Extended Isogeometric Analysis Research Articles

X-IGA Used for Orthotropic Material Crack Growth.

In this paper, we propose a new approach for numerically simulating the growth of cracks in unidirectional composite materials, termed extended isogeometric analysis, evaluating the maximum stress intensity factor and T-stress. To validate our approach, we used a small anisotropic plate with two edge cracks, beginning with formulating the governing equations based on the energy integral method, Stroh's Formula, and the Elastic Law describing the behaviour of anisotropic materials, while considering boundary conditions and initial states. A MATLAB code was developed to solve these equations numerically and to post-process the tensile stress and the stress intensity factor (SIF) in the first mode. The results for the SIF closely match those obtained using the extended finite element method (X-FEM), with a discrepancy of only 0.0021 Pa·m0.5. This finding underscores the credibility of our approach. The extended finite element method has demonstrated robustness in predicting crack propagation in composite materials in recent years, leading to its adoption by several widely used software packages in various industries.

Phase-field fracture analysis for implicit geometric model via adaptive extended isogeometric analysis

Mechanical parts produced via additive manufacturing method often needs a re-evaluation of their strength due to precision challenges. In this study, we introduce an adaptive extended isogeometric analysis (XIGA) method, grounded in an implicit geometric model, to assess structural strength. This method is seamlessly integrated with the phase field fracture theory to appraise part fracture strength and replicate crack propagation. The investigation encompasses both second-order and fourth-order phase-field models. The fourth-order phase-field model entails the computation of higher-order phase-field derivatives, necessitating the utilization of at least C1 continuous shape functions. This prerequisite is readily met through the application of isogeometric analysis. Moreover, we employ polynomial splines over hierarchical T-meshes (PHT-splines) within this framework to enable localized hierarchical refinement. The adaptive refinement of boundary elements efficiently furnishes precise stress solutions, while substantially enhancing the computational efficiency of phase-field fracture analysis through adaptively refining the element near the crack. Numerical illustrations validate the method's exceptional accuracy in stress field solutions and fracture crack predictions. Experimental outcomes underscore the method's proficiency in faithfully capturing the propagation of cracks within intricate porous structures. Furthermore, we furnish a schematic outlining the process of assessing fracture strength in additively manufactured components, thereby emphasizing the promising utility of the proposed methodology.

Stress-based evolutionary topology optimization via XIGA with explicit geometric boundaries

This paper presents a novel approach to stress-based topology optimization that employs NURBS representations of explicit geometric boundaries to address stress minimization and stress-constrained problems. The proposed methodology is compatible with CAD systems and utilizes Extended IsoGeometric Analysis (XIGA) to ensure an accurate representation of stress field in the evolving structure. A p-norm aggregation scheme is used to measure global stress levels, and the aggregated stress constraint is augmented to the objective (compliance) via a Lagrange multiplier in stress-constrained problems. The proposed approach introduces a novel and efficient Lagrange multiplier trail strategy aimed at reducing computational costs. Additionally, a normalization scheme and a partial differential equation filter are presented to stabilize the highly nonlinear stress-based optimization. Validation studies demonstrate the effectiveness and characteristics of the proposed approach. Moreover, control points of the structural design can be directly edited to further improve strength performance. Comparative studies show clear advantages of the proposed methodology in terms of computational efficiency compared to other Finite element analysis (FEA)-based optimization methods. Overall, this paper presents a practical and novel approach that addresses the limitations of existing FEA-based optimization methods in terms of accuracy, flexibility, and degree of freedom (DOF) cost, and makes a significant contribution to the field of stress-based topology optimization.

An enhanced extended isogeometric analysis with strong imposition of essential boundary conditions for crack problems using B++ splines

We present the extended isogeometric analysis (XIGA) based on B++ splines for linear elastic fracture problems. The proposed approach converts the control points associated with crack elements into collocation points on the crack surface. The opening of the crack surface is characterized by double-layer collocation points whereas the tip singularity is simulated by additional collocation points. In doing so, the additional control points associated with tip elements are replaced by the additional collocation points on the crack surface, which reduces the number of unknowns. The collocation points on the entire crack surface satisfy the Kronecker Delta property, which allows directly imposing Dirichlet boundary condition on the crack surface. A posteriori error estimation technique is also presented in this paper. The values of the normalised stress intensity factors (SIFs) are calculated by interaction integral technique. Finally, the suggested approach is verified by 2D benchmark problems.

XIGA: An eXtended IsoGeometric analysis approach for multi-material problems

Multi-material problems often exhibit complex geometries along with physical responses presenting large spatial gradients or discontinuities. In these cases, providing high-quality body-fitted finite element analysis meshes and obtaining accurate solutions remain challenging. Immersed boundary techniques provide elegant solutions for such problems. Enrichment methods alleviate the need for generating conforming analysis grids by capturing discontinuities within mesh elements. Additionally, increased accuracy of physical responses and geometry description can be achieved with higher-order approximation bases. In particular, using B-splines has become popular with the development of IsoGeometric Analysis. In this work, an eXtended IsoGeometric Analysis (XIGA) approach is proposed for multi-material problems. The computational domain geometry is described implicitly by level set functions. A novel generalized Heaviside enrichment strategy is employed to accommodate an arbitrary number of materials without artificially stiffening the physical response. Higher-order B-spline functions are used for both geometry representation and analysis. Boundary and interface conditions are enforced weakly via Nitsche’s method, and a new face-oriented ghost stabilization methodology is used to mitigate numerical instabilities arising from small material integration subdomains. Two- and three-dimensional heat transfer and elasticity problems are solved to validate the approach. Numerical studies provide insight into the ability to handle multiple materials considering sharp-edged and curved interfaces, as well as the impact of higher-order bases and stabilization on the solution accuracy and conditioning.

Fracture Modelling of a Cracked Pressurized Cylindrical Structure by Using Extended Iso-Geometric Analysis (X-IGA)

In this study, a NURBS basis function-based extended iso-geometric analysis (X-IGA) has been implemented to simulate a two-dimensional crack in a pipe under uniform pressure using MATLAB code. Heaviside jump and asymptotic crack-tip enrichment functions are used to model the crack’s behaviour. The accuracy of this investigation was ensured with the stress intensity factors (SIFs) and the J-integral. The X-IGA—based SIFs of a 2-D pipe are compared using MATLAB code with the conventional finite element method available in ABAQUS FEA, and the extended finite element method is compared with a user-defined element. Therefore, the results demonstrate the possibility of using this technique as an alternative to other existing approaches to modeling cracked pipelines.

Analysis of thermal effect on buckling of imperfect FG composite plates by adaptive XIGA

This paper presents an effective computational framework based on adaptive extended isogeometric analysis (XIGA) for analyzing thermal buckling stability behavior of functional graded plates (FGPs) with flaws (e.g., holes, cracks). In this model, the locally refined non-uniform rational B-splines (LR NURBS), which can exactly represent the structure geometry and carry out the local refinement of the mesh, are used as the basis functions; and the first-order shear deformation theory is adopted to describe the kinematics of plates. According to the enhanced stresses on the mid-plane under the first critical buckling mode, a posteriori error estimator is evaluated. In addition, based on the posteriori error estimator, the meshes are automatically locally refined through refining the basis functions with large error. The obtained results illustrate the accuracy and reliability of the developed method, and show that the adaptive local refinement has better convergence ratio and higher computational efficiency than those of the uniform global refinement. Addressing the influence of some numerical aspects such as gradient index, direction and location of crack, and hole size on the critical thermal buckling temperature is also given.

Multi-patch local mesh refinement XIGA based on LR NURBS and Nitsche’s method for crack growth in complex cracked plates

The objective of this article is to simulate crack growth of complex Mindlin–Reissner plates by developing an adaptive multi-patch extended isogeometric analysis (XIGA). Nitsche’s method is used to treat continuity between multi-patches or the coupling of non-conforming meshes, exactly describing the geometry of complex plates. The computational meshes in XIGA are independent of the cracks by introducing some enrichment functions into the displacement approximation based on the partition of unity, thus it is convenient in modeling evolution of crack. The locally refined non-uniform rational B-splines (LR NURBS), which have the properties of local refinement and exact description of geometry, are used to construct geometric model and taken as the shape functions in XIGA. To implement the adaptive local refinement, a method based on recovery technique by Zienkiewicz and Zhu is proposed. Based on the Mindlin–Reissner plate theory, interaction integral method is employed to calculate stress intensity factors (SIFs) and the maximum circumferential stress criterion is used to determine the crack propagation direction. Several numerical examples are carried out to demonstrate the performance of the adaptive multi-patch XIGA for simulating crack growth in complex plates.

Extended isogeometric analysis using B++ splines for strong discontinuous problems

Strong imposing Dirichlet boundary conditions remain a challenge for eXtended Finite Element Methods (XFEMs) or eXtended Isogeometric Analysis (XIGA) of discontinuous problems. Moreover, the physical meaning of the additional Degrees of Freedom (DOFs) in the displacement expression of XFEMs or XIGA is not clarified. To address these issues, we proposed a new method that combines XIGA and B++ splines for modeling the fracture behaviors of single and multiple cracks in 2D elasticity solids. We adopt the overlapping trimming curves to represent the crack curves. The control points that the basis functions are non-zeros for the penetrating elements in XIGA are changed into the double-layer collocation points on the crack curves in our proposed method. In doing so, the DOFs of control points of the elements penetrated by the crack in the XIGA are replaced by those of the double-layer collocation points on the crack curves. The movement of the crack edges is simulated by the displacement of collocation points. Thus, the physical meaning of the DOFs of the elements penetrated by the cracks is clarified. As the basis functions of the collocation points on the crack boundary satisfy the Kronecker delta property, the presented method allows strong imposing Dirichlet boundary conditions on the crack edges. The stress intensity factors (SIFs) are calculated by adopting the interaction integral technology. Numerical examples verify the accuracy of the proposed method.

Extended Isogeometric Analysis with strong imposing essential boundary conditions for weak discontinuous problems using B++ splines

Strong imposing Dirichlet boundary conditions remain a challenge for eXtended Isogeometric Analysis (XIGA) or eXtended Finite Element Methods (XFEM). This paper proposed a novel XIGA method of using B++ splines (Boundary plus plus splines) for bi-material weak discontinuous problems. Because B++ spline basis functions satisfy the Kronecker delta property, the presented method allows strong imposing Dirichlet boundary conditions on the boundaries and bi-material interfaces. In addition, compared with other XIGA methods, the number of the basis functions and degrees of freedom (DOFs) are significantly reduced in the process of conversion, which save the computational resources. Numerical examples verify the efficiency and accuracy of the proposed method.

Error-controlled adaptive LR B-splines XIGA for assessment of fracture parameters in through-cracked Mindlin-Reissner plates

This paper presents an error-controlled adaptive extended isogeometric analysis (XIGA) for assessment of fracture behavior for through-cracked Mindlin-Reissner plates. The locally refined (LR) B-splines that facilitate the local refinement are used as the basis functions for the approximations, while the kinematics of plates are described in terms of the Mindlin-Reissner plate theory. Appropriate enrichment functions for cracked plates are incorporated into the formulation to represent geometrically displacement discontinuities across the crack faces and singularity of the stress in the vicinity at the crack tips, thus cracks are modeled topologically independent of the computational mesh. Here we extend the Zienkiewicz-Zhu method-based error estimation to the XIGA formulation for cracks in Mindlin-Reissner plates such that non-smoothnesses and stress singularity are reflected accurately. We conduct the error-controlled adaptivity for local mesh refinement to enhance the accuracy and performance. Fracture parameters (i.e., mixed-mode intensity factors) are calculated through the contour interaction integral method in the context of Mindlin-Reissner plate theory. Several numerical examples for through-cracked plates are studied to show the accuracy and effectiveness of the proposed formulation in modeling fracture of cracked moderately thick plates.

Improved ANN technique combined with Jaya algorithm for crack identification in plates using XIGA and experimental analysis

This paper presents an effective method for crack identification to improve the training of Artificial Neural Networks (ANN) parameters using Jaya algorithm. Dynamic and static datasets are introduced using eXtended IsoGeometric Analysis (XIGA) to improve the accuracy of the proposed application based on the frequency and strain measurements. Based on the concept used in our previous works, XIGA provided more accurate results for fracture mechanics applications than other modelling techniques. Therefore, XIGA datasets of cracked plate are used to improve ANN technique for static and dynamic analyses. Model updating of the cracked plate is considered by introducing the mass of accelerometers and identifying Young’s modulus of the plate and stiffness of springs using Jaya algorithm. The difference between measured and calculated frequencies is used as an objective function to calibrate the XIGA model. The crack length is predicted using an adaptive approach without any previous knowledge based on the data provided from a numerical model. Jaya algorithm is used to optimize the most important parameters of ANN. Several numerical examples with different crack scenarios and different boundary conditions are studied in order to evaluate the proposed approach. The results show that the proposed application is able to predict all considered scenarios and accurately identify the crack length. Experimental data of cracked plates are used to validate the numerical predictions. Hence, this application is found to be robust and accurate for crack identification in plates.

Multi-material topology optimization for crack problems based on eXtended isogeometric analysis

This paper proposes a novel topology optimization method generating multiple materials for external linear plane crack structures based on the combination of IsoGeometric Analysis (IGA) and eXtended Finite Element Method (X-FEM). A so-called eXtended IsoGeometric Analysis (X-IGA) is derived for a mechanical description of a strong discontinuity state's continuous boundaries through the inherited special properties of X-FEM. In X-IGA, control points and patches play the same role with nodes and sub-domains in the finite element method. While being similar to X-FEM, enrichment functions are added to finite element approximation without any mesh generation. The geometry of structures based on basic functions of Non-Uniform Rational B-Splines (NURBS) provides accurate and reliable results. Moreover, the basis function to define the geometry becomes a systematic p-refinement to control the field approximation order without altering the geometry or its parameterization. The accuracy of analytical solutions of X-IGA for the crack problem, which is superior to a conventional X-FEM, guarantees the reliability of the optimal multi-material retrofitting against external cracks through using topology optimization. Topology optimization is applied to the minimal compliance design of two-dimensional plane linear cracked structures retrofitted by multiple distinct materials to prevent the propagation of the present crack pattern. The alternating active-phase algorithm with optimality criteria-based algorithms is employed to update design variables of element densities. Numerical results under different lengths, positions, and angles of given cracks verify the proposed method's efficiency and feasibility in using X-IGA compared to a conventional X-FEM.

A computational approach for crack identification in plate structures using XFEM, XIGA, PSO and Jaya algorithm

In this paper, a creative and intelligent approach based on an inverse problem that accurately predicts crack location in plate structures is presented. The eXtended Finite Element (XFEM) and the eXtended IsoGeometric Analysis (XIGA) are combined with two optimization techniques, namely Particle Swarm Optimization (PSO) and Jaya algorithm to predict the crack location. The superiority of XIGA is demonstrated by using various NURBS orders to reduce the number of elements, provide fast simulation and achieve best convergence compared with XFEM. Four numerical-optimization techniques are considered in this paper, namely XFEM-Jaya, XIGA-Jaya, XFEM-PSO and XIGA-PSO. In the optimization techniques, the objective function minimizes the difference between the calculated and measured displacements and strains. Convergence studies for various positions of a crack and a hole in plates are performed and the results show that Jaya algorithm significantly performs more accurate and faster than PSO. In addition, theproposedtechniques are validated using experimental data and another numerical-optimization technique, i.e. XFEM coupled with Genetic Algorithm (GA), presented in literature. The comparisons show that XIGA-Jaya performs the best of all considered techniques.

Stable Generalized Iso-Geometric Analysis (SGIGA) for problems with discontinuities and singularities

Numerical analysis of physical/mathematical problems based on generalized/extended isogeometric analysis suffers from the major drawbacks of sub optimal convergence rates and ill conditioning of system matrices. Blending elements and linear dependency of basis functions are some of the causes attributed to these drawbacks. The presence of blending elements reduces the overall convergence rates and the ill conditioning of system matrices results in either increasing computational time when iterative solvers are used or erroneous results when direct solvers are employed. In order to alleviate these drawbacks, three different Stable Generalized IsoGeometric Analysis (SGIGA) methods are proposed in this paper. In SGIGA, the enrichment functions are modified by shifting the enrichment function using linear or least square interpolant of the enrichment function. Problems with weak and strong discontinuities, singularities and combination of both discontinuities and singularities are considered as case studies to illustrate the performance of the proposed SGIGA methods. From the results, it is observed that SGIGA yields optimal convergence rates as well as better conditioning of system matrices. The results obtained from the proposed SGIGA methods are also compared with the results from the established methods, Stable Generalized Finite Element Method (SGFEM) and eXtended IsoGeometric Analysis (XIGA), to study the relative performances with respect to accuracy and conditioning.

Fast simulations for solving fracture mechanics inverse problems using POD-RBF XIGA and Jaya algorithm

An inverse algorithm based on Proper Orthogonal Decomposition (POD) and Radial Basis Functions (RBF) for single and multiple cracks identification in plate structures is presented. The inverse analyses combine experimental fracture mechanics tests with numerical models based on the eXtended IsoGeometric Analysis (XIGA) method. The proposed technique identifies the location of cracks in a plate using static strains as input data. Simple and powerful algorithms, namely Jaya and Cuckoo search algorithms are coupled with the proposed XIGA-POD-RBF technique. The objective function is expressed as the difference between the strains of the actual cracks and those of the estimated ones. It is found that the proposed methodology is able to accurately estimate the cracks' locations with high accuracy and much faster than other techniques used in the literature. The stability of the identification algorithm based on Jaya is tested against measurement uncertainty by introducing a white Gaussian noise in the input data. The results show high stability for noise levels up to 4%.

Fatigue life estimation and crack propagation analysis of orthotropic lamina using XIGA methodology

ABSTRACTIn the present work, a NURBS-based Extended Isogeometric Analysis (XIGA) code has been developed for fracture analysis of orthotropic lamina. The XIGA code is capable of an efficient analysis of crack problems using non-uniform rational B-spline basis functions for both the solution field and geometric description of model. The level set approach is used to track the crack geometry. Crack faces are enriched by Heaviside enrichment function and crack tip singularity is captured by asymptotic crack tip enrichment functions. The interaction integral (M-Integral) method is used to calculate mixed mode stress intensity factor for further stable crack growth. The Paris crack growth law is used to simulate the crack propagation and fatigue life estimation in orthotropic media using life cycle increment approach. The developed code has been validated for static and fatigue analysis. The effect of varying angle of orthotropy on fatigue life of 2D plate is investigated using XIGA methodology leading to an optimized ply orientation of orthotropic plate.

Fatigue crack growth analysis of an interfacial crack in heterogeneous materials using homogenized XIGA

In this work, the fatigue life of an interfacial cracked plate is evaluated in the presence of flaws by homogenized extended isogeometric analysis (XIGA). In XIGA, the crack faces are modeled by discontinuous Heaviside jump functions, whereas the singularity in the stress field at the crack tip is modeled by crack tip enrichment functions. Holes and inclusions are modeled by Heaviside function and distance function respectively. The discontinuities are modeled in a selected region near the major crack whereas the region away from the crack is modeled by an equivalent homogeneous material. The stress intensity factors (SIFs) for the interface cracks are numerically evaluated using the domain based interaction integral approach. Paris law of fatigue crack growth is employed for computing the fatigue life of an interfacial cracked plate. The results show that the defects/ discontinuities away from the main crack barely influence the fatigue life.

Extended isogeometric analysis for dynamic fracture in multiphase piezoelectric/piezomagnetic composites

A dynamic extended isogeometric analysis (XIGA) is developed for transient fracture of cracked magnetoelectroelastic (MEE) solids under coupled electro-magneto-mechanical loading, taking the advantages of high order NURBS basis functions and enrichment methods. The extended dynamic fracture parameters are estimated through the electro-magneto-mechanical interaction integral. Numerical examples of electrically and magnetically impermeable cracks are studied to demonstrate the accuracy of the proposed XIGA and its ability in reproducing important phenomenological characteristics and behaviors of transient dynamic cracks in MEE materials. Our main objective focuses on the numerical investigations of transient dynamic cracks computed by the XIGA, investigating the effects of some numerical aspects on the responses, presenting new numerical results of dynamic responses, and addressing the XIGA performance. Different degrees of NURBS, polarization, finite size effects, enrichments, loadings, multiple cracks, volume fraction, different multiphase compositions of piezoelectric/piezromagnetic, etc. affecting the responses are investigated. Illustration of scattered elastic waves propagating in the cracked MEE is depicted to take an insight look at the behavior of responses.

Fatigue crack growth in functionally graded material using homogenized XIGA

In this work, the effect of defects/flaws (holes, inclusions, cracks) on the fatigue life of functionally graded material (FGM) is analyzed by homogenized extended isogeometric analysis (XIGA). In FGM, the gradation in the material properties is taken along the length of the plate. In XIGA, the crack faces are modeled by discontinuous Heaviside jump function, whereas the singularity in stress field at the crack tip is modeled by crack tip enrichment functions. Holes and inclusions are modeled by Heaviside jump function and distance function, respectively. The values of stress intensity factor (SIF) are numerically evaluated using the domain form of interaction integral approach. Paris law of fatigue crack growth is employed for computing the fatigue life. The flaws are modeled in a 30% region near the main crack, while the rest of the region is modeled with an equivalent homogeneous material. Several problems involving discontinuities in 30% region of the domain are solved by XIGA, and the results are compared with those obtained by modeling discontinuities in the entire domain of the plate.