Point configurations that are asymmetric yet balanced

A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to show that every sufficiently symmetrical configuration is balanced, but the converse is far from obvious. In 1957 Leech completely classified the balanced configurations in R^3, and his classification is equivalent to the converse for R^3. In this paper we disprove the converse in high dimensions. We construct several counterexamples, including one with trivial symmetry group.Comment: 10 page

On the use of a truly–mixed formulation in topology optimization with global stress–constraints

The work refers to the field of topology optimization for bidimensional structures and addresses the case in which global stress–constraints are considered to improve final designs. Most of the previous research tackles this topic relying on classical displacement–based finite elements where stresses are recovered via post–processing techniques. The work conversely investigates the use of a truly–mixed formulation where stresses are independent variables of the problem while displacements play the secondary role of Lagrangian multiplier. The implemented discretization is based on a composite triangular element whose features may be advantageously exploited in stress–constrained topology optimization. The discretization is checkerboard–free and allows to tackle topology optimization with element–based constraints without introducing any additional filtering technique. The high accuracy in the evaluation of the average stresses is expected to improve the efficiency of the numerical procedure, especially in the case of a single global constraint that has to govern the whole domain. The adopted discretization also passes the robustness condition even in the case of incompressible materials and this allows to menage strength constraints also for rubber–like components. Basing on these ideas, numerical investigations are carried out to test preliminary applications of the truly–mixed technique coupled with topology optimization and global stress–constraints. To handle the well–known singularity problem, that affects the constraints imposition, an alternative scheme is herein adopted instead of a classical "–relaxation. An example where a homogenous stress distribution is expected is firstly tested, having the aim of pointing out the main features of the proposed procedure. Afterwards, numerical simulations address a classical L–shaped specimen, pointing out pros and cons of the approach

A simplified approach to the topology optimization of structures in case of unilateral material/supports

A simplified method to cope with the topology optimization of truss–like structures in case of unilateral behavior of material or supports is presented. The conventional formulation for volume–constrained compliance minimization is enriched with a set of stress constraints that enforce a suitable version of the Drucker–Prager strength criterion in order to prevent the arising of tensile (or compressive) members in the whole domain or within limited regions in the vicinity of the supports. The adopted numerical framework combines an ad hoc selection strategy along with the use of aggregation techniques that succeed in driving the energy–based minimization towards feasible designs through the enforcement of a limited number of stress constraints. Numerical simulations assess the proposed optimization framework in comparison with methods that are based on a full non–linear modeling of unilateral material/supports. An extension to the safety analysis of structures made of no–tension material is also highlighted

Shape optimization of microstructural designs subject to local stress constraints within an XFEM-level set framework

The present paper investigates the tailoring of bimaterial microstructures minimizing their local stress field exploiting shape optimization. The problem formulation relies on the extended finite element method (XFEM) combined with a level set representation of the geometry, to deal with complex microstructures and handle large shape modifications while working on fixed meshes. The homogenization theory, allowing extracting the behavior of periodic materials built from the repetition of a representative volume element (RVE), is applied to impose macroscopic strain fields and periodic boundary conditions to the RVE. Classical numerical homogenization techniques are adapted to the selected XFEM-level set framework. Following previous works on analytical sensitivity analysis [31], the scope of the developed approach is extended to tackle the problem of stress objective or constraint functions. Finally, the method is illustrated by revisiting 2D classical shape optimization examples: finding the optimal shapes of single or multiple inclusions in a microstructure while minimizing its local stress field.Peer reviewe

3D Shape Optimization with X-FEM and a Level Set Constructive Geometry Approach

This paper extends previous work on structural optimization with the eXtended Finite Element Method (X-FEM) and the Level Set description of the geometry. The proposed method takes advantage of fixed mesh approach by using an X-FEM structural analysis method and from the geometrical shape representation of the Level Set description. In order to allow the optimization of complex geometries represented with a Level Set description, we apply here a Constructive Solid Geometry (CSG) approach with the Level Set geometrical representation. Hence, this extension allows to optimize any boundary of the structure that is defined with a coumpound Level Set. Design variables are the parameters of basic geometric primitives which are described with a Level Set representation and/or the control points of the NURBS curves that act as the definition of an advanced Level Set primitive. The number of design variables of this formulation remains small whereas global (i.e. compliance or eigenfrequency) and local constraints (i.e. stresses) can be considered. Our results illustrate that fixed grid optimization with X-FEM coupled to a Level Set geometrical description is a promising technique to achieve structural shape optimization

Compressor and Turbine Blade Design by Optimization

Compressor and turbine blade design involves thermodynamical, aerodynamical and mechanical aspects, resulting in an important number of iterations. Inverse methods and optimization procedures help the designer in this long and eventually frustrating process. In this paper an optimization procedure is presented which solves two types of two-dimensional or quasi-three-dimensional problems: the inverse problem, for which a target velocity distribution is imposed, and a more global problem, in which the aerodynamic load is maximized

Analytical sensitivity analysis using the extended finite element method in shape optimization of bimaterial structures

peer reviewedThe present work investigates the shape optimization of bimaterial structures. The problem is formulated using a level set description of the geometry and the extended finite element method (XFEM) to enable an easy treatment of complex geometries. A key issue comes from the sensitivity analysis of the structural responses with respect to the design parameters ruling the boundaries. Even if the approach does not imply any mesh modification, the study shows that shape modifications lead to difficulties when the perturbation of the level sets modifies the set of extended finite elements. To circumvent the problem, an analytical sensitivity analysis of the structural system is developed. Differences between the sensitivity analysis using FEM or XFEM are put in evidence. To conduct the sensitivity analysis, an efficient approach to evaluate the so-called velocity field is developed within the XFEM domain. The proposed approach determines a continuous velocity field in a boundary layer around the zero level set using a local finite element approximation. The analytical sensitivity analysis is validated against the finite differences and a semi- analytical approach. Finally our shape optimization tool for bimaterial structures is illustrated by revisiting the classical problem of the shape of soft and stiff inclusions in plates

Note on the minimum length scale and its defining parameters. Analytical relationships for Topology Optimization based on uniform manufacturing uncertainties

The robust topology optimization formulation that introduces the eroded and dilated versions of the design has gained increasing popularity in recent years, mainly because of its ability to produce designs satisfying a minimum length scale. Despite its success in various topology optimization fields, the robust formulation presents some drawbacks. This paper addresses one in particular, which concerns the imposition of the minimum length scale. In the density framework, the minimum size of the solid and void phases must be imposed implicitly through the parameters that define the density filter and the smoothed Heaviside projection. Finding these parameters can be time consuming and cumbersome, hindering a general code implementation of the robust formulation. Motivated by this issue, in this article we provide analytical expressions that explicitly relate the minimum length scale and the parameters that define it. The expressions are validated on a density-based framework. To facilitate the reproduction of results, MATLAB codes are provided. As a side finding, this paper shows that to obtain simultaneous control over the minimum size of the solid and void phases, it is necessary to involve the 3 fields (eroded, intermediate and dilated) in the topology optimization problem. Therefore, for the compliance minimization problem subject to a volume restriction, the intermediate and dilated designs can be excluded from the objective function, but the volume restriction has to be applied to the dilated design in order to involve all 3 designs in the formulation

Microstructural design using stress-based topology optimization

New additive manufacturing techniques break the limitations encountered for years when producing components descending from topology optimization. Classical design procedures focus on macro-structural optimization to sustain given loads but today innovative manufacturing processes allow considering structures exhibiting tailored microstructures, i.e. the well known microstructural design. The practical applications of structures including material design is mainly motivated by the greater performances that can be achieved compared to classical solutions. Microstructural design has been shown a great interest as attested by recent works[1, 2]. However, stress-based topology optimization has not yet been extensively exploited when addressing microstructural design using numerical homogenization though stress constraints is an important feature and have gained in interest in the field of topology optimization as pointed out by [3].This contribution investigates the problem of material design enforcing stress constraints within periodic microstructures by considering a representative volume element (RVE) subject to prescribed strain fields. The SIMP approach is adopted as material interpolation law while the optimization problems are solved using a sequential convex programming approach. In particular the well known method of moving asymptotes (MMA) is considered. Numerical homogenization is used to assess the effective elastic properties of the microstructures. The Von Mises stress criterion is used to impose the constraints on the stress level. This work discusses the formulation of a well-posed design problem as well as some numerical issues encountered. The developed solution procedure is first validated by comparison against analytical results, e.g. the single inclusion of Vigdergauz microstructure. Finally the optimized layouts are fabricated using a multimaterial inkjet polymer printing (Connex by Stratasys) to test the actual performances of optimized designs