Manufacturing constraints in parameter–free shape optimization

Structural shape optimization has become an important tool for engineers when it comes to improving components with respect to a given goal function. During this process the designer has to ensure that the optimized part stays manufacturable. Depending on the manufacturing process several requirements could be relevant such as demolding or different kinds of symmetry. This work introduces two approaches on how to handle manufacturing constraints in parameter-free shape optimization. In the so–called explicit approach equality and inequality equations are formulated using the coordinates of the FE-nodes. These equations can be used to extend the optimization problem. Since the number of the additional constraint equations may be very large we apply aggregation formulations, e.g. the Kreisselmeier-Steinhauser function, if necessary. In the second approach, the so–called implicit method, the set of design nodes is split in two groups called optimization nodes and dependent nodes. The optimization nodes are now handled as design nodes but the dependent nodes are coupled to the optimization nodes in such a way that the manufacturing constraint is fulfilled. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spatial shape‐preserving interpolation using ν‐splines

We present a global iterative algorithm for constructing spatial G ²continuous interpolating splines, which preserve the shape of the polygonal line that interpolates the given points. Furthermore, the algorithm can handle data exhibiting two kinds of degeneracy, namely, coplanar quadruples and collinear triplets of points. The convergence of the algorithm stems from the asymptotic properties of the curvature, torsion and Frénet frame of splines for large values of the tension parameters, which are thoroughly investigated and presented. The performance of our approach is tested on two data sets, one of synthetic nature and the other of industrial interest.     

On the shape of random Pólya structures

Panagiotou and Stufler recently proved an important fact on their way to establish the scaling limits of random Pólya trees: a uniform random Pólya tree of size $n$ consists of a conditioned critical Galton–Watson tree $C_n$ and many small forests, where with probability tending to one, as $n$ tends to infinity, any forest $F_n\left(v\right)$, that is attached to a node $v$ in $C_n$, is maximally of size $\left|F_n\left(v\right)\right|=O(logn)$. Their proof used the framework of a Boltzmann sampler and deviation inequalities.In this paper, first, we employ a unified framework in analytic combinatorics to prove this fact with additional improvements for $\left|F_n\left(v\right)\right|$, namely $\left|F_n\left(v\right)\right|=\Theta (logn)$. Second, we give a combinatorial interpretation of the rational weights of these forests and the defining substitution process in terms of automorphisms associated to a given Pólya tree. Third, we derive the limit probability that for a random node $v$ the attached forest $F_n\left(v\right)$ is of a given size. Moreover, structural properties of those forests like the number of their components are studied. Finally, we extend all results to other Pólya structures.     

On curvature control in node–based shape optimization

We present and compare three approaches to control of the curvature of the domain boundary in two–dimensional shape optimization. Since the coordinates of the FE–nodes are used as design variables, objective and constraint functions have to be formulated in terms of the nodal coordinates. Therefore we investigate various formulations for the curvature of the domain boundary in terms of the coordinates of the boundary nodes. In addition, the sensitivities of these expressions with respect to the design nodes are required for the application of gradient–based optimization algorithms. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Can one hear the shape of a group?

Let Γ be a finitely generated group, and letS be a finite, non-necessarily symmetric, generating subset of Γ. Leth be the transition operator of the directed Cayley graphG(Γ,S), acting onl ² (Γ). Staring with Kesten’s seminal results, we give a survey of results linking group-theoretic properties of the pair (Γ,S) with spectral properties ofh.     

On condition numbers in hp-FEM with Gauss–Lobatto-based shape functions

Sharp bounds on the condition number of stiffness matrices arising in hp/spectral discretizations for two-dimensional problems elliptic problems are given. Two types of shape functions that are based on Lagrange interpolation polynomials in the Gauss–Lobatto points are considered. These shape functions result in condition numbers O(p) and $\text{O(p}\text{ln}\text{p)}$ for the condensed stiffness matrices, where p is the polynomial degree employed. Locally refined meshes are analyzed. For the discretization of Dirichlet problems on meshes that are refined geometrically toward singularities, the conditioning of the stiffness matrix is shown to be independent of the number of layers of geometric refinement.     

The cubic trigonometric Bézier curve with two shape parameters

A cubic trigonometric Bézier curve analogous to the cubic Bézier curve, with two shape parameters, is presented in this work. The shape of the curve can be adjusted by altering the values of shape parameters while the control polygon is kept unchanged. With the shape parameters, the cubic trigonometric Bézier curves can be made close to the cubic Bézier curves or closer to the given control polygon than the cubic Bézier curves. The ellipses can be represented exactly using cubic trigonometric Bézier curves.     

Can a ground-based vehicle hear the shape of a room?

Assume that a ground-based vehicle moves in a room with walls or other planar surfaces. Can the vehicle reconstruct the positions of the walls from the echoes of a single sound event? We assume that the vehicle carries some microphones and that a loudspeaker is either also mounted on the vehicle or placed at a fixed location in the room. We prove that the reconstruction is almost always possible if (1) no echoes are received from floors, ceilings, or sloping walls and the vehicle carries at least three noncollinear microphones, or if (2) walls of any inclination may occur, the loudspeaker is fixed in the room and there are four noncoplanar microphones. The difficulty lies in the echo-matching problem: How to determine which echoes come from the same wall. We solve this by using a Cayley–Menger determinant. Our proofs use methods from computational commutative algebra.     

A control point based curve with two exponential shape parameters

A generalization of a recently developed trigonometric Bézier curve is presented in this paper. The set of original basis functions are generalized also for non-trigonometric functions, and essential properties, such as linear independence, nonnegativity and partition of unity are proved. The new curve—contrary to the original one—can be defined by arbitrary number of control points meanwhile it preserves the properties of the original curve.     

Persistency of wellposedness of Ventcel’s boundary value problem under shape deformations

Ventcel boundary conditions are second order differential conditions that appear in asymptotic models. Like Robin boundary conditions, they lead to well-posed variational problems under a sign condition of the coefficient. This is achieved when physical situations are considered. Nevertheless, situations where this condition is violated appeared in several recent works where absorbing boundary conditions or equivalent boundary conditions on rough surfaces are sought for numerical purposes. The well-posedness of such problems was recently investigated: up to a countable set of parameters, existence and uniqueness of the solution for the Ventcel boundary value problem holds without the sign condition. However, the values to be avoided depend on the domain where the boundary value problem is set. In this work, we address the question of the persistency of the solvability of the boundary value problem under domain deformation.     

Numerical instabilities in structural optimization – analogy between topology & shape design problems

The formulation of structural optimization problems on the basis of the finite–element–method often leads to numerical instabilities resulting in non–optimal designs, which turn out to be difficult to realize from the engineering point of view. In the case of topology optimization problems the formation of designs characterized by oscillating density distributions such as the well–known “checkerboard–patterns” can be observed, whereas the solution of shape optimization problems often results in unfavourable designs with non–smooth boundary shapes caused by high–frequency oscillations of the boundary shape functions. Furthermore a strong dependence of the obtained designs on the finite–element–mesh can be observed in both cases. In this context we have already shown, that the topology design problem can be regularized by penalizing spatial oscillations of the density function by means of a penalty–approach based on the density gradient. In the present paper we apply the idea of problem regularization by penalizing oscillations of the design variable to overcome the numerical difficulties related to the shape design problem, where an analogous approach restricting the boundary surface can be introduced. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reconstruction of the shape of object with near field measurements in a half-plane

We consider a mathematical problem modelling some characteristics of near field optical microscope.We take a monofrequency line source to illuminate a sample with constant index of refraction and use the scattered field data measured near the sample to reconstruct the shape of it. Mixed reciprocity relation and factorization method are applied to solve our problem.Some numerical examples to show the feasibility of the method are presented.     

Classroom interactions that shape the nature of students’ appropriation of a complex mathematical practice

Scholars continue to emphasize the importance of fostering proficiency with mathematical practices as an educational outcome. As teachers attempt to support students in developing these practices, they communicate subtle messages about their nature. However, researchers lack a detailed understanding of the classroom interactions that communicate these messages. To begin to address this gap in the literature, we investigated the relationship between the types of classroom interactions around the mathematical practice of imposing structure and the ways students subsequently engaged in that practice. This led to the identification of three types of classroom interactions that shaped the nature of students’ appropriation of imposing structure: (a) engaging students in the practice, (b) providing different representations of the practice, and (c) reflecting on different instantiations of the practice. Our examination of the nature of these interactions suggests teachers must attend to details as they support students to appropriate mathematical practices in formal learning environments.     

On the shape–from–moments problem and recovering edges from noisy Radon data

We consider the problem of reconstructing a planar convex set from noisy observations of its moments. An estimation method based on pointwise recovering of the support function of the set is developed. We study intrinsic accuracy limitations in the shape–from–moments estimation problem by establishing a lower bound on the rate of convergence of the mean squared error. It is shown that the proposed estimator is near–optimal in the sense of the order. An application to tomographic reconstruction is discussed, and it is indicated how the proposed estimation method can be used for recovering edges from noisy Radon data.Mathematics Subject Classification (2000):62C20, 62G20, 94A12     

Real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster Reeb parallel shape operator

In a paper due to Jeong et al. (Kodai Math J 34(3):352–366, 2011) we have shown that there does not exist a hypersurface in $G_{2}({\mathbb{C }}^{m+2})$ with parallel shape operator in the generalized Tanaka–Webster connection (see Tanaka in Jpn J Math 20:131–190, 1976; Tanno in Trans Am Math Soc 314(1):349–379, 1989). In this paper, we introduce the notion of the Reeb parallel in the sense of generalized Tanaka–Webster connection for a hypersurface $M$ in $G_{2}({\mathbb{C }}^{m+2})$ and prove that $M$ is an open part of a tube around a totally geodesic $G_2(\mathbb{C }^{m+1})$ in $G_2(\mathbb{C }^{m+2})$ .