EFFICIENT SIZING OF STRUCTURES UNDER STRESS CONSTRAINTS

<p>Optimisation algorithms used to automatically size structural members commonly involve stress constraints to avoid material failure. Therefore the cost of optimisation grows rapidly as the number of structural members is increased due to the corresponding increase in the number of constraints. In this work, an efficient method for large scale stress constrained structural sizing optimisation problems is proposed. A convex, separable, and scalable approximation for stress constraints which splits the approximation into a local fully stressed term and a global load distribution term is introduced. Predictor-corrector interior point method, an excellent option for large scale optimization problem, is chosen to solve the approximate subproblems. The core idea in this work is to achieve computational efficiency in the optimization procedure by avoiding the construction and the solution of the Schur complement system generated by the interior point method. Avoiding the Schur complement, and explicit sensitivity analysis, eliminates the high cost of solving stress constrained problems within the interior point optimisation. This is achieved using the preconditioned conjugate gradient method, and a new preconditioner is proposed specifically for stress constrained problems. The proposed method is applied to a number of beam sizing problems. Numerical results show that optimal complexity is achieved by the algorithm, the computational cost being linearly proportional to the number of sizing variables.</p

Adapting the interior point method for the solution of linear programs on high performance computers

In this paper we describe a unified algorithmic framework for the interior point method (IPM) of solving Linear Programs (LPs) which allows us to adapt it over a range of high performance computer architectures. We set out the reasons as to why IPM makes better use of high performance computer architecture than the sparse simplex method. In the inner iteration of the IPM a search direction is computed using Newton or higher order methods. Computationally this involves solving a sparse symmetric positive definite (SSPD) system of equations. The choice of direct and indirect methods for the solution of this system and the design of data structures to take advantage of coarse grain parallel and massively parallel computer architectures are considered in detail. Finally, we present experimental results of solving NETLIB test problems on examples of these architectures and put forward arguments as to why integration of the system within sparse simplex is beneficial

Gradient-based estimation of Manning's friction coefficient from noisy data

We study the numerical recovery of Manning's roughness coefficient for the diffusive wave approximation of the shallow water equation. We describe a conjugate gradient method for the numerical inversion. Numerical results for one-dimensional model are presented to illustrate the feasibility of the approach. Also we provide a proof of the differentiability of the weak form with respect to the coefficient as well as the continuity and boundedness of the linearized operator under reasonable assumptions using the maximal parabolic regularity theory.Comment: 19 pages, 3 figure

A specialized interior-point algorithm for huge minimum convex cost flows in bipartite networks

Research Report UPC-DEIO DR 2018-01. November 2018The computation of the Newton direction is the most time consuming step of interior-point methods. This direction was efficiently computed by a combination of Cholesky factorizations and conjugate gradients in a specialized interior-point method for block-angular structured problems. In this work we apply this algorithmic approach to solve very large instances of minimum cost flows problems in bipartite networks, for convex objective functions with diagonal Hessians (i.e., either linear, quadratic or separable nonlinear objectives). After analyzing the theoretical properties of the interior-point method for this kind of problems, we provide extensive computational experiments with linear and quadratic instances of up to one billion arcs and 200 and five million nodes in each subset of the node partition. For linear and quadratic instances our approach is compared with the barriers algorithms of CPLEX (both standard path-following and homogeneous-self-dual); for linear instances it is also compared with the different algorithms of the state-of-the-art network flow solver LEMON (namely: network simplex, capacity scaling, cost scaling and cycle canceling). The specialized interior-point approach significantly outperformed the other approaches in most of the linear and quadratic transportation instances tested. In particular, it always provided a solution within the time limit and it never exhausted the 192 Gigabytes of memory of the server used for the runs. For assignment problems the network algorithms in LEMON were the most efficient option.Peer ReviewedPreprin