Solution to the large problem is attempted iteratively through repeated solutions to the modest subproblems.
This generator allows repeated solutions with iterated meshes so that it can be employed in a first-order optimization strategy exploiting its faster convergence.
Forming the Jacobian can be prohibitively expensive because it requires repeated solutions of the forward and adjoint time-dependent parabolic partial differential equations corresponding to multiple sources and receivers.
Due to this, and following the literature (Zitzler et al. 2003), this set was formed by all sets P F known obtained with different algorithms by removing dominated and repeated solutions.
The main advantage of the proposed approach is that if repeated solutions are needed (such as in Monte Carlo technique or in optimization problems requiring multiple solutions for a set of perturbed surfaces), computational resources considerably decrease because a single mesh is used and the global matrix is formed only once.
To calculate these indicators, generally three sets of solutions are used: 1) P F approx : formed by non-dominated solutions returned by one execution of the algorithm; 2) P F known : combination of all P F approx obtained through different executions of an algorithm, removing dominated and repeated solutions; 3) P F true : represents the Optimal Pareto Front to the problem.
Two different SBD versions are then presented and demonstrated on a complex industrial problem, namely the optimal shape redesign of a ship under real-world geometrical and functional constraints, whose evaluation during the optimization process involves repeated solutions of the Reynolds Averaged Navier Stokes equations.
For one thing, it avoids revisiting and repeated solution evaluation.
In an Appendix a numerical method is outlined for the repeated solution of large sparse sets of linear equations.
Solution space of ModEnPBT is compact with no repeated solution vectors, therefore it possesses relatively higher searching efficiency compared with EnPBT algorithm.
A robust algorithm is presented for the repeated solution of the design equations and the constrained optimisation of the solids residence time.
