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

A review of new fundamental principles in exact topology optimization

Abstract After reviewing briefly the history of exact topology optimization of structures, a number of fundamental principles for deriving new optimal structural layouts will be presented. These also throw some light on general properties of optimal topologies

On the prediction of topology and local properties for optimal trussed structures

A new formulation is presented for mathematical modelling to predict the distribution of material, material properties, and topology for the optimal design of trussed structures. The design problem is cast in a form to minimize a measure of generalized compliance , which is calculated as a sum over the structure of weighted displacement. Member stiffnesses appear as design variables and, starting with a given ground structure, the solution predicts the optimal layout and distribution of stiffness. The isoperimetric constraint in the reformulated problem measures total cost in generalized form , based on independently specified unit relative cost factors for each truss element. One or another form of optimal design is generated via a process where designated elements in the unit relative cost field are adjusted systematically at each cycle. The generalized cost feature provides as well for the introduction of certain technical constraints into the design problem, e.g. the facility to design around obstacles. Results for each cycle of an algorithm for computational treatment are identified as the solution to a properly posed optimization problem. Computational procedures are demonstrated by the prediction of optimal designs for a variety of truss problems in 2D.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46074/1/158_2005_Article_BF01197558.pd

liteITD a MATLAB Graphical User Interface (GUI) program for topology design of continuum structures

Over the past few decades, topology optimization has emerged as a powerful and useful tool for the design of structures, also exploiting the ever growing computational speed and power. The design process has also been affected by computers which changed the concept of form into the concept of formation and the emergence of digital design. Topology optimization can modify existing designs, incorporate explicit features into a design and generate completely new designs. This paper will show how topology optimization can be used as a digital tool. The liteITD (lite version of Isolines Topology Design) software package will be described with the purpose of providing a tool for topology design. The liteITD program solves the topology optimization of two-dimensional continuum structures using von Mises stress isolines under single or multiple loading conditions, with different material properties in tension and compression, and multiple materials. The liteITD program is fully implemented in the MATrix LABoratory (MATLAB) software environment of MathWorks under Windows operating system. GUIDE (Graphical User Interface Development Environment) was used to create a friendly Graphical User Interface (GUI). The usage of this application is directed to students mainly (educational purposes), although also to designers and engineers with experience. The liteITD program can be downloaded and used for free from the website: http://www.upct.es/goe/software/liteITD.php

Topology synthesis of multi-input-multi-output compliant mechanisms.

A generalized formulation to design Multi-Input-Multi-Output (MIMO) compliant mechanisms is presented in this work. This formulation also covers the simplified cases of the design of Multi-Input and Multi-Output compliant mechanisms, more commonly used in the literature. A Sequential Element Rejection and Admission (SERA) method is used to obtain the optimum design that converts one or more input works into one or more output displacements in predefined directions. The SERA procedure allows material to flow between two different material models: 'real' and 'virtual'. The method works with two separate criteria for the rejection and admission of elements to efficiently achieve the optimum design. Examples of Multi-Input, Multi-Output and MIMO compliant mechanisms are presented to demonstrate the validity of the proposed procedure to design complex complaint mechanisms

On transmissible load formulations in topology optimization

Transmissible loads are external loads defined by their line of action, with actual points of load application chosen as part of the topology optimization process. Although for problems where the optimal structure is a funicular, transmissible loads can be viewed as surface loads, in other cases such loads are free to be applied to internal parts of the structure. There are two main transmissible load formulations described in the literature: a rigid bar (constrained displacement) formulation or, less commonly, a migrating load (equilibrium) formulation. Here, we employ a simple Mohr’s circle analysis to show that the rigid bar formulation will only produce correct structural forms in certain specific circumstances. Numerical examples are used to demonstrate (and explain) the incorrect topologies produced when the rigid bar formulation is applied in other situations. A new analytical solution is also presented for a uniformly loaded cantilever structure. Finally, we invoke duality principles to elucidate the source of the discrepancy between the two formulations, considering both discrete truss and continuum topology optimization formulations

Dimensional accuracy of Electron Beam Melting (EBM) additive manufacture with regard to weight optimized truss structures

The Electron Beam (EBM) additive manufacturing process is well suited to fabricating complex structural designs in Ti–6Al–4V because of the design freedoms it offers combined with strong and consistent material properties. However it has been observed that complications may arise when manufacturing truss-like structures (such as those produced via structural topology optimization) in the form of undersized features on the finished part. The issue appears to affect truss members that are not aligned with the vertical build direction, with an apparent lack of material on the negative surfaces. This effect appears to worsen with a greater angle between the truss member and the build direction, even with the use of support structures. This investigation has characterized and measured the dimensional errors that result from this issue through 3D scanning techniques. Process modifications have then been made which result in significant improvements in dimensional accuracy. This investigation highlights the importance of heat management at features with negative surfaces to yield parts that are dimensionally accurate without introducing excessive internal melt defects in the form of voids and porosity