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

On the performance of non-conforming finite elements for the upper bound limit analysis of plates

This is a preprint version of an article accepted for publication in International Journal for Numerical Methods in Engineering Copyright © 2012 John Wiley & Sons, Ltd.International audienceIn this paper, the upper bound limit analysis of thin plates in bending is addressed using various types of triangular finite elements for the generation of velocity fields and second order cone programming (SOCP) for the minimization problem. Three different C1-discontinuous finite elements are considered : the quadratic 6 node Lagrange triangle (T6), an enhanced T6 element with a cubic bubble function at centroid (T6b) and the cubic Hermite triangle (H3). Through numerical examples involving Johansen and von Mises yield criteria, it is shown that cubic elements (H3) give far better results in terms of convergence rate and precision than fully conforming elements found in the literature, especially for problems involving clamped boundaries

A cell-based smoothed finite element method for kinematic limit analysis

This paper presents a new numerical procedure for kinematic limit analysis problems, which incorporates the cell-based smoothed finite element method with second-order cone programming. The application of a strain smoothing technique to the standard displacement finite element both rules out volumetric locking and also results in an efficient method that can provide accurate solutions with minimal computational effort. The non-smooth optimization problem is formulated as a problem of minimizing a sum of Euclidean norms, ensuring that the resulting optimization problem can be solved by an efficient second-order cone programming algorithm. Plane stress and plane strain problems governed by the von Mises criterion are considered, but extensions to problems with other yield criteria having a similar conic quadratic form or 3D problems can be envisaged

Multisurface plasticity for Cosserat materials: plate element implementation and validation

International audienceThe macroscopic behaviour of materials is affected by their inner micro-structure. Elementary considerations based on the arrangement, and the physical and mechanical features of the micro-structure may lead to the formulation of elastoplastic constitutive laws, involving hardening/softening mechanisms and non-associative properties. In order to model the non-linear behaviour of micro-structured materials, the classical theory of time-independent multisurface plasticity is herein extended to Cosserat continua. The account for plastic relative strains and curvatures is made by means of a robust quadratic-convergent projection algorithm, specifically formulated for non-associative and hardening/softening plasticity. Some important limitations of the classical implementation of the algorithm for multisurface plasticity prevent its application for any plastic surfaces and loading conditions. These limitations are addressed in this paper, and a robust solution strategy based on the Singular Value Decomposition technique is proposed. The projection algorithm is then implemented into a finite element formulation for Cosserat continua. A specific finite element is considered, developed for micropolar plates. The element is validated through illustrative examples and applications, showing able performance

Homogenized rigid-plastic model for masonry walls subjected to impact

A simple rigid-plastic homogenization model for the analysis of masonry structures subjected to out-of-plane impact loads is presented. The objective is to propose a model characterized by a few material parameters, numerically inexpensive and very stable. Bricks and mortar joints are assumed rigid perfectly plastic and obeying an associated flow rule. In order to take into account the effect of brickwork texture, out-of-plane anisotropic masonry failure surfaces are obtained by means of a limit analysis approach, in which the unit cell is subdivided into a fixed number of sub-domains and layers along the thickness. A polynomial representation of micro-stress tensor components is utilized inside each sub-domain, assuring both stress tensor admissibility on a regular grid of points and continuity of the stress vector at the interfaces between contiguous sub-domains. Limited strength (frictional failure with compressive cap and tension cutoff) of brick-mortar interfaces is also considered in the model, thus allowing the reproduction of elementary cell failures due to the possible insufficient resistance of the bond between units and joints. Triangular Kirchhoff-Love elements with linear interpolation of the displacement field and constant moment within each element are used at a structural level. In this framework, a simple quadratic programming problem is obtained to analyze entire walls subjected to impacts. In order to test the capabilities of the approach proposed, two examples of technical interest are discussed, namely a running bond masonry wall constrained at three edges and subjected to a point impact load and a masonry square plate constrained at four edges and subjected to a distributed dynamic pressure simulating an air-blast. Only for the first example, numerical and experimental data are available, whereas for the second example insufficient information is at disposal from the literature. Comparisons with standard elastic-plastic procedures conducted by means of commercial FE codes are also provided. Despite the obvious approximations and limitations connected to the utilization of a rigid-plastic model for masonry, the approach proposed seems able to provide results in agreement with alternative expensive numerical elasto-plastic approaches, but requiring only negligible processing time. Therefore, the proposed simple tool can be used (in addition to more sophisticated but expensive non-linear procedures) by practitioners to have a fast estimation of masonry behavior subjected to impact

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/

Remarks on some properties of conic yield restrictions in limit analysis

A major difficulty when applying the kinematic theorem in limit analysis is the derivation of expressions ofthe dissipation functions and the set of plastically admissible strains. At present, no standard methodologyexists. Here, it is shown that they can be readily obtained, provided that the yield restriction can berewritten as an intersection of cones, and that the expression defining the dual cones is available. This isalways possible for the case of self-dual cones and some other classes, and covers many of the well-knowncriteria. Therefore, a difficult obstacle with respect to the use of the kinematic theorem in conjunctionwith any numerical method can be overcome. The methodology is illustrated by giving the expressions ofthe dissipation functions for various conic yield restrictions. A special emphasis is given on upper boundfinite element limit analysis. Taking advantage of duality in conic programming, we can obtain the dualproblem, where knowledge of the dual cone is not necessary. Therefore, this formulation is feasible forany cone. Finally, it is interesting that the form of the dual problem, for varying yield strength within thefinite element, differs from that presented in other papers