Isogeometric Element Research Articles

Volumetric mesh generation from T-spline surface representations

A new approach to obtain a volumetric discretization from a T-spline surface representation is presented. A T-spline boundary zone is created beneath the surface, while the core of the model is discretized with Lagrangian elements. T-spline enriched elements are used as an interface between isogeometric and Lagrangian finite elements. The thickness of the T-spline zone and thereby the isogeometric volume fraction can be chosen arbitrarily large such that pure Lagrangian and pure isogeometric discretizations are included. The presented approach combines the advantages of isogeometric elements (accuracy and smoothness) and classical finite elements (simplicity and efficiency).Different heat transfer problems are solved with the finite element method using the presented discretization approach with different isogeometric volume fractions. For suitable applications, the approach leads to a substantial accuracy gain.

Free Vibrations of Plates and Shells with an Isogeometric RM Shell Element

Free vibration analysis of plates and shells is carried out by using isogeometric approach. For this purpose, an isogeometric shell element based on Reissner-Mindlin (RM) shell theory is developed. Non-uniform rational B-spline surface (NURBS) definition is introduced to represent the geometry of shell and it is also used to derive all terms required in the isogeometric element formulation. New anchor positions are proposed to calculate the shell normal vector. Gauss integration rule is used for the formation of stiffness and mass matrices. The proposed shell element is then used to examine vibrational behaviours of plate and shell structures. From numerical results, it is found to be that reliable natural frequencies and associated mode shapes can be predicted by the present isogeometric RM shell element.

Computational Wear and Contact Modeling for Fretting Analysis with Isogeometric Dual Mortar Methods

A finite element framework based on dual mortar methods is presented for simulating fretting wear effects in the finite deformation regime. The mortar finite element discretization is realized with Lagrangean shape functions as well as isogeometric elements based on non-uniform rational B-splines (NURBS) in two and three dimensions. Fretting wear effects are modeled in an incremental scheme with the help of Archard’s law and the worn material is considered as additional contribution to the gap function. Numerical examples demonstrate the robustness and accuracy of the presented algorithm.

Analysis of the dynamical behavior of piezoceramic actuators using piezoelectric isogeometric finite elements

In this paper an electromechanically coupled isogeometric finite element is utilized to analyse Lamb wave excitation with piezoceramic actuators. An effective actuator design reduces the energy needed for Lamb wave excitation, which is beneficial if a structural health monitoring system should be applied for a structure. For a better understanding of the actuator behavior the piezoeceramics are studied both free and bonded at a structure. The numerical part of the analysis is performed utilizing isogeometric finite elements. To obtain the optimal performance for the numerical analysis the effect of k-refinement of the isogeometric element with respect to the convergence is studied and discussed. The optimal numerical setup with the best convergence rate is proposed and is validated with free piezoeceramic actuators. The validated model is then utilized to study the impact of actuator shape and adhesive bondline effect to the wave amplitude. The study shows that simplified analytical equations do not predict the optimal excitation frequencies for all piezoceramic designs accurately.

An efficient computational approach for control of nonlinear transient responses of smart piezoelectric composite plates

An efficient computational approach based on a generalized unconstrained approach in conjunction with isogeometric analysis (IGA) are proposed for dynamic control of smart piezoelectric composite plates. In composite plates, the mechanical displacement field is approximated according to the proposal model using isogeometric elements and the nonlinear transient formulation for plates is formed in the total Lagrange approach based on the von Kármán strains and solved by Newmark time integration. Through the thickness of each piezoelectric layer, the electric potential is assumed linearly. For active control of the piezoelectric composite plates, a close-loop system is used. An optimization procedure using genetic algorithm (GA) is considered to search optimal design for actuator input voltages. Various numerical examples are investigated to show high accuracy and reliability of the proposed method.

Matched [formula omitted]-constructions always yield [formula omitted]-continuous isogeometric elements

Gk (geometrically continuous surface) constructions were developed to create surfaces that are smooth also at irregular points where, in a quad-mesh, three or more than four elements come together. Isogeometric elements were developed to unify the representation of geometry and of engineering analysis. We show how matched Gk constructions for geometry and analysis automatically yield Ck isogeometric elements. This provides a formal framework for the existing and any future isogeometric elements based on geometric continuity.

Igatools: An Isogeometric Analysis Library

We present the design of an object oriented general purpose library for isogeometric analysis, where the mathematical concepts of the isogeometric method and their relationships are directly mapped into classes and their interactions. The encapsulation of mathematical concepts into interacting building blocks gives flexibility to use the library in a wide range of scientific areas and applications. We provide a precise framework for a lot of loose, available information regarding the implementation of the isogeometric method, and also discuss the similarities and differences between this and the finite element method. We also describe how to implement this proposed design in a C++11 open source library, \textttigatools (http://www.igatools.org). The library uses advanced object oriented and generic programming techniques to ensure reusability, reliability, and maintainability of the source code. It includes isogeometric elements of the h-div and h-curl type, and supports the development of dimension independent code (including manifolds and tensor-valued spaces). We finally present a number of code examples to illustrate the flexibility and power of the library, including surface domains, nonlinear elasticity, and Navier--Stokes computations on nontrivial geometries.

Isogeometric analysis of functionally graded carbon nanotube-reinforced composite plates using higher-order shear deformation theory

This paper presents a simple and effective formulation based on isogeometric analysis (IGA) and higher-order shear deformation theory (HSDT) to investigate the static and dynamic behavior of functionally graded carbon nano-reinforced composite plates. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to be graded through the thickness direction according to several linear distributions of the volume fraction of carbon nanotubes. The governing equation is approximated according to the HSDT model using isogeometric elements based on Non-Uniform Rational B-Spline (NURBS) basis functions. This achieves naturally any desired degree of continuity through the choice of the interpolation order, so that the method easily fulfils the C1-continuity requirement of the HSDT model. The accuracy and reliability of the proposed method is verified by comparing its numerical predictions with those of other available numerical approaches.

Three-dimensional isogeometrically enriched finite elements for frictional contact and mixed-mode debonding

We present an isogeometric enrichment technique for three-dimensional finite element computations applied to frictional contact and mixed-mode debonding. This is an extension of previous work that focused on two-dimensional and frictionless problems. To offer a more complete view of the enriched element’s performance, a comparison of the results to tri-variate isogeometric discretizations and standard Lagrangian elements is also included here. The enrichment is applied by discretizing parts of the surface that require higher accuracy with isogeometric basis functions, while the rest of the body uses Lagrangian shape functions. By using an isogeometric surface representation, the higher continuity across element boundaries and higher order of interpolation can be exploited. At the same time, the generation of tri-variate isogeometric meshes is avoided.A convergence study without any surface effects, involving only volume integrals, shows that the enriched elements can also be beneficial for these problems. The major advantage of the isogeometric element enrichment over standard tri-linear elements is demonstrated in contact problems including normal and tangential tractions. For both, mixed-mode cohesive debonding and frictional contact, the enrichment increases robustness and leads to more accurate results than standard linear Lagrangian elements. All computations are also compared to results using tri-variate isogeometric discretizations to give a complete picture of the element’s performance. It is also shown that the proposed enrichment formulation has advantages in mesh generation.

Isogeometric Analysis of Laminated Plates under Free Vibration

A plate element is developed by using isogeometric approach in order to determine natural frequencies of laminated composite plates. Reissner-Mindlin (RM) assumptions is adopted to consider the shear deformation and rotatory inertia effect. All terms required in isogeometric element formulation are consistently derived by using Non-uniform rational B-spline surface (NURBS) definition. Gauss quadrature rule is used to form the element stiffness matrix and separately Lobatto quadrature rule is used to calculate element mass matrix. The capability of the present plate element is demonstrated by using numerical examples. From numerical tests, the present isogeometric element produces reliable numerical results for both thin and thick plate situations.

An extended isogeometric thin shell analysis based on Kirchhoff–Love theory

An extended isogeometric element formulation (XIGA) for analysis of through-the-thickness cracks in thin shell structures is developed. The discretization is based on Non-Uniform Rational B-Splines (NURBS). The proposed XIGA formulation can reproduce the singular field near the crack tip and the discontinuities across the crack. It is based on the Kirchhoff–Love theory where C1-continuity of the displacement field is required. This condition is satisfied by the NURBS basis functions. Hence, the formulation eliminates the need of rotational degrees of freedom or the discretization of the director field facilitating the enrichment strategy. The performance and validity of the formulation is tested by several benchmark examples.

On the numerical integration of trimmed isogeometric elements

A numerical algorithm is proposed to construct efficient quadrature rules for trimmed isogeometric elements of arbitrary shape and topology as part of the standard finite element preprocessing step. The constructed integration rule is unique to an element and it is considered to be optimal in the sense that the final quadrature points and weights satisfy the moment fitting equations within the trimmed domain up to a predefined tolerance. The resulting quadrature points are in the interior of the trimmed domain and the positivity of the weights is preserved. The accuracy and efficiency of the quadrature scheme are assessed and compared to traditionally employed integration techniques. Results indicate that the proposed integration rules are more precise and inhibit the proliferation of quadrature points observed in competing approaches. Selected problems of elastostatics, elastodynamics, and elasto-plastic dynamics are used to further demonstrate the validity and efficacy of the method.

Isogeometric bending analysis of composite plates based on a higher-order shear deformation theory

This research paper presents an isogeometric plate finite element formulation for analysis of thick composite plates. Isogeometric finite element method which is based on non-uniform rational B-splines (NURBS) basis functions, is a novel numerical procedure developed to bridge the gap between CAD and FEM modeling of structures. In order to investigate the behavior of isogeometric plate elements under static loading, plate kinematics is based on third order shear deformation theory (TSDT) of Reddy, which is free from transverse shear locking. This paper discusses accurate transverse stress recovery procedures for TSDT isogeometric finite elements. Numerical experiments with quadratic, cubic and quartic elements are presented and obtained results are compared to other available ones.

Plastic collapse analysis of cracked structures using extended isogeometric elements and second-order cone programming

We investigate a numerical procedure based on extended isogeometric elements in combination with second-order cone programming (SOCP) for assessing collapse limit loads of cracked structures. We exploit alternative basis functions, namely B-splines or non-uniform rational B-splines (NURBS) in the context of limit analysis. The optimization formulation of limit analysis is rewritten in the form of second-order cone programming (SOCP) such that interior-point solvers can be exploited efficiently. Numerical examples are given to demonstrate reliability and effectiveness of the present method.

Free Vibrations of Thin Shells with Isogeometric Approach

Free vibration analysis of thin shells is carried out by using isogeometric approach. For this purpose, a thin shell element based on Kirchhoff-Love shell theory is developed. Non-uniform rational B-spline surface (NURBS) definition is introduced to represent the geometry of shell and also used to derive all terms required in the isogeometric element formulation. Gauss integration rule is used for stiffness and mass matrices. The present shell element is then applied to examine vibrational behaviours of thin plate and shell structures. From numerical results, it is found be that reliable natural frequencies and associated mode shapes of thin shell structures can be predicted by the present isogeometric shell element.

Analysis of laminated composite plates integrated with piezoelectric sensors and actuators using higher-order shear deformation theory and isogeometric finite elements

This paper presents a simple and effective formulation based on Isogeometric Analysis (IGA) and Higher-order Shear Deformation Theory (HSDT) to investigate static, free vibration and dynamic control of piezoelectric composite plates integrated with sensors and actuators. In the composite plates, the mechanical displacement field is approximated according to the HSDT model using isogeometric elements based on Non-Uniform Rational B-Spline (NURBS) basis functions. These achieve naturally any desired degree of continuity through the choice of the interpolation order, so that the method easily fulfills the C1-continuity requirement of the HSDT model. The electric potential is assumed to vary linearly through the thickness for each piezoelectric sublayer. A displacement and velocity feedback control algorithm is used for the active control of the static deflection and of the dynamic response of the plates through a closed-loop control with bonded or embedded distributed piezoelectric sensors and actuators. The accuracy and reliability of the proposed method is verified by comparing its numerical predictions with those of other available numerical approaches.

A computational formulation for constrained solid and liquid membranes considering isogeometric finite elements

A geometrically exact membrane formulation is presented that is based on curvilinear coordinates and isogeometric finite elements, and is suitable for both solid and liquid membranes. The curvilinear coordinate system is used to describe both the theory and the finite element equations of the membrane. In the latter case this avoids the use of local cartesian coordinates at the element level. Consequently, no transformation of derivatives is required. The formulation considers a split of the in-plane and out-of-plane membrane contributions, which allows the construction of a stable formulation for liquid membranes with constant surface tension. The proposed membrane formulation is general, and accounts for dead and live loading, as well as enclosed volume, area, and contact constraints. The new formulation is illustrated by several challenging examples, considering linear and quadratic Lagrange elements, as well as isogeometric elements based on quadratic NURBS and cubic T-splines. It is seen that the isogeometric elements are much more accurate than standard Lagrange elements. The gain is especially large for the liquid membrane formulation since it depends explicitly on the surface curvature.

Vibration and Buckling of Thick Plates using Isogeometric Approach

A study on the free vibration and linear buckling analyses of thick plates is described in this article. In order to determine the natural frequencies and buckling loads of plates, a plate element is developed by using isogeometric approach. The Non-uniform B-spline surface (NURBS) is used to represent both plate geometry and the unknown displacement field of plate. All terms required in isogeometric formulation are consistently derived by NURBS definition. The capability of the present plate element is demonstrated by using several numerical examples. From numerical results, it is found to be that the present isogeometric element can predict accurate natural frequencies and buckling loads of plates.

2010 非一様B-Splineに基づくアイソジオメトリック要素を用いた2次元Helmholtz方程式に対する高速多重極境界要素法

2010 非一様B-Splineに基づくアイソジオメトリック要素を用いた2次元Helmholtz方程式に対する高速多重極境界要素法

Comparison of different higher order finite element schemes for the simulation of Lamb waves

Structural Health Monitoring (SHM) applications call for both efficient and powerful numerical tools to predict the behavior of ultrasonic guided waves. When considering waves in thin-walled structures, so called Lamb waves, conventional linear or quadratic pure displacement finite elements soon reach their limits. The spatial as well as temporal discretisation, required to obtain good quality results has to be very fine. This results in enormous computational costs (computational time and memory storage requirements) when ultrasonic wave propagation problems are solved in the time domain. To resolve this issue several higher order finite element methods with polynomial degrees p>2 are proposed. The objective of the current article is to develop such higher order schemes and to verify their capabilities with respect to accuracy and numerical performance. To the best of the authors’ knowledge such comparison has not been reported in literature, yet. Specifically, spectral elements based on Lagarange polynomials (SEM), p-elements using the normalized integrals of the Legendre polynomials (p-FEM) and isogeometric elements utilizing non-uniform rational B-splines (NURBS, N-FEM) are discussed in this paper. By solving a two-dimensional benchmark problem, their advantages and drawbacks with respect to Lamb wave propagation are highlighted. The results of the convergence studies are then used to derive guidelines for estimating the optimal element size for a given finite element type and polynomial degree template. These findings serve the purpose to determine the optimal mesh configuration a priori and thus, save a considerable amount of computational effort. The proposed guideline is then tested on a three-dimensional structure with a conical hole showing an excellent agreement with the predicted behaviour.