Flexible Polyhedron Research Articles

Modeling and economic optimization of an industrial site for natural gas processing: A nonlinear optimization approach

In this work an integrated simulation-optimization model to determine the optimal economic operating point of an industrial site for natural gas processing with multiple process units and feedstock. Natural gas industrial facilities have the intrinsic characteristic of dealing with dynamic scenarios, with no control over raw inlet gas, which might lead the plant to suboptimal conditions. In this context, an integrated simulation-optimization framework was proposed, using Aspen HYSYS for process simulation, Python for optimization and Microsoft Excel as data transfer interface. Process model comprises three slug catchers, five gas processing units, three liquid fractionating units, compression systems, auxiliary equipment and gas treating units. The optimization problem was formulated based on the maximum business profit, incorporating, as penalty functions, inequality constraints: product specifications, logistic bottlenecks and plant physical limits. Optimization model was validated against actual data from an industrial facility responsible for processing over one quarter of all natural gas used in Brazil. The nonlinear programming was successfully executed using Particle Swarm Optimization method in series with Nelder-Mead flexible polyhedron for ten normalized decision variables indicating a potential increase of 7.3 % in variable profit, as well as the existence of local minima in the optimization space.

Zero‐sum cycles in flexible polyhedra

We show that if a polyhedron in the three-dimensional affine space with triangular faces is flexible, i.e., can be continuously deformed preserving the shape of its faces, then there is a cycle of edges whose lengths sum up to zero once suitably weighted by 1 and -1. We do this via elementary combinatorial considerations, made possible by a well-known compactification of the three-dimensional affine space as a quadric in the four-dimensional projective space. The compactification is related to the Euclidean metric, and allows us to use a simple degeneration technique that reduces the problem to its one-dimensional analogue, which is trivial to solve.

The spectrum of the Laplacian in a domain bounded by a flexible polyhedron in $$\mathbb R^d$$ does not always remain unaltered during the flex

Being motivated by the theory of flexible polyhedra, we study the Dirichlet and Neumann eigenvalues for the Laplace operator in special bounded domains of Euclidean d-space. The boundary of such a domain is an embedded simplicial complex which allows a continuous deformation (a flex), under which each simplex of the complex moves as a solid body and the change in the spatial shape of the domain is achieved through a change of the dihedral angles only. The main result of this article is that both the Dirichlet and Neumann spectra of the Laplace operator in such a domain do not necessarily remain unaltered during the flex of its boundary.

Four-bar linkages, elliptic functions, and flexible polyhedra

We revisit Darboux' parametrization of the configuration space of a quadrilateral with fixed side lengths by elliptic functions, remind how it implies Darboux' folding porism, and relate it to Bottema's zigzag porism. Further, we review some mechanical applications of this parametrization giving rise to overconstrained linkages and polyhedra.

Necessary conditions for the extendibility of a first-order flex of a polyhedron to its flex

We derive fundamentally new equations that are satisfied by first-order flexes of a flexible polyhedron. Moreover, we indicate two sources of such new equations. These sources are the Dehn invariants and rigidity matrix. The equations derived provide us with fundamentally new necessary conditions for the extendibility of a first-order flex of a polyhedron to its flex.

A sufficient condition for a polyhedron to be rigid

We study oriented connected closed polyhedral surfaces with non-degenerate triangular faces in three-dimensional Euclidean space, calling them polyhedra for short. A polyhedron is called flexible if its spatial shape can be changed continuously by changing its dihedral angles only. We prove that for every flexible polyhedron some integer combination of its dihedral angles remains constant during the flex. The proof is based on a recent result of A. A. Gaifullin and L. S. Ignashchenko.

On the model flexibility of Siamese dipyramids

Polyhedra called Siamese dipyramids are known to be non-flexible, however their physical models behave like physical models of flexible polyhedra. We discuss a simple mathematical method for explaining the model flexibility of the Siamese dipyramids.

Dehn Invariant and Scissors Congruence of Flexible Polyhedra

We prove that the Dehn invariant of any flexible polyhedron in n-dimensional Euclidean space, where n ≥ 3, is constant during the flexion. For n = 3 and 4 this implies that any flexible polyhedron remains scissors congruent to itself during the flexion. This proves the Strong Bellows Conjecture posed by R. Connelly in 1979. It was believed that this conjecture was disproved by V. Alexandrov and R. Connelly in 2009. However, we find an error in their counterexample. Further, we show that the Dehn invariant of a flexible polyhedron in the n-dimensional sphere or n-dimensional Lobachevsky space, where n ≥ 3, is constant during the flexion whenever this polyhedron satisfies the usual Bellows Conjecture, i.e., whenever its volume is constant during every flexion of it. Using previous results of the first named author, we deduce that the Dehn invariant is constant during the flexion for every bounded flexible polyhedron in odd-dimensional Lobachevsky space and for every flexible polyhedron with sufficiently small edge lengths in any space of constant curvature of dimension at least 3.

The Bellows Conjecture for Small Flexible Polyhedra in Non-Euclidean Spaces

The bellows conjecture claims that the volume of any flexible polyhedron of dimension 3 or higher is constant during the flexion. The bellows conjecture was proved for flexible polyhedra in the Euclidean spaces of dimensions 3 and higher, and for bounded flexible polyhedra in the odd-dimensional Lobachevsky spaces. Counterexamples to the bellows conjecture are known in all open hemispheres of dimensions 3 and higher. The aim of this paper is to prove that, nonetheless, the bellows conjecture is true for all flexible polyhedra in either spheres or Lobachevsky spaces of dimensions greater than or equal to 3 with sufficiently small edge lengths.

The analytic continuation of volume and the Bellows conjecture in Lobachevsky spaces

A flexible polyhedron in an -dimensional space of constant curvature is a polyhedron with rigid -dimensional faces and hinges at -dimensional faces. The Bellows conjecture claims that, for , the volume of any flexible polyhedron is constant during the flexion. The Bellows conjecture in Euclidean spaces was proved by Sabitov for (1996) and by the author for (2012). Counterexamples to the Bellows conjecture in open hemispheres were constructed by Alexandrov for (1997) and by the author for (2015). In this paper we prove the Bellows conjecture for bounded flexible polyhedra in odd-dimensional Lobachevsky spaces. The proof is based on the study of the analytic continuation of the volume of a simplex in Lobachevsky space considered as a function of the hyperbolic cosines of its edge lengths. Bibliography: 37 titles.

Motion planning for an actuated flexible polyhedron manifold

Stepping or crawling translation manoeuvres are computationally complicated when applied to hyper-redundant mechanisms. We introduce a motion planner for a novel polyhedron-shaped Actuated Flexible Manifold (AFM). The AFM is a two-dimensional closed or open surface embedded in . The AFM can reshape itself up to a given radius of curvature at any given location on its surface. Motion can be made possible using a mesh of linear actuators. Our first goal was to calculate the forward and inverse kinematics in the configuration space for a discrete flat AFM. We demonstrate our solution on some actuated flexible grid-shaped manifolds using different actuation methods. We then simulate a translation of a hyper-redundant discrete, polyhedron-shaped AFMs. In addition, we show how the AFM can use its excessive degrees of freedom to avoid obstacles while moving. Our algorithm is applicable for a set of hyper-redundant mechanisms of various geometries.

The set of nondegenerate flexible polyhedra of a prescribed combinatorial structure is not always algebraic

We construct some example of a closed nondegenerate nonflexible polyhedron P in Euclidean 3-space that is the limit of a sequence of nondegenerate flexible polyhedra each of which is combinatorially equivalent to P. This implies that the set of nondegenerate flexible polyhedra combinatorially equivalent to P is not algebraic.

Continuous Flattening of a Regular Tetrahedron with Explicit Mappings

We use the terminology polyhedron for a closed polyhedral surface which is permitted to touch itself but not self-intersect (and so a doubly covered polygon is a polyhedron). A flat folding of a polyhedron is a folding by creases into a multilayered planar shape ([7], [8]). A. Cauchy [4] in 1813 proved that any convex polyhedron is rigid: precisely, if two convex polyhedra P, P ′ are combinatorially equivalent and their corresponding faces are congruent, then P and P ′ are congruent. By removing the condition of convexity, R. Connelly [5] in 1978 gave an example of a (non-convex) flexible polyhedron: precisely, there is a continuous family of polyhedra {Pt : 0 ≤ t ≤ 1} such that for every t 6= 0, the corresponding faces of P0 and Pt are congruent while polyhedra P0 and Pt are not congruent. (See also [6].) After then I. Sabitov [15] in 1998 proved that the volume of any polyhedron is invariant under flexing: precisely, if there is a continuous family of polyhedra {Pt : 0 ≤ t ≤ 1} such that, for every t, the corresponding faces of P0 and Pt are congruent, then the volumes P0 and Pt are equal for all 0 ≤ t ≤ 1. (See also [14].) A. Milka [12] in 1994 showed that any polyhedron admits a continuous (isometric) deformation by using moving edges, and that all Platonic polyhedra can be changed in

Most important phases of development optimization of complex structure

The paper is concerned with most important phases of formation and development optimization of complex structure. A formal approach to multidisciplinary optimization of complex devices is developed. The stages of optimization of complex devices: the choice of models of analysis, design variables, functional objectives and constraints. Optimization techniques are classified as direct; the first and second order; stochastic; linear and nonlinear programming. The applications of different types of Super Computer Simulation and Optimization Based Design / Engineering (multidimensional, structural, topological, parametric, multicriteria) are shown. On the basis of a mathematical model that takes into account the spatial and temporal fields of different nature (electrical, magnetic, elastic deformation, mechanical and thermal) with use of optimization genetic algorithms and the flexible polyhedron method, the complex structure is developed, which is characterized of reduced weight and size and high efficiency.

Embedded flexible spherical cross-polytopes with nonconstant volumes

We construct examples of embedded flexible cross-polytopes in the spheres of all dimensions. These examples are interesting from two points of view. First, in dimensions 4 and higher, they are the first examples of embedded flexible polyhedra. Notice that, in contrast to the spheres, in the Euclidean and Lobachevsky spaces of dimensions 4 and higher still no example of an embedded flexible polyhedron is known. Second, we show that the volumes of the constructed flexible cross-polytopes are nonconstant during the flexion. Hence these cross-polytopes give counterexamples to the Bellows Conjecture for spherical polyhedra. Earlier a counterexample to this conjecture was constructed only in dimension 3 (V.A. Alexandrov, 1997), and it was not embedded. For flexible polyhedra in spheres we suggest a weakening of the Bellows Conjecture, which we call the Modified Bellows Conjecture. We show that this conjecture holds for all flexible cross-polytopes of the simplest type, which includes our counterexamples to the ordinary Bellows Conjecture. Simultaneously, we obtain several geometric results on flexible cross-polytopes of the simplest type. In particular, we write down relations for the volumes of their faces of codimensions 1 and 2.

Generalization of Sabitov’s Theorem to Polyhedra of Arbitrary Dimensions

In 1996 Sabitov proved that the volume $$V$$ V of an arbitrary simplicial polyhedron $$P$$ P in the $$3$$ 3 -dimensional Euclidean space $$\mathbb {R}^3$$ R 3 satisfies a monic (with respect to $$V$$ V ) polynomial relation $$F(V,\ell )=0$$ F ( V , l ) = 0 , where $$\ell $$ l denotes the set of the squares of edge lengths of $$P$$ P . In 2011 the author proved the same assertion for polyhedra in $$\mathbb {R}^4$$ R 4 . In this paper, we prove that the same result is true in arbitrary dimension $$n\ge 3$$ n ? 3 . Moreover, we show that this is true not only for simplicial polyhedra, but for all polyhedra with triangular $$2$$ 2 -faces. As a corollary, we obtain the proof in arbitrary dimension of the well-known Bellows Conjecture posed by Connelly in 1978. This conjecture claims that the volume of any flexible polyhedron is constant. Moreover, we obtain the following stronger result. If $$P_t, t\in [0,1]$$ P t , t ? [ 0 , 1 ] , is a continuous deformation of a polyhedron such that the combinatorial type of $$P_t$$ P t does not change and every $$2$$ 2 -face of $$P_t$$ P t remains congruent to the corresponding face of $$P_0$$ P 0 , then the volume of $$P_t$$ P t is constant. We also obtain non-trivial estimates for the oriented volumes of complex simplicial polyhedra in $$\mathbb {C}^n$$ C n from their orthogonal edge lengths.

Deformations of Period Lattices of Flexible Polyhedral Surfaces

In the end of the 19th century Bricard discovered a phenomenon of flexible polyhedra, that is, polyhedra with rigid faces and hinges at edges that admit non-trivial flexes. One of the most important results in this field is a theorem of Sabitov asserting that the volume of a flexible polyhedron is constant during the flexion. In this paper we study flexible polyhedral surfaces in the 3-space two-periodic with respect to translations by two non-colinear vectors that can vary continuously during the flexion. The main result is that the period lattice of a flexible two-periodic surface homeomorphic to a plane cannot have two degrees of freedom.

Sabitov polynomials for volumes of polyhedra in four dimensions

In 1996 I.Kh. Sabitov proved that the volume of a simplicial polyhedron in a 3-dimensional Euclidean space is a root of certain monic polynomial with coefficients depending on the combinatorial type and on edge lengths of the polyhedron only. Moreover, the coefficients of this polynomial are polynomials in edge lengths of the polyhedron. This result implies that the volume of a simplicial polyhedron with fixed combinatorial type and edge lengths can take only finitely many values. In particular, this yields that the volume of a flexible polyhedron in a 3-dimensional Euclidean space is constant. Until now it has been unknown whether these results can be obtained in dimensions greater than 3. In this paper we prove that all these results hold for polyhedra in a 4-dimensional Euclidean space.

Application of QFA coupled with CFD analysis to predict the hardness of T6 heat treated Al6061 cylinder

QFA (quench factor analysis) method is widely used to predict the mechanical properties such as hardness and strength according to temperature of quenched material that has to be determined by experimental heat treatment. But, QFA coupled with CFD (computational fluid dynamics) analysis of this study can predict the mechanical properties without the experiment of heat treatment except the experiment to determine the material constants of QFA. First of all, Jominy test and FPS (flexible polyhedron search) are performed to determine the material constants of QFA. The CFD analysis is applied to predict the cooling temperature of Al6061 cylinder cooled by water of 25°C during quenching of solid solution heat treatment. Hardnesses of T6 heat treated Al6061 cylinder is predicted by the QFA coupled with CFD analysis without experiment of heat treatment and then the predicted hardnesses are compared with experimentally measured hardnesses according to positions of cylinder. The predicted hardnesses of cylinder are in good agreement with the measured ones within a maximum error of 8.45%.

Die shape design of tube drawing process using FE analysis and optimization method

The cold tube drawing has been generally regarded as an effective technology because of its high productivity and low production costs. It has been used instead of the swaging process to form monobloc tube shafts (MTSs) with a constant outer diameter or hollow shafts without defects. MTSs are used nowadays to reduce the weight of shafts and to improve power transmission in the auto industry nowadays. The objective of this study was to design an optimal die that can prevent material fracture considering critical damage value in the tube drawing process. To achieve this objective, a finite element simulation and the flexible polyhedron search method are used. The drawing die shape is represented by the Bezier curve that is able to generate all the possible die shapes. The tube drawing experiments were carried out using the optimized tool. In the experiments, it was possible to produce sound products without fracture.