Isogeometric Finite Element Analysis Research Articles

Isogeometric Finite Element Analysis with Machine Learning Integration for Piezoelectric Laminated Shells

Isogeometric Finite Element Analysis with Machine Learning Integration for Piezoelectric Laminated Shells

Surface elasticity and area incompressibility regulate fiber beading instability

A continuum body endowed with an energetic surface can exhibit different mechanical behavior than its bulk counterpart. Soft polymeric cylinders under surface effects become unstable and form surface undulations referred to as the elastic Plateau–Rayleigh (PR) instability, exclusively driven by competing surface and bulk properties. However, the impact of surface elasticity and area compressibility, along with bulk compressibility, on the PR instability of soft solids remains unexplored. Here we develop a theoretical, finite deformations framework to capture the onset of the PR instability in highly deformable solids with surface tension, surface elasticity, and surface compressibility, while retaining the compressibility of the bulk as a material parameter. In addition to the well-known elastocapillary number, surface compressibility and a dimensionless parameter related to the surface modulus are found to govern the instability behavior. The results of the theoretical framework are analyzed for an exhaustive list of bulk and surface parameters and loading scenarios, and it is found that increasing surface elasticity and surface incompressibility preclude PR instability. Theoretical results are compared with high-fidelity numerical simulation results from surface-enhanced isogeometric finite element analysis and an excellent agreement is observed across a broad range of material parameters and large deformation levels. Our results demonstrate how surface effects can be used to (i) render stable soft structures and prevent PR instability when it occurs as an unwanted by-product of manufacturing techniques or (ii) tune the instability behavior for possible applications involving polymer fibers.

Moving local mesh based on analysis-suitable T-splines and Bézier extraction for extended isogeometric finite element analysis - Application to two-dimensional crack propagation

We propose a moving local mesh for extended isogeometric analysis exploiting finite element data structures based on Bézier extraction of analysis-suitable T-splines. The local mesh, which is a refined subdomain, can be translocated over the physical space with no changes in its topological size and its polynomial functions, i.e. the number of elements, polynomial functions and their continuity remain the same for any potential change in the local mesh place. The idea is to consider the whole space as one or multiple domains of uniform rational B-splines, within which a subdomain is refined, in order to obtain two standardized local refinement and Bézier extraction operators. The local operators are compatible with every subdomain that has the same topological size. They are computed one single time and used for each iteration to find new refined control points of another specified subdomain. The computational time of isogeometric analysis is improved using extended sets of control points and parametric knots. These extended sets allow to use predetermined numbers in the connectivity arrays to describe the whole control net and physical mesh regardless of the local mesh place. It means there is no need to rebuild the element connectivity in each iteration for mapping the control points and physical nodes. A description for the implementation of this moving local mesh in isogeometric analysis is given, and three numerical examples in crack propagation analysis are studied. • Moving local mesh without knot reinsertion and element connectivity reconstruction. • A detailed description for the implementation + an updating algorithm. • Three numerical examples in crack propagation analysis for the validation.

Linear Independence of T-Spline Blending Functions of Degree One for Isogeometric Analysis

Linear independence of the blending functions is a necessary requirement for T-spline in isogeometric analysis. The main work in this paper focuses on the analysis about T-splines of degree one, we demonstrate that all the blending functions of such T-spline of degree one are linearly independent. The advantage owned by one degree T-spline is that it can avoid the problem of judging whether the model is analysis-suitable or not, especially for occasions that need a quick response from the analysis results. This may provide a new way of using T-spline for a CAD and CAE integrating scenario, since one degree T-spline still guarantees the topology flexibility and is compatible with the spline-based modeling system. In addition, we compare the numerical approximations of isogeometric analysis and finite element analysis, and the experiment indicates that isogeometric analysis using T-spline of degree one can reach a comparable result with classical method.

Thermal buckling of curvilinearly stiffened laminated composite plates with cutouts using isogeometric analysis

This research presents the Non-Uniform Rational B-Splines (NURBS) based isogeometric finite element analysis of stiffened laminated composite plates. A first-order shear deformation theory is used to derive the governing equations by employing a variational formulation. Isotropic, orthotropic and laminated composite plates stiffened with multiple curvilinear stiffeners of different profiles are investigated. A novel way to achieve displacement compatibility between the panel and stiffeners interfaces is introduced. An easy way of modeling plates with complicated cutouts by using edge curves and generating a ruled NURBS surface between them is described. Influence on the critical thermal buckling load due to the presence of circular and elliptical cutouts is also investigated. Results of parametric studies are presented which show the influence of ply orientation, size and orientation of the cutout, and the position and profile of the curvilinear stiffener. The numerical examples show high reliability and efficiency of the proposed formulation when compared to other published solutions and those obtained using ABAQUS, a commercial software.

Isogeometric analysis of FG-CNTRC plates in combination with hybrid type higher-order shear deformation theory

Abstract Bezier extraction based isogeometric finite element analysis method coupled with a new hybrid type higher-order shear deformation theory (HSDT) is presented for the static bending and buckling analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates. By mapping non-uniform rational B-spline basis functions to Bezier element in the forms of Bernstein polynomial basis via the Bezier extraction operator, the isogeometric analysis (IGA) can be implemented in the conventional finite element method (FEM) framework. This study focuses on the introduction of the Bezier extraction based IGA combined with the HSDT for the static analysis through the formulation and implementation of the second-order derivatives of the Bezier element basis functions, which has hardly been attempted in the open literature. Compared to the conventional FEM based on first-order shear deformation theory, the present IGA method based on the HSDT surmounts the shear locking without exploiting the shear correction factor. The proposed isogeometric approach is thus believed to be more accurate and effective and achieves higher convergence as the polynomial order increases. Performance accuracy of the Bezier extraction based isogeometric approach is first evidenced by comparing the computed results with the reference solutions, followed by the illustrative examples to further explore the flexural and buckling behavior of FG-CNTRC plates with various reinforcement patterns.

Review of Patient-Specific Vascular Modeling: Template-Based Isogeometric Framework and the Case for CAD

We review the literature on patient-specific vascular modeling, with particular attention paid to three-dimensional arterial networks. Patient-specific vascular modeling typically involves three main steps: image processing, analysis suitable model generation, and computational analysis. Analysis suitable model generation techniques that are currently utilized suffer from several difficulties and complications, which often necessitate manual intervention and crude approximations. Because the modeling pipeline spans multiple disciplines, the benefits of integrating a computer-aided design (CAD) component for the geometric modeling tasks has been largely overlooked. Upon completion of our review, we adopt this philosophy and present a CAD-integrated template-based modeling framework that streamlines the construction of solid non-uniform rational B-spline vascular models for performing isogeometric finite element analysis. Examples of arterial models for mouse and human circles of Willis and a porcine coronary tree are presented.

Isogeometric finite element analysis of mode I cracks within strain gradient elasticity

A variational formulation within an H2 Sobolev space setting is formulated for fourth-order plane strain/stress boundary value problems following a widely-used one parameter variant of Mindlin's strain gradient elasticity theory. A corresponding planar mode I crack problem is solved by isogeometric Cp-1-continuous discretizations for NURBS basis functions of order p \ge 2. Stress eld singularities of the classical elasticity are shown to be removed by the strain gradient formulation.

Convergence analysis of Laplacian-based gradient elasticity in an isogeometric framework

A convergence study is presented for a form of gradient elasticity where the enrichment is through the Laplacian of the strain, so that a fourth-order partial differential equation results. Isogeometric finite element analysis is used to accommodate the higher continuity required by the inclusion of strain gradients. A convergence analysis is carried out for the original system of a fourth-order partial differential equation. Both global refinement, using NURBS, and local refinement, using T-splines, have been applied. Theoretical convergence rates are recovered, except for a polynomial order of two, when the convergence rate is suboptimal, a result which also has been found for the (fourth-order) Cahn-Hilliard equation. The convergence analyses have been repeated for the case that an operator split is applied so that a set of two (one-way) coupled partial differential equations results. Differences occur with the results obtained for the original fourth-order equation, which is caused by the boundary conditions, which is the first time this effect has been substantiated.

CS-IGA: A new cell-based smoothed isogeometric analysis for 2D computational mechanics problems

In this work a new cell based strain smoothing formulation for isogeometric finite element analysis is proposed. The new CS-IGA method is formulated by incorporating cell-wise strain smoothing operation into IGA method. The compatible strain fields are smoothed based on smoothing domains associated with entities of isogeometric elements and the smoothed Galerkin weak form based on these smoothing domains is then applied to compute the system stiffness matrix. Unlike IGA, this new approach avoids gradient computation and the assembling of stiffness matrices. A transformation method is employed to accurately solve the problem and enforce properly the essential boundary conditions. Several numerical examples are investigated to verify the validity and accuracy of the proposed method. It has been found that numerical results obtained from CS-IGA can achieve very satisfactory accuracy and improved convergence rate.

Fluid flow in fractured and fracturing porous media: A unified view

Fluid flow in fractures that pre-exist or propagate in a porous medium can have a major influence on the deformation and flow characteristics. With the aim of carrying out large-scale calculations at reasonable computing costs, a sub-grid scale model has been developed. While this model was originally embedded in extended finite element methods, thereby exploiting some special properties of the enrichment functions, we will herein show that, using proper micro–macro relations, in particular for the mass balance, sub-grid scale models can be coupled to a range of discretisation methods at the macroscopic scale, from standard interface elements to isogeometric finite element analysis.

A generalized unconstrained theory and isogeometric finite element analysis based on Bézier extraction for laminated composite plates

This work presents an isogeometric finite element formulation based on Bezier extraction of the non-uniform rational B-splines (NURBS) in combination with a generalized unconstrained higher-order shear deformation theory (UHSDT) for laminated composite plates. The proposed approach relaxes zero-shear stresses at the top and bottom surfaces of the plates and no shear correction factors are required. A weak form of static, free vibration and transient response analyses for laminated composite plates is then established and is numerically solved using isogeometric Bezier finite elements. NURBS can be written in terms of Bernstein polynomials and the Bezier extraction operator. IGA is implemented with the presence of C°-continuous Bezier elements which allow to easily incorporate into existing finite element codes without adding many changes as the former IGA. As a result, all computations can be performed based on the basis functions defined previously as the same way in finite element method (FEM). Numerical results performed over static, vibration and transient analysis show high efficiency of the present method.

Isogeometric analysis of the Cahn–Hilliard equation – a convergence study

Herein, we present a numerical convergence study of the Cahn-Hilliard phase-field model within an isogeometric finite element analysis framework. Using a manufactured solution, a mixed formulation of the Cahn-Hilliard equation and the direct discretisation of the weak form, which requires a C 1 -continuous approximation, are compared in terms of convergence rates. For approximations that are higher than second-order in space, the direct discretisation is found to be superior. Suboptimal convergence rates occur when splines of order p = 2 are used. This is validated with a priori error estimates for linear problems. The convergence analysis is completed with an investigation of the temporal discretisation. Second-order accuracy is found for the generalised-α method. This ensures the functionality of an adaptive time stepping scheme which is required for the efficient numerical solution of the Cahn-Hilliard equation. The isogeometric finite element framework is eventually validated by two numerical examples of spinodal decomposition.

Adaptive isogeometric finite element analysis of steady‐state groundwater flow

SummaryNumerical challenges occur in the simulation of groundwater flow problems because of complex boundary conditions, varying material properties, presence of sources or sinks in the flow domain, or a combination of these. In this paper, we apply adaptive isogeometric finite element analysis using locally refined (LR) B‐splines to address these types of problems. The fundamentals behind isogeometric analysis and LR B‐splines are briefly presented. Galerkin's method is applied to the standard weak formulation of the governing equation to derive the linear system of equations. A posteriori error estimates are calculated to identify which B‐splines should be locally refined. The error estimates are calculated based on recovery of the L2‐projected solution. The adaptive analysis method is first illustrated by performing simulation of benchmark problems with analytical solutions. Numerical applications to two‐dimensional groundwater flow problems are then presented. The problems studied are flow around an impervious corner, flow around a cutoff wall, and flow in a heterogeneous medium. The convergence rates obtained with adaptive analysis using local refinement were, in general, observed to be of optimal order in contrast to simulations with uniform refinement. Copyright © 2015 John Wiley & Sons, Ltd.

Isogeometric finite element analysis of interior acoustic problems

Isogeometric Analysis (IGA) can bridge the gap between geometrical and numerical modeling. To this end, the same basis functions used in Computer Aided Design are applied to represent geometry and approximate physical field in analysis. In this paper, the IGA is firstly introduced to finite element method (FEM) for interior acoustic problems. The domain is parameterized by Non-Uniform Rational B-Spline (NURBS) in the algorithm, which simplifies the mesh generation greatly and furthermore supplies an exact representation of curved boundaries. In addition, the IGA FEM possesses a distinct feature of flexible order-elevation technique without modifying the geometry. Several numerical examples are presented to validate the accuracy and demonstrate the merits of the IGA FEM in the analysis of interior acoustic problems.

Isogeometric analysis of functionally graded plates using a refined plate theory

We present in this paper a simple and effective approach that incorporates isogeometric finite element analysis (IGA) with a refined plate theory (RPT) for static, free vibration and buckling analysis of functionally graded material (FGM) plates. A new inverse tangent distributed function through the plate thickness is proposed. The RPT enables us to describe the non-linear distribution of shear stresses through the plate thickness without any requirement of shear correction factors (SCF). IGA utilizes basis functions namely B-splines or non-uniform rational B-splines (NURBS) which reach easily the smoothness of any arbitrary order. It hence satisfies the C1 requirement of the RPT model. The present method approximates the displacement field with four degrees of freedom per each control point allowing an efficient solution process.

Isogeometric finite element analysis of composite sandwich plates using a higher order shear deformation theory

We present in this paper a simple and effective formulation based on a fifth-order shear deformation theory (FiSDT) in combination with isogeometric finite element analysis (IGA) for composite sandwich plates. The FiSDT yields non-linear distribution of the transverse shear stresses through the plate thickness and ensures a prior tangential stress-free boundary condition. The IGA uses same basis functions, namely B-splines or non-uniform rational B-splines (NURBS), for preserving the precisely geometric representation and providing the numerical solution. It enables to achieve easily the smoothness with arbitrary continuity order and in the present method the C1-continuity requirement of higher order shear deformation models is fulfilled. The static, dynamic and buckling analysis of rectangular and circular plates is investigated for different boundary conditions. Numerical examples are given to show high accuracy of the proposed method.

Interlaminar stress calculation in composite and sandwich plates in NURBS Isogeometric finite element analysis

This research paper describes the development of NURBS Isogeometric finite element analysis and post-processor for interlaminar stress calculation in composite and sandwich plates. First-order, shear-deformable laminate composite plate theory is utilized in deriving the governing equations using a variational formulation. Linear, quadratic, higher order and k-refined NURBS elements are constructed and numerical validation is performed for laminated composite and sandwich plates. Lagrange finite element suffers from higher order stress gradient oscillations due to Gibbs phenomenon and require alternative stress recovery procedures for accurate interlaminar stress calculations, especially interlaminar normal stress. In this paper, direct post-processing is performed which computes interlaminar shear and normal stresses from higher order gradients of NURBS basis in a single step procedure. Interlaminar stresses are found to be in an excellent agreement with 3D elasticity solution and FSDT along with k-refinement procedure of NURBS basis is found to compute equivalent or better interlaminar normal stress than higher-order shear deformation theory.

Isogeometric finite element analysis of poroelasticity

SUMMARYWe present an alternative numerical approach for predicting the behaviour of a deformable fluid‐saturated porous medium. The conventional finite element technology is replaced by isogeometric analysis that uses non‐uniform rational B‐splines. The ability of these functions to provide higher‐order continuity and to exactly represent complex geometries makes isogeometric analysis a suitable candidate for accurately modeling a poroelastic medium. After some preliminaries regarding the formulation of isogeometric finite elements using Bézier extraction and a concise outline of poroelasticity theory, we describe how isogeometric finite elements can be used for a mixed formulation that results in case of a porous medium. The manuscript concludes by one‐dimensional and two‐dimensional examples, which demonstrate the superiority of the results of isogeometric finite element analysis in terms of the smoothness of the results compared with conventional finite element analysis and suggest that the requirement on the minimum time step for consolidation problems can be mitigated using interpolation functions that possess a higher‐order continuity. Copyright © 2013 John Wiley & Sons, Ltd.

The cohesive band model: a cohesive surface formulation with stress triaxiality

In the cohesive surface model cohesive tractions are transmitted across a two-dimensional surface, which is embedded in a three-dimensional continuum. The relevant kinematic quantities are the local crack opening displacement and the crack sliding displacement, but there is no kinematic quantity that represents the stretching of the fracture plane. As a consequence, in-plane stresses are absent, and fracture phenomena as splitting cracks in concrete and masonry, or crazing in polymers, which are governed by stress triaxiality, cannot be represented properly. In this paper we extend the cohesive surface model to include in-plane kinematic quantities. Since the full strain tensor is now available, a three-dimensional stress state can be computed in a straightforward manner. The cohesive band model is regarded as a subgrid scale fracture model, which has a small, yet finite thickness at the subgrid scale, but can be considered as having a zero thickness in the discretisation method that is used at the macroscopic scale. The standard cohesive surface formulation is obtained when the cohesive band width goes to zero. In principle, any discretisation method that can capture a discontinuity can be used, but partition-of-unity based finite element methods and isogeometric finite element analysis seem to have an advantage since they can naturally incorporate the continuum mechanics. When using interface finite elements, traction oscillations that can occur prior to the opening of a cohesive crack, persist for the cohesive band model. Example calculations show that Poisson contraction influences the results, since there is a coupling between the crack opening and the in-plane normal strain in the cohesive band. This coupling holds promise for capturing a variety of fracture phenomena, such as delamination buckling and splitting cracks, that are difficult, if not impossible, to describe within a conventional cohesive surface model.