Boolean Intersection Research Articles

Structure and Validation of a Kinematic Surface Simulation Model for the Direct Laser Writing Process

The article introduces the structure and validation of a kinematic surface simulation model for the replication of laser‐structured surfaces and their impressions after hot embossing. The surface replication is accomplished by the stepwise Boolean intersection of a structured and digitized point ablation with the workpiece. The resulting surface is characterized according to ISO 25178. Resulting from the kinematic simulation principle, the model can be applied to a wide range of material–machine combinations in the field of laser structuring and enables a time–efficient, nanometer‐resolved replication of representative surfaces with a size up to several square millimeters. Within the validation process it was proven that the main surface characteristics can be predicted by the model. Compared to AI approaches and numerical or multiphysical simulations, preliminary studies are only needed to a very small extent. Furthermore, complex influences such as material and temperature‐related properties do not need separate consideration as they are taken into account by the digitized point ablation.

A new virtual functional element method for deviation prediction of assembled structures with parallel connection chain

The assembly precision prediction of industrial products with numerous tolerance information and complex assembly relations is a challenge for existing deviation analysis methods since their dimensional chains usually form parallel connections. The parallel connection chain is often transformed into a serial connection chain by discarding some tolerance information and physical constraints in practical application, which leads to amplifying the range of predicted results and reducing the predicted accuracy. In this paper, a virtual functional element method is proposed to simplify the parallel connection chain by integrating the tolerance information into the torsor parameters of specified virtual functional elements. A general Jacobian-Torsor model is developed to solve the simplified parallel connection chain, in which the Boolean intersection operation on torsor parameters is used to obtain comprehensive deviation results for multiple branches in a parallel connection chain. Meanwhile, the locking effect and leverage activation effect are proposed to characterize the effect of the invariant degree in parallel connection chain. Finally, an assembly case of a power plant is applied to validate the proposed method, and the results show that it has a high potential to improve the prediction accuracy of assembly deviation compared to three conventional methods.

Porous interbody fusion cage design via topology optimization and biomechanical performance analysis

The porous interbody fusion cage could provide space and stable mechanical conditions for postoperative intervertebral bone ingrowth. It is considered to be an important implant in anterior cervical discectomy and internal fixation. In this study, two types of unit cells were designed using topology optimization method and introduced to the interbody fusion cage to improve the biomechanical performances of the cage. Topology optimization under two typically loading conditions was first conducted to obtain two unit cells (O-unit cell and D-unit cell) with the same volume fraction. Porous structures were developed by stacking the obtained unit cells in space, respectively. Then, porous interbody fusion cages were obtained by the Boolean intersection between the global structural layout and the porous structures. Finite element models of cervical spine were created that C5-C6 segment was fused by the designed porous cages. The range of motion (ROM) of the cervical spine, the maximum stress on the cage and the bone graft, and the stress and displacement distributions of the cage were analyzed. The results showed the ROMs of C5-C6 segment in D-unit cell and O-unit cell models were range from 0.14° to 0.25° under different loading conditions; the cage composed of the D-unit cells had a more uniform stress distribution, smaller displacement on cage, a more reasonable internal stress transfer mode (transmission along struts of the unit cell), and higher stress on the internal bone graft (0.617 MPa). In conclusion, the optimized porous cage is a promising candidate for fusion surgery, which would avoid the cage subsidence, and promote the fusion of adjacent endplates.

Boolean intersection ideals of permutations in the Bruhat order

Motivated by recent work with Mazorchuk, we characterize the conditions under which the intersection of two principal order ideals in the Bruhat order is boolean. That characterization is presented in three versions: in terms of reduced words, in terms of permutation patterns, and in terms of permutation support. The equivalence of these properties follows from an analysis of what it means to have a specific letter repeated in a permutation's reduced words; namely, that a specific 321-pattern appears.

Determination of geometric information and radiation field overlaps on the skin in percutaneous coronary interventions with computer-aided design-based X-ray beam modeling.

PurposeThis study aimed to develop a method for the determination of the source‐to‐surface distance (SSD), the X‐ray beam area in a plane perpendicular to the beam axis at the entrance skin surface (Ap ), and the X‐ray beam area on the actual skin surface (As ) during percutaneous coronary intervention (PCI).Materials and MethodsMale and female anthropomorphic phantoms were scanned on a computed tomography scanner, and the data were transferred to a commercially available computer‐aided design (CAD) software. A cardiovascular angiography system with a 200 × 200 mm flat‐panel detector with a field‐of‐view of 175 × 175 mm was modeled with the CAD software. Both phantoms were independently placed on 40 mm thick pads, and the examination tabletop at the patient entrance reference point. Upon panning, the heart center was aligned to the central beam axis. The SSD, Ap , and As were determined with the measurement tool and Boolean intersection operations at 10 gantry angulations.ResultsThe means and standard deviations of the SSD, Ap , and As for the male and female phantoms were 573 ± 15 and 580 ± 15 mm, 8799 ± 1009 and 9661 ± 1152 mm2, 10495 ± 602 and 11913 ± 600 mm2, respectively. The number of As overlaps for the male and female phantoms were 15/45 and 21/45 view combinations, respectively.ConclusionsCAD‐based X‐ray beam modeling is useful for the determination of the SSD, Ap , and As . Furthermore, the knowledge of the As distribution helps to reduce the As overlap in PCI.

Development of the distinct lattice spring model with polyhedral particles

Distinct Lattice Spring Model (DLSM) is a microscopic model to overcome the mismatch problem of degrees of freedom when coupling between discrete element method (DEM) and finite element method (FEM). In the original DLSM, the lattice model is composed of a number of same spherical particles, which has the problem of inaccurate geometric description. A polyhedral 3D DLSM was developed to overcome some limitations of the original DLSM. A new pre- and post-processing method is introduced to extend the ability of DLSM on precise geometry description. Following this, a pre-processing method is proposed to generate the lattice model from a 3D Finite Element (FE) Mesh or a 3D Numerical Manifold Method (NMM) Mesh. It is required that the generated mesh can present the precise 3D geometry and consider the spatial morphology of joint surfaces. The 3D FE mesh is generated by using the general mesh software. The FE mesh is generated according to the free surfaces and the joint surfaces defined in the model. To get the 3D NMM mesh, a 3D surface mesh Boolean algorithm is developed. The surface mesh Boolean algorithm is used for solving the Boolean operations between mathematic covers and physical covers in NMM. By using the algorithm, the 3D closed geometry cut by 3D surface and the union of multiple 3D surfaces can be solved. The 3D NMM mesh is ultimately generated through the Boolean intersection of mathematical covers and physical covers, the physical covers include the free surfaces and joint faces. The polyhedral particles generated by the FE mesh or NMM mesh have different shapes and sizes, the springs between polyhedral particles will also have different lengths. The effectiveness, necessity, and correctness of these enrichments are demonstrated from a number of specially designed numerical examples.

Mollified finite element approximants of arbitrary order and smoothness

The approximation properties of the finite element method can often be substantially improved by choosing smooth high-order basis functions. It is extremely difficult to devise such basis functions for partitions consisting of arbitrarily shaped polytopes. We propose the mollified basis functions of arbitrary order and smoothness for partitions consisting of convex polytopes. On each polytope an independent local polynomial approximant of arbitrary order is assumed. The basis functions are defined as the convolutions of the local approximants with a mollifier. The mollifier is chosen to be smooth, to have a compact support and a unit volume. The approximation properties of the obtained basis functions are governed by the local polynomial approximation order and mollifier smoothness. The convolution integrals are evaluated numerically first by computing the boolean intersection between the mollifier and the polytope and then applying the divergence theorem to reduce the dimension of the integrals. The support of a basis function is given as the Minkowski sum of the respective polytope and the mollifier. The breakpoints of the basis functions, i.e. locations with non-infinite smoothness, are not necessarily aligned with polytope boundaries. Furthermore, the basis functions are not boundary interpolating so that we apply boundary conditions with the non-symmetric Nitsche method as in immersed/embedded finite elements. The presented numerical examples confirm the optimal convergence of the proposed approximation scheme for Poisson and elasticity problems.

Porous fusion cage design via integrated global-local topology optimization and biomechanical analysis of performance

Porous fusion cage is considered as a satisfactory substitute for solid fusion cage in transforaminal lumbar interbody fusion (TLIF) surgery due to its interconnectivity for bone ingrowth and appropriate stiffness reducing the risk of cage subsidence and stress shielding. This study presents an integrated global-local topology optimization approach to obtain porous titanium (Ti) fusion cage with desired biomechanical properties. Local topology optimizations are first conducted to obtain unit cells, and the numerical homogenization method is used to quantified the mechanical properties of unit cells. The preferred porous structure is then fabricated using selective laser melting, and its mechanical property is further verified via compression tests and numerical simulation. Afterward, global topology optimization is used for the global layout. The porous fusion cage obtained by the Boolean intersection between global structural layout and the porous structure decreases the solid volume of the cage by 9% for packing more bone grafts while achieving the same stiffness to conventional porous fusion cage. To eliminate stress concentration in the thin-wall structure, framework structures are constructed on the porous fusion cage. Although the alleviation of cage subsidence and stress shielding is decelerated, peak stress on the cage is significantly decreased, and more even stress distribution is demonstrated in the reinforced porous fusion cage. It promises long-term integrity and functions of the fusion cage. Overall, the reinforced porous fusion cage achieves a favorable mechanical performance and is a promising candidate for fusion surgery. The proposed optimization approach is promising for fusion cage design and can be extended to other orthopedic implant designs.

A CAD-oriented structural topology optimization method

In this paper, we develop a CAD-oriented topology optimization method incorporating directly the CAD model into the feature-driven optimization (FDO) framework. Practically, any free-form design domain model (FDDM), defined by the CAD model in stereolithography (STL) file format, is first represented by a level set function employing a radial basis function (RBF) interpolation technique. Feature-based topology variation model (TVM) consisting of design and non-design features is then constructed, with the former affording design flexibility whilst the latter preserving the components for engineering requirements. By coupling the FDDM with the TVM via Boolean intersection operation, the topology optimization framework is constructed, and each analysis problem is solved by the finite cell method (FCM). A boundary integral scheme is implemented for efficient sensitivity analysis. It is shown that the developed method shall pose almost no restriction on the expertise of modeling and meshing related to the conventional finite element method. Several examples of increasing complexities are dealt with to demonstrate the effectiveness and the generality of the proposed method.

WORKSPACE ANALYSIS AND OPTIMIZATION OF THE PARALLEL ROBOTS BASED ON COMPUTER-AIDED DESIGN APPROACH

This paper provides workspace determination and analysis based on the graphical technique of both spatial and planar parallel manipulators. The computation and analysis of workspaces will be carried out using the parameterization and three-dimensional representation of the workspace. This technique is implemented in CAD (Computer Aided Design) Software CATIA workbenches. In order to determine the workspace of the proposed manipulators, the reachable region by each kinematic chain is created as a volume/area; afterwards, the full reachable workspace is obtained by the application of a Boolean intersection function on the previously generated volumes/areas. Finally, the relations between the total workspace and the design parameters are simulated, and the Product Engineering Optimizer workbench is used to optimize the design variables in order to obtain a maximized workspace volume. Simulated annealing (SA) and Conjugate Gradient (CG) are considered in this study as optimization tools.

Dual binary discriminator varieties

Left normal bands, strongly distributive skew lattices, implicative BCS-algebras, skew Boolean algebras, skew Boolean intersection algebras, and certain other non-commutative structures occur naturally as term reducts in the study of ternary discriminator algebras and the varieties that they generate, giving rise thereby to various classes of pointed discriminator varieties that generalise the class of pointed ternary discriminator varieties. For each such class of varieties there is a corresponding pointed discriminator function that generalises the ternary discriminator. In this paper some of the classes of pointed discriminator varieties that are contained in the class of dual binary discriminator varieties are characterised. A key unifying property is that the principal ideals of an algebra in a dual binary discriminator variety are entirely determined by the dual binary discriminator term for that variety.

Reliability of a novel 3-dimensional computed tomography method for reverse shoulder arthroplasty postoperative evaluation

BackgroundLong-term function and survival of reverse shoulder arthroplasties (RSAs) are reliant on component positioning and fixation. Conventional postoperative analysis is performed using plain radiographs or 2-dimensional (2D) computed tomography (CT) images. Although 3-dimensional (3D) CT would be preferred, its use is limited by metal artifacts. This study proposes a new 3D CT method for postoperative RSA evaluation and compares its interobserver reliability with conventional methods.Materials and methodsPreoperative and postoperative CT scans, as well as postoperative radiographs, were obtained from 18 patients who underwent RSA implantation; the scapula, implant, and screws were reconstructed as 3D CT models. The postoperative 3D scapula and implant were imported into preoperative coordinates and matched to the preoperative scapula. Standardized scapula coordinates were defined, in which the glenoid baseplate version and inclination angle were measured. The percentage of screw volume in bone was measured from a Boolean intersection operation between the preoperative scapula and screw models. Four independent reviewers performed the measurements using 3D CT and conventional 2D methods. Intraclass correlation coefficients (ICCs) were used to compare the reliability of the methods.ResultsThe 3D CT method showed excellent reliability (ICC > 0.75) in baseplate inclination (ICC = 0.92), version (ICC = 0.97), and screw volume in bone (ICC = 0.99). Conventional 2D methods demonstrated poor reliability (ICC < 0.4). For radiographs, inclination showed poor reliability (ICC = 0.09) and the screw percentage in bone showed fair reliability (ICC = 0.54). Version was not measured with plain radiographs. For 2D CT slice measurements, inclination showed poor reliability (ICC = 0.02), version showed excellent reliability (ICC = 0.81), and the screw percentage in bone showed poor reliability (ICC = 0.28).ConclusionThe new 3D CT–based method for evaluating RSA glenoid implant positioning and screw volume in bone showed excellent reliability and overcame the metal-artifact limitation of postoperative CT and 3D CT reconstruction.

Workspace characterization and kinematic analysis of general spherical parallel manipulators revisited via graphical based approaches

This paper revisits some fundamental issues in kinematic analysis of spherical parallel robotic manipulators (SPRMs), such as the orientation capacities of the mobile platform (MPF), and assembly modes determination. The orientation capacities are analyzed by using several types of workspaces: total orientation workspace, full-spin orientation workspace, and constant-spin orientation workspace. The adopted representation of the MPF orientation brings up-to-date the use of the spherical coordinate system, which we find more appropriate when Euler angle orientation parameters are used. This representation is applied to each limb, considered independent from the rest of the mechanism and having the mobile platform as end-effector. The 3D orientation space reached by each limb, which is called also vertex space, is represented as a 3D solid volume. Thereafter, the operational space of the SPRM is directly determined by the Boolean intersection of the previously obtained volumes. Moreover, by fixing the joint space variables, the domain reached by each limb in the operational space is a surface. The Boolean intersection of the obtained surfaces is nothing but the set of points corresponding to the solutions of the forward kinematic problem. These operations are implemented in CAD software and the proposed approaches have been validated successfully by using several examples of application for both workspace determination and representation and FKP resolution.

Latent Network Features and Overlapping Community Discovery via Boolean Intersection Representations.

We propose a new latent Boolean feature model for complex networks that captures different types of node interactions and network communities. The model is based on a new concept in graph theory, termed the Boolean intersection representation of a graph, which generalizes the notion of an intersection representation. We mostly focus on one form of Boolean intersection, termed cointersection, and describe how to use this representation to deduce node feature sets and their communities. We derive several general bounds on the minimum number of features used in cointersection representations and discuss graph families for which exact cointersection characterizations are possible. Our results also include algorithms for finding optimal and approximate cointersection representations of a graph.

A comprehensive study of feature definitions with solids and voids for topology optimization

In this paper, we present a CAD-oriented and feature-driven topology optimization methodology for engineering structures with freeform design domains. Mixed solid–void features are adopted as basic design primitives to build a unified formulation of feature definition, while pure solid or pure void features are considered as two special cases. Topology optimization is achieved as a Boolean intersection process between the so-called topology variation modeler (TVM) and freeform design domain modeler (FDDM) in terms of level-set functions (LSFs). The TVMs constructed with solid and/or void features are applied to clip the FDDM through Boolean operations. It is shown that sensitivity analysis with boundary integral scheme is independent of the mathematical expressions of the LSF. Numerical problems with freeform and periodic design domains are dealt with to demonstrate the proposed topology optimization method.

Free Skew Boolean Intersection Algebras and Set Partitions

We show that atoms of the n-generated free left-handed skew Boolean intersection algebra are in a bijective correspondence with pointed partitions of non-empty subsets of \(\{1,2,\dots , n\}\). Furthermore, under the canonical inclusion into the k-generated free algebra, where k≥n, an atom of the n-generated free algebra decomposes into an orthogonal join of atoms of the k-generated free algebra in an agreement with the containment order on the respective partitions. As a consequence of these results, we describe the structure of finite free left-handed skew Boolean intersection algebras and express several their combinatorial characteristics in terms of Bell numbers and Stirling numbers of the second kind. We also look at the infinite case. For countably many generators, our constructions lead to the ‘partition analogue’ of the Cantor tree whose boundary is the ‘partition variant’ of the Cantor set.

Tolerance-Maps for Line-Profiles Formed by Intersecting Kinematically Transformed Primitive Tolerance-Map Elements

For the purposes of automating the assignment of tolerances during design, a math model, called the Tolerance-Map (T-Map), has been produced for most of the tolerance classes that are used by designers. Each T-Map is a hypothetical point-space that represents the geometric variations of a feature in its tolerance-zone. Of the six tolerance classes defined in the ASME/ANSI/ISO Standards, profile tolerances have received the least attention for representation in computer models. The objective of this paper is to provide a comprehensive treatment of T-Map construction for any line-profile by using primitive T-Map elements and their Boolean intersection. The method requires (a) decomposing a profile into segments, each of constant curvature; (b) creating a solid-model T-Map primitive for each in a common global reference frame; and (c) combining these by Boolean intersection to generate the T-Map for a complete line-profile of any shape. Freeform portions of a profile are modeled as a series of closely spaced points and subsequent formation of short circular arc-segments, each formed from the circle that osculates to three adjacent points.

Boolean Combinations of Implicit Functions for Model Clipping in Computer-Assisted Surgical Planning.

This paper proposes an interactive method of model clipping for computer-assisted surgical planning. The model is separated by a data filter that is defined by the implicit function of the clipping path. Being interactive to surgeons, the clipping path that is composed of the plane widgets can be manually repositioned along the desirable presurgical path, which means that surgeons can produce any accurate shape of the clipped model. The implicit function is acquired through a recursive algorithm based on the Boolean combinations (including Boolean union and Boolean intersection) of a series of plane widgets’ implicit functions. The algorithm is evaluated as highly efficient because the best time performance of the algorithm is linear, which applies to most of the cases in the computer-assisted surgical planning. Based on the above stated algorithm, a user-friendly module named SmartModelClip is developed on the basis of Slicer platform and VTK. A number of arbitrary clipping paths have been tested. Experimental results of presurgical planning for three types of Le Fort fractures and for tumor removal demonstrate the high reliability and efficiency of our recursive algorithm and robustness of the module.

CAD-based unified graphical methodology for solving the main problems related to geometric and kinematic analysis of planar parallel robotic manipulators

CAD environments provide powerful tools for graphical programming and geometric feature handling. This paper explores this potential for robotics applications by presenting a set of several original approaches for solving the main problems related to geometric and kinematic analysis of planar parallel robotic manipulators (PPRMs). These approaches rely on the use of CAD-based graphical programming enabling rapid resolution and geometric interpretation of the 3D total workspace characteristics. The novelty of the proposed approach resides in the association of an original, unique and natural 3D graphical representation of the end-effector poses and the use of geometric feature handling and graphical solver capabilities within a CAD environment, for determining the 3D total operational workspace as well as solving the forward kinematic problem (FKP) of PPRMs. The approach consists in considering the mobile platform separately attached to each limb that we disconnect from the rest of the mechanism. The geometric construction of the spaces reachable by the end-effector is performed by using Boolean intersections of the vertex volumes, respectively the vertex surfaces, relative to each limb for 3D total workspace determination, respectively for FKP resolution. By combining in the same 3D graphical environment several geometric entities associated to the PPRM, such as workspace volume, singularity surface, and the different solutions of the FKP, this approach allows designers, in a user-friendly way, to generate the singularity-free trajectories connecting different assembly modes. The approach presented in this paper is mostly useful for the architectures of 3-DOF parallel robotic manipulators for which an algebraic closed form solution does not exist for the forward kinematic model and the singularity-free trajectory planning between assembly modes is not a trivial task. It is applied for several PPRMs such as: 3-RPR, 3-RRR, 3-PPR, 3-RRP, 3-PRP, 3-PRR, 3-RRR, and 3-RPR. Display Omitted A new method based on CAD techniques is proposed to resolve FKP for PPRMs.CAD approach allows designer to locate all FKP solutions within 3D total workspace.Methodology presented is very powerful for planning singularity-free trajectory.CAD approaches permit a high precision for: workspace, FKP solutions, singularity.CAD methodology may be extended, with some adaptations for all types of PPRMs.

Tolerance-Maps for line-profiles constructed from Boolean intersection of T-Map primitives for arc-segments

For purposes of automating the assignment of tolerances during design, a math model, called the Tolerance-Map (T-Map), has been produced for most of the tolerance classes that are used by designers. Each T-Map is a hypothetical point-space that represents the geometric variations of a feature in its tolerance-zone. Of the six tolerance classes defined in the ASME/ANSI/ISO Standards, profile tolerances have received the least attention for representation in computer models. The objective of this paper is to describe a new method of construction, using computer-aided geometric design, which can produce the T-Map for any line-profile. The new method requires decomposing a profile into segments, creating a solid-model T-Map primitive for each, and then combining these by Boolean intersection to generate the T-Map for a complete line profile of any shape. To economize on length, the scope of this paper is limited to line-profiles formed from circular arc-segments. The parts containing the line-profile features are considered to be rigid.