HyBoDT: Hybrid Bounded Distance Transforms of Trimmed NURBS Models

Distance field representation of objects in 3D space has several applications such as shape manipulation, graphics rendering, path planning, etc. Distance transforms (DTs) are discrete representations of distance fields in a regular voxel grid. The two main limitations of using distance transforms are that they are compute-intensive, and there are errors introduced while representing the object using DTs. In this work, we develop an hybrid GPU-accelerated marching wavefront method for computing DTs of models composed of trimmed NURBS surfaces with theoretical bounds. Our hybrid marching approach eliminates the error due to calculating approximate distances by marching. We also calculate the bounds on the error introduced due to the tessellation of the trimmed NURBS surfaces and calculate the propagation of these bounds in computing the DT. Finally, we present computation times for both 2D and 3D GPU DTs of test objects. We show that our GPU-accelerated approach is significantly faster than existing CPU-based methods.

Quasi-Isotropic Initial Triangulation of NURBS Surfaces

Isotropic triangulation of NURBS surfaces provides high quality triangular meshes, where all triangles are equilateral. This isotropy increases representation quality and analysis accuracy. We introduce a new algorithm to generate quasi-isotropic triangulation on NURBS surfaces at once, with no prior meshing. The procedure consists of one front made of vertexes that advances in a divergence manner avoiding front collision. Vertexes are calculated by arcs intersection whose radius length is estimated by trapezoidal rule integration of directional derivatives. The parameter space is discretized in partitions such that the error of trapezoidal rule is controlled. A new space, called pattern space, is used to infer the direction of the arcs intersection. Derivatives, whose analytical computation is expensive, are estimated by NURBS surface fitting procedures, which raise the speed of the process. The resultant algorithm is robust and efficient. The mesh achieved possesses most of the triangles equilateral and with high uniformity of sizes. The performance is illustrated by measuring angles, vertex valences and size uniformity in numerical examples

BICGSTAB-FFT Method of Moments with NURBS for Analysis of Planar Generic Layouts Embedded in Large Multilayer Structures

BICGSTAB-FFT method of moment (MM) scheme is proposed to analyze several levels of planar generic layouts embedded in large multilayer structures when the layout geometries are modeled by NURBS surfaces. In this scheme, efficient computation of normalized error defined in iterative bi-conjugate gradient stabilized (BICGSTAB) method for large multilayer structure analysis problems is implemented. The efficient computation is based on pulse expansion with dense equi-spaced mesh of generalized rooftop basis functions (BFs) defined on NURBS surfaces and equivalent periodic problem (EPP) in order to apply fast Fourier transforms (FFT). Moreover, efficient computation of Green’s functions for multilayer structure is implemented for near and far field regions. Experimental and numerical validations of whole printed reflect array antennas of electrical size between 8 and 16 times the vacuum wavelengths are shown. In these validations, CPU time consumptions of the proposed method are obtained with results between few minutes and half an hour using a conventional laptop.

An Improvement on the Upper Bounds of the Partial Derivatives of NURBS Surfaces

The Non-Uniform Rational B-spline (NURBS) surface not only has the characteristics of the rational Bézier surface, but also has changeable knot vectors and weights, which can express the quadric surface accurately. In this paper, we investigated new bounds of the first- and second-order partial derivatives of NURBS surfaces. A pilot study was performed using inequality theorems and degree reduction of B-spline basis functions. Theoretical analysis provides simple forms of the new bounds. Numerical examples are performed to illustrate that our method has sharper bounds than the existing ones.

Generating NURBS cladding and structures with parametric programming and BIM

The process of designing and building curvilinear architectures is still challenging. The use of multiple applications with distinctive design paradigms are unlikely to disappear. The interoperability used here was not only the conventional one. It was also ‘live’, in ‘real time’, with two of the applications involved opened and running simultaneously. A design workflow based on the use of form-forming applications connected via parametric programming to building information modeling, BIM, was proposed. The main objective was to facilitate designing and building curvilinear architectures and their supporting structures using simultaneously two different design paradigms. The tools needed in our research can be summarized as follows: NURBS Lofting for surface creation, contouring for modular slicing and structural axis grid definition, sweeping along axes for surface creation of the curved beams of I profile and paneling for the subdivision of curved surfaces into planar fractions. Parametric programming was used to automate sweeping along axes to generating curved I-beams and paneling to subdivide the NURBS surfaces into planar fractions. To the best of our knowledge, our major contribution resides in defining a workflow and developing new algorithms for facilitating designing NURBS surfaces and corresponding supporting structures through ‘live’ interoperability among different applications.

Geometric Modeling of Compound NURBS Surfaces

The design of sculptured surfaces occupies an essential area in the field of modern industrial, aerospace, and medical applications. The challenge is to design products that have complex features efficiently with great flexibility of editing in certain regions without affecting other regions, which the designer has no intent to modify. In this paper, we propose a surface design method based on compound NURBS surface to model automotive parts with 400 control points. First, a Non-Uniform B-Spline basis function is derived with a cubic degree and 20 control points. This method is utilized to design car posterior door, car hood, and rear car door as case studies.

Method of Moments Based on Equivalent Periodic Problem and FFT with NURBS Surfaces for Analysis of Multilayer Periodic Structures

In this paper, an efficient technique of computation of method of moments (MM) matrix entries for multilayer periodic structures with NURBS surface and Bézier patches modelling is proposed. An approximation in terms of constant pulses of generalized rooftop basis functions (BFs) defined on Bézier patches is proposed. This approximation leads discrete convolutions instead of usual continuous convolution between Green’s functions and BFs obtained by the direct mixed potential integral equation (MPIE) approach. An equivalent periodic problem (EPP) which contains the original problem is proposed to transform the discrete convolutions in discrete cyclic convolutions. The resultant discrete cyclic convolutions are computed by efficiently using the Fast Fourier Transform (FFT) procedure. The performance of the proposed method and direct computation of the MM entries are compared for phases of reflection coefficient. The proposed method is between 9 and 50 times faster than the direct computation for phase errors less than 1 deg. The proposed method exhibits a behaviour of CPU time consumption of O(NbLog10Nb) as the number Nb of BFs increases. This behaviour provides significant CPU time savings with respect to the expected behaviour of O(Nb2) provided by the direct computation of the MM matrix entries.