A New Isogeometric Finite Element Method for Analyzing Structures

High-performance finite element research has always been a major focus of finite element method studies.This article introduces isogeometric analysis into the finite element method and proposes a new isogeometric finite element method.Firstly,the physical field is approximated by uniform B-spline interpolation,while geometry is represented by non-uniform rational B-spline interpolation.By introducing a transformation matrix,elements of types C^(0)and C^(1)are constructed in the isogeometric finite element method.Subsequently,the corresponding calculation formats for one-dimensional bars,beams,and two-dimensional linear elasticity in the isogeometric finite element method are derived through variational principles and parameter mapping.The proposed method combines element construction techniques of the finite element method with geometric construction techniques of isogeometric analysis,eliminating the need for mesh generation and maintaining flexibility in element construc-tion.Two elements with interpolation characteristics are constructed in the method so that boundary conditions and connections between elements can be processed like the finite element method.Finally,the test results of several examples show that:(1)Under the same degree and element node numbers,the constructed elements are almost consistent with the results obtained by traditional finite element method;(2)For bar problems with large local field variations and beam problems with variable cross-sections,high-degree and multi-nodes elements constructed can achieve high computational accuracy with fewer degrees of freedom than finite element method;(3)The computational efficiency of isogeometric finite element method is higher than finite element method under similar degrees of freedom,while as degrees of freedom increase,the computational efficiency between the two is similar.

A Deep Learning Approach to Shape Optimization Problems for Flexoelectric Materials Using the Isogeometric Finite Element Method

A new approach for flexoelectricmaterial shape optimization is proposed in this study.In this work,a proxymodel based on artificial neural network(ANN)is used to solve the parameter optimization and shape optimization problems.To improve the fitting ability of the neural network,we use the idea of pre-training to determine the structure of the neural network and combine different optimizers for training.The isogeometric analysis-finite element method(IGA-FEM)is used to discretize the flexural theoretical formulas and obtain samples,which helps ANN to build a proxy model from the model shape to the target value.The effectiveness of the proposed method is verified through two numerical examples of parameter optimization and one numerical example of shape optimization.

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.

A Hybrid Parallel Strategy for Isogeometric Topology Optimization via CPU/GPU Heterogeneous Computing

This paper aims to solve large-scale and complex isogeometric topology optimization problems that consumesignificant computational resources. A novel isogeometric topology optimization method with a hybrid parallelstrategy of CPU/GPU is proposed, while the hybrid parallel strategies for stiffness matrix assembly, equationsolving, sensitivity analysis, and design variable update are discussed in detail. To ensure the high efficiency ofCPU/GPU computing, a workload balancing strategy is presented for optimally distributing the workload betweenCPU and GPU. To illustrate the advantages of the proposedmethod, three benchmark examples are tested to verifythe hybrid parallel strategy in this paper. The results show that the efficiency of the hybrid method is faster thanserial CPU and parallel GPU, while the speedups can be up to two orders of magnitude.

Quantifying Uncertainty in Dielectric Solids’ Mechanical Properties Using Isogeometric Analysis and Conditional Generative Adversarial Networks

Accurate quantification of the uncertainty in the mechanical characteristics of dielectric solids is crucial for advancing their application in high-precision technological domains,necessitating the development of robust com-putational methods.This paper introduces a Conditional Generation Adversarial Network Isogeometric Analysis(CGAN-IGA)to assess the uncertainty of dielectric solids’mechanical characteristics.IGA is utilized for the precise computation of electric potentials in dielectric,piezoelectric,and flexoelectric materials,leveraging its advantage of integrating seamlessly with Computer-Aided Design(CAD)models to maintain exact geometrical fidelity.The CGAN method is highly efficient in generating models for piezoelectric and flexoelectric materials,specifically adapting to targeted design requirements and constraints.Then,the CGAN-IGA is adopted to calculate the electric potential of optimum models with different parameters to accelerate uncertainty quantification processes.The accuracy and feasibility of this method are verified through numerical experiments presented herein.

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.

Full-Scale Isogeometric Topology Optimization of Cellular Structures Based on Kirchhoff-Love Shells

Cellular thin-shell structures are widely applied in ultralightweight designs due to their high bearing capacity and strength-to-weight ratio.In this paper,a full-scale isogeometric topology optimization(ITO)method based on Kirchhoff-Love shells for designing cellular tshin-shell structures with excellent damage tolerance ability is proposed.This method utilizes high-order continuous nonuniform rational B-splines(NURBS)as basis functions for Kirchhoff-Love shell elements.The geometric and analysis models of thin shells are unified by isogeometric analysis(IGA)to avoid geometric approximation error and improve computational accuracy.The topological configurations of thin-shell structures are described by constructing the effective density field on the controlmesh.Local volume constraints are imposed in the proximity of each control point to obtain bone-like cellular structures.To facilitate numerical implementation,the p-norm function is used to aggregate local volume constraints into an equivalent global constraint.Several numerical examples are provided to demonstrate the effectiveness of the proposed method.After simulation and comparative analysis,the results indicate that the cellular thin-shell structures optimized by the proposed method exhibit great load-carrying behavior and high damage robustness.

Interpolating Isogeometric Boundary Node Method and Isogeometric Boundary Element Method Based on Parameter Space

In this paper,general interpolating isogeometric boundary node method(IIBNM)and isogeometric boundary element method(IBEM)based on parameter space are proposed for 2D elasticity problems.In both methods,the integral cells and elements are defined in parameter space,which can reproduce the geometry exactly at all the stages.In IIBNM,the improved interpolating moving leastsquare method(IIMLS)is applied for field approximation and the shape functions have the delta function property.The Lagrangian basis functions are used for field approximation in IBEM.Thus,the boundary conditions can be imposed directly in both methods.The shape functions are defined in 1D parameter space and no curve length needs to be computed.Besides,most methods for the treatment of the singular integrals in the boundary element method can be applied in IIBNM and IBEM directly.Numerical examples have demonstrated the accuracy of the proposed methods.

Isogeometric Collocation:A Mixed Displacement-Pressure Method for Nearly Incompressible Elasticity Dedicated to Professor Karl Stark Pister for his 95th birthday

We investigate primal and mixed u−p isogeometric collocation methods for application to nearly-incompressible isotropic elasticity.The primal method employs Navier’s equations in terms of the displacement unknowns,and the mixed method employs both displacement and pressure unknowns.As benchmarks for what might be considered acceptable accuracy,we employ constant-pressure Abaqus finite elements that are widely used in engineering applications.As a basis of comparisons,we present results for compressible elasticity.All the methods were completely satisfactory for the compressible case.However,results for low-degree primal methods exhibited displacement locking and in general deteriorated in the nearly-incompressible case.The results for the mixed methods behaved very well for two of the problems we studied,achieving levels of accuracy very similar to those for the compressible case.The third problem,which we consider a“torture test”presented a more complex story for the mixed methods in the nearly-incompressible case.

A Comprehensive Review of Isogeometric Topology Optimization:Methods,Applications and Prospects

Topology Optimization(TO)is a powerful numerical technique to determine the optimal material layout in a design domain,which has accepted considerable developments in recent years.The classic Finite Element Method(FEM)is applied to compute the unknown structural responses in TO.However,several numerical deficiencies of the FEM significantly influence the effectiveness and efficiency of TO.In order to eliminate the negative influence of the FEM on TO,IsoGeometric Analysis(IGA)has become a promising alternative due to its unique feature that the Computer-Aided Design(CAD)model and Computer-Aided Engineering(CAE)model can be unified into a same mathematical model.In the paper,the main intention is to provide a comprehensive overview for the developments of Isogeometric Topology Optimization(ITO)in methods and applications.Finally,some prospects for the developments of ITO in the future are also presented.

Structural Design Optimization Using Isogeometric Analysis: A Comprehensive Review

Isogeometric analysis(IGA),an approach that integrates CAE into conventional CAD design tools,has been used in structural optimization for 10 years,with plenty of excellent research results.This paper provides a comprehensive review on isogeometric shape and topology optimization,with a brief coverage of size optimization.For isogeometric shape optimization,attention is focused on the parametrization methods,mesh updating schemes and shape sensitivity analyses.Some interesting observations,e.g.the popularity of using direct(differential)method for shape sensitivity analysis and the possibility of developing a large scale,seamlessly integrated analysis-design platform,are discussed in the framework of isogeometric shape optimization.For isogeometric topology optimization(ITO),we discuss different types of ITOs,e.g.ITO using SIMP(Solid Isotropic Material with Penalization)method,ITO using level set method,ITO using moving morphable com-ponents(MMC),ITO with phase field model,etc.,their technical details and applications such as the spline filter,multi-resolution approach,multi-material problems and stress con-strained problems.In addition to the review in the last 10 years,the current developmental trend of isogeometric structural optimization is discussed.

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.

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.

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.

T-Splines Based Isogeometric Topology Optimization with Arbitrarily Shaped Design Domains

In this paper,a new isogeometric topology optimization(ITO)method is proposed by using T-splines based isogeometric analysis(IGA).The arbitrarily shaped design domains,directly obtained from CAD,are represented by a single T-spline surface which overcomes the topological limitations of Non-Uniform Rational B-Spline(NURBS).The coefficients correlated with control points are directly used as design variables.Therefore,the T-spline basis functions applied for geometry description and calculation of structural response are simultaneously introduced to represent the density distribution.Several numerical examples show that the proposed approach leads to a coherent workflow to handle design problems of complicated structures.The optimized results are free of checkerboard patterns without additional stabilization and filtering techniques due to the properties of T-splines,which also simplified the post-processing.In addition,through performing local refinement,we can easily achieve multiresolution optimization and infill optimization within the T-splines based framework.In general,the proposed method provides a possibility to design,analyze,and optimize engineering structures in a uniform model,which has the potential to improve design efficiency and reduce the cost of product development.

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.

Multiresolution Isogeometric Topology Optimisation Using Moving Morphable Voids

A general and new explicit isogeometric topology optimisation approach with moving morphable voids(MMV)is proposed.In this approach,a novel multiresolution scheme with two distinct discretisation levels is developed to obtain high-resolution designs with a relatively low computational cost.Ersatz material model based on Greville abscissae collocation scheme is utilised to represent both the Young’s modulus of the material and the density field.Two benchmark examples are tested to illustrate the effectiveness of the proposed method.Numerical results show that high-resolution designs can be obtained with relatively low computational cost,and the optimisation can be significantly improved without introducing additional DOFs.

Isogeometric Boundary Element Analysis for 2D Transient Heat Conduction Problem with Radial Integration Method

This paper presents an isogeometric boundary element method(IGABEM)for transient heat conduction analysis.The Non-Uniform Rational B-spline(NURBS)basis functions,which are used to construct the geometry of the structures,are employed to discretize the physical unknowns in the boundary integral formulations of the governing equations.Bezier extraction technique is employed to accelerate the evaluation of NURBS basis functions.We adopt a radial integration method to address the additional domain integrals.The numerical examples demonstrate the advantage of IGABEM in dimension reduction and the seamless connection between CAD and numerical analysis.

Multiscale Isogeometric Topology Optimization with Unified Structural Skeleton

This paper proposes a multiscale isogeometric topology optimization(ITO)method where the configuration and layout of microstructures are optimized simultaneously.At micro scale,a shape deformation method is presented to transform a prototype microstructure(PM)for obtaining a series of graded microstructures(GMs),where microstructural skeleton based on the level set framework is applied to retain more topology features and improve the connectability.For the macro scale calculation,the effective mechanical properties can be estimated by means of the numerical homogenization method.By adopting identical non-uniform rational basis splines(NURBS)as basis functions for both parameterized level set model and isogeometric calculation model,the isogeometric analysis(IGA)is integrated into the level set method,which contributes to improving the accuracy and efficiency.Numerical examples demonstrate that,the proposed method is effective in improving the performance and manufacturability.

Monte Carlo Simulation of Fractures Using Isogeometric Boundary Element Methods Based on POD-RBF

This paper presents a novel framework for stochastic analysis of linear elastic fracture problems.Monte Carlo simulation(MCs)is adopted to address the multi-dimensional uncertainties,whose computation cost is reduced by combination of Proper Orthogonal Decomposition(POD)and the Radial Basis Function(RBF).In order to avoid re-meshing and retain the geometric exactness,isogeometric boundary element method(IGABEM)is employed for simulation,in which the Non-Uniform Rational B-splines(NURBS)are employed for representing the crack surfaces and discretizing dual boundary integral equations.The stress intensity factors(SIFs)are extracted by M integral method.The numerical examples simulate several cracked structures with various uncertain parameters such as load effects,materials,geometric dimensions,and the results are verified by comparison with the analytical solutions.