Extended isogeometric analysis based on PHT‐splines for crack propagation near inclusions

Adaptive local refinement is one of the main issues for isogeometric analysis (IGA). In this paper, an adaptive extended IGA (XIGA) approach based on polynomial splines over hierarchical T‐meshes (PHT‐splines) for modeling crack propagation is presented. The PHT‐splines overcome certain limitations of nonuniform rational B‐splines–based formulations; in particular, they make local refinements feasible. To drive the adaptive mesh refinement, we present a recovery‐based error estimator for the proposed method. The method is based on the XIGA method, in which discontinuous enrichment functions are added to the IGA approximation and this method does not require remeshing as the cracks grow. In addition, crack propagation is modeled by successive linear extensions that are determined by the stress intensity factors under linear elastic fracture mechanics. The proposed method has been used to analyze numerical examples, and the stress intensity factors results were compared with reference results. The findings demonstrate the accuracy and efficiency of the proposed method.     

A NURBS‐based interface‐enriched generalized finite element method for problems with complex discontinuous gradient fields

A non‐uniform rational B‐splines (NURBS)‐based interface‐enriched generalized finite element method is introduced to solve problems with complex discontinuous gradient fields observed in the structural and thermal analysis of the heterogeneous materials. The presented method utilizes generalized degrees of freedom and enrichment functions based on NURBS to capture the solution with non‐conforming meshes. A consistent method for the generation and application of the NURBS‐based enrichment functions is introduced. These enrichment functions offer various advantages including simplicity of the integration, possibility of different modes of local solution refinement, and ease of implementation. In addition, we show that these functions well capture weak discontinuities associated with highly curved material interfaces. The convergence, accuracy, and stability of the method in the solution of two‐dimensional elasto‐static problems are compared with the standard finite element scheme, showing improved accuracy. Finally, the performance of the method for solving problems with complex internal geometry is highlighted through a numerical example. Copyright © 2015 John Wiley & Sons, Ltd.     

T‐spline finite element method for the analysis of shell structures

A T‐spline surface is a nonuniform rational B‐spline (NURBS) surface with T‐junctions, and is defined by a control grid called T‐mesh. The T‐mesh is similar to a NURBS control mesh except that in a T‐mesh, a row or column of control points is allowed to terminate in the inner parametric space. This property of T‐splines makes local refinement possible. In the present study, shell formulation based on the T‐spline finite element method (FEM) is presented. Shell formulation based on NURBS or T‐splines has fundamental limitations because rotational DOFs, which are necessary in the shell formulation, cannot be defined on control points. In this study, the simple mapping scheme, in which every control point is mapped into one geometric point on the surface, is employed to eliminate the limitations. Using this mapping scheme, T‐spline FEM can be easily extended to the analysis of shells. The proposed shell formulation is verified through various benchmarking problems. This study is a part of the efforts by the authors for the integration of CAD–CAE processes. Copyright © 2009 John Wiley & Sons, Ltd.     

Weakening the tight coupling between geometry and simulation in isogeometric analysis: From sub‐ and super‐geometric analysis to Geometry‐Independent Field approximaTion (GIFT)

This paper presents an approach to generalize the concept of isogeometric analysis by allowing different spaces for the parameterization of the computational domain and for the approximation of the solution field. The method inherits the main advantage of isogeometric analysis, ie, preserves the original exact computer‐aided design geometry (for example, given by nonuniform rational B‐splines), but allows pairing it with an approximation space, which is more suitable/flexible for analysis, for example, T‐splines, LR‐splines, (truncated) hierarchical B‐splines, and PHT‐splines. This generalization offers the advantage of adaptive local refinement without the need to reparameterize the domain, and therefore without weakening the link with the computer‐aided design model. We demonstrate the use of the method with different choices of geometry and field spaces and show that, despite the failure of the standard patch test, the optimum convergence rate is achieved for nonnested spaces.     

Singular‐ended spline interpolation for two‐dimensional boundary element analysis

A method of interpolation of the boundary variables that uses spline functions associated with singular elements is presented. This method can be used in boundary element method analysis of 2‐D problems that have points where the boundary variables present singular behaviour. Singular‐ended splines based on cubic splines and Overhauser splines are developed. The former provides C²‐continuity and the latter C¹‐continuity across element edges. The potentialities of the methodology are demonstrated analysing the dynamic response of a 2‐D rigid footing interacting with a half‐space. It is shown that, for a given number of elements at the soil–foundation interface, the singular‐ended spline interpolation increases substantially the displacement convergence rate and delivers smoother traction distributions. Copyright © 2000 John Wiley & Sons, Ltd.     

A Kriging‐based error‐reproducing and interpolating kernel method for improved mesh‐free approximations

An error‐reproducing and interpolating kernel method (ERIKM), which is a novel and improved form of the error‐reproducing kernel method (ERKM) with the nodal interpolation property, is proposed. The ERKM is a non‐uniform rational B‐splines (NURBS)‐based mesh‐free approximation scheme recently proposed by Shaw and Roy (Comput. Mech. 2007; 40 (1):127–148). The ERKM is based on an initial approximation of the target function and its derivatives by NURBS basis functions. The errors in the NURBS approximation and its derivatives are then reproduced via a family of non‐NURBS basis functions. The non‐NURBS basis functions are constructed using a polynomial reproduction condition and added to the NURBS approximation obtained in the first step. In the ERKM, the interpolating property at the boundary is achieved by repeating the knot (open knot vector). However, for most problems of practical interest, employing NURBS with open knots is not possible because of the complex geometry of the domain, and consequently ERKM shape functions turn out to be non‐interpolating. In ERIKM, the error functions are obtained through localized Kriging based on a minimization of the squared variance of the estimate with the reproduction property as a constraint. Interpolating error functions so obtained are then added to the NURBS approximant. While enriching the ERKM with the interpolation property, the ERIKM naturally possesses all the desirable features of the ERKM, such as insensitivity to the support size and ability to reproduce sharp layers. The proposed ERIKM is finally applied to obtain strong and weak solutions for a class of linear and non‐linear boundary value problems of engineering interest. These illustrations help to bring out the relative numerical advantages and accuracy of the new method to some extent. Copyright © 2007 John Wiley & Sons, Ltd.     

Evaluation of power-logarithmic singularities, T-stresses and higher order terms of in-plane singular stress fields at cracks and multi-material corners

The scaled boundary finite-element method is extended to analyze the in-plane singular stress fields at cracks and multi-material corners. A complete singular stress field is represented semi-analytically as a series of matrix power functions of the radial coordinate originating from the singular point. This method is capable of directly evaluating orders of singularity, stress intensity factors, T-stresses, higher order terms, angular distributions of stresses for each term and, if present, power-logarithmic singularities. In this method, the singular functions are represented analytically and are not evaluated close to the singular point. Numerical examples are calculated to demonstrate the simplicity and to evaluate the accuracy of the scaled boundary finite-element method.     

Multiresolution topology optimization using isogeometric analysis

The paper introduces a novel multiresolution scheme to topology optimization in the framework of the isogeometric analysis. A new variable parameter space is added to implement multiresolution topology optimization based on the Solid Isotropic Material with Penalization approach. Design density variables defined in the variable space are used to approximate the element analysis density by the bivariate B‐spline basis functions, which are easily obtained using k‐refinement strategy in the isogeometric analysis. While the nonuniform rational B‐spline basis functions are used to exactly describe geometric domains and approximate unknown solutions in finite element analysis. By applying a refined sensitivity filter, optimized designs include highly discrete solutions in terms of solid and void materials without using any black and white projection filters. The Method of Moving Asymptotes is used to solve the optimization problem. Various benchmark test problems including plane stress, compliant mechanism inverter, and 2‐dimensional heat conduction are examined to demonstrate the effectiveness and robustness of the present method.     

An X‐FEM‐based numerical–asymptotic expansion for simulating a Stokes flow near a sharp corner

A numerical technique that is based on the integration of the asymptotic solution in the numerical framework for computing the local singular behavior of Stokes flow near a sharp corner is presented. Moffat's asymptotic solution is used, and special enriched shape functions are developed and integrated in the extended finite element method (X‐FEM) framework to solve the Navier–Stokes equations. The no‐slip boundary condition on the walls of the corner is enforced via the use of Lagrange multipliers. Flows around corners with different angles are simulated, and the results are compared with both those of the known analytic solution and the X‐FEM with no special enrichment near the corner. The results of the present technique are shown to greatly reduce the error made in computing the pressure and velocity fields near a corner tip without the need for mesh refinement near the corner. The method is then applied to the estimation of the permeability of a network of fibers, where it is shown that the local small‐scale pressure singularities have a large impact on the large‐scale network permeability. Copyright © 2014 John Wiley & Sons, Ltd.     

Powell–Sabin B‐splines and unstructured standard T‐splines for the solution of the Kirchhoff–Love plate theory exploiting Bézier extraction

The equations that govern Kirchhoff–Love plate theory are solved using quadratic Powell–Sabin B‐splines and unstructured standard T‐splines. Bézier extraction is exploited to make the formulation computationally efficient. Because quadratic Powell–Sabin B‐splines result in ‐continuous shape functions, they are of sufficiently high continuity to capture Kirchhoff–Love plate theory when cast in a weak form. Unlike non‐uniform rational B‐splines (NURBS), which are commonly used in isogeometric analysis, Powell–Sabin B‐splines do not necessarily capture the geometry exactly. However, the fact that they are defined on triangles instead of on quadrilaterals increases their flexibility in meshing and can make them competitive with respect to NURBS, as no bending strip method for joined NURBS patches is needed. This paper further illustrates how unstructured T‐splines can be modified such that they are ‐continuous around extraordinary points, and that the blending functions fulfil the partition of unity property. The performance of quadratic NURBS, unstructured T‐splines, Powell–Sabin B‐splines and NURBS‐to‐NURPS (non‐uniform rational Powell–Sabin B‐splines, which are obtained by a transformation from a NURBS patch) is compared in a study of a circular plate. Copyright © 2015 John Wiley & Sons, Ltd.     

NURBS based least‐squares finite element methods for fluid and solid mechanics

This contribution investigates the performance of a least‐squares finite element method based on non‐uniform rational B‐splines (NURBS) basis functions. The least‐squares functional is formulated directly in terms of the strong form of the governing equations and boundary conditions. Thus, the introduction of auxiliary variables is avoided, but the order of the basis functions must be higher or equal to the order of the highest spatial derivatives. The methodology is applied to the incompressible Navier–Stokes equations and to linear as well as nonlinear elastic solid mechanics. The numerical examples presented feature convective effects and incompressible or nearly incompressible material. The numerical results, which are obtained with equal‐order interpolation and without any stabilisation techniques, are smooth and accurate. It is shown that for p and h refinement, the theoretical rates of convergence are achieved. Copyright © 2014 John Wiley & Sons, Ltd.     

Blending moving least squares techniques with NURBS basis functions for nonlinear isogeometric analysis

IsoGeometric Analysis (IGA) is increasing its popularity as a new numerical tool for the analysis of structures. IGA provides: (i) the possibility of using higher order polynomials for the basis functions; (ii) the smoothness for contact analysis; (iii) the possibility to operate directly on CAD geometry. The major drawback of IGA is the non-interpolatory characteristic of the basis functions, which adds a difficulty in handling essential boundary conditions. Nevertheless, IGA suffers from the same problems depicted by other methods when it comes to reproduce isochoric and transverse shear strain deformations, especially for low order basis functions. In this work, projection techniques based on the moving least square (MLS) approximations are used to alleviate both the volumetric and the transverse shear lockings in IGA. The main objective is to project the isochoric and transverse shear deformations from lower order subspaces by using the MLS, alleviating in this way the volumetric and the transverse shear locking on the fully-integrated space. Because in IGA different degrees in the approximation functions can be used, different Gauss integration rules can also be employed, making the procedures for locking treatment in IGA very dependent on the degree of the approximation functions used. The blending of MLS with Non-Uniform Rational B-Splines (NURBS) basis functions is a methodology to overcome different locking pathologies in IGA which can be also used for enrichment procedures. Numerical examples for three-dimensional NURBS with only translational degrees of freedom are presented for both shell-type and plane strain structures.     

Direct evaluation of stress intensity factors for curved cracks using Irwin's integral and XFEM with high‐order enrichment functions

This paper presents a comprehensive study on the use of Irwin's crack closure integral for direct evaluation of mixed‐mode stress intensity factors (SIFs) in curved crack problems, within the extended finite element method. The approach employs high‐order enrichment functions derived from the standard Williams asymptotic solution, and SIFs are computed in closed form without any special post‐processing requirements. Linear triangular elements are used to discretize the domain, and the crack curvature within an element is represented explicitly. An improved quadrature scheme using high‐order isoparametric mapping together with a generalized Duffy transformation is proposed to integrate singular fields in tip elements with curved cracks. Furthermore, because the Williams asymptotic solution is derived for straight cracks, an appropriate definition of the angle in the enrichment functions is presented and discussed. This contribution is an important extension of our previous work on straight cracks and illustrates the applicability of the SIF extraction method to curved cracks. The performance of the method is studied on several circular and parabolic arc crack benchmark examples. With two layers of elements enriched in the vicinity of the crack tip, striking accuracy, even on relatively coarse meshes, is obtained, and the method converges to the reference SIFs for the circular arc crack problem with mesh refinement. Furthermore, while the popular interaction integral (a variant of the J‐integral method) requires special auxiliary fields for curved cracks and also needs cracks to be sufficiently apart from each other in multicracks systems, the proposed approach shows none of those limitations. Copyright © 2017 John Wiley & Sons, Ltd.     

The enhanced assumed strain method for the isogeometric analysis of nearly incompressible deformation of solids

Isogeometric analysis has recently become very popular for the numerical modeling of structures and fluids. Among other potential features, advantages of using a non‐uniform rational B‐splines (NURBS)‐based isogeometric analysis over the traditional finite element method include the possibility of using higher‐order polynomials for the basis functions of the approximation space, which may be easily built on a recursive (hierarchical) fashion as well as higher convergence ratio. Nevertheless, NURBS‐based isogeometric analysis suffers from the same problems depicted by other methods when it comes to reproduce isochoric deformations, that is, it shows volumetric locking, especially for low‐order basis functions. Similar remedies as those that have been proposed for the finite element method may be appropriate for integration in the NURBS‐based isogeometric analysis and some have already been tried with success. In this work, the analysis of the underlying space of incompressible deformations of a NURBS‐based isogeometric approximation is performed with the main objective of understanding the likelihood of volumetric locking. As a remedy, the enhanced assumed strain methodology is blended with the NURBS‐based isogeometric analysis to alleviate the volumetric locking associated with incompressible deformations. The solution includes a stabilization term derived directly from a penalized form of the classical Veubeke–Hu–Washizu three‐field variational principle. Copyright © 2012 John Wiley & Sons, Ltd.     

三维实体结构NURBS等几何分析

等几何分析是近年来在有限元法基础上发展起来的一种新的数值方法,它消除了有限元的几何误差,具有高阶连续性。该文研究了三维结构等几何分析中NURBS几何体的表示方式及载荷、约束的施加方法,分别从计算精度和仿真效率两个方面对比了等几何分析的计算结果与有限元法一阶单元和二阶单元的计算结果,展示了等几何分析相对于标准有限元法的优势,并以厚壁圆筒模型算例验证了等几何分析的实用性。将NURBS单元应用于几何形状精度要求高的齿轮和变截面圆筒,数值结果表明三维NURBS等几何分析方法在复杂三维结构的仿真计算中具有较好的灵活性和适用性,可得到连续的应力场,有望在工程中得到广泛应用。     

Matrix form near tip solutions of interface corners

In this paper, the stresses near the tip of interface corners are expressed in matrix power function form. To make this matrix power function form valid for all possible cases of interface corners, all kinds of singularities including oscillatory and logarithmic singularity are discussed in this paper. The coefficient vector of this matrix power function form solution is then defined as a vector of stress intensity factors along the radial direction. Since the stress intensity factors are functions of radial direction, the maximum stress intensity factor of a certain radial direction may be useful for the fracture prediction of interface corners. This new definition of stress intensity factors is applicable for all possible singular orders??repeated or distinct, real or complex, and keeps the same unit for all possible interface corners. Therefore, it will be helpful for bridging the problems of cracks, corners, interface cracks and interface corners. Finally, the matrix form solutions are reduced to the scalar form solutions for two special cases. One is a special case of corner angles??cracks, and the other is a special case of anisotropic materials??isotropic materials. Through this reduction, new scalar form analytical solutions are obtained and specialized further to compare with the existing analytical solutions.     

An isogeometric‐meshfree coupling approach for analysis of cracks

This paper presents a new concurrent simulation approach to couple isogeometric analysis (IGA) with the meshfree method for studying of crack problems. In the present method, the overall physical domain is divided into 2 subdomains that are formulated with the IGA and meshfree method, respectively. In the meshfree subdomain, the moving least squares shape function is adopted for the discretization of the area around crack tips, and the IGA subdomain is adopted in the remaining area. Meanwhile, the interface region between the 2 subdomains is represented by coupled shape functions. The resulting shape function, which comprises both IGA and meshfree shape functions, satisfies the consistency condition, thus ensuring convergence of the method. Moreover, the meshfree shape functions augmented with the enriched basis functions to predict the singular stress fields near a crack tip are presented. The proposed approach is also applied to simulate the crack propagation under a mixed‐mode condition. Several numerical examples are studied to demonstrate the use and robustness of the proposed method.