There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The MATLAB toolbox YALMIP is introduced. It is described how YALMIP can be used to model and solve optimization problems typically occurring in systems and control theory. In this paper, free MATLAB toolbox YALMIP, developed initially to model SDPs and solve these by interfacing eternal solvers. The toolbox makes development of optimization problems in general, and control oriented SDP problems in particular, extremely simple. In fact, learning 3 YALMIP commands is enough for most users to model and solve the optimization problems

YALMIP : a toolbox for modeling and optimization in MATLAB

Topology optimization has undergone a tremendous development since its introduction in the seminal paper by Bendsoe and Kikuchi in 1988. By now, the concept is developing in many different directions, including “density”, “level set”, “topological derivative”, “phase field”, “evolutionary” and several others. The paper gives an overview, comparison and critical review of the different approaches, their strengths, weaknesses, similarities and dissimilarities and suggests guidelines for future research.

Topology optimization approaches: A comparative review

The past few decades have seen substantial growth in Additive Manufacturing (AM) technologies. However, this growth has mainly been process-driven. The evolution of engineering design to take advantage of the possibilities afforded by AM and to manage the constraints associated with the technology has lagged behind. This paper presents the major opportunities, constraints, and economic considerations for Design for Additive Manufacturing. It explores issues related to design and redesign for direct and indirect AM production. It also highlights key industrial applications, outlines future challenges, and identifies promising directions for research and the exploitation of AM's full potential in industry.

/pdf/design-for-additive-manufacturing-trends-opportunities-22heccleea.pdf

Design for additive manufacturing: trends, opportunities, considerations, and constraints

Topology optimization is the process of determining the optimal layout of material and connectivity inside a design domain. This paper surveys topology optimization of continuum structures from the year 2000 to 2012. It focuses on new developments, improvements, and applications of finite element-based topology optimization, which include a maturation of classical methods, a broadening in the scope of the field, and the introduction of new methods for multiphysics problems. Four different types of topology optimization are reviewed: (1) density-based methods, which include the popular Solid Isotropic Material with Penalization (SIMP) technique, (2) hard-kill methods, including Evolutionary Structural Optimization (ESO), (3) boundary variation methods (level set and phase field), and (4) a new biologically inspired method based on cellular division rules. We hope that this survey will provide an update of the recent advances and novel applications of popular methods, provide exposure to lesser known, yet promising, techniques, and serve as a resource for those new to the field. The presentation of each method's focuses on new developments and novel applications.

/pdf/a-survey-of-structural-and-multidisciplinary-continuum-3lbeegkisv.pdf

A survey of structural and multidisciplinary continuum topology optimization: post 2000

We introduce a computer program PENNON for the solution of problems of convex Nonlinear and Semidefinite Programming (NLP-SDP). The algorithm used in PENNON is a generalized version of the Augmented Lagrangian method, originally introduced by Ben-Tal and Zibulevsky for convex NLP problems. We present generalization of this algorithm to convex NLP-SDP problems, as implemented in PENNON and details of its implementation. The code can also solve second-order conic programming (SOCP) problems, as well as problems with a mixture of SDP, SOCP and NLP constraints. Results of extensive numerical tests and comparison with other optimization codes are presented. The test examples show that PENNON is particularly suitable for large sparse problems.

http://www.optimization-online.org/DB_FILE/2002/04/471.pdf

PENNON: A code for convex nonlinear and semidefinite programming

The design of auxetic structures is traditionally based on engineering experience, see, for instance, refs. [1,2]. The goal of this article is to demonstrate how an existing auxetic structure can be improved by structural optimization techniques and manufactured by a selective electron beam melting (SEBM) system. Auxetic materials possess negative Poisson’s ratios ν which is in contrast to almost all engineering materials (e. g. metals ν ≈ 0.3). The negative Poisson’s ratio translates into the unusual geometric effect that these materials expand when stretched and contract when compressed. This behavior is of interest for example for fastener systems and fi lters. [ 2 , 3 ] It also results in interesting mechanical behavior like increased penetration resistance for large negative Poisson’s ratios [ 4 ] and in the case of isotropy for higher shear strength relative to the Young’s modulus [ 2 ] of the structure. The auxetic behavior can always be traced back to complex mesoor microscopic geometric mechanisms, e. g. unfolding of inverted elements or rotation of star shaped rigid units relative to each other. [ 5 – 7 ] These geometries can be realized on vastly different scales from specifi cally designed polymers to reactor cores. [ 2 , 8 ]

Design of Auxetic Structures via Mathematical Optimization

An algebraic formulation is proposed for the static output feedback (SOF) problem: the Hermite stability criterion is applied on the closed-loop characteristic polynomial, resulting in a non-convex bilinear matrix inequality (BMI) optimization problem for SIMO or MISO systems. As a result, the BMI problem is formulated directly in the controller parameters, without additional Lyapunov variables. The publicly available solver PENBMI 2.0 interfaced with YALMIP 3.0 is then applied to solve benchmark examples. Implementation and numerical aspects are widely discussed.

/pdf/solving-polynomial-static-output-feedback-problems-with-371ec3lvp1.pdf

Solving polynomial static output feedback problems with PENBMI

PENLAB is an open source software package for nonlinear optimization, linear and nonlinear semidefinite optimization and any combination of these. It is written entirely in MATLAB. PENLAB is a young brother of our code PENNON \cite{pennon} and of a new implementation from NAG \cite{naglib}: it can solve the same classes of problems and uses the same algorithm. Unlike PENNON, PENLAB is open source and allows the user not only to solve problems but to modify various parts of the algorithm. As such, PENLAB is particularly suitable for teaching and research purposes and for testing new algorithmic ideas. 
In this article, after a brief presentation of the underlying algorithm, we focus on practical use of the solver, both for general problem classes and for specific practical problems.

PENLAB: A MATLAB solver for nonlinear semidefinite optimization

We present a thorough investigation of the mechanical behaviour of a non-stochastic cellular auxetic structure. A combination of experimental and numerical methods is used to gain a deeper understanding of the mechanical behaviour and its dependence on the geometric properties of the cellular structure. The experimental samples are built from Ti-6Al-4V using selective electron beam melting, an additive manufacturing process giving the possibility to vary the geometry of the structure in a highly controlled manner. The use of finite element simulations and mathematical homogenisation allows us also to investigate off-axis properties of the cellular material. This leads to a more comprehensive understanding of the mechanical behaviour of the auxetics. Ultimately, the gained knowledge can be used to tailor auxetic materials to specific applications.

Michael Stingl

Papers

PENNON: A code for convex nonlinear and semidefinite programming

Design of Auxetic Structures via Mathematical Optimization

Solving polynomial static output feedback problems with PENBMI

PENLAB: A MATLAB solver for nonlinear semidefinite optimization

Mechanical characterisation of a periodic auxetic structure produced by SEBM