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This paper presents a new isogeometric topology optimization (TO) method based on moving morphable components (MMC), where the R-functions are used to represent the topology description functions (TDF) to overcome the C 1 discontinuity problem of the overlapping regions of components. Three new ersatz material models based on uniform, Gauss and Greville abscissae collocation schemes are used to represent both the Young’s modulus of material and the density field based on the Heaviside values of collocation points. Three benchmark examples are tested to evaluate the proposed method, where the collocation schemes are compared as well as the difference between isogeometric analysis (IGA) and finite element method (FEM). The results show that the convergence rate using R-functions has been improved in a range of 17%–60% for different cases in both FEM and IGA frameworks, and the Greville collocation scheme outperforms the other two schemes in the MMC-based TO. [ABSTRACT FROM AUTHOR]

In this paper, we devise cell-based maximum-entropy (max-ent) basis functions that are used in a Galerkin method for the solution of partial differential equations. The motivation behind this work is the construction of smooth approximants with controllable support on unstructured meshes. In the variational scheme to obtain max-ent basis functions, the nodal prior weight function is constructed from an approximate distance function to a polygonal curve in R 2 . More precisely, we take powers of the composition of R-functions via Boolean operations. The basis functions so constructed are nonnegative, smooth, linearly complete, and compactly-supported in a neighbor-ring of segments that enclose each node. The smoothness is controlled by two positive integer parameters: the normalization order of the approximation of the distance function and the power to which it is raised. The properties and mathematical foundations of the new compactly-supported approximants are described, and its use to solve two-dimensional elliptic boundary-value problems (Poisson equation and linear elasticity) is demonstrated. The sound accuracy and the optimal rates of convergence of the method in Sobolev norms are established. [ABSTRACT FROM AUTHOR]

Abstract: The popularity of medial axis in shape modeling and analysis comes from several of its well known fundamental properties. For example, medial axis captures the connectivity of the domain, has a lower dimension than the space itself, and is closely related to the distance function constructed over the same domain. We propose the new concept of a medial zone of an n-dimensional semi-analytic domain that subsumes the medial axis of the same domain as a special case, and can be thought of as a ‘thick’ skeleton having the same dimension as that of . We show that by transforming the exact, non-differentiable, distance function of domain into approximate but differentiable distance functions, and by controlling the geodesic distance to the crests of the approximate distance functions of domain , one obtains families of medial zones of that are homeomorphic to the domain and are supersets of . We present a set of natural properties for the medial zones of and discuss practical approaches to compute both medial axes and medial zones for 3-dimensional semi-analytic sets with rigid or evolving boundaries. Due to the fact that the medial zones fuse some of the critical geometric and topological properties of both the domain itself and of its medial axis, re-formulating problems in terms of medial zones affords the ‘best of both worlds’ in applications such as geometric reasoning, robotic and autonomous navigation, and design automation. [Copyright &y& Elsevier]

general method is proposed to derive equations of irregular curves O ( x, y) = 0 and of irregular surfaces O ( x, y, z) =0 in implicit form, where the functions O ( x, y) and O ( x, y, z) belong to a prescribed differentiability class. The method essentially involves interlineation and interflation of functions. An example is considered. [ABSTRACT FROM AUTHOR]

Abstract: Shape skeletons are fundamental concepts for describing the shape of geometric objects, and have found a variety of applications in a number of areas where geometry plays an important role. Two types of skeletons commonly used in geometric computations are the straight skeleton of a (linear) polygon, and the medial axis of a bounded set of points in the -dimensional Euclidean space. However, exact computation of these skeletons of even fairly simple planar shapes remains an open problem. In this paper we propose a novel approach to construct exact or approximate (continuous) distance functions and the associated skeletal representations (a skeleton and the corresponding radius function) for solid 2D semi-analytic sets that can be either rigid or undergoing topological deformations. Our approach relies on computing constructive representations of shapes with R-functions that operate on real-valued halfspaces as logic operations. We use our approximate distance functions to define a new type of skeleton, i.e, the C-skeleton, which is piecewise linear for polygonal domains, generalizes naturally to planar and spatial domains with curved boundaries, and has attractive properties. We also show that the exact distance functions allow us to compute the medial axis of any closed, bounded and regular planar domain. Importantly, our approach can generate the medial axis, the straight skeleton, and the C-skeleton of possibly deformable shapes within the same formulation, extends naturally to 3D, and can be used in a variety of applications such as skeleton-based shape editing and adaptive motion planning. [Copyright &y& Elsevier]