Xu Guo's blog

     Solving topology optimization problem is very computationally demanding especially when high resolution results are sought for. In the present work, a problem-independent machine learning (PIML) technique is proposed to reduce the computational time associated with finite element analysis (FEA) which constitutes the main bottleneck of the solution process.

    Structural topology optimization, which aims at placing available material within a prescribed design domain appropriately in order to achieve optimized structural performances, has received considerable research attention since the pioneering work of Bendsoe and Kikuchi. Many approaches have been proposed for structural topology optimization and it now has been extended to a wide range of physical disciplines such as acoustics, electromagnetics and optics.

   Recently, we have developed a so-called higher order Cauchy-Born rule based
   constitutive model for the analysis of large deformation of CNTs [1]. It is
   a generalized quasi-continuum model for nano-structures.

Dear Prof.Barber,

     This is the proof of the theorem I mentioned before. This proof is recorded in a book

  (titled as variational principle in elasticity and its applications)  written by Prof.Hai Chang Hu, one of the inventors of the well-known Hu-Washizu generalized variational principle. It is written in Chinese. I translated it into English. Of course, all errors and misunderstandings are due to me. I think this proof is elegant and accessible to most of the students with engineering background.

  Hope it helps

The essential assumption of Cauchy Stress
is the existence of the limit  (\Delta f)/ (\Delta A) as  \Delta A-->0. 
This means that a infinitesimal small area can only sustain infinitesimal
small surface force. Therefore in the framework of conventional continuum
mechanics, we can only talk about the "force density (stress)" at a point.
The expression int_A (f.n)dA has no physical meaning. Only int_A (stress.n)dA
can give the total force exerting on a closed surface A.