Second-order cone programming formulations for a class of problems in structural optimization

This paper provides efficient and easy to implement formulations for two problems in structural optimization as second-order cone programming (SOCP) problems based on the minimum compliance method and derived using the principle of complementary energy. In truss optimization both single and multiple loads (where we optimize the worst-case compliance) are considered. By using a heuristic which is based on the SOCP duality we can consider a simple ground structure and add only the members which improve the compliance of the structure. It is also shown that thickness optimization is a problem similar to truss optimization. Examples are given to illustrate the method developed in this pape

A formulation of thickness optimization for plane stress

Thickness optimization can be considered as a case of sizing optimization for plane structures. It canalso be used as an intermediate step for topology problems, i.e. we can eliminate the parts where thethickness tends to be zero. This paper is concerned with the case of plane stress structures coupled withthe finite element method. The aim is to present a formulation of this problem as a case of second-ordercone programming which is a standard form of mathematical programming. The advantage is that,on the one hand, all that the engineer has to do is to compute elemental data, and on the other, largediscretized structures can be optimized accurately due to the efficiency of the proposed formulation.Different types of elements regarding the thickness field are considered

A compliance based design problem of structures under multiple load cases

There are two popular methods concerning the optimal design of structures. The first is the minimization of the volume of the structure under stress constraints. The second is the minimization of the compliance for a given volume. For multiple load cases an arising issue is which energy quantity should be the objective function. Regarding the sizing optimization of trusses, Rozvany proved that the solution of the established compliance based problems leads to results which are awkward and not equivalent to the solutions of minimization of the volume under stress constraints, unlike under single loading 1. In this paper, we introduce the "envelope strain energy" problem where we minimize the volume integral of the worst case strain energy of each point of the structure. We also prove that in the case of sizing optimization of statically non-indeterminate2 trusses, this compliance method gives the same optimal design as the stress based design method.<br/

A formulation for optimizing the young’s modulus of a structure

Young’s modulus optimization can be used as an intermediate step for topology problems. This can be achieved by eliminating the parts of a structure where the Young’s modulus tends to be zero. This paper is concerned with the case of the compliance method coupled with the finite element method.We present a formulation to turn this problem in a standard form of mathematical programming - in our case it is the second-order cone programming. The advantage here is that on the one hand all that theengineer has to do is to compute elemental data, and on the other, large discretized structures can be optimized due to the efficiency of the proposed formulation by the use of standard solvers

Lower bound limit analysis of cohesive-frictional materials using second-order cone programming

The formulation of limit analysis by means of the finite element method leads to an optimization problem with a large number of variables and constraints. Here we present a method for obtaining strict lower bound solutions using second-order cone programming (SOCP), for which efficient primaldual interior-point algorithms have recently been developed. Following a review of previous work, we provide a brief introduction to SOCP and describe how lower bound limit analysis can be formulated in this way. Some methods for exploiting the data structure of the problem are also described, including an efficient strategy for detecting and removing linearly dependent constraints at the assembly stage. The benefits of employing SOCP are then illustrated with numerical examples. Through the use of an effective algorithm/software, very large optimization problems with up to 700 000 variables are solved in minutes on a desktop machine. The numerical examples concern plane strain conditions and the Mohr-Coulomb criterion, however we show that SOCP can also be applied to any other problem of lower bound limit analysis involving a yield function with a conic quadratic form (notable examples being the Drucker-Prager criterion in 2D or 3D, and Nielsen's criterion for plate. Copyright © 2005 John Wiley and Sons, Ltd

A novel formulation of upper bound limit analysis as a second-order cone programming problem

Here we present a dual kinematic formulation of limit analysis as a second-order cone programming problem, employing linear strain finite elements with a continuous displacement field. The result is a powerful tool for obtaining rigorous and tight upper bounds for very large discretized structures

Upper bound limit analysis using simplex strain elements and second-order cone programming

In geomechanics, limit analysis provides a useful method for assessing the capacity of structures such as footings and retaining walls, and the stability of slopes and excavations. This paper presents a finite element implementation of the kinematic (or upper bound) theorem that is novel in two main respects. First, it is shown that conventional linear strain elements (6-node triangle, 10-node tetrahedron) are suitable for obtaining strict upper bounds even in the case of cohesive-frictional materials, provided that the element sides are straight (or the faces planar) such that the strain field varies as a simplex. This is important because until now, the only way to obtain rigorous upper bounds has been to use constant strain elements combined with a discontinuous displacement field. It is well known (and confirmed here) that the accuracy of the latter approach is highly dependent on the alignment of the discontinuities, such that it can perform poorly if an unstructured mesh is employed. Second, the optimization of the displacement field is formulated as a standard second-order cone programming (SOCP) problem. Using a state-of-the-art SOCP code developed by researchers in mathematical programming, very large example problems are solved with outstanding speed. The examples concern plane strain and the Mohr-Coulomb criterion, but the same approach can be used in 3D with the Drucker-Prager criterion, and can readily be extended to other yield criteria having a similar conic quadratic form

Comments on ‘Rigid plastic model of incremental sheet deformation using second-order cone programming’

This is a discussion for the paper'Rigid plastic model of incremental sheetdeformation using second-order cone programming’ by A. Raithatha and S. R. Duncan, International Journal for Numerical Methods in Engineering 2009; 78:955–97