An improved isogeometrical analysis approach to functionally graded plane elasticity problems

The isogeometric analysis method is extended for addressing the plane elasticity problems with functionally graded materials. The proposed method which employs an improved form of the isogeometric analysis approach allows gradation of material properties through the patches and is given the name Generalized Iso-Geometrical Analysis (GIGA). The gradations of materials, which are considered as imaginary surfaces over the computational domain, are defined in a fully isoparametric formulation by using the same NURBS basis functions employed for the construction of the geometry and the approximation of the solution. The basic concept of the developed approach is concisely explained and its relation to the standard isogeometric analysis method is pointed out. It is shown that the difficulties encountered in the finite element analysis of the functionally graded materials are alleviated to a large degree by employing the mentioned method. Different numerical examples are presented and compared with available analytical solutions as well as the conventional and graded finite element methods to demonstrate the performance and accuracy of the proposed approach. The presented procedure can also be employed for solving other partial differential equations with non-constant coefficients.     

Patch coupling with the isogeometric dual mortar approach

The use of a common set of basis functions for design and analysis is the main paradigm of isogeometric analysis. The characteristics of the commonly used non-uniform rational B-splines (NURBS) surfaces require methods to handle non-conforming meshes to attain an efficient computational framework. The isogeometric mortar method uses constrained approximation spaces to enforce a coupling of deformations at the interface between patches in a weak manner. This method neither requires additional degrees of freedom nor the choice of empirical parameters. The main drawback of the standard isogeometric mortar approach is the non-local support of the mortar basis functions along the interface. This yields a large number of nodes per element for elements adjacent to the interface. Thus, the computational costs increase significantly for mesh refinement. This issue is remedied by the use of dual basis functions for the mortar method, which is referred to as dual mortar method. In this contribution several choices for the dual basis functions for B-splines are proposed and compared. A special focus is set on the support of the dual basis functions and on the support of the resulting mortar basis functions. Numerical examples show the influence of the choice for the dual basis functions on the accuracy of the global stress distribution, on the fulfillment of the interface conditions and on numerical efficiency. The use of approximate dual basis functions is shown to be competitive to computations of conforming meshes in terms of accuracy and efficiency. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application of domain decomposition methods to isogeometric analysis

During the past decade various new spatial discretization techniques have been developed. In particular, the usage of NURBS based shape functions, well known to the CAD community, has been adapted to finite element technology. In the present work we use the mortar finite element method for the coupling of nonconforming discretized sub-domains in the framework of nonlinear elasticity. We show that the method can be applied to isogeometric analysis with little effort, once the framework of NURBS based shape functions has been implemented. Furthermore, a specific coordinate augmentation technique allows the design of an energy-momentum scheme for the constrained mechanical system under consideration. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling and homogenization of textile reinforced composites based on the isogeometric analysis

In this contribution, the isogeometric analysis is used to compute the effective material properties of textile reinforced composites. The isogeometric analysis based on non-uniform rational B-splines (NURBS) provides an efficient approach for numerical modeling because there is no need for a mesh generation. There are further advantages such as the availability of a geometry representation based on NURBS in computer-aided design software and the possibility to apply different refinement methods which do not change the geometry of the numerical model. These properties motivate the combination of the isogeometric analysis with the homogenization method. Therefor, the unit cell model representing the inner architecture of a textile reinforced composite is defined using NURBS. In order to compute the effective mechanical properties of the heterogeneous material, the homogenization method with periodic boundary conditions is applied. Finally, two examples demonstrate the advantages of this approach. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A surface oriented solid formulation based on a hybrid Galerkin-collocation method

The contribution is concerned with a numerical method to analyze the mechanical behavior of 3D solids. The method employs directly the geometry defined by the boundary representation modeling technique, which is frequently used in CAD to define solids. It combines the benefits of the isogeometric analysis methodology with the scaled boundary finite element method. In the present approach, only the boundary surfaces of the solid are discretized. No tensor-product structure of three-dimensional objects is exploited to parametrize the physical domain. The weak form is applied only on the boundary surfaces. The governing partial differential equations of elasticity are transformed to an ordinary differential equation (ODE) of Euler type. The isogeometric Galerkin approach is employed to approximate the displacement response at the boundary surfaces. It exploits the two-dimensional NURBS objects to parametrize the boundary surfaces. To solve the Euler type ODE, the NURBS based collocation approach is applied. The accuracy of the method is validated against the analytical solutions. The presented method is able to analyze solids, which are bounded by an arbitrary number of surfaces. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Free vibration analysis of spatial Bernoulli–Euler and Rayleigh curved beams using isogeometric approach

This paper deals with the linear free vibration analysis of Bernoulli–Euler and Rayleigh curved beams using isogeometric approach. The geometry of the beam as well as the displacement field are defined using the NURBS basis functions which present the basic concept of the isogeometric analysis. A novel approach based on the fundamental relations of the differential geometry and Cauchy continuum beam model is presented and applied to derive the stiffness and consistent mass matrices of the corresponding spatial curved beam element. In the Bernoulli–Euler beam element only translational and torsional inertia are taken into account, while the Rayleigh beam element takes all inertial terms into consideration. Due to their formulation, isogeometric beam elements can be used for the dynamic analysis of spatial curved beams. Several illustrative examples have been chosen in order to check the convergence and accuracy of the proposed method. The results have been compared with the available data from the literature as well as with the finite element solutions.     

Dynamic responses of Euler–Bernoulli beam subjected to moving vehicles using isogeometric approach

Dynamic analysis of beam structures subjected to moving vehicles using an isogeometric Euler–Bernoulli formulation is presented in this paper. The method utilizes B-Splines or Non-Uniform Rational–Splines (NURBS) as the basis functions for both geometric and analysis implementation. The rotation-free technique has been incorporated into the formulation by using only one deflection variable with excluding the rotational degrees of freedom adopted for each control point. Then, it enables to use a few degrees of freedom (Dofs) to achieve a highly accurate solution. The validations of the proposed method included a complicated moving vehicle and rough pavement effects are compared to the precisely analytical results. Compared with most existing methods of finite element method (FEM) and readily analytical solutions, the present technique indicated the effectiveness of present isogeometric method and its well accurate prediction for suitable simulating the interaction model of the bridge structures and complicated vehicles.     

基于FETI的非协调等几何分析

基于非均匀有理B样条的等几何分析方法是一种无需网格划分的新的计算方法,旨在实现直接利用CAD模型进行分析,有望取代目前传统有限元技术．等几何分析已被成功应用在固体力学,流固耦合及拓扑优化等诸多领域．等几何分析方法要求CAD曲面或者实体高阶连续,而绝大多数CAD模型内多个曲面不但无法保持高阶连续,而且在公共界面处是几何非协调的．这一缺陷严重制约了等几何分析技术的进一步发展和应用．另外,由于采用高阶单元,等几何分析计算量较等自由度传统有限元要耗时．为解决这些难题,笔者在先前工作基础之上,提出了基于FETI方法的非协调等几何分析．新方法较以往的零空间解法更加快捷,适用于大规模数据的并行计算．数值算例表明本方法无需修改CAD模型,实施简单,精度满足要求,可处理复杂CAD模型．     

Isogeometric structural shape optimization using a fictitious energy regularization

In isogeometric analysis, NURBS basis functions are used as shape functions in an isoparametric finite-element-type discretization. Among other advantageous features, this approach is able to provide exact and smooth representations of a broad class of computational domains with curved boundaries. Therefore, this discretization method seems to be especially convenient for computational shape optimization, where a smooth and CAD-like parametrization of the optimal geometry is desired. Choosing boundary control point coordinates of an isogeometric discretization as design variables, an additional design model can be avoided. However, for a higher number of design variables, typical drawbacks like oscillating boundaries as known from early node-based shape optimization methods appear. To overcome this problem, we propose to use a fictitious energy regularization: the strain energy of a fictitious deformation, which maps the initial to the optimized domain, is employed as a regularizing term in the optimization problem. Moreover, this deformation is used for efficiently moving the dependent nodes within the domain in each step of the optimization process. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Explicit trace inequalities for isogeometric analysis and parametric hexahedral finite elements

We derive new trace inequalities for NURBS-mapped domains. In addition to Sobolev-type inequalities, we derive discrete trace inequalities for use in NURBS-based isogeometric analysis. All dependencies on shape, size, polynomial degree, and the NURBS weighting function are precisely specified in our analysis, and explicit values are provided for all bounding constants appearing in our estimates. As hexahedral finite elements are special cases of NURBS, our results specialize to parametric hexahedral finite elements, and our analysis also generalizes to T-spline-based isogeometric analysis. We compare the bounding constants appearing in our explicit trace inequalities with numerically computed optimal bounding constants, and we discuss application of our results to a Laplace problem. We finish this paper with a brief exploration of so-called patch-wise trace inequalities for isogeometric analysis.     

An isogeometric boundary element approach for topology optimization using the level set method

Among the various types of structural optimization, topology has been occupying a prominent place over the last decades. It is considered the most versatile because it allows structural geometry to be determined taking into account only loading and fixing constraints. This technique is extremely useful in the design phase, which requires increasingly complex computational modeling. Modern geometric modeling techniques are increasingly focused on the use of NURBS basis functions. Consequently, it seems natural that topology optimization techniques also use this basis in order to improve computational performance. In this paper, we propose a way to integrate the isogeometric boundary techniques to topology optimization through the level set function. The proposed coupling occurs by describing the normal velocity field from the level set equation as a function of the normal shape sensitivity. This process is not well behaved in general, so some regularization technique needs to be specified. Limiting to plane linear elasticity cases, the numerical investigations proposed in this study indicate that this type of coupling allows to obtain results congruent with the current literature. Moreover, the additional computational costs are small compared to classical techniques, which makes their advantage for optimization purposes evident, particularly for boundary element method practitioners.     

Static and dynamic analyses of isogeometric curvilinearly stiffened plates

The isogeometric analysis (IGA) is a new approach which builds a seamless connection between Computer Aided Design (CAD) and Computer Aided Engineering (CAE). This approach which uses the B-Splines or the Non-Uniform Rational B-Splines (NURBS) as a geometric representation of the object is a discretization technology for numerical analysis. The IGA has advantages of capturing exact geometry and making the flexibility of refinement, which results in higher calculation accuracy. To study the static and dynamic characteristics of curvilinearly stiffened plates, the NURBS based isogeometric analysis approach is developed in this paper. We use this approach to analyze the static deformation, the free vibration and the vibration behavior in the presence of in-plane loads of curvilinearly stiffened plates. Furthermore, the large deformation and the large amplitude vibration of the curvilinearly stiffened plates are also studied based on the von Karman's large deformation theory. One of the superiorities of the present method in the analysis of the stiffened plates is that the element number is much less than commercial finite element software, whereas another advantage is that the mesh refinement process is much more convenient compared with traditional finite element method (FEM). Some numerical examples are shown to validate the correctness and superiority of the present method by comparing with the results from commercial software and finite element analysis.     

Development of an inverse isogeometric methodology and its application in sheet metal forming process

This paper proposes an inverse isogeometric analysis to estimate the blank and predict the strain distribution in sheet metal forming processes. In this study, the same NURBS basis functions are used for drawing a final part and analysis of the forming process. In other words, this approach requires only one modeling and analysis representation, in contrast to inverse FEM. This model deals with minimization of potential energy, deformation theory of plasticity, and infinitesimal deformation relations with considering a new non-uniform friction model. One advantage of the presented methodology is that the governing equations are solved in two-dimensional space without concerning about pre-estimation results. As a result, the convergence is guaranteed and the computation time decreases significantly which is important at the initial stages of design. Furthermore, by employing this model at the forming design stage, the effects of changing the final part geometry and material property can be simultaneously observed on the formability of the part. Moreover, the effects of isogeometric element size can be automatically studied on the solution accuracy. The capability of this method is demonstrated by presenting three examples including blank estimation of cylindrical cup, square box, and weld line movement in forming of tailor welded blanks. The results obtained by the presented model and those obtained by the forward FEM reveal reasonable accuracy with decreased computational costs.     

Adaptive extended isogeometric analysis based on PHT-splines for thin cracked plates and shells with Kirchhoff–Love theory

In this paper, a posteriori error estimation and mesh adaptation approach for thin plate and shell structures of through-the-thickness crack is presented. This method uses the extended isogeometric analysis (XIGA) based on PHT-splines (Polynomial splines over Hierarchical T-meshes), which is abbreviated as XIGA-PHT. In XIGA-PHT, the isogeometric displacement approximation is locally enriched with enrichment functions, which efficiently capture the displacement discontinuity across the crack face as well as the stress singularity in the vicinity of the crack tip. On the one hand, the rotational degrees of freedom (RDOFs) are not required in Kirchhoff–Love theory, which drastically reduces the complexity of enrichment mode and computational scale for crack analysis. On the other hand, the PHT-splines basis functions can automatically satisfy the requirement of C¹-continuity for the Kirchhoff–Love theory. Moreover, the PHT-splines facilitate the local refinement, which is the deficiency of NURBS-based isogeometric formulations. The local refinement is highly suitable for adaptive analysis. The stress recovery-based posteriori error estimator combined with the superconvergent patch recovery (SPR) technique is used to evaluate the approximate local discretization error. A new strategy for selecting enriched recovered functions in the enriched areas was proposed. Special functions extracted from the asymptotic stress solutions are applied to obtain the recovered stress field in the enriched area. The results of stress intensity factors or J-integral values obtained by the adaptive XIGA-PHT are compared with reference solutions. Several thin plate and shell illustrative examples demonstrate the effectiveness and accuracy of the proposed adaptive XIGA-PHT.     

A polyconvex strain-energy split for a high-order phase-field approach to fracture

This contribution focuses on a novel phase-field model for a high-order phase-field approach to brittle fracture in the range finite deformation. In particular, two different challenges are tackled in this study: First, we want to establish a polyconvex free energy density to ensure the existence of a minimizer for the coupled problem, second, we have to deal with a fourth-order Cahn-Hilliard type equation for the approximation of the phase-field. Phase-field methods employ a variational framework for brittle fracture and have proven to predict complex fracture patterns in two and three dimensional examples. Basis of the model are the conjugate stresses of the three strain measures deformation gradient (line map), its cofactor (area map) and its determinant (volume map). The introduction of the tensor cross product simplifies the presentation of the first Piola-Kirchhoff stress tensor and its derivatives in elegant manner. The proposed Cahn-Hilliard type equation requires global -continuity. Therefore, we apply an isogeometric framework using NURBS basis functions. Moreover, a general hierarchical refinement scheme based on subdivision projection is used here for one, two and three dimensional simulations. This technique allows to enhance the approximation space using finer splines on each level but preserves the partition of unity as well as the continuity properties of the original discretization. We finally demonstrate the accuracy and the robustness with a series of benchmark problems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Weak coupling of thin-walled multi-patch NURBS structures in the framework of isogeometric analysis

In the framework of isogeometric analysis, models are typically derived by the use of computer-aided geometric design (CAGD) tools which often results in a large number of non-conforming NURBS patches that are connected along arbitrary curved boundaries. A strong coupling on the basis of matching control meshes is rarely possible and limits the modeling process for practical applications. Weak coupling according to the principles introduced by Nitsche is the method of choice if a stable and variationally consistent method is favored which does not require the solution of additional equations to enforce the coupling constraints. We concentrate on the weak coupling of thin-walled shell structures modeled according to the theory of Kirchhoff-Love. The proposed concept is free of ad-hoc decisions for stabilization thus truly supporting a design-through-analysis idea. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Material optimization of tri-directional functionally graded plates by using deep neural network and isogeometric multimesh design approach

The paper is aimed at enhancing computational performance for optimizing the material distribution of tri-directional functionally graded (FG) plates. We exploit advantages of using a non-uniform rational B-spline (NURBS) basis function for describing material distribution varying through all three directions of functionally graded (FG) plates. Two-dimensional free vibration and buckling behaviors of multi-directional (1D, 2D and 3D) FG plates analyzed by using a combination of generalized shear deformation theory (GSDT) and isogeometric analysis (IGA) is first proposed. This approach can help to save a significant amount of computational cost while still ensure the accuracy of the solutions. The effectiveness and reliability of the present method are demonstrated by comparing it to other methods in the literature. The obtained results are in excellent agreement with the reference ones. More importantly, data sets consisting of input-output pairs are randomly generated from the analysis process through iterations for the training process in deep neural networks (DNN). DNN is utilized as an analysis tool to supplant finite element analysis to reduce computational cost. By using DNN, behaviors of the multi-directional FG plates are directly predicted from those material distributions. Optimal material distributions of tri-directional FG plates under free vibration or compression in various volume fraction constraints are found by using modified symbiotic organisms search (mSOS) algorithm for the first time. Moreover, an isogeometric multimesh design technique is also used to diminish a large number of design variables in optimization. Optimal results obtained by DNN are compared with those of IGA to verify the effectiveness of the proposed method.     

T-splines discretizations for large deformation contact problems

A T-spline-based isogeometric analysis is applied to frictional contact problems between deformable bodies in the context of large deformations. The continuum is discretized with cubic T-splines and cubic NURBS (Non-Uniform Rational B-Splines) for comparison purposes. A Gauss-point-to-surface (GPTS) formulation is combined with the penalty method to treat the normal and friction contact constraints in the discretized setting. It is demonstrated that the proposed formulation combined with analysis-suitable T-spline interpolations, is a computationally accurate and efficient technology for local and global solutions of contact problems. T-spline analysis models are generated using commercially available T-spline modeling software without intermediate mesh generation or geometry clean-up steps. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)