The weak substitution method – an application of the mortar method for patch coupling in NURBS‐based isogeometric analysis

In this contribution, a mortar‐type method for the coupling of non‐conforming NURBS (Non‐Uniform Rational B‐spline) surface patches is proposed. The connection of non‐conforming patches with shared degrees of freedom requires mutual refinement, which propagates throughout the whole patch due to the tensor‐product structure of NURBS surfaces. Thus, methods to handle non‐conforming meshes are essential in NURBS‐based isogeometric analysis. The main objective of this work is to provide a simple and efficient way to couple the individual patches of complex geometrical models without altering the variational formulation. The deformations of the interface control points of adjacent patches are interrelated with a master‐slave relation. This relation is established numerically using the weak form of the equality of mutual deformations along the interface. With the help of this relation, the interface degrees of freedom of the slave patch can be condensated out of the system. A natural connection of the patches is attained without additional terms in the weak form. The proposed method is also applicable for nonlinear computations without further measures. Linear and geometrical nonlinear examples show the high accuracy and robustness of the new method. A comparison to reference results and to computations with the Lagrange multiplier method is given. Copyright © 2015 John Wiley & Sons, Ltd.     

A Nitsche‐type formulation and comparison of the most common domain decomposition methods in isogeometric analysis

This paper provides a detailed elaboration and assessment of the most common domain decomposition methods for their application in isogeometric analysis. The methods comprise a penalty approach, Lagrange multiplier methods, and a Nitsche‐type method. For the Nitsche method, a new stabilized formulation is developed in the context of isogeometric analysis to guarantee coercivity. All these methods are investigated on problems of linear elasticity and eigenfrequency analysis in 2D. In particular, focus is put on non‐uniform rational B‐spline patches which join nonconformingly along their common interface. Thus, the application of isogeometric analysis is extended to multi‐patches, which can have an arbitrary parametrization on the adjacent edges. Moreover, it has been shown that the unique properties provided by isogeometric analysis, that is, high‐order functions and smoothness across the element boundaries, carry over for the analysis of multiple domains. Copyright © 2013 John Wiley & Sons, Ltd.     

Isogeometric contact analysis using mortar method

In the present work, an isogeometric contact analysis scheme using mortar method is proposed. Because the isogeometric analysis is employed for contact analysis, the geometric exactness of the contact region is maintained without any loss of geometric data because of geometry approximation. Thus, the proposed method can overcome underlying shortcomings that result from the geometric approximation of contact surfaces in the conventional finite element (FE)‐based contact analysis. For an isogeometric contact analysis, the schemes for treating the contact conditions and detecting the real contact surfaces are essentially required. In the proposed method, the mortar method is adopted as a nonconforming contact treatment scheme because it is expected to be in good harmony with the useful characteristics of nonuniform rational B‐spline A new matching algorithm is proposed to combine the mortar method with the isogeometric analysis to guarantee consistent contact surface information with the nonuniform rational B‐spline curve. The present scheme is verified by patch test and the well‐known problems which have theoretical solutions such as interference fit and the Hertzian contact problem. It is shown that the problems with curved contact surfaces which are difficult to treat by conventional approaches can be easily dealt with. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

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

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

X‐FEM in isogeometric analysis for linear fracture mechanics

The extended finite element method (X‐FEM) has proven to be an accurate, robust method for solving problems in fracture mechanics. X‐FEM has typically been used with elements using linear basis functions, although some work has been performed using quadratics. In the current work, the X‐FEM formulation is incorporated into isogeometric analysis to obtain solutions with higher order convergence rates for problems in linear fracture mechanics. In comparison with X‐FEM with conventional finite elements of equal degree, the NURBS‐based isogeometric analysis gives equal asymptotic convergence rates and equal accuracy with fewer degrees of freedom (DOF). Results for linear through quartic NURBS basis functions are presented for a multiplicity of one or a multiplicity equal the degree. Copyright © 2011 John Wiley & Sons, Ltd.     

An isogeometric analysis approach to gradient‐dependent plasticity

Gradient‐dependent plasticity can be used to achieve mesh‐objective results upon loss of well‐posedness of the initial/boundary value problem because of the introduction of strain softening, non‐associated flow, and geometric nonlinearity. A prominent class of gradient plasticity models considers a dependence of the yield strength on the Laplacian of the hardening parameter, usually an invariant of the plastic strain tensor. This inclusion causes the consistency condition to become a partial differential equation, in addition to the momentum balance. At the internal moving boundary, one has to impose appropriate boundary conditions on the hardening parameter or, equivalently, on the plastic multiplier. This internal boundary condition can be enforced without tracking the elastic‐plastic boundary by requiring ‐continuity with respect to the plastic multiplier. In this contribution, this continuity has been achieved by using nonuniform rational B‐splines as shape functions both for the plastic multiplier and for the displacements. One advantage of this isogeometric analysis approach is that the displacements can be interpolated one order higher, making it consistent with the interpolation of the plastic multiplier. This is different from previous approaches, which have been exploited. The regularising effect of gradient plasticity is shown for 1‐ and 2‐dimensional boundary value problems.     

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.     

The semianalytical analysis of nearly singular integrals in 2D potential problem by isogeometric boundary element method

Benefited from the accuracy improvement in modeling physical problem of complex geometry and integrating the discretization and simulation, the isogeometric analysis in boundary element method (IGABEM) has been drawn a great deal of attention. The nearly singular integrals of 2D potential problem in the IGABEM are addressed by a semianalytical scheme in the present work. We use the subtraction technique to separate the integrals to singular and nonsingular parts, where the singular parts can be calculated by the analytical formulae derived by utilizing a series of integration by parts, while the nonsingular parts are calculated numerically with fewer quadrature points. Comparing the present semianalytical results with the ones of exact solutions, we find that the present method can obtain precise potential and flux densities of inner points much closer to the boundary without refining the elements nearby. Sufficient comparisons with other regularization schemes, such as the exponential and sinh transformation methods, are also conducted. The results in the numerical examples show the competitiveness of the present method, especially when calculating the nearly strongly and highly singular integrals during the simulation of the flux density.     

An isogeometric‐meshfree coupling approach for analysis of cracks

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

Mixed isogeometric analysis of strongly coupled diffusion in porous materials

In this paper, we develop a mixed isogeometric analysis approach based on subdivision stabilization to study strongly coupled diffusion in solids in both small and large deformation ranges. Coupling the fluid pressure and the solid deformation, the mixed formulation suffers from numerical instabilities in the incompressible and the nearly incompressible limit due to the violation of the inf‐sup condition. We investigate this issue using subdivision‐stabilized nonuniform rational B‐spline (NURBS) elements, as well as different families of mixed isogeometric analysis techniques, and assess their stability through a numerical inf‐sup test. Furthermore, the validity of the inf‐sup stability test in poromechanics is supported by a mathematical proof concerning the corresponding stability estimate. Finally, two numerical examples involving a rigid strip foundation on saturated soil and a swelling hydrogel structure are presented to validate the stability and to demonstrate the robustness of the proposed approach.     

Enriched isogeometric analysis of elliptic boundary value problems in domains with cracks and/or corners

The mapping method was introduced in Jeong et al. (2013) for highly accurate isogeometric analysis (IGA) of elliptic boundary value problems containing singularities. The mapping method is concerned with constructions of novel geometrical mappings by which push‐forwards of B‐splines from the parameter space into the physical space generate singular functions that resemble the singularities. In other words, the pullback of the singularity into the parameter space by the novel geometrical mapping (a non‐uniform rational basis spline (NURBS) surface mapping) becomes highly smooth. One of the merits of IGA is that it uses NURBS functions employed in designs for the finite element analysis. However, push‐forwards of rational NURBS may not be able to generate singular functions. Moreover, the mapping method is effective for neither the k‐refinement nor the h‐refinement. In this paper, highly accurate stress analysis of elastic domains with cracks and ∕ or corners are achieved by enriched IGA, in which push‐forwards of NURBS via the design mapping are combined with push‐forwards of B‐splines via the novel geometrical mapping (the mapping technique). In a similar spirit of X‐FEM (or GFEM), we propose three enrichment approaches: enriched IGA for corners, enriched IGA for cracks, and partition of unity IGA for cracks. Copyright © 2013 John Wiley & Sons, Ltd.     

Parameter‐free,weak imposition of Dirichlet boundary conditions and coupling of trimmed and non‐conforming patches

We present a parameter‐free domain sewing approach for low‐order as well as high‐order finite elements. Its final form contains only primal unknowns; that is, the approach does not introduce additional unknowns at the interface. Additionally, it does not involve problem‐dependent parameters, which require an estimation. The presented approach is symmetry preserving; that is, the resulting discrete form of an elliptic equation will remain symmetric and positive definite. It preserves the order of the underlying discretization, and we demonstrate high‐order accuracy for problems of non‐matching discretizations concerning the mesh size h as well as the polynomial degree of the order of discretization p. We also demonstrate how the method may be used to model material interfaces, which may be curved and for which the interface does not coincide with the underlying mesh. This novel approach is presented in the context of the p‐version and B‐spline version of the finite cell method, an embedded domain method of high order, and compared with more classical methods such as the penalty method or Nitsche's method. Copyright © 2014 John Wiley & Sons, Ltd.     

An isogeometric locking‐free NURBS‐based solid‐shell element for geometrically nonlinear analysis

In this work, we develop an isogeometric non‐uniform rational B‐spline (NURBS)‐based solid‐shell element for the geometrically nonlinear static analysis of elastic shell structures. A single layer of continuous 3D elements through the thickness of the shell is considered, and the order of approximation in that direction is chosen to be equal to two. A complete 3D constitutive relation is assumed. The objective is to develop a highly accurate low‐order element for coarse meshes. We propose an extension of the mixed method of Bouclier et al. [11] to deal with locking in the context of large rotations and large displacements. The main idea is to modify the interpolation of the average through the thickness of the stress components. It is also necessary to stabilize the element in order to avoid the occurrence of spurious zero‐energy modes. This was achieved, for the quadratic version, through the adjunction of artificial elementary stabilization stiffnesses. The result is an element of order 2, which is at least as accurate as standard NURBS shell elements of order 4. Linear and nonlinear test calculations have been carried out along with comparisons with other published NURBS and classical techniques in order to assess the performance of the element. Copyright © 2014 John Wiley & Sons, Ltd.     

Implementation of regularized isogeometric boundary element methods for gradient‐based shape optimization in two‐dimensional linear elasticity

The present work addresses shape sensitivity analysis and optimization in two‐dimensional elasticity with a regularized isogeometric boundary element method (IGABEM). Non‐uniform rational B‐splines are used both for the geometry and the basis functions to discretize the regularized boundary integral equations. With the advantage of tight integration of design and analysis, the application of IGABEM in shape optimization reduces the mesh generation/regeneration burden greatly. The work is distinct from the previous literatures in IGABEM shape optimization mainly in two aspects: (1) the structural and sensitivity analysis takes advantage of the regularized form of the boundary integral equations, eliminating completely the need of evaluating strongly singular integrals and jump terms and their shape derivatives, which were the main implementation difficulty in IGABEM, and (2) although based on the same Computer Aided Design (CAD) model, the mesh for structural and shape sensitivity analysis is separated from the geometrical design mesh, thus achieving a balance between less design variables for efficiency and refined mesh for accuracy. This technique was initially used in isogeometric finite element method and was incorporated into the present IGABEM implementation. Copyright © 2016 John Wiley & Sons, Ltd.     

A framework for implementation of RVE‐based multiscale models in computational homogenization using isogeometric analysis

This study presents an isogeometric framework for incorporating representative volume element–based multiscale models into computational homogenization. First‐order finite deformation homogenization theory is derived within the framework of the method of multiscale virtual power, and Lagrange multipliers are used to illustrate the effects of considering different kinematical constraints. Using a Lagrange multiplier approach in the numerical implementation of the discrete system naturally leads to a consolidated treatment of the commonly employed representative volume element boundary conditions. Implementation of finite deformation computational strain‐driven, stress‐driven, and mixed homogenization is detailed in the context of isogeometric analysis (IGA), and performance is compared to standard finite element analysis. As finite deformations are considered, a numerical multiscale stability analysis procedure is also detailed for use with IGA. Unique implementation aspects that arise when computational homogenization is performed using IGA are discussed, and the developed framework is applied to a complex curved microstructure representing an architectured material.     

Static,free vibration,and buckling analysis of laminated composite Reissner–Mindlin plates using NURBS‐based isogeometric approach

This paper presents a novel numerical procedure based on the framework of isogeometric analysis for static, free vibration, and buckling analysis of laminated composite plates using the first‐order shear deformation theory. The isogeometric approach utilizes non‐uniform rational B‐splines to implement for the quadratic, cubic, and quartic elements. Shear locking problem still exists in the stiffness formulation, and hence, it can be significantly alleviated by a stabilization technique. Several numerical examples are presented to show the performance of the method, and the results obtained are compared with other available ones. Copyright © 2012 John Wiley & Sons, Ltd.     

Certified real-time shape optimization using isogeometric analysis,PGD model reduction,and a posteriori error estimation

In this paper, we address the effective and accurate solution of problems with parameterized geometry. Considering the attractive framework of isogeometric analysis, which enables a natural and flexible link between computer-aided design and simulation tools, the parameterization of the geometry is defined on the mapping from the isogeometric analysis parametric space to the physical space. From the subsequent multidimensional problem, model reduction based on the proper generalized decomposition technique with off-line/online steps is introduced in order to describe the resulting manifold of parametric solutions with reduced CPU cost. Eventually, a posteriori estimation of various error sources inheriting from discretization and model reduction is performed in order to control the quality of the approximate solution, for any geometry, and feed a robust adaptive algorithm that optimizes the computational effort for prescribed accuracy. The overall approach thus constitutes an effective and reliable numerical tool for shape optimization analyses. Its performance is illustrated on several two- and three-dimensional numerical experiments.