Boolean operations based on the planar polyhedral representation

Boolean operations based on the planar polyhedral representation: Comput. & Graph. Vol 12 No 1 (1988) pp 59–64

This paper presents a unified differentiable boolean operator for implicit solid shape modeling using Constructive Solid Geometry (CSG). Traditional CSG relies on min, max operators to perform boolean operations on implicit shapes. But because these boolean operators are discontinuous and discrete in the choice of operations, this makes optimization over the CSG representation challenging. Drawing inspiration from fuzzy logic, we present a unified boolean operator that outputs a continuous function and is differentiable with respect to operator types. This enables optimization of both the primitives and the boolean operations employed in CSG with continuous optimization techniques, such as gradient descent. We further demonstrate that such a continuous boolean operator allows the modeling of both sharp mechanical objects and smooth organic shapes with the same framework. Our proposed boolean operator opens up new possibilities for future research toward fully continuous CSG optimization.

PurposeThe purpose of this paper is to develop robust metamodels, which allow propagating parametric uncertainties, in the presence of localized nonlinearities, with reduced cost and without significant loss of accuracy.Design/methodology/approachThe proposed metamodels combine the generalized polynomial chaos expansion (gPCE) for the uncertainty propagation and reduced order models (ROMs). Based on the computation of deterministic responses, the gPCE requires prohibitive computational time for large-size finite element models, large number of uncertain parameters and presence of nonlinearities. To overcome this issue, a first metamodel is created by combining the gPCE and a ROM based on the enrichment of the truncated Ritz basis using static residuals taking into account the stochastic and nonlinear effects. The extension to the Craig–Bampton approach leads to a second metamodel.FindingsImplementing the metamodels to approximate the time responses of a frame and a coupled micro-beams structure containing localized nonlinearities and stochastic parameters permits to significantly reduce computation cost with acceptable loss of accuracy, with respect to the reference Latin Hypercube Sampling method.Originality/valueThe proposed combination of the gPCE and the ROMs leads to a computationally efficient and accurate tool for robust design in the presence of parametric uncertainties and localized nonlinearities.

Some important trends in geometric modeling are the reliance on solid models rather than surface-based models and the enhancement of the expressive power of models, by using free-form objects in addition to the usual geometric primitives and by incorporating physical principles. An additional trend is the emphasis on interactive performance. In this paper, we integrate all of these requirements into a single geometric primitive by endowing the tri-variate tensor-product free-form solid with several important physical properties, including volume and internal deformation energy. Volume preservation is of benefit in several application areas of geometric modeling, including computer animation, industrial design and mechanical engineering. However, previous physics-based methods, which have usually used some form of "energy", have neglected the issue of volume (or area) preservation. We present a novel method for modeling an object composed of several tensor-product solids while preserving the desired volume of each primitive and ensuring high-order continuity constraints between the primitives. The method utilizes the Uzawa algorithm for non-linear optimization, with objective functions based on deformation energy or least squares. We show how the algorithm can be used in an interactive environment by relaxing exactness requirements while the user interactively manipulates free-form solid primitives. On current workstations, the algorithm runs in real-time for tri-quadratic volumes and close to real-time for tri-cubic volumes.

In data mining, mining high utility itemset (HUI) is one among the recent thrust area that receives several approaches for solving it in an effective manner. In the past decade, addressing optimization problems using evolutionary algorithms are an unavoidable strategy due to its convergence towards optimal solution within the stipulated time. The results of evolutionary algorithms on various optimization problems are far effective when compared to the exhaustive approaches with respect to computational time. The problem with HUI is discovering a set of items from a transactional database that possess high level of utility when compared with other distinctive sets. This problem becomes harder while addressing the count of items in the database while its higher and computational time to solve this problem using exhaustive search becomes exponential as proposition of items in transaction database increases. In this paper, an optimization model based on the biological behaviour of grey wolf is proposed; the model namely grey wolf optimization algorithm is used to solve HUI using five different Boolean operations. The proposed model is evaluated using standard performance metrics over synthetic datasets and real-world datasets. The proposed model results are then compared with recent HUIM models to show the significance.

Solid modeling is the technique for representing and manipulating a complete description of physical objects. Valid representations of solid are developed base on a mathematical description of rigid object. A valid solid representation scheme provides sufficient information for geometric operations on objects, e.g. Boolean operations, collision detection. The description here introduces the Fundamental concepts in solid modeling are introduced. Characteristics of a solid representation scheme are discussed. Two popular solid representation schemes, the constructive solid geometry and the boundary representation schemes, are described. The fundamental algorithms for solid modeling, point membership classification, and Boolean operations, are also discussed. Validity of the boundary representation scheme, the Euler operators, as well as the Winged‐edge data structure for the implementation of boundary representation scheme are also presented.

PurposeAutonomous orbit determination using geomagnetic measurements is an important backup technique for safe spacecraft navigation with a mere magnetometer. The geomagnetic model is used for the state estimation of orbit elements, but this model is highly nonlinear. Therefore, many efforts have been paid to developing nonlinear filters based on extended Kalman filter (EKF) and unscented Kalman filter (UKF). This paper aims to analyze whether to use UKF or EKF in solving the geomagnetic orbit determination problem and try to give a general conclusion.Design/methodology/approachThis paper revisits the problem and from both the theoretical and engineering results, the authors show that the EKF and UKF show identical estimation performances in the presence of nonlinearity in the geomagnetic model.FindingsWhile EKF consumes less computational time, the UKF is computationally inefficient but owns better accuracy for most nonlinear models. It is also noted that some other navigation techniques are also very similar with the geomagnetic orbit determination.Practical implicationsThe intrinsic reason of such equivalence is because of the orthogonality of the spherical harmonics which has not been discovered in previous studies. Thus, the applicability of the presented findings are not limited only to the major problem in this paper but can be extended to all those schemes with spherical harmonic models.Originality/valueThe results of this paper provide a fact that there is no need to choose UKF as a preferred candidate in orbit determination. As UKF achieves almost the same accuracy as that of EKF, its loss in computational efficiency will be a significant obstacle in real-time implementation.

For large-scale hydraulic and hydroelectric project, it has complex underground tunnel group. Traditional design for tunnel was usually based on the method of CAD in 2D which could not embody the geological features where the tunnel was passed, and also could not describe the space relations of the tunnel networks. With the development of graphics and GIS, 3D Design for hydraulic tunnel is to be possible. This paper proposes one method for modeling hydraulic tunnel in 3D and analyzing its space distance in 3D based on distance analysis of GIS. The space distance is important for construction safety protection. The design of tunnel can be divided into two parts: the section and the axis. The design of section is based on its function of tunnel, such as non-pressure tunnel and pressure tunnel. The design of axis is based on space distribution. The main procedure of modeling the tunnel: determining the axis, and then choosing the corresponding section. The section needs to be performed the translated and rotated matrix to the proper position along the axis. The next step is to extrude the section along the axis, so the entity model of one tunnel is created. Two entities are the outer one and the inner one. The tunnel model can be acquired through the outer tunnel subtracting the inner one by Boolean operations. The tunnel model is one entity by complex geometric transformations and Boolean operations. Performing the space distance analysis directly is quite difficult, and it needs find an indirect mean. The axis of tunnel has relatively simple expression, so, the link between the tunnel model and its corresponding axis and section is established. Here, the C++ as the program design language, the map class is used to implement the link. Thus, it can find the axis and section equation from the tunnel. In virtue of such method, analyzing the space distance of complex solid model becomes easy. The calculating model is created and discussed how to calculate the minimum distance between tunnels. At first, the minimum distance between axes is calculated, which formed a minimum distance Vector. Second, the parts between the axes and sidewall of tunnel are subtracted. Finally, this method is applied to one large-scale hydraulic and hydroelectric project.

The system identification technique is applied to the FDTD Diakopties method. The computational time and computer storage are significantly reduced. This technique is successfully used in the FDTD Diakoptics simulation of active microwave circuits.

This paper proposes the structural-acoustic coupled analytical method. The special feature of this method is that the structural vibration is approximated by the superposition of the vibration modes which are included in the required frequency band, and the truncated higher-and lower-frequency band are compensated by the residual stiffness and mass, respectively. As a numerical example, a flexible rectangular prism model is used and the sound pressure in the model is calculated using the proposed method. It is shown that the calculated results agree with the exact ones which are obtained by using the full frequency band, and computer storage and calculation time are saved. As an application of the proposed method, simple acoustic control is carried out numerically. It is shown that the steady-state sound pressure at a certain point can be controlled using the secondary sound source operated with the adjusted signal based on the transfer function in the acoustic field.

The symplectic schemes are extended to the solution of one-dimensional time-independent Schrödinger equation. The Schrödinger equation is first transformed into a Hamiltonian canonical equation by means of the Legendre transformation, and then two methods are developed to solve the numerical solution of the one-dimensional time-independent Schrödinger equation: the symplectic scheme-matrix eigenvalue method and the symplectic scheme-shooting method. Both methods are applied to the calculations of a one-dimensional harmonic oscillator, the hydrogen atom, and a double-well anharmonic oscillator. It is shown that the numerical results of the two methods are nearly the same and are in good agreement with the exact ones when the step length is taken to be properly small. The computation with the symplectic scheme-shooting method spends less computer time than that with the symplectic scheme-matrix eigenvalue method. And thus the symplectic scheme-shooting method is a better numerical method for the calculation of the eigenvalue problem. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 343–349, 2000

An algorithm for finding intersection between ball B-spline curves

Conic splines play an important role in engineering geometry and computer graphics, in particular, in geometric and solids modeling where the majority of objects are bounded by simple surfaces such as natural quadrics (plane, cone, cylinder, and sphere). The integration of analytic and free‐form curves into solids modeling systems is one of the most important tasks in today's solids modelers. This paper presents algorithms that represent conic splines in nonuniform rational B‐spline (NURBS) form, thus incorporating them into a system based on free‐form geometry. The algorithms take as input the end points and end tangents of the segment, and one arbitrary point somewhere along the arc, and return the precise NURBS representation of the curve. It was originally developed for supporting Boolean operations on trimmed surfaces, i.e., to compute conic arc surface‐surface intersections in true 3‐D space. Since during Boolean operations all kinds of segments can pop up, special attention is paid to such cases as parallel end tangents and arcs contained outside the convex hull. Knot insertion is applied to obtain a representation that is numerically as stable as possible. The input required to generate a curve is very general, therefore the spline conversion has different kinds of applications apart from surface‐surface intersection. Some examples are font design, shape generation, geometry conversion and processing, and curve approximation.

Using dexels to make hollow models for rapid prototyping

We propose a new robust algorithm for Boolean operations on solid models. The algorithm produces a consistent intersection graph between two input solids whose geometrical data are represented in floating point numbers. In order to prevent numerical calculation errors and inaccuracy of input data from causing inconsistency of the output, we put higher priority on symbolical connectivity of the edge‐face intersection points than their numerical nearness. Each edge‐face intersection point is symbolically represented using face names, which generate connectivity relations between the intersection points and the intersection line segments. The symbols with the same connectivity are made into clusters. The intersection line segments connected together at their end clusters form the intersection graph of two solids. Inconsistency of the connectivity of the clusters is detected and the intersection graph is corrected automatically. We describe the algorithm in detail for polyhedral solids, discuss extension to curves solids, and show its effectiveness by some examples of Boolean operations for two solids whose faces intersect at a very small angle.