Rational B-spline Research Articles

Isogeometric Analysis for Buckling and Wrinkling of Variable-Stiffness Sandwich Plates Under Compression

In this paper, an isogeometric analysis (IGA) framework based on the high-order sandwich plate theory is developed for the first time to deal with both buckling and wrinkling problems of variable-stiffness sandwich plates under compression. The present sandwich plate model is formulated based on the extended high-order sandwich plate theory (EHSAPT). The weak-form governing equations are firstly derived from the principle of virtual work, and afterward the IGA formulations based on the Non-Uniform Rational B-Splines (NURBS) are applied to predict the instability loads and the corresponding instability modes. Both the membrane and non-membrane conditions are integrated into the linear buckling analysis. The novelty of this work lies in that the employment of high-order continuous NURBS basis functions can directly fulfill the [Formula: see text] continuity requirement inherent in the EHSAPT theory, which enables the EHSAPT-based IGA model to be formulated with fewer degrees of freedom. Besides, the developed EHSAPT-based IGA model has also the ability to capture the overall buckling or wrinkling modes of the sandwich plates. The accuracy and effectiveness of the proposed EHSAPT-based IGA model are verified by comparison with previously published results and those obtained using ABAQUS. Effects of the core thickness and fiber angle on the buckling behaviors of the sandwich plates are also studied in numerical examples. It is observed that the non-uniform in-plane pre-stress over the skin layers is the main reason for the non-periodic wrinkling of constant-stiffness sandwich constructions, whereas the non-uniformity of both the in-plane pre-stress and out-of-plane bending stiffness is responsible for the non-periodic wrinkling of variable-stiffness sandwich constructions. The results presented herein may be beneficial for the design of variable-stiffness sandwich constructions under compression.

Modular gas-cooled reactor core optimal design and additive manufacturing performance based on SiC matrix dispersed with TRi-structural ISOtropic fuel

High-temperature gas-cooled reactors present significant technical competitiveness in various fields, such as low-carbon electricity, thermal energy, and hydrogen supply. Silicon Carbide (SiC) matrix dispersed with TRi-structural ISOtropic (TRISO) particle fuel is characterized by high-temperature tolerance, radiation resistance, and good neutron economy, which has emerged as a pivotal technological pathway for reactor core design. With the rapid advancement of SiC Additive Manufacturing (AM) technology, there is a substantial increase in core design flexibility, and an innovative gas-cooled reactor design was proposed by Monte Carlo (MC) method in this paper. The Non-uniform Rational B-spline (NURBS), Latin Hypercube Sampling (LHS), second-order polynomial response surface, as well as Genetic Algorithm (GA) were coupled to realize the optimization design for the coolant channel. Besides, a modular fuel design was proposed, and the AM SiC shell was fabricated using Binder Jetting (BJ) technology. Furthermore, the densification of the AM SiC shell and the matrix with SiC powder and SiC microspheres (TRISO simulator) was investigated by Chemical Vapor Infiltration (CVI). Finally, the microscopic morphology analysis and thermal-mechanical performance characterization of the samples were conducted, which demonstrated the technical feasibility and advanced capabilities of the SiC matrix dispersed with TRISO fuel based on AM technology.

Automatic heliostat learning for in situ concentrating solar power plant metrology with differentiable ray tracing

Concentrating solar power plants are a clean energy source capable of competitive electricity generation even during night time, as well as the production of carbon-neutral fuels, offering a complementary role alongside photovoltaic plants. In these power plants, thousands of mirrors (heliostats) redirect sunlight onto a receiver, potentially generating temperatures exceeding 1000°C. Practically, such efficient temperatures are never attained. Several unknown, yet operationally crucial parameters, e.g., misalignment in sun-tracking and surface deformations can cause dangerous temperature spikes, necessitating high safety margins. For competitive levelized cost of energy and large-scale deployment, in-situ error measurements are an essential, yet unattained factor. To tackle this, we introduce a differentiable ray tracing machine learning approach that can derive the irradiance distribution of heliostats in a data-driven manner from a small number of calibration images already collected in most solar towers. By applying gradient-based optimization and a learning non-uniform rational B-spline heliostat model, our approach is able to determine sub-millimeter imperfections in a real-world setting and predict heliostat-specific irradiance profiles, exceeding the precision of the state-of-the-art and establishing full automatization. The new optimization pipeline enables concurrent training of physical and data-driven models, representing a pioneering effort in unifying both paradigms for concentrating solar power plants and can be a blueprint for other domains.

Automated 3D Cobb Angle Measurement Using U-Net in CT Images of Preoperative Scoliosis Patients.

To propose a deep learning framework "SpineCurve-net" for automated measuring the 3D Cobb angles from computed tomography (CT) images of presurgical scoliosis patients.A total of 116 scoliosis patients were analyzed, divided into a training set of 89 patients (average age 32.4 ± 24.5 years) and a validation set of 27 patients (average age 17.3 ± 5.8 years). Vertebral identification and curve fitting were achieved through U-net and NURBS-net and resulted in a Non-Uniform Rational B-Spline (NURBS) curve of the spine. The 3D Cobb angles were measured in two ways: the predicted 3D Cobb angle (PRED-3D-CA), which is the maximum value in the smoothed angle map derived from the NURBS curve, and the 2D mapping Cobb angle (MAP-2D-CA), which is the maximal angle formed by the tangent vectors along the projected 2D spinal curve.The model segmented spinal masks effectively, capturing easily missed vertebral bodies. Spoke kernel filtering distinguished vertebral regions, centralizing spinal curves. The SpineCurve Network method's Cobb angle (PRED-3D-CA and MAP-2D-CA) measurements correlated strongly with the surgeons' annotated Cobb angle (ground truth, GT) based on 2D radiographs, revealing high Pearson correlation coefficients of 0.983 and 0.934, respectively.This paper proposed an automated technique for calculating the 3D Cobb angle in preoperative scoliosis patients, yielding results that are highly correlated with traditional 2D Cobb angle measurements. Given its capacity to accurately represent the three-dimensional nature of spinal deformities, this method shows potential in aiding physicians to develop more precise surgical strategies in upcoming cases.

Shape and size optimization framework of stiffened shell using isogeometric analysis

A shape and size optimization framework is developed to improve the structural performance of the stiffened shell using isogeometric analysis. To accurately model the stiffened shell, the Lagrange multiplier method is employed to establish the coupling relationship of isogeometric degenerated shell elements and isogeometric degenerated beam elements. Due to the advantages of the non-uniform rational B-spline (NURBS), the unit normal direction vectors of the shell can be defined analytically, which guarantees stiffeners perpendicular to the skin. Additionally, the stiffeners can be easily generated based on the mapping relationship of the parametric space and the physical space of the shell. Analytical sensitivity is derived in detail, and the gradient-based optimization method can be used to solve the minimum compliance optimization problem owing to the high order continuity of NURBS. Three numerical illustrative examples are presented to verify the effectiveness of the proposed framework for the shape and size optimization of stiffened shells, consisting of a cylindrical shell patch, a hyperbolic shell, and a complex engineering segment. It is worth noting that the proposed framework is especially suitable for optimization problems of the stiffened shell, eliminating the need of complex feature extraction.

Numerical Analysis of the Cylindrical Shell Pipe with Preformed Holes Subjected to a Compressive Load Using Non-Uniform Rational B-Splines and T-Splines for an Isogeometric Analysis Approach

In this paper, we implement the finite detail technique primarily based on T-Splines for approximating solutions to the linear elasticity equations in the connected and bounded Lipschitz domain. Both theoretical and numerical analyses of the Dirichlet and Neumann boundary problems are presented. The Reissner–Mindlin (RM) hypothesis is considered for the investigation of the mechanical performance of a 3D cylindrical shell pipe without and with preformed hole problems under concentrated and compression loading in the linear elastic behavior for trimmed and untrimmed surfaces in structural engineering problems. Bézier extraction from T-Splines is integrated for an isogeometric analysis (IGA) approach. The numerical results obtained, particularly for the displacement and von Mises stress, are compared with and validated against the literature results, particularly with those for Non-Uniform Rational B-Spline (NURBS) IGA and the finite element method (FEM) Abaqus methods. The obtained results show that the computation time of the IGA based on the T-Spline method is shorter than that of the IGA NURBS and FEM Abaqus/CAE (computer-aided engineering) methods. Furthermore, the highlighted results confirm that the IGA approach based on the T-Spline method shows more success than numerical reference methods. We observed that the NURBS IGA method is very limited for studying trimmed surfaces. The T-Spline method shows its power and capability in computing trimmed and untrimmed surfaces.

Nitsche's method enhanced isogeometric homogenization of unidirectional composites with cylindrically orthotropic carbon/graphite fibers

An isogeometric homogenization (IGH) technique is constructed for the homogenization and localization of unidirectional composites with radially or circumferentially orthotropic carbon/graphite fibers. The proposed theory employs multiple non-conforming Non-Uniform Rational B-Splines (NURBS) patches to depict repeating unit cells (RUCs) representative of composite microstructures. Displacements are formulated using a two-scale expansion that integrates macroscopic and microscopic contributions, with the latter addressed through the isogeometric analysis technique. Nitsche's method is utilized to apply the interfacial traction and displacement continuity and periodicity conditions. The capability and accuracy of the IGH theory were validated upon comparison with the elasticity solutions that take into account explicitly fiber morphologies, along with classical micromechanics solutions based on equivalent fiber moduli. A comparative analysis with conventional finite-element techniques showcases the developed theory's ability to accurately replicate the singular stress field at the fiber center and to capture smooth stress distributions where significant stress gradients are encountered.

Isogeometric Topology Optimization of Multi-Material Structures under Thermal-Mechanical Loadings Using Neural Networks

An isogeometric topology optimization (ITO) model for multi-material structures under thermal-mechanical loadings using neural networks is proposed. In the proposed model, a non-uniform rational B-spline (NURBS) function is employed for geometric description and analytical calculation, which realizes the unification of the geometry and computational models. Neural networks replace the optimization algorithms of traditional topology optimization to update the relative densities of multi-material structures. The weights and biases of neural networks are taken as design variables and updated by automatic differentiation without derivation of the sensitivity formula. In addition, the grid elements can be refined directly by increasing the number of refinement nodes, resulting in high-resolution optimal topology without extra computational costs. To obtain comprehensive performance from ITO for multi-material structures, a weighting coefficient is introduced to regulate the proportion between thermal compliance and compliance in the loss function. Some numerical examples are given and the validity is verified by performance analysis. The optimal topological structures obtained based on the proposed model exhibit both excellent heat dissipation and stiffness performance under thermal-mechanical loadings.

Isogeometric collocation method to simulate phase-field crystal model

PurposeThe purpose of this study is to develop a new numerical algorithm to simulate the phase-field model.Design/methodology/approachFirst, the derivative of the temporal direction is discretized by a second-order linearized finite difference scheme where it conserves the energy stability of the mathematical model. Then, the isogeometric collocation (IGC) method is used to approximate the derivative of spacial direction. The IGC procedure can be applied on irregular physical domains. The IGC method is constructed based upon the nonuniform rational B-splines (NURBS). Each curve and surface can be approximated by the NURBS. Also, a map will be defined to project the physical domain to a simple computational domain. In this procedure, the partial derivatives will be transformed to the new domain by the Jacobian and Hessian matrices. According to the mentioned procedure, the first- and second-order differential matrices are built. Furthermore, the pseudo-spectral algorithm is used to derive the first- and second-order nodal differential matrices. In the end, the Greville Abscissae points are used to the collocation method.FindingsIn the numerical experiments, the efficiency and accuracy of the proposed method are assessed through two examples, demonstrating its performance on both rectangular and nonrectangular domains.Originality/valueThis research work introduces the IGC method as a simulation technique for the phase-field crystal model.

Two-dimensional electromagnetic scattering analysis based on the boundary element method

An effective formula for the shape-sensitivity analysis of electromagnetic scattering is presented in this paper. First, based on the boundary element method, a new electromagnetic scattering formula is derived by combining the traditional electromagnetic scattering formula with the non-uniform rational B-spline (NURBS) curve, and the geometric model is represented by NURBS, which ensures the geometric accuracy, avoids the heavy grid division in the optimization process, and realizes the fast calculation of high-fidelity numerical solutions. Second, by deducing the sensitivity variables, the electromagnetic scattering equation of shape optimization is obtained, which can provide reliable data references for shape optimization. Finally, the effectiveness and accuracy of the algorithm are demonstrated by an example, and the sensitivity data of some examples are given.

Geometry-independent spline finite element method to analyze two-dimensional heat conduction and elasticity problems

Traditional isogeometric analysis involves redundant geometric calculations and is more complex than the finite element method when imposing boundary conditions. To address this, a geometry-independent spline finite element method is proposed, which integrates the geometric construction techniques of isogeometric analysis and the element construction techniques of finite element method. Firstly, the geometrical precision element is constructed by using non-uniform rational B-spline and B-splines to describe the geometry and the physical field, where a transformation matrix is introduced to construct shape functions with interpolation properties. Secondly, calculation formats for two-dimensional heat conduction and elasticity problems are derived by parameter mapping. Finally, the feasibility and accuracy of the method are verified through several numerical examples. Unlike in isogeometric analysis, in the proposed method, geometry and physics are separated, and the shape functions have interpolation properties, which reduce redundant geometry calculations and allow the boundary conditions to be applied directly to the nodes. The numerical results indicate that the method has a higher convergence rate compared to the original isogeometric analysis.

Density-Based Isogeometric Topology Optimization of Shell Structures

Shell structures with high stiffness-to-weight ratios are desirable in various engineering applications. Topology optimization serves as a popular and effective tool for generating optimal shell structures. The solid isotropic material with penalization (SIMP) method is often chosen because of its simplicity and convenience. However, SIMP method is typically integrated with conventional Finite Element Analysis (FEA) which has limitations in computational accuracy. Achieving high accuracy with FEA necessitates a substantial number of elements, leading to computational burdens. In addition, the discrete representation of the material distribution function may result in rough boundaries. Owing to these limitations, this paper proposes an Isogeometric Analysis (IGA) based SIMP method for optimizing the topology of shell structures based on Reissner–Mindlin theory. This method uses Non-Uniform Rational B-Splines (NURBS) to represent both the shell structure and the material distribution function with the same basis functions, allowing for higher accuracy and smoother boundaries. The optimization model takes compliance as the objective function with a volume fraction constraint and the coefficients of the density function as design variables, resulting in an optimized shell structure defined by the material distribution function. To obtain fairing boundaries of the holes in the optimized shell structure, further process is conducted by fitting the boundaries with fair B-spline curves automatically. Furthermore, the proposed IGA-SIMP framework is applied to generate porous shell structures by imposing different local volume fraction constraints. Numerical examples are provided to demonstrate the feasibility and efficiency of the IGA-SIMP method, showing that it outperforms the FEA-SIMP method and produces smoother boundaries.

Comparative post-therapeutic dosimetry between 2D planar-based and hybrid-based methods for personalized Lu-177 treatment.

This study aims to compare the calculated absorbed dose in target organs and tumors obtained using the different imaging protocols and the calculation methodologies implemented by HERMES HybridViewer dosimetry software for 177Lu-PSMA I&T and 177Lu-DOTATATE therapy. Multiple time-point whole-body planar images and one SPECT/CT image were acquired from 18 patients including 177Lu-PSMA I&T (13 patients) and 177Lu-DOTATATE treatment (5 patients) after administration of 3.80-8.58GBq injected activity. The regions of interest were drawn in the whole body, kidneys, liver, urinary bladder, salivary glands, and tumors to determine the time-integrated activity (TIA) in source organs. Absorbed doses in target organs were calculated according to the Medical Internal Radiation Dose (MIRD) scheme using the HERMES HybridViewer dosimetry integrated with OLINDA/EXM V.2.1 that utilizes the non-uniform rational B-splines (NURBS) for computational digital phantom. The planar-based dosimetry showed a higher dose per injected activity compared to the hybrid-based dosimetry, primarily due to organ overlap. The highest difference in absorbed dose between the imaging scenarios was observed in the spleen with a variation of up to 51.6%, while the difference for other target organs and tumors was less than 40%. The dosimetry calculation derived from the 2D planar-based method consistently demonstrates a significantly higher absorbed dose in organs and tumors compared with the hybrid-based method. However, the hybrid method outperforms the planar method in terms of tumor visualization and overlap-free organ delineation.

A novel tool path planning method for 5-axis single-point diamond turning

5-Axis single-point diamond turning adds two linked axes based on 3-axis single-point diamond turning, which can smoothly and continuously process complex surfaces. Compared to 3-axis turning, tool path planning for 5-axis single-point diamond turning is a complex decision-making process, which is a novel problem in the existing studies and a difficulty in multi-axis turning. In this study, a novel path planning method for 5-axis single-point diamond turning is proposed. Firstly, the mapping relation from scallop height to the distribution of cutter contact (CC) points is established in the non-uniform rational B-spline (NURBS) parameter space, thereby the planning of CC points is quickly achieved under fewer constraints. Furthermore, the local interference and global interference generation mechanism of 5-axis single-point diamond turning are revealed by geometrical analysis, combining diamond tool parameters and tool shank parameters. By transforming the motion of the cutter axis posture in 3D space into swings in two planes, the posture accessibility diagram of the cutter axis is efficiently constructed. Finally, the feasibility and effectiveness of the tool path planning method are verified by 5-axis single-point diamond turning experiment. And the contour accuracy is improved through the quadratic planning of the CC points (the contour accuracy after compensation is 3.2 μm compared to normal one 9.6 μm). This study extends the traditional 3-axis turning to 5-axis turning, providing a new perspective for the machining of complex surfaces.

On the isogeometric analysis of vibration and buckling of bioinspired composite plates using inverse hyperbolic shear deformation theory

Inspired by the remarkable energy absorption and damage tolerance capabilities observed in mantis shrimp crustaceans, this study delves into the intricate realm of biological composites with helicoidal structures. However, the effect of the relative orientation of fibers within the helicoidal structure matrix on the buckling and vibration characteristics still warrants comprehensive investigation for the better design of the bioinspired structures. For the first time, a Non-Uniform Rational B-Splines (NURBS)-based isogeometric analysis (IGA) framework termed NURBio is proposed to investigate the vibration and buckling characteristics of bioinspired laminated composite structures. The framework employs NURBS basis functions for exact complex geometric representation and field approximation and offers superior computational accuracy and efficiency as compared to conventional finite element method (FEM). The present analysis is underpinned by an inverse hyperbolic shear deformation theory (IHSDT) capable of accurately capturing the interlaminar stress distribution. This research investigates the performance of various bioinspired layup configurations encompassing recursive, exponential, semicircular, and Fibonacci helicoidal designs and compares them with those of unidirectional, and cross-ply laminates. A series of numerical examples on the buckling and vibration response of bioinspired composite plates are presented to demonstrate the versatility and efficacy of the proposed isogeometric framework. The influence of different layups, loading and boundary conditions, and geometric properties on the response of different bioinspired plate structures are meticulously examined. The study offers valuable insights into the design and optimization of high-performance bioinspired structures.

Discretization of Non-uniform Rational B-Spline (NURBS) Models for Meshless Isogeometric Analysis

We present an algorithm for fast generation of quasi-uniform and variable-spacing nodes on domains whose boundaries are represented as computer-aided design (CAD) models, more specifically non-uniform rational B-splines (NURBS). This new algorithm enables the solution of partial differential equations within the volumes enclosed by these CAD models using (collocation-based) meshless numerical discretizations. Our hierarchical algorithm first generates quasi-uniform node sets directly on the NURBS surfaces representing the domain boundary, then uses the NURBS representation in conjunction with the surface nodes to generate nodes within the volume enclosed by the NURBS surface. We provide evidence for the quality of these node sets by analyzing them in terms of local regularity and separation distances. Finally, we demonstrate that these node sets are well-suited (both in terms of accuracy and numerical stability) for meshless radial basis function generated finite differences discretizations of the Poisson, Navier-Cauchy, and heat equations. Our algorithm constitutes an important step in bridging the field of node generation for meshless discretizations with isogeometric analysis.

Inverse Geometric Reconstruction Based on MW-NURBS Curves

Currently, rational curves such as the Non-Uniform Rational B-Spline (NURBS) play a significant role in both shape representation and shape reconstruction. NURBS weights are often real in nature and are referred to as challenging to assign, with the exception of conics. ‘Matrix Weighted Rational Curves’ are the expanded form of rational curves that result from replacing these real weights with matrices, or matrix weights. The only difference between these curves and conventional curves is the geometric definition of the matrix weights. In this paper, MW-NURBS curves are used to reconstruct space curves from their stereo perspectives. In particular, MW-NURBS fitting is carried out in stereo views, and the weight matrices for the MW-NURBS curves are produced using the normal vectors provided at the control points. Instead of needing to solve a complicated system, the MW-NURBS model can reconstruct curves by choosing control points and control normals from the input data. The efficacy of the proposed strategy is verified by using many examples based on both synthetic and real images. The various error types are compared to those of conventional methods like point-based and NURBS-based approaches. The results demonstrate that the errors acquired from the proposed approach are much fewer than those obtained from the point-based method and the NURBS-based method.

DE3D-NURBS: A differential evolution-based 3D path-planner integrating kinematic constraints and obstacle avoidance

This work presents a novel path planner, DE3D-NURBS, for 3D path planning considering the maximum and minimum climb/dive angle and the maximum curvature imposed by a robot, such as an autonomous fixed-wing aircraft. To deal with this constrained problem, we propose a Differential Evolution (DE) based algorithm, assuming the partitioning of feasible and infeasible individuals and a novel mutation operator. This solution can provide a short differentiable smooth path, directly parameterized by a kth-order Non-Uniform Rational B-Spline (NURBS) curve, a generalization of B-spline curves. This representation provides more flexibility to satisfy essential kinematic constraints and to take into account obstacles and topography. This path can guide an Unmanned Aerial Vehicle (UAV) through a set of desired robot configurations (position and orientation) in 3D space. The simulation results provide a more comprehensive insight into the capabilities of the proposed planner to generate feasible paths, considering simple scenarios in a free obstacle environment and also complex scenarios. Such scenarios include topographic environments, narrow passages such as a tunnel, and a solution for closed-loop missions in a modular approach. We also provide a comparative benchmark with other nature-inspired algorithms. These results serve as a baseline for evaluating the effectiveness of the proposed DE algorithm in complex and challenging case studies.

Preform design to reduce forging load and grain size simultaneously in disk forging of IN718

In this study, we propose a new methodology for designing preform shapes for turbofan blisk and disk forgings of IN718 superalloy using a CNN (Convolutional Neural Network). The objective function of the preform design model is set to minimize grain size and reduce forging load. Various preform shape candidates were generated using NURBS (Non-Uniform Rational B-Splines) curves, defined by points on the target shape outline, and the grain size was calculated through FE (Finite Element) analysis based on the JMAK (Johnson-Mehl-Avrami-Kolmogorov) model to train the CNN model. The preform shapes derived from the proposed load and grain size-aware CNN model were compared and validated against those from previous studies regarding grain size and load reduction. An objective function f, which is capable of quantitatively expressing the load and grain size improvement, was developed for comparison and validation. The results showed an improvement of 19.64% for the blisk and 18.96% for the disk compared to previous studies. This research is expected to significantly contribute to the forging of aerospace components.

Sensitivity analysis of non-uniform rational B-splines–based finite element/boundary element coupling in structural-acoustic design

For the purpose of modeling the acoustic fluid-structure interaction using direct differentiation method and conducting a structural-acoustic sensitivity analysis, a coupling approach based on the finite element method and the fast multipole boundary element method is suggested. Non-uniform rational B-splines isogeometric analysis bypasses the difficult volume parameterization procedure in the isogeometric finite element method and the time-consuming meshing process in classical finite element/boundary element method, allowing numerical analysis on computer-aided design models to be completed directly. The finite element/fast multipole boundary element method based on non-uniform rational B-splines isogeometric analysis enables the numerical prediction of the effects of arbitrarily formed vibrating structures on the sound field. Several numerical examples are shown to demonstrate the usefulness and efficiency of the proposed method.