Dev2PQ: Planar Quadrilateral Strip Remeshing of Developable Surfaces

We introduce an algorithm to remesh triangle meshes representing developable surfaces to planar quad dominant meshes. The output of our algorithm consists of planar quadrilateral (PQ) strips that are aligned to principal curvature directions and closely approximate the curved parts of the input developable, and planar polygons representing the flat parts of the input. Developable PQ-strip meshes are useful in many areas of shape modeling, thanks to the simplicity of fabrication from flat sheet material. Unfortunately, they are difficult to model due to their restrictive combinatorics and locking issues. Other representations of developable surfaces, such as arbitrary triangle or quad meshes, are more suitable for interactive freeform modeling, but generally have non-planar faces or are not aligned to principal curvatures. Our method leverages the modeling flexibility of non-ruling based representations of developable surfaces, while still obtaining developable, curvature aligned PQ-strip meshes. Our algorithm optimizes for a scalar function on the input mesh, such that its level sets are extrinsically straight and align well to the locally estimated ruling directions. The condition that guarantees straight level sets is nonlinear of high order and numerically difficult to enforce in a straightforward manner. We devise an alternating optimization method that makes our problem tractable and practical to compute. Our method works automatically on any developable input, including multiple patches and curved folds, without explicit domain decomposition. We demonstrate the effectiveness of our approach on a variety of developable surfaces and show how our remeshing can be used alongside handle based interactive freeform modeling of developable shapes

Interactive design exploration for constrained meshes

In architectural design, surface shapes are commonly subject to geometric constraints imposed by material, fabrication or assembly. Rationalization algorithms can convert a freeform design into a form feasible for production, but often require design modifications that might not comply with the design intent. In addition, they only offer limited support for exploring alternative feasible shapes, due to the high complexity of the optimization algorithm. We address these shortcomings and present a computational framework for interactive shape exploration of discrete geometric structures in the context of freeform architectural design. Our method is formulated as a mesh optimization subject to shape constraints. Our formulation can enforce soft constraints and hard constraints at the same time, and handles equality constraints and inequality constraints in a unified way. We propose a novel numerical solver that splits the optimization into a sequence of simple subproblems that can be solved efficiently and accurately. Based on this algorithm, we develop a system that allows the user to explore designs satisfying geometric constraints. Our system offers full control over the exploration process, by providing direct access to the specification of the design space. At the same time, the complexity of the underlying optimization is hidden from the user, who communicates with the system through intuitive interfaces

Geometric computing for freeform architecture

Geometric computing has recently found a new field of applications, namely the various geometric problems which lie at the heart of rationalization and construction-aware design processes of freeform architecture. We report on our work in this area, dealing with meshes with planar faces and meshes which allow multilayer constructions (which is related to discrete surfaces and their curvatures), triangles meshes with circle-packing properties (which is related to conformal uniformization), and with the paneling problem. We emphasize the combination of numerical optimization and geometric knowledge.

Planar hexagonal meshing for architecture

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Numerical wave propagation for the triangular P1DGP1_{DG}-P2P2 finite element pair

Inertia-gravity mode and Rossby mode dispersion properties are examined for discretisations of the linearized rotating shallow-water equations using the $P1_{DG}$-$P2$ finite element pair on arbitrary triangulations in planar geometry. A discrete Helmholtz decomposition of the functions in the velocity space based on potentials taken from the pressure space is used to provide a complete description of the numerical wave propagation for the discretised equations. In the $f$-plane case, this decomposition is used to obtain decoupled equations for the geostrophic modes, the inertia-gravity modes, and the inertial oscillations. As has been noticed previously, the geostrophic modes are steady. The Helmholtz decomposition is used to show that the resulting inertia-gravity wave equation is third-order accurate in space. In general the \pdgp finite element pair is second-order accurate, so this leads to very accurate wave propagation. It is further shown that the only spurious modes supported by this discretisation are spurious inertial oscillations which have frequency $f$, and which do not propagate. The Helmholtz decomposition also allows a simple derivation of the quasi-geostrophic limit of the discretised $P1_{DG}$-$P2$ equations in the $\beta$-plane case, resulting in a Rossby wave equation which is also third-order accurate.Comment: Revised version prior to final journal submissio

An improved quadrilateral flat element with drilling degrees of freedom for shell structural analysis

This paper reports the development of a simple and efficient 4-node flat shell element with six degrees of freedom per node for the analysis of arbitrary shell structures. The element is developed by incorporating a strain smoothing technique into a flat shell finite element approach. The membrane part is formulated by applying the smoothing operation on a quadrilateral membrane element using Allman-type interpolation functions with drilling DOFs. The plate-bending component is established by a combination of the smoothed curvature and the substitute shear strain fields. As a result, the bending and a part of membrane stiffness matrices are computed on the boundaries of smoothing cells which leads to very accurate solutions, even with distorted meshes, and possible reduction in computational cost. The performance of the proposed element is validated and demonstrated through several numerical benchmark problems. Convergence studies and comparison with other existing solutions in the literature suggest that the present element is efficient, accurate and free of lockings

Modelling the induced magnetic signature of naval vessels

In the construction of naval vessels stealth is an important design feature. With recent advances in electromagnetic sensor technology the war time threat to shipping posed by electromagnetically triggered mines is becoming more significant and consequently the need to understand, predict and reduce the electromagnetic signature of ships is growing. There are a number of components to the electromagnetic field surrounding a ship, with each component originating from different physical processes. The work presented in this study is concerned with the magnetic signature resulting from the magnetisation of the ferromagnetic material of the ship, under the influence of the earth's magnetic field. The detection threat arising from this induced magnetic signature has been known for many years, and consequently, warships are generally fitted with degaussing coils which aim to generate a masking field to counteract this signature. In this work computational models are developed to enable the induced magnetic signature and the effects of degaussing coils to be studied. The models are intended to provide a tool set, to aid the electromagnetic signature analyst in ensuring that pre-production designs of a vessel lie within specified induced magnetic signature targets. Techniques presented where also allow the rapid calculation of currents in degaussing coils. This is necessary because the induced magnetisation of a vessel changes with orientation. Three models are presented within this work. The first model represents a ship as a simple geometric shape, a prolate spheroidal shell, of a given relative permeability. Analytical expressions are derived which characterise the magnetic perturbation to a previously uniform magnetic field, the earth's magnetic field, when the spheroid is placed within its influence. These results provide a quantitative insight into the shielding of large internal magnetic sources by the hull. This model is intended for use in preliminary design studies. A second model is described which is based on the finite element method. This is a numerical model which has the capability of accurately reproducing the relatively complex geometry of a ship and of including the effects of degaussing coils. For these reasons this model is intended for detailed quantitative studies of the induced magnetic signature. A method is described to calculate the optimal set of degaussing coil currents required to minimise the induced magnetic signature. The induced signature without and with degaussing is presented. For the successful application of the finite element method the generation of a mesh is of extreme importance. In this work a mesh generation procedure is described which permits meshes to be generated around a collection of planar surfaces. The relatively complex geometry of a ship can be easily specified as a number of planar surfaces and from this, the finite element mesh can be automatically generated. The automatic mesh generation detailed in this work eliminates an otherwise labour intensive step in the analysis procedure. These techniques are sufficiently powerful to allow meaningful calculations for real ships to be performed on desk-top computers of modest power. An example is presented which highlights the application of this model to a hypothetical ship structure. The third model detailed is specifically designed to study the induced magnetic signature of mine countermeasures vessels. Here the induced magnetic signature is no longer dominated by the gross structure of the ship, which is constructed from non-magnetic materials, but arises from the combined effect of the individual items of machinery onboard the craft

On organizing principles of Discrete Differential Geometry. Geometry of spheres

Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. In this survey we discuss the following two fundamental Discretization Principles: the transformation group principle (smooth geometric objects and their discretizations are invariant with respect to the same transformation group) and the consistency principle (discretizations of smooth parametrized geometries can be extended to multidimensional consistent nets). The main concrete geometric problem discussed in this survey is a discretization of curvature line parametrized surfaces in Lie geometry. We find a discretization of curvature line parametrization which unifies the circular and conical nets by systematically applying the Discretization Principles.Comment: 57 pages, 18 figures; In the second version the terminology is slightly changed and umbilic points are discusse