Patch coupling in isogeometric analysis of solids in boundary representation using a mortar approach

This contribution is concerned with a coupling approach for nonconforming NURBS patches in the framework of an isogeometric formulation for solids in boundary representation. The boundary representation modeling technique in CAD is the starting point of this approach. We parameterize the solid according to the scaled boundary finite element method and employ NURBS basis functions for the approximation of the solution. Therefore, solid surfaces consist of several sections, which can be regarded as patches and discretized independently. The main objective of this study is to derive an approach for the connection of independent sections in order to allow for local refinement and thus an accurate and efficient discretization of the computational domain. Nonconforming sections are coupled with a mortar approach within a master-slave framework. The coupling of adjacent sections ensures the equality of mutual deformations along the interface in a weak sense and is enforced by constraining the NURBS basis functions on the interface. We apply this approach to nonlinear problems in two dimensions and compare the results with conforming discretizations.     

A NURBS‐based interface‐enriched generalized finite element method for problems with complex discontinuous gradient fields

A non‐uniform rational B‐splines (NURBS)‐based interface‐enriched generalized finite element method is introduced to solve problems with complex discontinuous gradient fields observed in the structural and thermal analysis of the heterogeneous materials. The presented method utilizes generalized degrees of freedom and enrichment functions based on NURBS to capture the solution with non‐conforming meshes. A consistent method for the generation and application of the NURBS‐based enrichment functions is introduced. These enrichment functions offer various advantages including simplicity of the integration, possibility of different modes of local solution refinement, and ease of implementation. In addition, we show that these functions well capture weak discontinuities associated with highly curved material interfaces. The convergence, accuracy, and stability of the method in the solution of two‐dimensional elasto‐static problems are compared with the standard finite element scheme, showing improved accuracy. Finally, the performance of the method for solving problems with complex internal geometry is highlighted through a numerical example. Copyright © 2015 John Wiley & Sons, Ltd.     

Reissner–Mindlin shell implementation and energy conserving isogeometric multi‐patch coupling

A shear‐flexible isogeometric Reissner–Mindlin shell formulation using non‐uniform rational B‐splines basis functions is introduced, which is used for the demonstration of a coupling approach for multiple non‐conforming patches. The six degrees of freedom formulation uses the exact surface normal vectors and curvature. The shell formulation is implemented in an isogeometric analysis framework for computation of structures composed of multiple geometric entities. To enable local model refinement as well as non‐matching domains coupling, a conservative multi‐patch approach using Lagrange multipliers for structured non‐uniform rational B‐splines patches is presented. Here, an additional border frame mesh is used to couple geometries during structural analyses. This frame interface approach avoids the problem of excessive constraints when multiple patches are coupled at one point. First, the shell formulation is verified with several reference cases. Then the influence of the frame interface discretization and frame penalty stiffness on the smoothness of the results is investigated. The effects of the perturbed Lagrangian method in combination with the frame interface approach is shown. In addition, results of models with T‐joint interface connections and perpendicular stiffener patches are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

A refined non‐linear non‐conforming triangular plate/shell element

A refined non‐conforming triangular plate/shell element for geometric non‐linear analysis of plates/shells using the total Lagrangian/updated Lagrangian approach is constructed in this paper based on the refined non‐conforming element method for geometric non‐linear analysis. The Allman's triangular plane element with vertex degrees of freedom and the refined triangular plate‐bending element RT9 are used to construct the present element. Numerical examples demonstrate that the accuracy of the new element is quite high in the geometric non‐linear analysis of plates/shells. Copyright © 2003 John Wiley & Sons, Ltd.     

Powell–Sabin B‐splines and unstructured standard T‐splines for the solution of the Kirchhoff–Love plate theory exploiting Bézier extraction

The equations that govern Kirchhoff–Love plate theory are solved using quadratic Powell–Sabin B‐splines and unstructured standard T‐splines. Bézier extraction is exploited to make the formulation computationally efficient. Because quadratic Powell–Sabin B‐splines result in ‐continuous shape functions, they are of sufficiently high continuity to capture Kirchhoff–Love plate theory when cast in a weak form. Unlike non‐uniform rational B‐splines (NURBS), which are commonly used in isogeometric analysis, Powell–Sabin B‐splines do not necessarily capture the geometry exactly. However, the fact that they are defined on triangles instead of on quadrilaterals increases their flexibility in meshing and can make them competitive with respect to NURBS, as no bending strip method for joined NURBS patches is needed. This paper further illustrates how unstructured T‐splines can be modified such that they are ‐continuous around extraordinary points, and that the blending functions fulfil the partition of unity property. The performance of quadratic NURBS, unstructured T‐splines, Powell–Sabin B‐splines and NURBS‐to‐NURPS (non‐uniform rational Powell–Sabin B‐splines, which are obtained by a transformation from a NURBS patch) is compared in a study of a circular plate. Copyright © 2015 John Wiley & Sons, Ltd.     

A Kriging‐based error‐reproducing and interpolating kernel method for improved mesh‐free approximations

An error‐reproducing and interpolating kernel method (ERIKM), which is a novel and improved form of the error‐reproducing kernel method (ERKM) with the nodal interpolation property, is proposed. The ERKM is a non‐uniform rational B‐splines (NURBS)‐based mesh‐free approximation scheme recently proposed by Shaw and Roy (Comput. Mech. 2007; 40 (1):127–148). The ERKM is based on an initial approximation of the target function and its derivatives by NURBS basis functions. The errors in the NURBS approximation and its derivatives are then reproduced via a family of non‐NURBS basis functions. The non‐NURBS basis functions are constructed using a polynomial reproduction condition and added to the NURBS approximation obtained in the first step. In the ERKM, the interpolating property at the boundary is achieved by repeating the knot (open knot vector). However, for most problems of practical interest, employing NURBS with open knots is not possible because of the complex geometry of the domain, and consequently ERKM shape functions turn out to be non‐interpolating. In ERIKM, the error functions are obtained through localized Kriging based on a minimization of the squared variance of the estimate with the reproduction property as a constraint. Interpolating error functions so obtained are then added to the NURBS approximant. While enriching the ERKM with the interpolation property, the ERIKM naturally possesses all the desirable features of the ERKM, such as insensitivity to the support size and ability to reproduce sharp layers. The proposed ERIKM is finally applied to obtain strong and weak solutions for a class of linear and non‐linear boundary value problems of engineering interest. These illustrations help to bring out the relative numerical advantages and accuracy of the new method to some extent. Copyright © 2007 John Wiley & Sons, Ltd.     

POD‐based model reduction with empirical interpolation applied to nonlinear elasticity

The constantly rising demands on finite element simulations yield numerical models with increasing number of degrees‐of‐freedom. Due to nonlinearity, be it in the material model or of geometrical nature, the computational effort increases even further. For these reasons, it is today still not possible to run such complex simulations in real time parallel to, for example, an experiment or an application. Model reduction techniques such as the proper orthogonal decomposition method have been developed to reduce the computational effort while maintaining high accuracy. Nonetheless, this approach shows a limited reduction in computational time for nonlinear problems. Therefore, the aim of this paper is to overcome this limitation by using an additional empirical interpolation. The concept of the so‐called discrete empirical interpolation method is translated to problems of solid mechanics with soft nonlinear elasticity and large deformations. The key point of the presented method is a further reduction of the nonlinear term by an empirical interpolation based on a small number of interpolation indices. The method is implemented into the finite element method in two different ways, and it is extended by using different solution strategies including a numerical as well as a quasi‐Newton tangent. The new method is successfully applied to two numerical examples concerning hyperelastic as well as viscoelastic material behavior. Using the extended discrete empirical interpolation method combined with a quasi‐Newton tangent enables reductions in computational time of factor 10 with respect to the proper orthogonal decomposition method without empirical interpolation. Negligibly, orders of error can be reached. Copyright © 2015 John Wiley & Sons, Ltd.     

Direct modification for non‐conforming elements with drilling DOF

A unified approach to eliminate the undesirable lockings in distorted meshes for both non‐conforming quadrilateral membrane and hexahedral elements with drilling (or rotational) degrees of freedom is presented. The direct modification scheme is utilized in the formulation of non‐conforming modes to improve the general behaviour of isoparametric‐based elements. It is shown that the direct modification is very effective in eliminating the locking and this improvement of element behaviour may be doubled if the selective integration scheme is used simultaneously. To verify the validity of elements formulated by the proposed schemes and to evaluate their effectiveness, several numerical tests are carried out. The combined use of additional non‐conforming modes with the direct modification method and the selective integration technique plays an important role to improve the behaviour of elements, especially for distorted mesh cases. Test results also show that the results for both non‐conforming quadrilateral membrane and hexahedral elements obtained by the proposed schemes are equally satisfactory. Copyright © 2002 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.     

On the modelling of incompressibility in linear and non‐linear elasticity with the master–slave approach

The master–slave approach is adapted to model the kinematic constraints encountered in incompressibility. The method presented here allows us to obtain discrete displacement and pressure fields for arbitrary finite element formulations that have discontinuous pressure interpolations. The resulting displacements satisfy exactly the incompressibility constraints in a weak sense, and are obtained by solving a system of equations with the minimum (independent) degrees of freedom. In linear analysis, the method reproduces the well‐known stability results for inf–sup compliant elements, and permits to compute the pressure modes (physical or spurious) when they exist. By rewriting the equilibrium equations of a hyperelastic material, the method is extended to non‐linear elasticity, while retaining the exact fulfilment of the incompressibility constraints in a weak sense. Problems with analytical solution in two and three dimensions are tested and compared with other solution methods. Copyright © 2007 John Wiley & Sons, Ltd.     

Blending in the extended finite element method by discontinuous Galerkin and assumed strain methods

In the extended finite element method (XFEM), errors are caused by parasitic terms in the approximation space of the blending elements at the edge of the enriched subdomain. A discontinuous Galerkin (DG) formulation is developed, which circumvents this source of error. A patch‐based version of the DG formulation is developed, which decomposes the domain into enriched and unenriched subdomains. Continuity between patches is enforced with an internal penalty method. An element‐based form is also developed, where each element is considered a patch. The patch‐based DG is shown to have similar accuracy to the element‐based DG for a given discretization but requires significantly fewer degrees of freedom. The method is applied to material interfaces, cracks and dislocation problems. For the dislocations, a contour integral form of the internal forces that only requires integration over the patch boundaries is developed. A previously developed assumed strain (AS) method is also developed further and compared with the DG method for weak discontinuities and linear elastic cracks. The DG method is shown to be significantly more accurate than the standard XFEM for a given element size and to converge optimally, even where the standard XFEM does not. The accuracy of the DG method is similar to that of the AS method but requires less application‐specific coding. Copyright © 2007 John Wiley & Sons, Ltd.     

Stability analysis of cylindrical shells using refined non‐conforming rectangular cylindrical shell elements

The accuracy of stability analysis depends on the accuracy of both the element stiffness matrix and geometry stiffness matrix. Therefore, when carrying out the stability analysis of thin cylindrical shells using the finite element methods will require, firstly, a refined non‐conforming rectangular curved cylindrical shell element RCSR4 is proposed according to the refined non‐conforming FE method, in which both the C¹ and C⁰ weak continuity conditions are satisfied and as a result, can ensure the convergence of computation. At the same time, a refined geometrical stiffness matrix is introduced to replace the standard consistent geometrical stiffness matrix. Simple expressions of the refined constant strain matrices with adjustable constants are introduced with respect to the weak continuity conditions. Numerical examples are presented to show that the present method can indeed improve the performance and the accuracy in stability analysis. Copyright © 2001 John Wiley & Sons, Ltd.     

Development of a finite element contact analysis algorithm for charged‐hydrated soft tissues with large sliding

This paper focuses on a finite element analysis of contact phenomena with large sliding between charged‐hydrated biological soft tissues, such as articular cartilages, based on the triphasic theory. The impenetrability constraint between the contacting bodies and the continuity of the interstitial fluid and ion phases at the contact surfaces are imposed by applying a Lagrange multiplier approach with the contact pressure, chemical potential of the fluid and electrochemical potentials of ions as Lagrange multipliers. A node‐to‐segment one‐pass approach is adopted to cope with large deformations and sliding between the contact surfaces. To pass the contact patch test, contact boundary integrations are performed on both the master and slave contact surfaces. On the other hand, the degrees of freedom of the multipliers at the master nodes are eliminated by projecting the master nodes onto the slave surface to avoid overconstraint. The effectiveness of the proposed algorithm is verified by a couple of numerical examples, in which continuous distributions of displacement, fluid flow, ionic molar flow and Lagrange multipliers on or across the contact surface are confirmed. Copyright © 2008 John Wiley & Sons, Ltd.     

Stabilized conforming nodal integration in the natural‐element method

A stabilized conforming nodal integration scheme is implemented in the natural neighbour method in conjunction with non‐Sibsonian interpolation. In this approach, both the shape functions and the integration scheme are defined through use of first‐order Voronoi diagrams. The method illustrates improved performance and significant advantages over previous natural neighbour formulations. The method also shows substantial promise for problems with large deformations and for the computation of higher‐order gradients. Copyright © 2004 John Wiley & Sons, Ltd.     

Variational principles for non‐conforming finite element methods

Variational principles with relaxed inter‐element continuity requirement for non‐conforming element methods in linear and non‐linear analyses are developed. Based on the principles, any non‐conforming element displacement can be used directly to derive the explicit expressions of non‐conforming displacement function, which can ensure the passage of the patch test C for the requirement of convergence Copyright © 2001 John Wiley & Sons, Ltd.     

Coupling of non‐conforming meshes in a component mode synthesis method

A common mesh refinement‐based coupling technique is embedded into a component mode synthesis method, Craig–Bampton. More specifically, a common mesh is generated between the non‐conforming interfaces of the coupled structures, and the compatibility constraints are enforced on that mesh via L2‐minimization. This new integrated method is suitable for structural dynamic analysis problems where the substructures may have non‐conforming curvilinear and/or surface interface meshes. That is, coupled substructures may have different element types such as shell, solid, and/or beam elements. The proposed method is implemented into a commercial finite element software, B2000++, and its demonstration is carried out using an academic and industry oriented test problems. Copyright © 2013 John Wiley & Sons, Ltd.     

Methods for connecting dissimilar three‐dimensional finite element meshes

Two methods are presented for connecting dissimilar three‐dimensional finite element meshes. The first method combines the concept of master and slave surfaces with the uniform strain approach for finite elements. By modifying the boundaries of elements on a slave surface, corrections are made to element formulations such that first‐order patch tests are passed. The second method is based entirely on constraint equations, but only passes a weaker form of the patch test for non‐planar surfaces. Both methods can be used to connect meshes with different element types. In addition, master and slave surfaces can be designated independently of relative mesh resolutions. Example problems in three‐dimensional linear elasticity are presented. Published in 2000 by 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.