Free-form Shape Design Research Articles

Sketch2PQ: Freeform Planar Quadrilateral Mesh Design via a Single Sketch.

The freeform architectural modeling process often involves two important stages: concept design and digital modeling. In the first stage, architects usually sketch the overall 3D shape and the panel layout on a physical or digital paper briefly. In the second stage, a digital 3D model is created using the sketch as a reference. The digital model needs to incorporate geometric requirements for its components, such as the planarity of panels due to consideration of construction costs, which can make the modeling process more challenging. In this work, we present a novel sketch-based system to bridge the concept design and digital modeling of freeform roof-like shapes represented as planar quadrilateral (PQ) meshes. Our system allows the user to sketch the surface boundary and contour lines under axonometric projection and supports the sketching of occluded regions. In addition, the user can sketch feature lines to provide directional guidance to the PQ mesh layout. Given the 2D sketch input, we propose a deep neural network to infer in real-time the underlying surface shape along with a dense conjugate direction field, both of which are used to extract the final PQ mesh. To train and validate our network, we generate a large synthetic dataset that mimics architect sketching of freeform quadrilateral patches. The effectiveness and usability of our system are demonstrated with quantitative and qualitative evaluation as well as user studies.

Approximation of photovoltaic characteristics curves using Bézier Curve

Accuracy of solar cell model parameters is now a requirement for the manufacturing of appropriate photovoltaic models. Consequently, the methodologies of the parameters had already attracted immense interest among researchers over the years. This paper deals about the approximation of the characteristics curves of a solar cell using the Bézier curve method. This technique is selected to overcome the problems of the non-linearity, complexity, multivariate and multimodal of the characteristic's curves. The Bézier curves method is a parametric curve used throughout computer graphics and related disciplines. It is a robust methodology for constructing free-form curves, surfaces, shape design, and geometric representation of various products. Six different cases are carried out to test the validity of the proposed method, they are: (i) organic flexible dual junction amorphous silicon solar cell, (ii) silicon solar cell, (iii) Schutten Solar STP6-120/36 polycrystalline photovoltaic module, (iv) Schutten Solar STM6-40/36 monocrystalline photovoltaic module, (v) Photowatt-PWP 201 polycrystalline photovoltaic module and (vi) 230-W multi-silicon photovoltaic module under partial shading conditions. The technique is implemented and adapted by the mean of the De Casteljo algorithm to optimize the control points and minimize the distance between the experimental and calculated points. It is based on subdividing the curve into various segments and controlled by control points using the De Casteljo algorithm to minimize the distance between the experimental and theoretical points. The performance of the approximated characteristics curves is compared with experimental data as well as recent algorithms and techniques from the literature. For more validity, statistical error metrics from literature are computed and compared to ensure the accuracy of the approximated results and the performance of the proposed method. For instance, the comparison is carried out by resorting to the Individual Absolute Error (IAE) and the Relative Error (RE). Further, the Bias metrics and accuracy are calculated using the Autocorrelation Function (ACF), Tracking Signal (TS), and Normalized Forecast Metric (NFM). The proposed method provides a low error for the six case studies and better than that obtained by the other algorithms.

Analysis of the construction of a reinforced-concrete free-form roof formwork and the development of a unit-construction method

Since the advent of the modernist movement in architecture, organic shapes have attracted designers. Digital fabrication technology and three-dimensional computer-aided design have made the architectural design of free-form shapes much easier. However, the construction of these shapes still has many risks in terms of cost and time management. This study aims to develop a new unit construction method by integrating digital fabrication with the conventional wooden construction technique on a large architectural scale and compare its productivity throughout the construction process to the conventional method. This paper first illustrates the current situation of free-form shape production by surveying a reinforced-concrete roof-construction project using a conventional construction method in Japan. We then develop a new construction method in which curved surfaces are divided into portable units in order to reduce on-site labor and ensure a high degree of accuracy. Finally, we analyze its productivity by conducting some assembly experiments. The new method reduces on-site labor costs by 93% compared with the conventional method.

Mechanics of kerf patterns for creating freeform structures

A relief cutting method, or kerfing, is considered to create flexible freeform surfaces from relatively stiff and thick panels. The flexibility and moldability are achieved by introducing slender components within the panel, forming kerf patterns, and hence reducing the second moment and polar moment of an area of the solid panel. This paper presents a systematic study on the deformations of kerf unit-cells and of kerf panels. Two different kerf patterns, i.e., square and hexagon, with various cut densities are studied. The effects of different cutting density and kerf patterns on the stretching, bending, and twisting deformations are examined. Understanding the influence of kerf patterns and cut densities on various deformation mechanisms will guide the design of freeform complex shapes out of kerf panels. Experimental tests were performed on unit-cells under different boundary conditions, e.g., uniaxial and biaxial stretching and bending. The tests were also performed on kerf panels with different kerf patterns and varying cut densities. We used a nonlinear beam element in order to describe the deformations of the slender components within the kerf patterns. We compared the overall deformations in the kerf unit-cells and panels from the beam element model and experimental tests. Using the kerfing technique allows for generating flexible structures with complex geometries from mass-produced panels of standard shape and size. When using the kerfing method to achieve the desired surface topology, the stresses, strains, and displacements in the surface will depend on the kerf pattern, cut density, and constituent behavior.

CASTonCAST: Architectural freeform shapes from precast stackable components

<div data-canvas-width="505.6672666666668">This article introduces the CASTonCAST system for the design and production of architectural freeform shapes from precast stackable components. This system is composed of two complementary parts: a novel manufacturing technique of precast stackable building components and a new geometric method for the design of freeform shapes by means of stackable solid tiles. This paper describes both parts of the system by means of physical prototypes and geometric studies.</div><p class="IJACAbstractText"> </p>

Novel free-form hohlraum shape design and optimization for laser-driven inertial confinement fusion

The hohlraum shape attracts considerable attention because there is no successful ignition method for laser-driven inertial confinement fusion at the National Ignition Facility. The available hohlraums are typically designed with simple conic curves, including ellipses, parabolas, arcs, or Lame curves, which allow only a few design parameters for the shape optimization, making it difficult to improve the performance, e.g., the energy coupling efficiency or radiation drive symmetry. A novel free-form hohlraum design and optimization approach based on the non-uniform rational basis spline (NURBS) model is proposed. In the present study, (1) all kinds of hohlraum shapes can be uniformly represented using NURBS, which is greatly beneficial for obtaining the optimal available hohlraum shapes, and (2) such free-form uniform representation enables us to obtain an optimal shape over a large design domain for the hohlraum with a more uniform radiation and higher drive temperature of the fuel capsule. Finally, a hohlraum is optimized and evaluated with respect to the drive temperature and symmetry at the Shenguang III laser facility in China. The drive temperature and symmetry results indicate that such a free-form representation is advantageous over available hohlraum shapes because it can substantially expand the shape design domain so as to obtain an optimal hohlraum with high performance.

Partial shape-preserving splines

A complex geometric shape is often a composition of a set of simple ones, which may differ from each other in terms of their mathematical representations and the ways in which they are constructed. One of the necessary requirements in combining these simple shapes is that their original shapes can be preserved as much as possible. In this paper, a set of partial shape-preserving (PSP) spline basis functions is introduced to smoothly combine a collection of shape primitives with flexible blending range control. These spline basis functions can be considered as a kind of generalization of traditional B -spline basis functions, where the shape primitives used are control points or control polygons. The PSP-spline basis functions have all the advantages of the conventional B -spline technique in the sense that they are nonnegative, piecewise polynomial and of property of partition of unity. However, PSP-spline is a more powerful freeform geometric shape design technique in the sense that it is also a kind of shape-preserving spline. In addition, the PSP-spline technique implicitly integrates the weights of shape control primitives into its basis functions, which allows users to design a required geometric shape based on weighted control primitives. Though its basis functions are simply piecewise polynomial functions, it has the same shape design strengths as the rational piecewise polynomial based spline techniques such as NURBS. In particular, when control shape primitives are specified as a set of control points, PSP-spline behaves like a polygon smoother, with which a shape can be designed to approximate the specified control polygon or control mesh smoothly with any required precision. Consequently, a richer set of geometric shapes can be built using a relatively smaller set of control points.

Parametric free-form shape design with PDE models and reduced basis method

We present a coupling of the reduced basis methods and free-form deformations for shape optimization and design of systems modelled by elliptic PDEs. The free-form deformations give a parameterization of the shape that is independent of the mesh, the initial geometry, and the underlying PDE model. The resulting parametric PDEs are solved by reduced basis methods. An important role in our implementation is played by the recently proposed empirical interpolation method, which allows approximating the non-affinely parameterized deformations with affinely parameterized ones. These ingredients together give rise to an efficient online computational procedure for a repeated evaluation design environment like the one for shape optimization. The proposed approach is demonstrated on an airfoil inverse design problem.

A simple method for interpolating meshes of arbitrary topology by Catmull–Clark surfaces

Interpolating an arbitrary topology mesh by a smooth surface plays important role in geometric modeling and computer graphics. In this paper we present an efficient new algorithm for constructing Catmull–Clark surface that interpolates a given mesh. The control mesh of the interpolating surface is obtained by one Catmull–Clark subdivision of the given mesh with modified geometric rule. Two methods—push-back operation based method and normal-based method—are presented for the new geometric rule. The interpolation method has the following features: (1) Efficiency: we obtain a generalized cubic B-spline surface to interpolate any given mesh in a robust and simple manner. (2) Simplicity: we use only simple geometric rule to construct control mesh for the interpolating subdivision surface. (3) Locality: the perturbation of a given vertex only influences the surface shape near this vertex. (4) Freedom: for each edge and face, there is one degree of freedom to adjust the shape of the limit surface. These features make interpolation using Catmull–Clark surfaces very simple and thus make the method itself suitable for interactive free-form shape design.

Geometry of multi-layer freeform structures for architecture

The geometric challenges in the architectural design of freeform shapes come mainly from the physical realization of beams and nodes. We approach them via the concept of parallel meshes, and present methods of computation and optimization. We discuss planar faces, beams of controlled height, node geometry, and multilayer constructions. Beams of constant height are achieved with the new type of edge offset meshes. Mesh parallelism is also the main ingredient in a novel discrete theory of curvatures. These methods are applied to the construction of quadrilateral, pentagonal and hexagonal meshes, discrete minimal surfaces, discrete constant mean curvature surfaces, and their geometric transforms. We show how to design geometrically optimal shapes, and how to find a meaningful meshing and beam layout for existing shapes.

Geometry of multi-layer freeform structures for architecture

The geometric challenges in the architectural design of freeform shapes come mainly from the physical realization of beams and nodes. We approach them via the concept of parallel meshes, and present methods of computation and optimization. We discuss planar faces, beams of controlled height, node geometry, and multilayer constructions. Beams of constant height are achieved with the new type of edge offset meshes. Mesh parallelism is also the main ingredient in a novel discrete theory of curvatures. These methods are applied to the construction of quadrilateral, pentagonal and hexagonal meshes, discrete minimal surfaces, discrete constant mean curvature surfaces, and their geometric transforms. We show how to design geometrically optimal shapes, and how to find a meaningful meshing and beam layout for existing shapes.

Interpolation over arbitrary topology meshes using a two-phase subdivision scheme

The construction of a smooth surface interpolating a mesh of arbitrary topological type is an important problem in many graphics applications. This paper presents a two-phase process, based on a topological modification of the control mesh and a subsequent Catmull-Clark subdivision, to construct a smooth surface that interpolates some or all of the vertices of a mesh with arbitrary topology. It is also possible to constrain the surface to have specified tangent planes at an arbitrary subset of the vertices to be interpolated. The method has the following features: 1) It is guaranteed to always work and the computation is numerically stable, 2) there is no need to solve a system of linear equations and the whole computation complexity is O(K) where K is the number of the vertices, and 3) each vertex can be associated with a scalar shape handle for local shape control. These features make interpolation using Catmull-Clark surfaces simple and, thus, make the new method itself suitable for interactive free-form shape design.

Warp sculpting

The task of computer-based free-form shape design is fraught with practical and conceptual difficulties. Incorporating elements of traditional clay sculpting has long been recognized as a means of shielding the user from these complexities. We present warp sculpting, a variant of spatial deformation, which allows deformations to be initiated by the rigid body transformation or uniform scaling of volumetric tools. This is reminiscent of a tool imprinting, flexing, and molding clay. Unlike previous approaches, the deformation is truly interactive. Tools, encoded in a distance field, can have arbitrarily complex shapes. Although individual tools have a static shape, several tools can be applied simultaneously. We enhance the basic formulation of warp sculpting in two ways. First, deformation is toggled to automatically overcome the problem of "sticky" tools, where the object's surface clings to parts of a tool that are moving away. Second, unlike many other spatial deformations, we ensure that warp sculpting remains foldover-free and, hence, prevent self-intersecting objects.

A User Interface for Free-Form Shape Design at the Responsive Workbench

This paper presents a virtual environment based user interface for the conceptual design of free form surface models from scratch. The user performs sketching and elaboration directly within a projection-based, table-like environment. He uses head-tracked stereo glasses and simple input devices. We describe user interface components for creation, manipulation, and application control, which were specifically designed for use in a 3D environment. These components are part of a two-handed interaction scheme. In our modeling approach, the user draws curves and constructs a curve network that forms the skeleton of the surface. Automatic surfacing methods generate shapes that correspond to the outlined boundary, thus freeing the designer from specifying all surface parameters by hand. We demonstrate how the use of a virtual environment benefits such creation and manipulation tasks.

Smoothing rational B-spline curves using the weights in an optimization procedure

Freeform shape design is typically accomplished in an interactive manner and shapes generated by a computer are rarely immediately acceptable. The available techniques for any subsequent modifications depend on the chosen representation for the geometry. In many computer aided styling and design systems which use nonuniform rational B-splines (NURBS) for representation of geometry, the use of the weights as a shape control tool is very inadequately supported. In fact they are often hidden from the user and therefore remain unused. This paper investigates the possibilities of entering the weights in an automatic fairing process. In order to produce a curve with a more gradual change in curvature and the smallest deviation from its initial shape the perturbation of the weights is stated as an optimization problem. Examples of applications to automotive shape design are presented and discussed.

DEFORMATION OF N-DIMENSIONAL OBJECTS

This paper presents a new technique for computing space deformations that interpolate a set of user-defined constraints. Constraints are specified by indicating the images of selected points. The deformation is formulated as the product of a polynomial function f of ℝn into a higher-dimensional space, ℝm, with a linear projection from ℝm back to ℝn. The projection matrix is computed using a pseudo-inverse technique. For sufficient m, the degrees of freedom may be used to optimize potential functions controlled by attracting and repulsing points. A prototype implementation is presented, which demonstrates the application of this technique to the interactive design of free-form shapes (when n=3) and of deformation processes in space-time domain (when n=4).

Deformable curve and surface finite-elements for free-form shape design

The finite element method is applied to generate primitives that build continuous deformable shapes designed to support a new free-form modeling paradigm. The primitives autonomously deform to minimize an energy functional subject to user controlled geometric constraints and loads. The approach requires less user input than conventional free-form modeling approaches because the shape can be parameterized independently of the number of degrees of freedom needed to describe the shape.Both a curve and a surface finite element are developed. The properties of these geometric primitives have been engineered to support an interactive three phase approach for defining very fair free-form shapes as found in automobiles, ship hulls and car bodies. The shape's character lines or folds and edges are defined with deformable curve segments. These character lines are then "skinned" with a deformable surface. The final shape is sculpted interactively by applying loads to the surface to control the surface shape between character lines. Shapes created with this technique enjoy the advantage that they are already meshed for further finite element analysis.

A CAD/CAM system for free-form shape design.

It is necessary to construct useful CAD/CAM systems which can design attractive, high-quality objects in short periods of time. An interactive CAD/CAM system for free-form shape design, which can make an attractive object such as a brooch or a ring in only few hours from the first to the last step of the design processes, is presented. This system consists of the following three parts. (1)Interactive polyhedral modeller, (2) Translator which translates polyhedra into free-form curved shapes, (3) Stereo lithography device which can make 3D objects from free-form curved shape data. Illustrative examples of shape models generated interactively and a photograph of a real object made by using the stereo lithography device are shown.

意匠的に特徴ある自由形状の設計およびその工具経路自動生成システムの開発

An integrated freeform shape design and manufacturing system in which collision-free tool paths can be generated directly from the designed shape has been developed. The system employs a new surface patch representation method which has shape control points and a method of connecting surface patches even if their boundaries are symmetrical along a common boundary. This system runs on an engineering workstation and has a menu-driven human interface. With this system, designers are able to represent aesthetically pleasing shapes in the computer with greater precision than previously and display them on a color graphic terminal using a variety of new display techniques so that the designed shapes can be thoroughly examined. These features together with the system's automatic tool path generation reduce leadtime by 80%, compared with conventional procedures. A camera handle designed and manufactured using this system is shown.

Design of solids with free-form surfaces

We propose a unified method of generating a wide range of three dimensional objects from polyhedra to solids with free-form surfaces. Modeling systems for polyhedra and systems for free-form surfaces have been developed independently in the past because of the difference in their underlying theory and practices. However, this is not desirable for a designer. So in this paper, we have shown a method in which a wide range of shapes are generated in one system by using local modifications. Local modifications are procedures used to change the shape of solids locally. The construction and the modification of three dimensional shapes by these procedures are natural and easy for a designer in many cases. The implementation of these procedures in a computer is easy and their execution does not require much time. Our method to construct a solid with free-form surfaces consists of following three phases. 1) A solid which serves as a basis of free-form shape design is generated by local modifications. Edges of this solid are straight lines but its faces are not necessarily flat planes. 2) From this model, a curve model which adequately represents the characteristics of a free-form shape is generated. 3) Surface equations interpolating over the curve model are generated. We have made a geometric modeling system MODIF. Using this system, a complicated solid with free-form surfaces can be designed easily. MODIF can generate color shaded pictures and cutter path data for making a real object model by NC machining tool.