classified the possible interface in all dimensions, 2 or 3 in particular.
The only reason for my rant is to point out that even for simple questions things are pretty complicated and also to show you what I really also had in mind regarding my post above.
Some of the code used in the simulations is discussed in Paper 1. But the simulation is just a tool to see what theorems are rigorously proveable. For example symmetry breaking was a total surprise for this problem.Paper 1Paper 2