Geometric Approximation Errors Research Articles

Marching Windows: Scalable Mesh Generation for Volumetric Data With Multiple Materials.

Volumetric data abounds in medical imaging and other fields. With the improved imaging quality and the increased resolution, volumetric datasets are getting so large that the existing tools have become inadequate for processing and analyzing the data. Here we consider the problem of computing tetrahedral meshes to represent large volumetric datasets with labeled multiple materials, which are often encountered in medical imaging or microscopy optical slice tomography. Such tetrahedral meshes are a more compact and expressive geometric representation so are in demand for efficient visualization and simulation of the data, which are impossible if the original large volumetric data are used directly due to the large memory requirement. Existing methods for meshing volumetric data are not scalable for handling large datasets due to their sheer demand on excessively large run-time memory or failure to produce a tet-mesh that preserves the multi-material structure of the original volumetric data. In this article we propose a novel approach, called Marching Windows, that uses a moving window and a disk-swap strategy to reduce the run-time memory footprint, devise a new scheme that guarantees to preserve the topological structure of the original dataset, and adopt an error-guided optimization technique to improve both geometric approximation error and mesh quality. Extensive experiments show that our method is capable of processing very large volumetric datasets beyond the capability of the existing methods and producing tetrahedral meshes of high quality.

Low Grating Lobe Cable-Network Reflector Antenna Design Using Integrated Structural-Electromagnetic Optimization Method

Geometric approximation errors of periodically distributed surface facets of cable-network reflector antennas can result in grating lobes undesired in some space missions. As the surface is shaped by the tensioned flexible cable-network structure, the electromagnetic (EM) and structural parameters are highly correlated, and the well-designed surface obtained only concerning the EM performance may not be shaped by the tension-only cable network with specified cable-tension constraints. To address the problem, this study proposes an integrated structural-EM optimization method to enable low grating lobe cable-network antenna design. In this method, force densities of the cables, which can determine both the cable tensions and surface node positions that decide the boundary conditions of the EM field, are selected as the design variables. An adaptive objective/constraint function is proposed, which enables the grating lobe level (GLL) to be minimized with constrained cable tensions, antenna gain, and first sidelobe level (SLL) in a simultaneous, feasible and relatively fast way. To demonstrate the feasibility and effectiveness of the proposed method, it was implemented on a 12 m-diameter cable-network antenna. The results indicate that the grating lobes are greatly degraded and the achieved surface can be formed by well tensioned cables.

Error analysis and 3D reconstruction using airborne array InSAR images

Multi-temporal synthetic aperture radar interferometry (MT-InSAR) is able to detect surface deformation and reconstruct a 3D surface model with high precision but requires a long observation period to accumulate the multi-baseline SAR images. The airborne array InSAR system is able to acquire a stack of multi-baseline SAR images in a single acquisition, which significantly improves the 3D modeling capability. However, processing the images obtained by the low-altitude platform using the conventional model will lead to geometric approximation (GA) errors, such as flattened phase error and reference error, which degrade the precision of the 3D reconstruction. In this paper, we quantitatively analyze the error sources of the array-InSAR interferograms and design a hybrid 3D phase unwrapping approach for 3D reconstruction. A hypothesis test is developed to identify the phase ambiguity by comparing the initial solution of the least-square with that of the beam-forming. Three indicators are proposed to identify the reliable arcs, achieving a reliable phase unwrapping. The L1 norm approach is adopted to detect the unwrapping errors during the spatial unwrapping and a two-tie network strategy is used to process the data in the individual blocks from the perspective of global optimization. Furthermore, an iterative scheme is recommended to compensate the geometric approximation errors. The main advantage of the proposed algorithm is the compensation of the GA error and reliable phase unwrapping. The experimental results by both simulated and real SAR data show that the proposed algorithm can eliminate the GA error and provide a viable solution to rapid 3D SAR imaging with an airborne platform.

Divergence/Connection Preservation Scheme in the Curvilinear Domain with a Small Geometric Approximation Error

Additional grid points are often introduced for the higher-order polynomial of a numerical solution with curvilinear elements. However, those points are likely to be located slightly outside the domain, even when the vertices of the curvilinear elements lie within the curved domain. This misallocation of grid points generates a mesh error, called geometric approximation error. This error is smaller than the discretization error but large enough to significantly degrade a long-time integration. Moreover, this mesh error is considered to be the leading cause of conservation error. Two novel schemes are proposed to improve conservation error and/or discretization error for long-time integration caused by geometric approximation error: The first scheme retrieves the original divergence of the original domain; the second scheme reconstructs the original path of differentiation, called connection, thus retrieving the original connection. The increased accuracies of the proposed schemes are demonstrated by the conservation error for various partial differential equations with moving frames on the sphere.

Hybrid IGA-FEA of fiber reinforced thermoplastic composites for forward design of AI-enabled 4D printing

Fused deposition modeling (FDM)-based 4D printing uses thermoplastics to produce artifacts and requires computational analysis to assist its design processes of complex geometries. Previously, finite element analysis (FEA) has been used to simulate 4D printing deformations, and its accuracy has been computationally and experimentally verified. However, using FEA also leads to several limitations, such as geometric approximation error and the computational time-cost due to the high degrees of freedom. To address these issues, this paper introduces isogeometric analysis (IGA) into the deformation simulations and propounds a composite design by hybridizing FEA and IGA elements to reduce the number of degrees of freedom while maintaining the simulation accuracy. Moreover, since the hybrid IGA-FEA method used for modeling 4D printing structure deformation excludes real-time interactivity, we develop a polycube-based random forest regressor machine learning (ML) model to learn the IGA-FEA-based structural mechanics simulations and provide fast deformation predictions. Given the input actuator block distribution and geometry configurations, our well-trained model can predict the residual stress-induced deformation behaviors of mesh-like thermoplastic composite structures. With an error less than 0.11% and computation speed 20 times faster than hybrid IGA-FEA simulations, our model can create real-time (0.93 s) and truthful (99.89% accuracy) results. The effectiveness of the proposed model is demonstrated with several complex design examples. We believe the presented workflow effectively combines IGA, FEA, ML, and 4D printing to provide a powerful computational tool that enriches the 4D printing design tool box, and brings huge application potentials.

Nanostructural interface and strength of polymer composite scaffolds applied to intervertebral bone

The pores of bone tissue that play an important part in bone regeneration can emulate the areas of nanoparticles from porous scaffolds. This work evaluated a novel designed and developed nanostructure surface of polyetheretherketone-reduced graphene oxide, calcium hydroxyapatite (PEEK-rGO-cHAp) composite scaffolds of four different lattice structures. The scaffolds were fabricated through fused deposition modeling (FDM), as rGO-cHAp composite was coated on PEEK. The composite scaffolds’ mechanical strength and surface microstructure were studied, using different nanostructure methods of unit cell homogenization and tensile test. The homogenization method for the four lattice structures was designed and analyzed to mimic spine bone structure. A new approach was introduced to homogenize the mechanical characteristics of a periodical lattice of 3D printing structures based on a semi-rigid frame unit cell. They were taking into consideration the impact of geometric approximation errors and joint tightening. A typical frame element with semi-rigid is integrated to assess combined stiffening effects in a discrete homogenization process. The analysis was performed by considering the fundamental unit cell as a scaffold that defined the periodic pattern. Also, this study created an avenue to examine and improve the interfacial bonding between PEEK and rGO-cHAp scaffolds for biocompatibility and degradability, using surface functionalization techniques. The purpose of this research is to compare the manufacturing processes in a model of intervertebral spacer, describe the characteristics of PEEK biomaterial, and explain some parameters related to its processing. In addition to its manufacturing part, a brief theory on the anatomy of the spine region was also presented. To establish its practical applications and benefits in tissue engineering, this study focused on the cervical region via a simulation approach using an anterior method. • Evaluation of novel designed/developed nanostructured surface of PEEK-rGO-cHAp composites. • AM/FDM and characterisation of 3D-printed PEEK-rGO-cHAp scaffolds for spine implants. • Study designed microstructures with programable homogenization in PEEK/rGO/cHAp composite. • PEEK-cHAp performed better in strength than PEEK with 2–3 MPa difference in their yield strengths. • With an in vitro test, the scaffolds exhibited better adhesion, proliferation and cell activity than pure PEEK.

Guaranteed-quality higher-order triangular meshing of 2D domains

We present a guaranteed quality mesh generation algorithm for the curvilinear triangulation of planar domains with piecewise polynomial boundary. The resulting mesh consists of higher-order triangular elements which are not only regular (i.e., with injective geometric map) but respect strict bounds on quality measures like scaled Jacobian and MIPS distortion. This also implies that the curved triangles' inner angles are bounded from above and below. These are key quality criteria, for instance, in the field of finite element analysis. The domain boundary is reproduced exactly, without geometric approximation error. The central idea is to transform the curvilinear meshing problem into a linear meshing problem via a carefully constructed transformation of bounded distortion, enabling us to leverage key results on guaranteed-quality straight-edge triangulation. The transformation is based on a simple yet general construction and observations about convergence properties of curves under subdivision. Our algorithm can handle arbitrary polynomial order, arbitrarily sharp corners, feature and interface curves, and can be executed using rational arithmetic for strict reliability.

An effective energy-conserving contact modelling strategy for spherical harmonic particles represented by surface triangular meshes with automatic simplification

An effective discrete element modelling strategy for triangular mesh represented spherical harmonic particles is proposed. It features: (1) using a golden spiral lattice on the unit sphere to generate an initial triangular mesh with any number of vertices/triangles for a star-shaped surface; (2) applying an edge contraction mesh simplification algorithm to reduce the mesh size to any desired level; and (3) adopting an energy-conserving linear normal contact model to compute the contact geometric and force features of contacting particles. In particular, the edge contraction algorithm is applicable to any triangular mesh. It is algorithmically very simple and highly effective, and can be easily incorporated into existing discrete element frameworks. Numerical experiments are conducted to demonstrate that the simplified mesh by the edge contraction can not only have a very low geometric approximation error but also achieve expected mechanical responses. Thus this mesh simplification approach can serve as an ideal pre-processing tool to optimise a large input triangular mesh in order to significantly reduce the computational cost associated with discrete element simulations without compromising the modelling accuracy.

3D mesh simplification with feature preservation based on Whale Optimization Algorithm and Differential Evolution

Large-scale 3D models consume large computing and storage resources. To address this challenging problem, this paper proposes a new method to obtain the optimal simplified 3D mesh models with the minimum approximation error. First, we propose a feature-preservation edge collapse operation to maintain the feature edges, in which the collapsing cost is calculated in a novel way by combining Gauss curvature and Quadratic Error Metrics (QEM). Second, we introduce the edge splitting operation into the mesh simplification process and propose a hybrid ‘undo/redo’ mechanism that combines the edge splitting and edge collapse operation to reduce the number of long and narrow triangles. Third, the proposed ‘undo/redo’ mechanism can also reduce the approximation error; however, it is impossible to manually choose the best operation sequence combination that can result in the minimum approximation error. To solve this problem, we formulate the proposed mesh simplification process as an optimization model, in which the solution space is composed of the possible combinations of operation sequences, and the optimization objective is the minimum of the approximation error. Finally, we propose a novel optimization algorithm, WOA-DE, by replacing the exploration phase of the original Whale Optimization Algorithm (WOA) with the mutate and crossover operations of Differential Evolution (DE) to compute the optimal simplified mesh model more efficiently. We conduct numerous experiments to test the capabilities of the proposed method, and the experimental results show that our method outperforms the previous methods in terms of the geometric feature preservation, triangle quality, and approximation error.

Homogenization of Mechanical Properties for Material Extrusion Periodic Lattice Structures Considering Joint Stiffening Effects

Many engineering applications utilize periodic lattice structures to take advantage of their favorable and tailorable mechanical properties. However, manufacturing the structures and evaluating their mechanical properties are still challenging. Additive manufacturing (AM) processes offer an alternative method to fabricate periodic lattice structures but the processes only approximate bounding part surfaces. Periodic lattice structures generally have two important geometrical characteristics, large bounding surfaces, and a large number of joints. Since geometric approximation errors on large bounding surfaces critically affect mechanical properties of the structures, designers and engineers should incorporate this degradation into mechanical property estimation procedures. In addition, the effects of joints should be analyzed in the estimation process, because joints reduce struts lengths, and as a result, they add stiffness to lattice structures. This paper presents a new homogenization approach to estimate mechanical properties of additively manufactured periodic lattice structures that is based on semirigid joint frame elements, and it takes into account effects of geometric approximation errors and joint stiffening. Effective structural parameters of a semirigid joint frame element are calculated from an as-fabricated voxel model to incorporate the geometric approximation errors. The semirigid joint frame element is integrated into a discrete homogenization process to evaluate joint stiffening effects. This paper reports results of parametric studies that investigate effects of AM process and joint properties on periodic lattice structures fabricated by material extrusion. This paper also compares estimates from the proposed approach and conventional homogenization approaches with test results. The comparison shows that the proposed method provides estimates that are more accurate.

Designing and Evaluating a Mesh Simplification Algorithm for Virtual Reality

With the increasing accessibility of the mobile head-mounted displays (HMDs), mobile virtual reality (VR) systems are finding applications in various areas. However, mobile HMDs are highly constrained with limited graphics processing units (GPUs) and low processing power and onboard memory. Hence, VR developers must be cognizant of the number of polygons contained within their virtual environments to avoid rendering at low frame rates and inducing simulator sickness. The most robust and rapid approach to keeping the overall number of polygons low is to use mesh simplification algorithms to create low-poly versions of pre-existing, high-poly models. Unfortunately, most existing mesh simplification algorithms cannot adequately handle meshes with lots of boundaries or nonmanifold meshes, which are common attributes of many 3D models. In this article, we present QEM 4VR , a high-fidelity mesh simplification algorithm specifically designed for VR. This algorithm addresses the deficiencies of prior quadric error metric (QEM) approaches by leveraging the insight that the most relevant boundary edges lie along curvatures while linear boundary edges can be collapsed. Additionally, our algorithm preserves key surface properties, such as normals, texture coordinates, colors, and materials, as it preprocesses 3D models and generates their low-poly approximations offline. We evaluated the effectiveness of our QEM 4VR algorithm by comparing its simplified-mesh results to those of prior QEM variations in terms of geometric approximation error, texture error, progressive approximation errors, frame rate impact, and perceptual quality measures. We found that QEM 4VR consistently yielded simplified meshes with less geometric approximation error and texture error than the prior QEM variations. It afforded better frame rates than QEM variations with boundary preservation constraints that create unnecessary lower bounds on overall polygon count reduction. Our evaluation revealed that QEM 4VR did not fair well in terms of existing perceptual distance measurements, but human-based inspections demonstrate that these algorithmic measurements are not suitable substitutes for actual human perception. In turn, we present a user-based methodology for evaluating the perceptual qualities of mesh simplification algorithms.

High-order evolving surface finite element method for parabolic problems on evolving surfaces

High-order spatial discretisations and full discretisations of parabolic partial differential equations on evolving surfaces are studied. We prove convergence of the high-order evolving surface finite element method, by showing high-order versions of geometric approximation errors and perturbation error estimates and by the careful error analysis of a modified Ritz map. Furthermore, convergence of full discretisations using backward difference formulae and implicit Runge-Kutta methods are also shown.

An adaptive method to reduce the effect of geometric approximation error on 3-D DC resistivity finite element numerical modeling

The computational area is divided into disjoint elements before 3-D DC finite element numerical simulation. If these elements cannot completely approximate the geoelectric model to be simulated, some of them have geometric approximation errors. The elements with geometric approximation errors may decrease the accuracy of the simulation. If the effect of geometric approximation errors on simulation is reduced purely by minimizing geometric approximation errors, a large number of grid resources are wasted. As a result, the corresponding simulation complexity is limited. We propose an adaptive method to reduce the effect of geometric approximation errors on 3-D DC finite element resistivity numerical modeling. This method focuses on the effect of geometric approximation errors rather than geometric approximation errors themselves. For the elements with geometric approximation errors, only those with large negative effect on simulation are refined in the adaptive process. We simulate five geoelectric models to verify the feasibility of this method. The experiments show that despite many elements with geometric approximation errors, high accuracy numerical result can be still obtained, and the element number of simulation decreases significantly. With this method, many more complex geoelectric models can be simulated using the same amount of computer resources.

Surface Configuration Design of Cable-Network Reflectors Considering the Radiation Pattern

Since the curved surface of a cable-network reflector is generally approximated by regular flat facets, the electromagnetic performance of the cable-network reflector is determined by the size and shape of the facets. In many cases, the geometric approximation errors of the facets are periodically distributed on the surface, which can result in strong grating lobes. First, in this paper, the periodicity of the geometric approximation error and the effects of this periodicity on the EM performance are investigated. Second, the periodic geometric approximation errors of the regularly faceted configurations are disturbed by three proposed approaches, and three new types of surface configurations are thus formed. The first approach is to introduce radial direction disturbance parameters, the second one is to change the shape of the outlines and the last one is to use a Penrose-tiling-like configuration. The first two approaches are carried out in combination with the optimization strategies. Finally, the effects of the mentioned disturbance parameters on the EM performance are simulated, and some numerical examples are given to show that the surface configurations proposed in this paper can reduce the grating lobes level obviously, without either decreasing the gain or increasing the structural complexity of the cable-network reflectors.

Real-time PH-based interpolation algorithm for high speed CNC machining

Various methods for parametric interpolation of non-uniform rational B-spline (NURBS) curves have been proposed in the past. However, the errors caused by the approximate nature of the NURBS interpolator were rarely taken into account. This paper proposes an integrated look-ahead algorithm for parametric interpolation along NURBS curves. The algorithm interpolates the sharp corners on the curve with the Pythagorean-hodograph (PH) interpolation. This will minimize the geometric and interpolator approximation errors simultaneously. The algorithm consists of four different modules: a sharp corner detection module, a PH construction module, a feedrate planning module, and a dynamics module. Simulations are performed to show correctness of the proposed algorithm. Experiments on an X–Y table confirm that the developed method improves tracking and contour accuracies significantly compared to previously proposed algorithms.

Automated adaptive 3D forming simulation processes

In this study, an automated adaptive mesh control scheme, based on local mesh modifications, is developed for the finite element simulations of 3D metal-forming processes. Error indicators are used to control the mesh discretization errors, and an h-adaptive procedure is conducted. The mesh size field used in the h-adaptive procedure is processed to control the discretization and geometric approximation errors of the evolving workpiece mesh. Industrial problems are investigated to demonstrate the capabilities of the developed scheme.

Incremental view-dependent multiresolution triangulation of terrain

A view-dependent multiresolution triangulation algorithm is presented for a real-time fly-through. The triangulation of the terrain is generated incrementally on-the-fly during the rendering time. We show that since the view changes smoothly, only a few incremental modifications are required to update the triangulation to a new view. The resulting triangles form a multiresolution Delaunay triangulation which satisfies a predetermined view-dependent error tolerance. The presented method provides a guaranteed-quality mesh since it has control over the global geometric approximation error of the multiresolution view-dependent triangulation. © 1998 John Wiley & Sons, Ltd.