12 research outputs found
OPTIMIZATION-BASED APPROACH TO TILING OF FINITE AREAS WITH ARBITRARY SETS OF WANG TILES
Wang tiles proved to be a convenient tool for the design of aperiodic tilings in computer graphics and in materials engineering. While there are several algorithms for generation of finite-sized tilings, they exploit the specific structure of individual tile sets, which prevents their general usage. In this contribution, we reformulate the NP-complete tiling generation problem as a binary linear program, together with its linear and semidefinite relaxations suitable for the branch and bound method. Finally, we assess the performance of the established formulations on generations of several aperiodic tilings reported in the literature, and conclude that the linear relaxation is better suited for the problem
On bounded Wang tilings
Wang tiles enable efficient pattern compression while avoiding the
periodicity in tile distribution via programmable matching rules. However, most
research in Wang tilings has considered tiling the infinite plane. Motivated by
emerging applications in materials engineering, we consider the bounded version
of the tiling problem and offer four integer programming formulations to
construct valid or nearly-valid Wang tilings: a decision, maximum-rectangular
tiling, maximum cover, and maximum adjacency constraint satisfaction
formulations. To facilitate a finer control over the resulting tilings, we
extend these programs with tile-based, color-based, packing, and variable-sized
periodic constraints. Furthermore, we introduce an efficient heuristic
algorithm for the maximum-cover variant based on the shortest path search in
directed acyclic graphs and derive simple modifications to provide a
approximation guarantee for arbitrary tile sets, and a guarantee for tile
sets with cyclic transducers. Finally, we benchmark the performance of the
integer programming formulations and of the heuristic algorithms showing that
the heuristics provides very competitive outputs in a fraction of time. As a
by-product, we reveal errors in two well-known aperiodic tile sets: the Knuth
tile set contains a tile unusable in two-way infinite tilings, and the Lagae
corner tile set is not aperiodic
Global weight optimization of frame structures with polynomial programming
Weight optimization of frame structures with continuous cross-section
parametrization is a challenging non-convex problem that has traditionally been
solved by local optimization techniques. Here, we exploit its inherent
semi-algebraic structure and adopt the Lasserre hierarchy of relaxations to
compute the global minimizers. While this hierarchy generates a natural
sequence of lower bounds, we show, under mild assumptions, how to project the
relaxed solutions onto the feasible set of the original problem and thus
construct feasible upper bounds. Based on these bounds, we develop a simple
sufficient condition of global -optimality. Finally, we prove that
the optimality gap converges to zero in the limit if the set of global
minimizers is convex. We demonstrate these results by means of two academic
illustrations.Comment: 10 pages, 2 figure
Modular-topology optimization of structures and mechanisms with free material design and clustering
Topology optimization of modular structures and mechanisms enables balancing
the performance of automatically-generated individualized designs, as required
by Industry 4.0, with enhanced sustainability by means of component reuse. For
optimal modular design, two key questions must be answered: (i) what should the
topology of individual modules be like and (ii) how should modules be arranged
at the product scale? We address these challenges by proposing a bi-level
sequential strategy that combines free material design, clustering techniques,
and topology optimization. First, using free material optimization enhanced
with post-processing for checkerboard suppression, we determine the
distribution of elasticity tensors at the product scale. To extract the
sought-after modular arrangement, we partition the obtained elasticity tensors
with a novel deterministic clustering algorithm and interpret its outputs
within Wang tiling formalism. Finally, we design interiors of individual
modules by solving a single-scale topology optimization problem with the design
space reduced by modular mapping, conveniently starting from an initial guess
provided by free material optimization. We illustrate these developments with
three benchmarks first, covering compliance minimization of modular structures,
and, for the first time, the design of non-periodic compliant modular
mechanisms. Furthermore, we design a set of modules reusable in an inverter and
in gripper mechanisms, which ultimately pave the way towards the rational
design of modular architectured (meta)materials.Comment: 30 page
OPTIMIZATION-BASED APPROACH TO TILING OF FINITE AREAS WITH ARBITRARY SETS OF WANG TILES
Wang tiles proved to be a convenient tool for the design of aperiodic tilings in computer graphics and in materials engineering. While there are several algorithms for generation of finite-sized tilings, they exploit the specific structure of individual tile sets, which prevents their general usage. In this contribution, we reformulate the NP-complete tiling generation problem as a binary linear program, together with its linear and semidefinite relaxations suitable for the branch and bound method. Finally, we assess the performance of the established formulations on generations of several aperiodic tilings reported in the literature, and conclude that the linear relaxation is better suited for the problem
On optimum design of frame structures
International audienceOptimization of frame structures is formulated as a non-convex optimization problem,which is currently solved to local optimality. In this contribution, we investigate four optimizationapproaches: (i) general non-linear optimization, (ii) optimality criteria method, (iii) non-linear semidefi-nite programming, and (iv) polynomial optimization. We show that polynomial optimization solves theframe structure optimization to global optimality by building the (moment-sums-of-squares) hierarchyof convex linear semidefinite programming problems, and it also provides guaranteed lower and upperbounds on optimal design. Finally, we solve three sample optimization problems and conclude that thelocal optimization approaches may indeed converge to local optima, without any solution quality mea-sure, or even to infeasible points. These issues are readily overcome by using polynomial optimization,which exhibits a finite convergence, at the prize of higher computational demands
Global optimality in minimum compliance topology optimization of frames and shells by moment-sum-of-squares hierarchy
International audienceThe design of minimum-compliance bending-resistant structures with continuous cross-section parameters is a challenging task because of its inherent non-convexity. Our contribution develops a strategy that facilitates computing all guaranteed globally optimal solutions for frame and shell structures under multiple-load cases and self-weight. To this purpose, we exploit the fact that the stiffness matrix is usually a polynomial function of design variables, allowing us to build an equivalent non-linear semidefinite programming formulation over a semi-algebraic feasible set. This formulation is subsequently solved using the Lasserre moment-sum-of-squares hierarchy, generating a sequence of outer convex approximations that monotonically converges from below to the optimum of the original problem. Globally optimal solutions can subsequently be extracted using the Curto-Fialkow flat extension theorem. Furthermore, we show that a simple correction to the solutions of the relaxed problems establishes a feasible upper bound, thereby deriving a simple sufficient condition of global ε-optimality. When the original problem possesses a unique minimum, we show that this solution is found with a zero optimality gap in the limit. These theoretical findings are illustrated on several examples of topology optimization of frames and shells, for which we observe that the hierarchy converges in a finite (rather small) number of steps
Global optimality in minimum compliance topology optimization of frames and shells by moment-sum-of-squares hierarchy
International audienceThe design of minimum-compliance bending-resistant structures with continuous cross-section parameters is a challenging task because of its inherent non-convexity. Our contribution develops a strategy that facilitates computing all guaranteed globally optimal solutions for frame and shell structures under multiple-load cases and self-weight. To this purpose, we exploit the fact that the stiffness matrix is usually a polynomial function of design variables, allowing us to build an equivalent non-linear semidefinite programming formulation over a semi-algebraic feasible set. This formulation is subsequently solved using the Lasserre moment-sum-of-squares hierarchy, generating a sequence of outer convex approximations that monotonically converges from below to the optimum of the original problem. Globally optimal solutions can subsequently be extracted using the Curto-Fialkow flat extension theorem. Furthermore, we show that a simple correction to the solutions of the relaxed problems establishes a feasible upper bound, thereby deriving a simple sufficient condition of global ε-optimality. When the original problem possesses a unique minimum, we show that this solution is found with a zero optimality gap in the limit. These theoretical findings are illustrated on several examples of topology optimization of frames and shells, for which we observe that the hierarchy converges in a finite (rather small) number of steps