We present a method for approximating surface data of arbitrary topology by a model of smoothly connected B-spline surfaces. Most of the existing solutions for this problem use constructions with limited degrees-of-freedom or they address smoothness between surfaces in a post-processing step, often leading to undesirable surface behavior in proximity of the boundaries. Our contribution is the design of a local method for the approximation process. We compute a smooth B-spline surface approximation without imposing restrictions on the topology of a quadrilateral base mesh defining the individual B-spline surfaces, the used B-spline knot vectors, or the number of B-spline control points. Exact tangent plane continuity can generally not be achieved for a set of B-spline surfaces for an arbitrary underlying quadrilateral base mesh. Our method generates a set of B-spline surfaces that lead to a nearly tangent plane continuous surface approximation and is watertight, i.e., continuous. The presented examples demonstrate that we can generate B-spline approximations with differences of normal vectors along shared boundary curves of less than one degree. Our approach can also be adapted to locally utilize other approximation methods leading to higher orders of continuity.

In many manufacturing applications, robotic manipulators need to operate in cluttered environments. Quickly finding high-quality paths is very important in applications that require high part fix and frequent setup changes. This paper presents a point-to-point path planning framework for manipulators operating in cluttered environments. It uses a bi-directional tree-search to find path and facilitates finding a balance between path quality and planning time. The framework dynamically switches between search strategies based on the search progress to produce high-quality paths quickly. This paper three main contributions. First, we extend a previously developed sampling-based modular tree-search. Specifically, we present a strategy that can sample effectively in challenging regions of the search-space by using local approximations of the configuration space. Second, we add new strategies and scheduling logic that decreases the failure rate as well as the planning time compared to the prior work. We also present an inter-tree connection strategy that adapts to collision information gathered over time. We introduce a scheduling rule that regulates the exploitation of focusing hints derived from the workspace obstacles. Third, we present theoretical reasoning behind strategy switching and how it can help decrease planning times and increase path quality. Together, these new features the reduce average failure rate by a factor of 4 and improve the average planning time by 22 % over the previous work.

We present a new method for computing volume integrals based on data sampled on a regular Cartesian grid. We treat the case where the domain is defined implicitly by an inequality, and the input data include sampled values of the defining function and the integrand. The method employs Federer’s coarea formula (Federer, 1969, Geometric Measure Theory, Grundlehren der mathematischen Wissenschaften, Springer) to convert the volume integral to a one-dimensional quadrature over level set values where the integrand is an integral over a level set surface. Application of any standard quadrature method produces an approximation of the integral over the continuous range as a weighted sum of integrals over level sets corresponding to a discrete set of values. The integral over each level set is evaluated using the grid-based approach presented by Yurtoglu et al. (2018, “Treat All Integrals as Volume Integrals: A Unified, Parallel, Grid-Based Method for Evaluation of Volume, Surface, and Path Integrals on Implicitly Defined Domains,” J. Comput. Inf. Sci. Eng., 18 , p. 3). The new coarea method fills a need for computing volume integrals whose integrand cannot be written in terms of a vector potential. We present examples with known results, specifically integration of polynomials over the unit sphere. We also present Saye’s (2015, “High-Order Quadrature Methods for Implicitly Defined Surfaces and Volumes in Hyperrectangles,” SIAM J. Sci. Comput., 37 ) example of integrating a logarithmic integrand over the intersection of a bounding box with an open domain implicitly defined by a trigonometric polynomial. For the final examples, the input data is a grid of mixture ratios from a direct numerical simulation of fluid mixing, and we demonstrate that the grid-based coarea method applies to computing volume integrals when no analytical form of the implicit defining function is given. The method is highly parallelizable, and the results presented are obtained using a parallel implementation capable of producing results at interactive rates.

Stability control is critical to the exoskeleton robot controller design. Considering the complex structural characteristics of lower limb exoskeleton robots, the major challenge of the controller design is the accuracy and uncertainty of the dynamics model. To fill in this research gap, this study proposes successive approximation-based radial basis function (RBF) neural networks (NNs). The proposed model simplifies the lower limb exoskeleton robot as three degrees-of-freedom (3-DOF) model with the human hip joints for adduction/extension, bending/extension, and internal/external rotation. To minimize the gait tracking errors and stabilize the closed-loop system, a gait trajectory-based control and approximation model was proposed in this study. To verify the proposed method, a validation experiment was conducted for typical lower limb motions. The experiment results demonstrated the effectiveness of the proposed method.

Mesh generation for traditional finite element analysis has proven to be very difficult to fully automate especially using hexahedral elements for complex 3D geometry. Several modifications to the finite element method (FEM), such as the meshless methods, have been proposed for avoiding mesh generation. An alternative approach has recently gained popularity where the geometry, created as a solid model in cad software, is embedded or immersed in a nonconforming background mesh for analysis. In this approach, referred to here as the immersed boundary approach, a background mesh that is independent of the geometry is used for piecewise interpolation or approximation of the solution. Therefore, a uniform mesh with regular-shaped or undistorted elements can be used, and such a mesh is easy to generate automatically. When the geometry is immersed in the background mesh, the boundary elements are often only partly inside the geometry and the nodes of the mesh may not be on the boundaries. Many new methods have been developed to integrate over partial elements and to apply boundary and interface conditions when the boundaries of the geometries do not conform to the background mesh. These methods are reviewed in this article with particular emphasis on the implicit boundary method and step boundary method for applying boundary conditions. In addition, B-spline elements and several applications of the immersed boundary approach are surveyed including composite microstructures and structural elements for plates and shells.

Parametric finite element analysis (FEA) models are commonly used in iterative design processes to obtain an optimum model given a set of loads, constraints, objectives, and design parameters to vary. In some instances, it is desirable for a designer to obtain some intuition about how changes in design parameters can affect the FEA solution of interest, before simply sending the model through the optimization loop. For example, designers who wish to explore the design space and understand how each variable changes the output in a visual way, looking at the whole model and not just numbers or a response surface of a single FEA node. This could be accomplished by running the FEA on the parametric model for a set of part family members, but this can be very time consuming and only gives snapshots of the model's real behavior. This paper presents a method of visualizing the FEA solution of the parametric model as design parameters are changed in real-time by approximating the FEA solution using parametric FEA modeling, surrogate modeling methods, and visualization methods. The implementation develops a parametric FEA mode that includes mesh morphing algorithms that allow the mesh to change parametrically along with the model geometry. This allows the surrogate models assigned to each individual node to use the nodal solution of multiple finite element analyses as regression points to approximate the FEA solution. The surrogate models can then be mapped to their respective geometric locations in real-time. The results of the FEA calculations are updated in real-time as the parameters of the design model change allowing real-time visualization.

This paper formulates a new explicit expression for the generalized Jacobi polynomials (GJPs) in terms of Bernstein basis. We also establish and prove the basis transformation between the GJPs basis and Bernstein basis and vice versa. This transformation embeds the perfect least-square performance of the GJPs with the geometrical insight of the Bernstein form. Moreover, the GJPs with indexes corresponding to the number of endpoint constraints are the natural basis functions for least-square approximation of Bézier curves and surfaces. Application to multidegree reduction (MDR) of Bézier curves and surfaces in computer aided geometric design (CAGD) is given.

Modeling using developable surfaces plays an important role in computer graphics and computer aided design. In this paper, we investigate a new problem called variational developable surface interpolation (VDSI). For a polyline boundary P, different from previous work on interpolation or approximation of discrete developable surfaces from P, the VDSI interpolates a subset of the vertices of P and approximates the rest. Exactly speaking, the VDSI allows to modify a subset of vertices within a prescribed bound such that a better discrete developable surface interpolates the modified polyline boundary. Therefore, VDSI could be viewed as a hybrid of interpolation and approximation. Generally, obtaining discrete developable surfaces for given polyline boundaries are always a time-consuming task. In this paper, we introduce a dynamic programming method to quickly construct a developable surface for any boundary curves. Based on the complexity of VDSI, we also propose an efficient optimization scheme to solve the variational problem inherent in VDSI. Finally, we present an adding point condition, and construct a G1 continuous quasi-Coons surface to approximate a quadrilateral strip which is converted from a triangle strip of maximum developability. Diverse examples given in this paper demonstrate the efficiency and practicability of the proposed methods.

When digitally realized, virtual environments (VEs) do not perfectly match the physical environments they are supposed to emulate. This paper deals with energy aspects of such a mismatch, i.e., artificial energy leaks. A methodology is developed that employs smooth correction (SC) and leak dissipation (LD) to achieve a stable interconnection of the VE with the haptic device. The SC-LD naturally blends with the original laws for rendering the VE and gives rise to modified force feedback laws. These laws can be regarded as energy-consistent discretizations of their continuous-time counterparts. For some fundamental examples including virtual springs and masses, these laws are analytically reduced to simple closed-form equations. The methodology is then generalized to the multivariable case. Several experiments are conducted including a 2-DOF coupled nonlinear VE example, and a scenario leading to a sequence of contacts with a virtual object. Besides the conceptual advantage, simulation and experimental results demonstrate some other advantages of the SC-LD over well-known time-domain passivity methods. These advantages include improved fidelity, simpler implementation, and less susceptibility to produce impulsive/chattering response.

We present a robust and efficient approach to directly slicing implicit solids. Different from prior slicing techniques that reconstruct contours on the slicing plane by tracing the topology of intersected line segments, which is actually not robust, we generate contours by a topology guaranteed contour extraction on binary images sampled from given solids and a subsequent contour simplification algorithm which has the topology preserved and the geometric error controlled. The resultant contours are free of self-intersection, topologically faithful to the given r-regular solids and with shape error bounded. Therefore, correct objects can be fabricated from them by rapid prototyping. Moreover, since we do not need to generate the tessellated B-rep of given solids, the memory cost our approach is low—only the binary image and the finest contours on one particular slicing plane need to be stored in-core. Our method is general and can be applied to any implicit representations of solids.

B-spline surfaces are widely used in engineering practices as a flexible and efficient mathematical model for product design, analysis, and assessment. In this paper, we propose a new sequential B-spline surface construction procedure using multiresolution measurements. At each iterative step of the proposed procedure, we first update knots vectors based on bias and variance decomposition of the fitting error and then incorporate new data into the current surface approximation to fit the control points using Kalman filtering technique. The asymptotical convergence property of the proposed procedure is proved under the framework of sieves method. Using numerical case studies, the effectiveness of the method under finite sample is tested and demonstrated.

This paper presents an algorithm for exact contact prediction between moving objects bounded by curved surfaces. The algorithm uses hierarchies of oriented bounding boxes (HOBBs) and local numerical methods for finding contact. Objects need not be convex and are described using the B-rep scheme. The bounding faces are represented by nonuniform rational B-splines (NURBS). The collision time is sought in short time intervals during the motion, during which time is one of the problem variables. HOBBs are based on curvature regions of the surfaces. This criterion ensures that local numerical methods will converge to the contact points if they exist. The patches enclosed in overlapping leaf nodes are tested for contact by solving a system of nonlinear equations, based on the type of collision expected. The types of collision studied are cusp–cusp, cusp–ridge, cusp–face, ridge–ridge, ridge–face, and face–face collisions. The current algorithm is implemented and compared to an efficient polyhedral collision package, and results appear promising.

This paper presents a unified framework for computing a surface to approximate a target shape defined by discrete data points. A signed point-to-surface distance function is defined, and its properties are investigated, especially, its second-order Taylor approximant is derived. The intercorrelations between the signed and the squared distance functions are clarified, and it is demonstrated that the squared distance function studied in the previous works is just the Type I squared distance function deduced from the signed distance function. It is also shown that surface approximations under different criteria and constraints can all be formulated as optimization problems with specified requirements on the residual errors represented by the signed distance functions, and that classical numerical optimization algorithms can be directly applied to solve them since the derivatives of the involved objective functions and constraint functions are all available. Examples of global cutter position optimization for flank milling of ruled surface with a cylindrical tool, which requires approximating the tool envelope surface to the point cloud on the design surface following the minimum zone criterion, are given to confirm the validity of the proposed approach.

Faithfully obtaining design specifications from customer requirements is essential for successful designs. The natural lingual, inexact, incomplete, and vague attributes of customer requirements make it very difficult to map customer requirements to design specifications. In general design process, the design specifications are determined by designers based on their experience and intuition, and often a certain target value is set for a specification. However, it is on one hand very difficult, on the other hand unreasonable, so a suitable limit range rather than a certain value is preferred at the beginning of design, especially at the concept design process. In this paper, a simplified systematic approach of transforming customer requirements to design specifications is proposed. First, a two-stepped clustering approach for grouping customer requirements and design specifications based on the house of quality matrix is presented, by which the mapping is limited to within each group. To further simplify the inference mapping rules of customer requirements and design specifications, the minimal condition inference mapping rules for each design specification are extracted based on rough set theory. In the end, a suitable value range is determined for a specification by applying the fuzzy rule matrix.

This paper presents an approach to automatically recover mesh surfaces with sharp edges for solids from their binary volumetric discretizations (i.e., voxel models). Our method consists of three steps. The topology singularity is first eliminated on the binary grids so that a topology correct mesh M 0 can be easily constructed. After that, the shape of M 0 is refined, and its connectivity is iteratively optimized into M n . The shape refinement is governed by the duplex distance fields derived from the input binary volume model. However, the refined mesh surface lacks sharp edges. Therefore, we employ an error-controlled variational shape approximation algorithm to segment M n into nearly planar patches and then recover sharp edges by applying a novel segmentation-enhanced bilateral filter to the surface. Using the technique presented in this paper, smooth regions and sharp edges can be automatically recovered from raw binary volume models without scalar field or Hermite data Compared to other related surface recovering methods on binary volume, our algorithm needs less heuristic coefficients.

Volumetric models of 3D objects have recently been introduced into the reverse engineering (RE) process. Grid-based methods are considered as the major technique for reconstructing surfaces from these volumetric models. This is mainly due to the efficiency and simplicity of these methods. However, these grid-based methods suffer from a number of inherent drawbacks, resulting from the fact that the imposed Cartesian grid in general is not well adapted to the surface, neither in size nor in orientation. In order to overcome the above obstacles a new iso-surface extraction method is proposed for volumetric models. The main idea is first to construct a geometrical field that is induced by the object’s shape. This geometrical field represents the natural directions and a grid cell size for each point in the domain. Then, the imposed volumetric grid is deformed by the produced geometrical field toward the object’s shape. The iso-surface meshes can be extracted from the resulting adaptive grid by any conventional grid-based contouring technique. The proposed method provides better approximation of the unknown surface and exhibits anisotropy, which is present inherently in the surface. Moreover, since the produced meshes are quad-dominant, Catmull-Clark subdivision surfaces are directly constructed from these meshes.

We introduce a smooth approximation of the m i n ∕ m a x operations, called signed approximate real distance function (SARDF), for maintaining an approximate signed distance function in constructive shape modeling. We apply constructive distance-based shape modeling to design objects with heterogeneous material distribution in the constructive hypervolume model framework. The introduced distance approximation helps intuitively model material distributions parametrized by distances to so-called material features. The smoothness of the material functions, provided here by the smoothness of the defining function for the shape, helps to avoid undesirable singularities in the material distribution, like stress or concentrations. We illustrate application of the SARDF operations by two- and three-dimensional heterogeneous object modeling case studies.

The widespread availability of affordable high-performance personal computers and commercial software has prompted the integration of structural analyses with numerical optimization, reducing the need for design iterations by human designers. Despite its acceptance as a design tool, however, structural optimization seems yet to gain mainstream popularity in industry. To remedy this situation, this paper reviews past literatures on structural optimization with emphasis on their relation to mechanical product development, and discusses open research issues that would further enhance the industry acceptance of structural optimization. The past literatures are categorized based on their major research focuses: geometry parameterization, approximation methods, optimization algorithms, and the integration with nonstructural issues. Open problems in each category and anticipated future trends briefly are discussed.

Fiber placement is a modern method of constructing composite structures with complex curved surfaces. Modeling individual tows in a fiber placed part constitutes the challenge addressed by methods presented in this paper. One method is presented to approximate offset curves on a free form surface using the geometric constraints of the fiber placement process. A second method is presented to approximate a curve on a free form surface that can be used to generate a laminate family ply. We demonstrate that these approximation methods are sufficient for the accuracy of the fiber placement machine.

The presented approach to discretization of functionally defined heterogeneous objects is oriented towards applications associated with numerical simulation procedures, for example, finite element analysis (FEA). Such applications impose specific constraints upon the resulting surface and volume meshes in terms of their topology and metric characteristics, exactness of the geometry approximation, and conformity with initial attributes. The function representation of the initial object is converted into the resulting cellular representation described by a simplicial complex. We consider in detail all phases of the discretization algorithm from initial surface polygonization to final tetrahedral mesh generation and its adaptation to special FEA needs. The initial object attributes are used at all steps both for controlling geometry and topology of the resulting object and for calculating new attributes for the resulting cellular representation.