Abstract: At the beginning of EU-FP6's research project PLATO-N ('a PLAtform for Topology Optimisation incorporating Novel large-scale, free material optimisation and mixed integer programming methods') the industrial partners prepared a use case definition which comprises the industrial requirements that should be considered in the project. Using the, in aircraft industries commonly employed, design process as a basis, the engineer's tasks are described and his needs are derived. Single design criteria out of a huge amount of aircraft requirements are selected, prioritised and, from an industrial point of view, tailored to the scope of the project.
Keywords: PLATO-N, software platform, large scale, topology optimisation, aircraft industry, industrial requirements, design process, optimisation process, free material optimisation, mixed integer programming, visualisation
Authors: Markus J.D. Wagner and Lars Krog
Date: March 26, 2007
Title: Specification of Aircraft Topology Optimization System, PLATO-N [Download pdf version]
Abstract: This document details the software specification requirements for PLATO-N in accordance with deliverable D4 of Annex I of the project description of work. Details are provided on what is required of the PLATO-N software system to satisfy performance, function, and industry needs. These needs have been discussed and agreed through consultation within the PLATO-N consortium. Assumptions and risks associated with the development of the software system are also documented. Details on the PLATO-N system configuration, dependencies, formulations, organisation and usage are also documented.
The contributions from all partners within the PLATO-N consortium are acknowledged.
Keywords: PLATO-N, free material optimization, finite element, topology optimisation, large-scale convex optimization, sequential convex programming, aerospace
Author: Richard Boyd
Date: June 26, 2007
Title: Large Scale Methods for Convex FMO-Type Problems [Download pdf version]
Abstract: In this report we review several first order methods suited for problems arising from large-scale free material optimization (FMO) problems. Our focus is on the Non-Euclidean Restricted Level (NERML) and on the reduced gradient methods. Both algorithms have demonstrated encouraging empirical results in preliminary numerical testing. Finally, we describe the CONERML method, which is a variation of the NERML method capable of dealing with constraints.
Keywords: First-order methods, large-scale convex optimization, bundle methods, free-material problems
Authors: Amir Beck, Aharon Ben-Tal and Luba Tetruashvili
Date: September 15, 2007
Title: Sequential Convex Programming Methods for Free Material Optimization [Download pdf version]
Abstract: We consider a numerical method for constrained nonlinear programming, that is widely used in mechanical engineering and that is known under the name SCP for sequential convex programming. The algorithm consists of solving a sequence of convex and separable subproblems, where an augmented Lagrangian merit function is used for guaranteeing convergence. Originally, SCP methods were developed in structural mechanical optimization, and are particularly applied to solve topology optimization problems. A new challenge for SCP methods is the solution of free material optimization (FMO) problems which contain additional semidefinite variables and even nonlinear semi-definite matrix constraints. A few formulations are investigated in more details and possible solution approaches are outlined.
Keywords: Nonlinear Programming, sequential convex programming, method of moving asymptotes, free material optimization, semi-definite programming, interior point methods
Authors: Sonja Ertel, Klaus Schittkowski and Christian Zillober
Date: July 23, 2007
Title: Efficient Algorithms for Visualising FMO Results [Download pdf version]
Abstract: In this report we discuss the possibilities of FMO data visualisation, providing an inventory of the techniques to be supported by the FMS visualisation component of the PLATO-N software system. Furthermore we discuss the currently identified computationally critical subtasks that might pose performance problems, and we suggest solutions to cope with them in the form of efficient data representation techniques and algorithms, as well as in terms of software design.
Keywords: scientific visualisation, evenly spaced streamlines, skeletonization, pipeline visualisation architecture
Author: Gábor Bodnár
Date: June 22, 2007
Title: Convex approximations for complex non-convex constraints [Download pdf version]
Abstract: We consider three algorithmic approaches to deal with displacement and/or stress constraints in free material optimization (FMO) models. The first approach is a simple sequential linear programming (SLP) approach and we describe how to adopt it to FMO-type problems with semidefinite constraints. We then provide another approximation scheme based on [1] to a general form of nonconvex constraints that includes (among other) stress or displacement constraints. At each iteration the constraint function is approximated by another overestimate function and thus the approximated
solution is guaranteed to be feasible. Finally, we briefly review a sequential convex semidefinite programming method recently developed by Stingl, Kocvara and Leugering.
Authors: A. Beck, A. Ben-Tal and L. Tetruashvili
Date: October 13, 2007
Title: Free Material Optimization with Control of the fundamental Eigenfrequency [Download pdf version]
Abstract: The goal of this paper is to formulate and solve free material optimization problems with constraints on the minimal eigenfrequency of a structure. A natural formulation of this problem as linear semidefinite program turns out to be numerically intractable. As alternative, we propose a new approach, which is based on a nonlinear semidefinite low-rank approximation of the semidefinite dual. Throughout this article, an algorithm is introduced and convergence properties are investigated. The article is concluded by numerical experiments proving the effectiveness of the new approach.
Keywords: structural optimization, material optimization, semidefinite programming, nonlinear programming
Authors: M. Stingl, M. Kocvara, and G. Leugering
Date: September 27, 2007
Title: Lower bounding problems for stress constrained discrete structural topology optimization problems [Download pdf version]
Abstract: The multiple load structural topology design problem is modeled as a minimization of the weight of the structure subject to equilibrium
constraints and restrictions on the local stresses and nodal displacements. The problem involves a large number of discrete design variables and is modeled as a non-convex mixed 0–1 program. For this problem, several convex and mildly non-convex continuous relaxations are presented. Reformulations of these relaxations, obtained by using duality results from semi-definite and second order cone programming, are also presented. The reformulated problems are suitable for implementation in a nonlinear branch and bound framework for solving the considered class of problems to global optimality.
Key words: Topology optimization, Stress constraints, Relaxations, Global optimization
Authors: Mathias Stolpe, Roman Stainko, and Michal Kocvara
Date: September 28, 2007
Title: FMO Models with Displacement Constraints [Download pdf version]
Abstract: Free material design deals with the question of finding the lightest structure subject to one or more given loads when both the distribution of material and the material itself can be freely varied. We additionally consider constraints on displacements of the optimal structure.
Authors: Michal Kočvara, and Michael Stingl
Date: April 9, 2008
Title: Sequential Convex Programming for Free Material Optimization with Displacement and Stress Constraints [Download pdf version]
Abstract: We consider free material optimization (FMO) problems, which are defined in form of semidefinite programs. The objective is to compute the stiffest structure subject to given loads. Constraints are bounds for compliances, displacements or stresses. Optimization variables are the entries of element material matrices in two or three dimensions. FMO problems are solved by a sequential convex programming algorithm, which is frequently applied in mechanical structural design optimization. The idea is to construct convex and separable subproblems, which are either solved by an interior point method or by an external solver, which is able to treat positive semidefinite variables. In the case of isotropic materials, additional linear constraints guarantee positive definite material matrices. For anisotropic materials, we propose to optimize over Cholesky factors of the elasticity matrices. Some preliminary numerical results are presented for these two situations. Finally we discuss the integration of two semidefinite subproblem solvers called PENNON and CONERML and the possibility to add stress and displacement constraints.
Keywords: nonlinear programming; sequential convex programming; method of moving asymptotes; free material optimization; semidefinite programming; displacement constraints; stress constraints; SCPIP; PENNON; CONERML
Authors: Sonja Ertel, Klaus Schittkowski, and Christian Zillober
Date: March, 2008
Title: Methods for Computer Aided Interpretation of FMO Results [Download pdf version]
Abstract: It is a challenging task to interpret FMO-materials by investigating their properties and eventually trying to substitute them with suitable existing materials. In this report we discuss possibilities how computer algorithms can help engineers in this process. There are various approaches to the problem, for instance visualization, material property discovery or automatic substitution of FMO-materials with real ones. From each of these branches we discuss selected methods, their applicability to FMO results interpretation and their combinability, if applicable.
Keywords: scientific visualization, free material optimisation, elastic material classification, laminate realizations of materials
Authors: Gábor Bodnár, Peter Stadelmeyer, and Michael Bogomolny
Date: April 17, 2008
Title: Free Material Optimization for Plates and Shells: the Single Load Case [Download pdf version]
Abstract: In this article, we present the Free Material Optimization (FMO) problem for plates and shells based on Naghdi’s shell model. In FMO – a branch of structural optimization – we search for the ultimately best material properties in a given design domain loaded by a set of given forces. The optimization variable is the full material tensor at each point of the design domain. We give a basic formulation of the problem and prove existence of an optimal solution. Lagrange duality theory allows to identify the basic problem as the dual of an infinite-dimensional convex nonlinear semidefinite program. After discretization by the finite element method the latter problem can be solved using a nonlinear SDP code. The article is concluded by several numerical studies.
Keywords: Free Material Optimization, Structural Optimization, Shells, Continuum Mechanics, Elasticity
Author: Stefanie Gaile
Date: July 16, 2008
Title: Efficient methods for solving discrete topology design problems in the PLATO-N project [Download pdf version]
Abstract: This paper considers the general multiple load structural topology design problems in the framework of the PLATO-N project. The problems involve a large number of discrete design variables and were modeled as a non-convex mixed 0–1 program. For the class of problems considered, a global optimization method based on the branchand-cut concept was developed and implemented. In the method a large number of continuous relaxations were solved. We also present an algorithm for generating cuts to strengthen the quality of the relaxations. Several heuristics were also investigated to obtain efficient algorithms. The branch and cut method is used to solve benchmark examples which can be used to validate other methods and heuristics.
Keywords: Topology optimization, Branch and cut, Stress constraints, Reformulations, Relaxations, Heuristics.
Authors: Nguyen Canh Nam and Mathias Stople
Date: October 1, 2008
Title: Test Case Library PLATOlib and Documentation
Abstract: The main idea of this report is to describe a test-case library (PLATOlib) of aeronautic conceptual design problems as a challenge and benchmarking data base for researchers and software vendors. We provide the academic and industrial test cases, which can be used as benchmarks for real world applications, in order to test the results using Topology and Free Material Optimization.
This report includes two parts:
a. The first part explains the structure and scientific meaning of the test-case library.
b. The second part includes the 32 Academic Test Cases and 8 Industrial Test Cases.
Keywords:
Authors: M. Bogomolny
Date: October 1, 2008
Title: Free material optimization with multidisciplinary optimization constraints for plates and shells [Download pdf version]
Abstract: Free Material Optimization (FMO) is a branch of structural optimization and deals with the problem of finding the stiffest structure for a given design domain and a given set of loads, where the amount of available material is limited. The choice of material in FMO is not restricted to a certain material symmetry or to material that already exists in nature. Instead the entire material tensor is taken as design variable yielding not only the optimal material distribution, but also the optimal material properties at each point of the design domain. The FMO formalism has been extended to Naghdi shells and Reissner-Mindlin plates in previous contributions of the authors. In this article we consider additional constraints on the FMO problem for shells – namely displacement, stress, vibration and global stability constraints – to find a design more suited to realistic problems and their requirements.
Keywords: Free Material Optimization, Naghdi Shells, Displacement Constraints, Stress Constraints, Vibration Constraints, Global Stability Constraints.
Author: Stefanie Gaile and Günter Leugering and Michael Stingl
Date: February 2, 2009
Title: Mathematical Models of FMO with Stress Constraints [Download pdf version]
Abstract: Free material design deals with the question of finding the lightest structure subject to one or more given loads when both the distribution of material and the material itself can be freely varied. We consider additional constraints on local stresses of the optimal structure.
Keywords:
Author: Michal Kočvara and Michael Stingl
Date: October 13, 2009