An isogeometric analysis approach to gradient damage models

Continuum damage formulations are commonly used for the simulation of diffuse fracture processes. Implicit gradient damage models are employed to avoid the spurious mesh dependencies associated with local continuum damage models. The C⁰‐continuity of traditional finite elements has hindered the study of higher order gradient damage approximations. In this contribution we use isogeometric finite elements, which allow for the construction of higher order continuous basis functions on complex domains. We study the suitability of isogeometric finite elements for the discretization of higher order gradient damage approximations. Copyright © 2011 John Wiley & Sons, Ltd.     

A geometrically exact isogeometric Kirchhoff plate: Feature‐preserving automatic meshing and C1 rational triangular Bézier spline discretizations

The analysis of the Kirchhoff plate is performed using rational Bézier triangles in isogeometric analysis coupled with a feature‐preserving automatic meshing algorithm. Isogeometric analysis employs the same basis function for geometric design as well as for numerical analysis. The proposed approach also features an automatic meshing algorithm that admits localized geometric features (eg, small geometric details and sharp corners) with high resolution. Moreover, the use of rational triangular Bézier splines for domain triangulation significantly increases the flexibility in discretizing spaces bounded by complicated nonuniform rational B‐spline curves. To raise the global continuity to C¹ for the solution of the plate bending problem, Lagrange multipliers are leveraged to impose continuity constraints. The proposed approach also manipulates the control points at domain boundaries in such a way that the geometry is exactly described. A number of numerical examples consisting of static bending and free vibration analysis of thin plates bounded by complicated nonuniform rational B‐spline curves are used to demonstrate the advantage of the proposed approach.     

B-spline goal-oriented error estimators for geometrically nonlinear rods

We consider goal-oriented a posteriori error estimators for the evaluation of the errors on quantities of interest associated with the solution of geometrically nonlinear curved elastic rods. For the numerical solution of these nonlinear one-dimensional problems, we adopt a B-spline based Galerkin method, a particular case of the more general isogeometric analysis. We propose error estimators using higher order “enhanced” solutions, which are based on the concept of enrichment of the original B-spline basis by means of the “pure” k-refinement procedure typical of isogeometric analysis. We provide several numerical examples for linear and nonlinear output functionals, corresponding to the rotation, displacements and strain energy of the rod, and we compare the effectiveness of the proposed error estimators.     

An adaptive finite element method for second‐order plate theory

We present a discontinuous finite element method for the Kirchhoff plate model with membrane stresses. The method is based on P₂‐approximations on simplices for the out‐of‐plane deformations, using C⁰‐continuous approximations. We derive a posteriori error estimates for linear functionals of the error and give some numerical examples. Copyright © 2009 John Wiley & Sons, Ltd.     

Generalized Robin–Neumann explicit coupling schemes for incompressible fluid‐structure interaction: Stability analysis and numerics

We introduce a new class of explicit coupling schemes for the numerical solution of fluid‐structure interaction problems involving a viscous incompressible fluid and an elastic structure. These methods generalize the arguments reported in [Comput. Methods Appl. Mech. Engrg., 267:566–593, 2013, Numer. Math., 123(1):21–65, 2013] to the case of the coupling with thick‐walled structures. The basic idea lies in the derivation of an intrinsic interface Robin consistency at the space semi‐discrete level, using a lumped‐mass approximation in the structure. The fluid–solid splitting is then performed through appropriate extrapolations of the solid velocity and stress on the interface. Based on these methods, a new, parameter‐free, Robin–Neumann iterative procedure is also proposed for the partitioned solution of implicit coupling. A priori energy estimates, guaranteeing the stability of the schemes and the convergence of the iterative procedure, are established within a representative linear setting. The accuracy and performance of the methods are illustrated in several numerical examples. Copyright © 2014 John Wiley & Sons, Ltd.     

Isogeometric analysis of 2D gradient elasticity

In the present contribution the concept of isogeometric analysis is extended towards the numerical solution of the problem of gradient elasticity in two dimensions. In gradient elasticity the strain energy becomes a function of the strain and its derivative. This assumption results in a governing differential equation which contains fourth order derivatives of the displacements. The numerical solution of this equation with a displacement-based finite element method requires the use of C ¹-continuous elements, which are mostly limited to two dimensions and simple geometries. This motivates the implementation of the concept of isogeometric analysis for gradient elasticity. This NURBS based interpolation scheme naturally includes C ¹ and higher order continuity of the approximation of the displacements and the geometry. The numerical approach is implemented for two-dimensional problems of linear gradient elasticity and its convergence behavior is studied.     

Petrov–Galerkin finite element methods with a hinged test space for singularly perturbed problems

In this paper we introduce finite element methods of Petrov–Galerkin type for the approximate solution of two-point boundary-value problems for singularly perturbed, second-order, ordinary, linear differential equations. We write down Petrov–Galerkin methods on a uniform mesh which have asymptotic error estimates, as the mesh size tends to zero, whose magnitude is independent of the singular perturbation parameter. This is in marked contrast to standard finite element methods which do not possess such a property on a uniform mesh. For these, typically, the error on a fixed uniform mesh blows up as the singular perturbation parameter tends to zero. This robust behaviour of these Petrov–Galerkin methods for singularly perturbed problems is achieved by choosing trial spaces of standard piecewise polynomial type, while the test spaces consist of hinged piecewise polynomials. We consider self-adjoint and non-self-adjoint two-point boundary-value problems with Dirichlet boundary conditions. We define hinged test spaces for both types of problem. We then introduce a number of sample problems and we present numerical solutions of these sample problems using a Petrov–Galerkin method with the appropriate hinged test space.     

A large deformation frictional contact formulation using NURBS‐based isogeometric analysis

This paper focuses on the application of NURBS‐based isogeometric analysis to Coulomb frictional contact problems between deformable bodies, in the context of large deformations. A mortar‐based approach is presented to treat the contact constraints, whereby the discretization of the continuum is performed with arbitrary order NURBS, as well as C⁰‐continuous Lagrange polynomial elements for comparison purposes. The numerical examples show that the proposed contact formulation in conjunction with the NURBS discretization delivers accurate and robust predictions. Results of lower quality are obtained from the Lagrange discretization, as well as from a different contact formulation based on the enforcement of the contact constraints at every integration point on the contact surface. Copyright © 2011 John Wiley & Sons, Ltd.     

Isogeometric FEM‐BEM coupled structural‐acoustic analysis of shells using subdivision surfaces

We introduce a coupled finite and boundary element formulation for acoustic scattering analysis over thin‐shell structures. A triangular Loop subdivision surface discretisation is used for both geometry and analysis fields. The Kirchhoff‐Love shell equation is discretised with the finite element method and the Helmholtz equation for the acoustic field with the boundary element method. The use of the boundary element formulation allows the elegant handling of infinite domains and precludes the need for volumetric meshing. In the present work, the subdivision control meshes for the shell displacements and the acoustic pressures have the same resolution. The corresponding smooth subdivision basis functions have the C¹ continuity property required for the Kirchhoff‐Love formulation and are highly efficient for the acoustic field computations. We verify the proposed isogeometric formulation through a closed‐form solution of acoustic scattering over a thin‐shell sphere. Furthermore, we demonstrate the ability of the proposed approach to handle complex geometries with arbitrary topology that provides an integrated isogeometric design and analysis workflow for coupled structural‐acoustic analysis of shells.     

Implicit dynamic analysis using an isogeometric Reissner–Mindlin shell formulation

In isogeometric analysis, identical basis functions are used for geometrical representation and analysis. In this work, non‐uniform rational basis splines basis functions are applied in an isoparametric approach. An isogeometric Reissner–Mindlin shell formulation for implicit dynamic calculations using the Galerkin method is presented. A consistent as well as a lumped matrix formulation is implemented. The suitability of the developed shell formulation for natural frequency analysis is demonstrated by a numerical example. In a second set of examples, transient problems of plane and curved geometries undergoing large deformations in combination with nonlinear material behavior are investigated. Via a zero‐thickness stress algorithm for arbitrary material models, a J₂‐plasticity constitutive law is implemented. In the numerical examples, the effectiveness, robustness, and superior accuracy of a continuous interpolation method of the shell director vector is compared with experimental results and alternative numerical approaches. Copyright © 2016 John Wiley & Sons, Ltd.     

On the spectrum of the electric field integral equation and the convergence of the moment method

Existing convergence estimates for numerical scattering methods based on boundary integral equations are asymptotic in the limit of vanishing discretization length, and break down as the electrical size of the problem grows. In order to analyse the efficiency and accuracy of numerical methods for the large scattering problems of interest in computational electromagnetics, we study the spectrum of the electric field integral equation (EFIE) for an infinite, conducting strip for both the TM (weakly singular kernel) and TE polarizations (hypersingular kernel). Due to the self‐coupling of surface wave modes, the condition number of the discretized integral equation increases as the square root of the electrical size of the strip for both polarizations. From the spectrum of the EFIE, the solution error introduced by discretization of the integral equation can also be estimated. Away from the edge singularities of the solution, the error is second order in the discretization length for low‐order bases with exact integration of matrix elements, and is first order if an approximate quadrature rule is employed. Comparison with numerical results demonstrates the validity of these condition number and solution error estimates. The spectral theory offers insights into the behaviour of numerical methods commonly observed in computational electromagnetics. Copyright © 2001 John Wiley & Sons, Ltd.     

An isogeometric collocation method for efficient random field discretization

This paper presents an isogeometric collocation method for a computationally expedient random field discretization by means of the Karhunen-Loève expansion. The method involves a collocation projection onto a finite-dimensional subspace of continuous functions over a bounded domain, basis splines (B-splines) and nonuniform rational B-splines (NURBS) spanning the subspace, and standard methods of eigensolutions. Similar to the existing Galerkin isogeometric method, the isogeometric collocation method preserves an exact geometrical representation of many commonly used physical or computational domains and exploits the regularity of isogeometric basis functions delivering globally smooth eigensolutions. However, in the collocation method, the construction of the system matrices for a d-dimensional eigenvalue problem asks for at most d-dimensional domain integrations, as compared with 2d-dimensional integrations required in the Galerkin method. Therefore, the introduction of the collocation method for random field discretization offers a huge computational advantage over the existing Galerkin method. Three numerical examples, including a three-dimensional random field discretization problem, illustrate the accuracy and convergence properties of the collocation method for obtaining eigensolutions.     

An efficient isostatic mixed shell element for coarse mesh solution

A novel mixed shell finite element (FE) is presented. The element is obtained from the Hellinger–Reissner variational principle and it is based on an elastic solution of the generalized stress field, which is ruled by the minimum number of variables. As such, the new FE is isostatic because the number of stress parameters is equal to the number of kinematical parameters minus the number of rigid body motions. We name this new FE MISS-8. MISS-8 has generalized displacements and rotations interpolated along its contour and drilling rotation is also considered as degree of freedom. The element is integrated exactly on its contour, it does not suffer from rank defectiveness and it is locking-free. Furthermore, it is efficient for recovering both stress and displacement fields when coarse meshes are used. The numerical investigation on its performance confirms the suitability, accuracy, and efficiency to recover elastic solutions of thick- and thin-walled beam-like structures. Numerical results obtained with the proposed FE are also compared with those obtained with isogeometric high-performance solutions. Finally, numerical results show a rate of convergence between h² and h⁴ .     

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.     

Adaptive isogeometric analysis based on a combined r-h strategy

In the present work, an r-h adaptive isogeometric analysis is proposed for plane elasticity problems. For performing the r-adaption, the control net is considered to be a network of springs with the individual spring stiffness values being proportional to the error estimated at the control points. While preserving the boundary control points, relocation of only the interior control points is made by adopting a successive relaxation approach to achieve the equilibrium of spring system. To suit the noninterpolatory nature of the isogeometric approximation, a new point-wise error estimate for the h-refinement is proposed. To evaluate the point-wise error, hierarchical B-spline functions in Sobolev spaces are considered. The proposed adaptive h-refinement strategy is based on using De-Casteljau’s algorithm for obtaining the new control points. The subsequent control meshes are thus obtained by using a recursive subdivision of reference control mesh. Such a strategy ensures that the control points lie in the physical domain in subsequent refinements, thus making the physical mesh to exactly interpolate the control mesh and thereby allowing the exact imposition of essential boundary conditions in the classical isogeometric analysis (IGA). The combined r-h adaptive refinement strategy results in better convergence characteristics with reduced errors than r- or h-refinement. Several numerical examples are presented to illustrate the efficiency of the proposed approach.     

Superconvergence of quadratic optimal control problems by triangular mixed finite element methods

The aim of this work is to investigate the discretization of a quadratic convex optimal control problem using the mixed finite element method. The state and co‐state are approximated by the order k?1 Raviart–Thomas mixed finite element spaces, and the control is approximated by piecewise constant functions. We construct an interpolation of the exact control and a projection of the discrete scalar co‐state to be the approximated solution of the control variable for the continuous optimal control problem. As a result, it can be proved that the difference between the interpolation and the piecewise constant approximation has superconvergence property for the control of order h^3/2 for k=0 and of order h² for k=1. Moreover, only for the order k=1 Raviart–Thomas mixed finite element approximation does the postprocessing technique possess the superconvergence property of order h². Finally, some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence. Copyright © 2008 John Wiley & Sons, Ltd.     

eXtended Stochastic Finite Element Method for the numerical simulation of heterogeneous materials with random material interfaces

An eXtended Stochastic Finite Element Method has been recently proposed for the numerical solution of partial differential equations defined on random domains. This method is based on a marriage between the eXtended Finite Element Method and spectral stochastic methods. In this article, we propose an extension of this method for the numerical simulation of random multi‐phased materials. The random geometry of material interfaces is described implicitly by using random level set functions. A fixed deterministic finite element mesh, which is not conforming to the random interfaces, is then introduced in order to approximate the geometry and the solution. Classical spectral stochastic finite element approximation spaces are not able to capture the irregularities of the solution field with respect to spatial and stochastic variables, which leads to a deterioration of the accuracy and convergence properties of the approximate solution. In order to recover optimal convergence properties of the approximation, we propose an extension of the partition of unity method to the spectral stochastic framework. This technique allows the enrichment of approximation spaces with suitable functions based on an a priori knowledge of the irregularities in the solution. Numerical examples illustrate the efficiency of the proposed method and demonstrate the relevance of the enrichment procedure. Copyright © 2010 John Wiley & Sons, Ltd.     

Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation

When applying numerical methods for the computation of stationary waves from the Helmholtz equation, one obtains ‘numerical waves’ that are dispersive also in non-dispersive media. The numerical wave displays a phase velocity that depends on the parameter k of the Helmholtz equation. In dispersion analysis, the phase difference between the exact and the numerical solutions is investigated. In this paper, the authors' recent result on the phase difference for one-dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic. It is then shown that previous error estimates in H¹-norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only. On the other hand, recently proven error estimates for constant resolution contain a pollution term. With certain assumptions on the exact solution, this term is of the order of the phase difference. Thus a link is established between the results of dispersion analysis and the results of numerical analysis. Throughout the paper, the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems. Special attention is given to the performance of the Galerkin method with a higher order of polynomial approximation p(h-p-version).     

Numerical solution of second‐order,two‐point boundary value problems using continuous genetic algorithms

Second‐order, two‐point boundary‐value problems are encountered in many engineering applications including the study of beam deflections, heat flow, and various dynamic systems. Two classical numerical techniques are widely used in the engineering community for the solution of such problems; the shooting method and finite difference method. These methods are suited for linear problems. However, when solving the non‐linear problems, these methods require some major modifications that include the use of some root‐finding technique. Furthermore, they require the use of other basic numerical techniques in order to obtain the solution. In this paper, the author introduces a novel method based on continuous genetic algorithms for numerically approximating a solution to this problem. The new method has the following characteristics; first, it does not require any modification while switching from the linear to the non‐linear case; as a result, it is of versatile nature. Second, this approach does not resort to more advanced mathematical tools and is thus easily accepted in the engineering application field. Third, the proposed methodology has an implicit parallel nature which points to its implementation on parallel machines. However, being a variant of the finite difference scheme with truncation error of the order O(h²), the method provides solutions with moderate accuracy. Numerical examples presented in the paper illustrate the applicability and generality of the proposed method. Copyright © 2004 John Wiley & Sons, Ltd.