Application of Isogeometric Analysis Method in Three-Dimensional Gear Contact Analysis

Gears are pivotal in mechanical drives,and gear contact analysis is a typically difficult problem to solve.Emerging isogeometric analysis(IGA)methods have developed new ideas to solve this problem.In this paper,a threedimensional body parametric gear model of IGA is established,and a theoretical formula is derived to realize single-tooth contact analysis.Results were benchmarked against those obtained from commercial software utilizing the finite element analysis(FEA)method to validate the accuracy of our approach.Our findings indicate that the IGA-based contact algorithmsuccessfullymet theHertz contact test.When juxtaposed with the FEA approach,the IGAmethod demonstrated fewer node degrees of freedomand reduced computational units,all whilemaintaining comparable accuracy.Notably,the IGA method appeared to exhibit consistency in analysis accuracy irrespective of computational unit density,and also significantlymitigated non-physical oscillations in contact stress across the tooth width.This underscores the prowess of IGA in contact analysis.In conclusion,IGA emerges as a potent tool for addressing contact analysis challenges and holds significant promise for 3D gear modeling,simulation,and optimization of various mechanical components.

Isogeometric Analysis of Hyperelastic Material Characteristics for Calcified Aortic Valve

This study explores the implementation of computed tomography(CT)reconstruction and simulation techniques for patient-specific valves,aiming to dissect the mechanical attributes of calcified valves within transcatheter heart valve replacement(TAVR)procedures.In order to facilitate this exploration,it derives pertinent formulas for 3D multi-material isogeometric hyperelastic analysis based on Hounsfield unit(HU)values,thereby unlocking foundational capabilities for isogeometric analysis in calcified aortic valves.A series of uniaxial and biaxial tensile tests is executed to obtain an accurate constitutive model for calcified active valves.To mitigate discretization errors,methodologies for reconstructing volumetric parametric models,integrating both geometric and material attributes,are introduced.Applying these analytical formulas,constitutive models,and precise analytical models to isogeometric analyses of calcified valves,the research ascertains their close alignment with experimental results through the close fit in displacement-stress curves,compellingly validating the accuracy and reliability of the method.This study presents a step-by-step approach to analyzing themechanical characteristics of patient-specific valves obtained fromCT images,holding significant clinical implications and assisting in the selection of treatment strategies and surgical intervention approaches in TAVR procedures.

New Perspective to Isogeometric Analysis:Solving Isogeometric Analysis Problem by Fitting Load Function

Isogeometric analysis(IGA)is introduced to establish the direct link between computer-aided design and analysis.It is commonly implemented by Galerkin formulations(isogeometric Galerkin,IGA-G)through the use of nonuniform rational B-splines(NURBS)basis functions for geometric design and analysis.Another promising approach,isogeometric collocation(IGA-C),working directly with the strong form of the partial differential equation(PDE)over the physical domain defined by NURBS geometry,calculates the derivatives of the numerical solution at the chosen collocation points.In a typical IGA,the knot vector of the NURBS numerical solution is only determined by the physical domain.A new perspective on the IGAmethod is proposed in this study to improve the accuracy and convergence of the solution.Solving the PDE with IGA can be regarded as fitting the load function defined on the NURBS geometry(right-hand side)with derivatives of the NURBS numerical solution(left-hand side).Moreover,the design of the knot vector has a close relationship to theNURBS functions to be fitted in the area of data fitting in geometric design.Therefore,the detected feature points of the load function are integrated into the initial knot vector of the physical domainto construct thenewknot vector of thenumerical solution.Then,they are connected seamlessly with the IGA-C framework for its great potential combining the accuracy and smoothness merits with the computational efficiency,which we call isogeometric collocation by fitting load function(IGACL).In numerical experiments,we implement our method to solve 1D,2D,and 3D PDEs and demonstrate the improvement in accuracy by comparing it with the standard IGA-C method.We also verify the superiority in the accuracy of our knot selection scheme when employed in the IGA-G method,which we call isogeometric Galerkin by fitting load function(IGA-GL).

Parameterization Transfer for a Planar Computational Domain in Isogeometric Analysis

In this paper,we propose a parameterization transfer algorithm for planar domains bounded by B-spline curves,where the shapes of the planar domains are similar.The domain geometries are considered to be similar if their simplified skeletons have the same structures.One domain we call source domain,and it is parameterized using multi-patch B-spline surfaces.The resulting parameterization is C1 continuous in the regular region and G1 continuous around singular points regardless of whether the parameterization of the source domain is C1/G1 continuous or not.In this algorithm,boundary control points of the source domain are extracted from its parameterization as sequential points,and we establish a correspondence between sequential boundary control points of the source domain and the target boundary through discrete sampling and fitting.Transfer of the parametrization satisfies C1/G1 continuity under discrete harmonic mapping with continuous constraints.The new algorithm has a lower calculation cost than a decomposition-based parameterization with a high-quality parameterization result.We demonstrate that the result of the parameterization transfer in this paper can be applied in isogeometric analysis.Moreover,because of the consistency of the parameterization for the two models,this method can be applied in many other geometry processing algorithms,such as morphing and deformation.

Isogeometric Analysis of Longitudinal Displacement of a Simplified Tunnel Model Based on Elastic Foundation Beam

Serious uneven settlement of the tunnel may directly cause safety problems.At this stage,the deformation of the tunnel is predicted and analyzed mainly by numerical simulation,while the commonly used finite element method(FEM)uses low-order continuous elements.Therefore,the accuracy of tunnel settlement prediction is not enough.In this paper,a method is proposed to study the vertical deformation of the tunnel by using the combination of isogeometric analysis(IGA)and Bézier extraction operator.Compared with the traditional IGA method,this method can be easily integrated into the existing FEM framework,and ensure the same accuracy.A numerical example of an elastic foundation beam subjected to uniformly distributed load and an engineering example of an equivalent elastic foundation beamof the tunnel are given.The results show that the solution of the IGA method is closer to the theoretical solution of the initial-parameter method than the FEM,and the accuracy and reliability of the proposedmodel are verified.Moreover,it not only provides some theoretical support for the longitudinal design of the tunnel,but also provides a new way for the application and popularization of IGA in tunnel engineering.

Isogeometric analysis of free-form Timoshenko curved beams including the nonlinear effects of large deformations

In the present paper, the isogeometric analysis（IGA） of free-form planar curved beams is formulated based on the nonlinear Timoshenko beam theory to investigate the large deformation of beams with variable curvature. Based on the isoparametric concept, the shape functions of the field variables（displacement and rotation） in a finite element analysis are considered to be the same as the non-uniform rational basis spline（NURBS） basis functions defining the geometry. The validity of the presented formulation is tested in five case studies covering a wide range of engineering curved structures including from straight and constant curvature to variable curvature beams. The nonlinear deformation results obtained by the presented method are compared to well-established benchmark examples and also compared to the results of linear and nonlinear finite element analyses. As the nonlinear load-deflection behavior of Timoshenko beams is the main topic of this article, the results strongly show the applicability of the IGA method to the large deformation analysis of free-form curved beams. Finally, it is interesting to notice that, until very recently, the large deformations analysis of free-form Timoshenko curved beams has not been considered in IGA by researchers.

Data-Driven Structural Design Optimization for Petal-Shaped Auxetics Using Isogeometric Analysis

Focusing on the structural optimization of auxetic materials using data-driven methods,a back-propagation neural network(BPNN)based design framework is developed for petal-shaped auxetics using isogeometric analysis.Adopting a NURBSbased parametric modelling scheme with a small number of design variables,the highly nonlinear relation between the input geometry variables and the effective material properties is obtained using BPNN-based fitting method,and demonstrated in this work to give high accuracy and efficiency.Such BPNN-based fitting functions also enable an easy analytical sensitivity analysis,in contrast to the generally complex procedures of typical shape and size sensitivity approaches.

T-Splines for Isogeometric Analysis of Two-Dimensional Nonlinear Problems

Nonlinear behaviors are commonplace in many complex engineering applications,e.g.,metal forming,vehicle crash test and so on.This paper focuses on the T-spline based isogeometric analysis of two-dimensional nonlinear problems including general large deformation hyperelastic problems and small deformation elastoplastic problems,to reveal the advantages of local refinement property of T-splines in describing nonlinear behavior of materials.By applying the adaptive refinement capability of T-splines during the iteration process of analysis,the numerical simulation accuracy of the nonlinear model could be increased dramatically.The Bézier extraction of the T-splines provides an element structure for isogeometric analysis that can be easily incorporated into existing nonlinear finite element codes.In addition,T-splines show great superiority of modeling complex geometries especially when the model is irregular and with hole features.Several numerical examples have been tested to validate the accuracy and convergence of the proposed method.The obtained results are compared with those from NURBS-based isogeometric analysis and commercial software ABAQUS.

Three-Dimensional Isogeometric Analysis of Flexoelectricity with MATLAB Implementation

Flexoelectricity is a general electromechanical phenomenon where the electric polarization exhibits a linear dependency to the gradient of mechanical strain and vice versa.The truncated pyramid compression test is among the most common setups to estimate the flexoelectric effect.We present a three-dimensional isogeometric formulation of flexoelectricity with its MATLAB implementation for a truncated pyramid setup.Besides educational purposes,this paper presents a precise computational model to illustrate how the localization of strain gradients around pyramidal boundary shapes contributes in generation of electrical energy.The MATLAB code is supposed to help learners in the Isogeometric Analysis and Finite Elements Methods community to learn how to solve a fully coupled problem,which requires higher order approximations,numerically.The complete MATLAB code which is available as source code distributed under a BSD-style license,is provided in the part of Supplementary Materials of the paper.

A High-Accuracy Single Patch Representation of Multi-Patch Geometries with Applications to Isogeometric Analysis

This paper presents a novel approximating method to construct highprecision single-patch representation of B-spline surface from a multi-patch representation for isogeometric applications.In isogeometric analysis,multi-patch structure is not easy to achieve high continuity between neighboring patches which will reduce the advantage of isogeometric analysis in a sense.The proposed method can achieve high continuity at surface stitching region with low geometric error,and this technique exploits constructing the approximate surface with several control points are from original surfaces,which guarantees the local feature of the surface can be well-preserved with high precision.With the proposed approximating method,isogeometric analysis results using the new single-patch can be obtained efficiently compared with the original multi-patch structure.Several examples are presented to illustrate the effectiveness,accuracy and efficiency of the proposed method.

Isogeometric Analysis and Shape Optimization of Holed Structures via the Patch Removing Technique

In this study,a patch removing based Isogeometric analysis(PR-IGA)method is proposed to conduct the holed structural analysis with only one parametric domain,in which there are also no trimmed elements.The theoretical foundation of this novel patch removing approach is that any holed structure can be obtained by removing sub-patches(i.e.,the holes)from an intact base patch.Since the parametric domains of these patches are all meshed by rectangular grids,the elements in the resulted holed structural parametric domain could all be untrimmed rectangles under certain mapping conditions.To achieve the special condition,a systematic technique consisting of T-spline local refinement and control points substitution/adjustment is provided.Due to the intactness of parametric elements,the analysis procedure of holed structures based on the proposed PRIGA is quite simplified and efficient compared to traditional multi-patch and trimming schemes.Moreover,after the deduction of analytical sensitivities related to structural mass and mechanical responses,the PR-IGA is directly employed in the holed structural shape optimization to successfully eliminate the need for model transformation during modeling,analysis and optimization processes.Numerical examples involving analysis and shape optimization of complex holed structures are presented to demonstrate the effectiveness of the proposed method.

Parametric Structural Optimization of 2D Complex Shape Based on Isogeometric Analysis

The geometric model and the analysis model can be unified together through the isogeometric analysis method,which has potential to achieve seamless integration of CAD and CAE.Parametric design is a mainstream and successful method in CAD field.This method is not continued in simulation and optimization stage because of the model conversion in conventional optimization method based on the finite element analysis.So integration of the parametric modeling and the structural optimization by using isogeometric analysis is a natural and interesting issue.This paper proposed a method to realize a structural optimization of parametric complex shapes by using isogeometric analysis.By the given feature curves and the constraints,a feature frame model is built.Based on the feature frame model,a parametric representation of complex shape is obtained.After adding some auxiliary curves,the feature frame model is divided into many box-like patches in three dimension or four-sided patches in two dimension.These patches are built into parametric patches by using volume interpolation methods such as Coons method.Based on the parametric patches,isogeometic analysis is applied.Thus,the relationships are constructed among the size parameters,the control points and the physical performance parameters.Then the sensitivity matrix could be derived based on the relationships.The size optimization is carried out in the first stage by taking the size parameters as variables.Based on the result of size optimization,shape optimization with the constraints of stress is carried out in the second stage by taking the control points as variables.Serval planar complex shapes are taken as example to verify our method.The results verify that the parametric modeling and structural optimization can be united together without model conversion.Benefit from this,the optimization design can be executed as a dark box operation without considering the concrete modeling and analysis by input of the sizes,constraints and loads.

Isogeometric Analysis with Local Adaptivity for Vibration of Kirchhoff Plate

Based on our proposed adaptivity strategy for the vibration of Reissner-Mindlin plate,we develop it to apply for the vibration of Kirchhoff plate.The adaptive algorithm is based on the Geometry-Independent Field approximaTion(GIFT),generalized from Iso-Geometric Analysis(IGA),and it can characterize the geometry of the structure with NURBS(Non-Uniform Rational B-Splines),and independently apply PHT-splines(Polynomial splines over Hierarchical T-meshes)to achieve local refinement in the solution field.TheMAC(Modal AssuranceCriterion)is improved to locate unique,as well as multiple,modal correspondence between different meshes,in order to deal with error estimation.Local adaptivity is carried out by sweeping modes from low to high frequency.Numerical examples showthat a proper choice of the spline space in solution field(with GIFT)can deliver better accuracy than using NURBS solution field.In addition,for vibration of heterogeneous Kirchhoff plates,our proposed method indicates that the adaptive local h-refinement achieves a better solution accuracy than the uniform h-refinement.

Analysis-Aware Modelling of Spacial Curve for Isogeometric Analysis of Timoshenko Beam

Geometric fitting based on discrete points to establish curve structures is an important problem in numerical modeling.The purpose of this paper is to investigate the geometric fitting method for curved beam structure from points,and to get high-quality parametric model for isogeometric analysis.ATimoshenko beam element is established for an initially curved spacial beam with arbitrary curvature.The approximation and interpolation methods to get parametric models of curves from given points are examined,and three strategies of parameterization,meaning the equally spaced method,the chord length method and the centripetal method are considered.The influences of the different geometric approximation algorithms on the precision of isogeometric analysis are examined.The static analysis and the modal analysis with the established parametric models are carried out.Three examples with different complexities,the quarter arc curved beam,the Tschirnhausen beam and the Archimedes spiral beam are examined.The results show that for the geometric approximation the interpolation method performs good and maintains high precision.The fitting algorithms are able to provide parametric models for isogeometric analysis of spacial beam with Timoshenko model.The equally spaced method and centripetal method perform better than the chord length method for the algorithm to carry out the parameterization for the sampling points.

Isogeometric analysis for free vibration of bidirectional functionally graded plates in the fluid medium

This paper for first time proposes an isogeometric analysis (IGA) for free vibration response of bi-directional functionally graded (BDFG) rectangular plates in the fluid medium. Material properties of the BDFG plate change in both the thickness and length directions via power-law distributions and Mori-Tanaka model. The governing equation of motion of BDFG plate in the fluid-plate system is formulated basing on Hamilton's principle and the refined quasi three-dimensional (3D) plate theory with improved function f(z). The fluid velocity potential is derived from the boundary conditions of the fluid-plate system and is used to determine the added mass. The discrete system of equations is derived from the Galerkin weak form and numerically analyzed by IGA. The accuracy and reliability of the proposed solutions are verified by comparing the obtained results with those published in the literature. Moreover, the effects of the various parameters such as the interaction boundary condition, geometric parameter, submerged depth of plate, fluid density, fluid level, and the material volume control coefficients on the free vibration behavior of BDFG plate in the fluid medium are investigated in detail. Some major findings regarding the numerical results are withdrawn in conclusions.

An XBi-CFAO Method for the Optimization of Multi-Layered Variable Stiffness Composites Using Isogeometric Analysis

This paper presents an effective fiber angle optimization method for two and multi-layered variable stiffness composites.A gradient-based fiber angle optimization method is developed based on isogeometric analysis(IGA).Firstly,the element densities and fiber angles for two and multi-layered composites are synchronously optimized using an extended Bi-layered continuous fiber angle optimization method(XBi-CFAO).The densities and fiber angles in the base layer are attached to the control points.The structure response and sensitivity analysis are accomplished using the non-uniform rational B-spline(NURBS)based IGA.By the benefit of the B-spline space,this method is free from checkerboards,and no additional filtering is needed to smooth the sensitivity numbers.Then the curved fiber paths are generated using the streamline method and the discontinuous fiber paths are smoothed using a partitioned selection process.The proposed method in the paper can alleviate the phenomenon of fiber discontinuity,enhance information retention for the optimized fiber angles of the singular points and save calculating resources effectively.

Thermally Induced Vibration Analysis of Flexible Beams Based on Isogeometric Analysis

Spacecraft flexible appendages may experience thermally induced vibrations(TIV)under sudden heating loads,which in consequence will be unable to complete their intended missions.Isogeometric analysis(IGA)utilizes,in an isoparametric concept,the same high order and high continuity non-uniform rational B-splines(NURBS)to represent both the geometry and the physical field of the structure.Compared to the traditional Lagrange polynomial based finite element method where only C0-continuity across elements can be achieved,IGA is geometrically exact and naturally fulfills the C1-continuity requirement of Euler–Bernoulli(EB)beam elements,therefore,does not need extra rotational degrees-of-freedom.In this paper,we present a thermally induced vibration analysis framework based on the isogeometric method where thermal and structural behaviors are coupled.We fully exploited the higher order,higher continuous and geometric exactness of the NURBS basis with both benchmarks and sophisticated problems.In particular,we studied the thermally induced vibrations of the Hubble Space Telescope(HST)solar panel where main factors influencing thermal flutters are studied,and where possible improvements of the analytical reference methods are discussed.Additionally,thermally induced vibrations of the thin-walled lenticular tubes are studied and two new configurations of the tube are proposed to effectively suppress the thermally induced vibrations.Numerical examples of both benchmarks and sophisticated problems confirm the accuracy and efficiency of the isogeometric analysis framework for thermally induced vibration analysis of space structures.

Reusing the Evaluations of Basis Functions in the Integration for Isogeometric Analysis

We propose a new approach to reuse the basis function evaluations in the numerical integration of isogeometric analysis.The concept of reusability of the basis functions is introduced according to their symmetrical,translational and proportional features on both the coarse and refined levels.Based on these features and the parametric domain regularity of each basis,we classify the bases on the original level and then reuse them on the refined level,which can reduce the time for basis calculations at integration nodes.By using the sum factorization method and the mean value theorem for the integrals,a new integration method with high integral efficiency is proposed.We validate the proposed method by some structural analysis problems in domains with different dimensionality.Comparing the numerical result accuracy and the time cost of the proposed integration method with the full Gauss integration quadrature,it turns out to be very promising.

Isogeometric Analysis for the Arbitrary AFG Microbeam with Two-Phase Nonlocal Stress-Driven Model

Scale effects play critical roles in the mechanical responses of microstructures.An isogeometric analysis was developed here to investigate the mechanical responses of an axially functionally graded microbeam.The Euler–Bernoulli beam model was utilized,and size effects in the structure were modeled with a stress-driven two-phase local/nonlocal integral constitution.The governing equation of microstructures was given in an equivalent differential form with two additional constitutive boundary conditions.The framework was verified and utilized to analyze the microbeam’s static and dynamic mechanical responses.The present work showed great potential for modeling various types of functionally graded microstructures.

Isogeometric Analysis-Based Topological Optimization for Heterogeneous Parametric Porous Structures

Porous structures widely exist in nature and artifacts,which can be exploited to reduce structural weight and material usage or improve damage tolerance and energy absorption.In this study,the authors develop an approach to design optimized porous structures with Triply Periodic Minimal Surfaces(TPMSs)in the framework of isogeometric analysis(IGA)-based topological optimization.In the developed method,by controlling the density distribution,the designed porous structures can achieve the optimal mechanical performance without increasing the usage of materials.First,the implicit functions of the TPMSs are adopted to design several types of porous elements parametrically.Second,to reduce the cost of computation,the authors propose an equivalent method to forecast the elastic modulus of these porous elements with different densities.Subsequently,the relationships of different porous elements between the elastic modulus and the relative density are constructed.Third,the IGA-based porous topological optimization is developed to obtain an optimal density distribution,which solves a volume constrained compliance minimization problem based on IGA.Finally,an optimum heterogeneous porous structure is generated based on the optimized density distribution.Experimental results demonstrate the effectiveness and efficiency of the proposed method.