Implicit boundary method for analysis using uniform B‐spline basis and structured grid

Several analysis techniques such as extended finite element method (X‐FEM) have been developed recently, which use structured grid for the analysis. Implicit boundary method uses implicit equations of the boundary to apply boundary conditions in X‐FEM framework using structured grids. Solution structures for test and trial functions are constructed using implicit equations such that the boundary conditions are satisfied even if there are no nodes on the boundary. In this paper, this method is applied for analysis using uniform B‐spline basis defined over a structured grid. Solution structures that are C¹ or C² continuous throughout the analysis domain can be constructed using B‐spline basis functions. As a structured grid does not conform to the geometry of the analysis domain, the boundaries of the analysis domain are defined independently using equations of the boundary curves/surfaces. Compared with conforming mesh, it is easier to generate structured grids that overlap the geometry and the elements in the grid are regular shaped and undistorted. Numerical examples are presented to demonstrate the performance of these B‐spline elements. The results are compared with analytical solutions as well as with traditional finite element solutions. Convergence studies for several examples show that B‐spline elements provide accurate solutions with fewer elements and nodes compared with traditional FEM. They also provide continuous stress and strain in the analysis domain, thus eliminating the need for smoothing stress/strain results. Copyright © 2008 John Wiley & Sons, Ltd.     

Assessment of nonlinear exact geometry sampling surfaces solid-shell elements and ANSYS solid elements for 3D stress analysis of piezoelectric shell structures

In this work, the finite rotation exact geometry four-node solid-shell element using the sampling surfaces (SaS) method is developed for the analysis of the second Piola-Kirchhoff stresses in laminated piezoelectric shells. The SaS method is based on choosing inside the layers the arbitrary number of SaS parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the displacements and electric potentials of these surfaces as fundamental shell unknowns. The outer surfaces and interfaces are also included into a set of SaS. To circumvent shear and membrane locking, the hybrid-mixed solid-shell element on the basis of the Hu-Washizu variational principle is proposed. The tangent stiffness matrix is evaluated by 3D analytical integration throughout the finite element. This novelty provides a superior performance in the case of coarse meshes. A comparison with the SOLID226 element showed that the developed exact geometry SaS solid-shell element allows the use of load increments, which are much larger than possible with existing displacement-based finite elements. Thus, it can be recommended for the 3D stress analysis of doubly-curved laminated piezoelectric shells because the SaS formulation gives the opportunity to obtain the 3D solutions of electroelasticity with a prescribed accuracy.     

Non‐linear exact geometry 12‐node solid‐shell element with three translational degrees of freedom per node

This paper presents the finite rotation exact geometry (EG) 12‐node solid‐shell element with 36 displacement degrees of freedom. The term ‘EG’ reflects the fact that coefficients of the first and second fundamental forms of the reference surface and Christoffel symbols are taken exactly at each element node. The finite element formulation developed is based on the 9‐parameter shell model by employing a new concept of sampling surfaces (S‐surfaces) inside the shell body. We introduce three S‐surfaces, namely, bottom, middle and top, and choose nine displacements of these surfaces as fundamental shell unknowns. Such choice allows one to represent the finite rotation higher order EG solid‐shell element formulation in a very compact form and to derive the strain–displacement relationships, which are objective, that is, invariant under arbitrarily large rigid‐body shell motions in convected curvilinear coordinates. The tangent stiffness matrix is evaluated by using 3D analytical integration and the explicit presentation of this matrix is given. The latter is unusual for the non‐linear EG shell element formulation. Copyright © 2011 John Wiley & Sons, Ltd.     

Stress recovery and error estimation for the scaled boundary finite‐element method

The scaled boundary finite‐element method is a novel semi‐analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. This paper develops a stress recovery procedure based on a modal interpretation of the scaled boundary finite‐element method solution process, using the superconvergent patch recovery technique. The recovered stresses are superconvergent, and are used to calculate a recovery‐type error estimator. A key feature of the procedure is the compatibility of the error estimator with the standard recovery‐type finite element estimator, allowing the scaled boundary finite‐element method to be compared directly with the finite element method for the first time. A plane strain problem for which an exact solution is available is presented, both to establish the accuracy of the proposed procedures, and to demonstrate the effectiveness of the scaled boundary finite‐element method. The scaled boundary finite‐element estimator is shown to predict the true error more closely than the equivalent finite element error estimator. Unlike their finite element counterparts, the stress recovery and error estimation techniques work well with unbounded domains and stress singularities. Copyright © 2002 John Wiley & Sons, Ltd.     

From medical images to anatomically accurate finite element grids

The successful application of computational modelling of blood flow for the planning of surgical and interventional procedures to treat cardiovascular diseases strongly depends on the rapid construction of anatomical models. The large individual variability of the human vasculature and the strong dependence of blood flow characteristics on the vessel geometry require modelling on a patient‐specific basis. Various image processing and geometrical modelling techniques are integrated for the rapid construction of geometrical surface models of arteries starting from medical images. These discretely defined surfaces are then used to generate anatomically accurate finite element grids for hemodynamic simulations. The proposed methodology operates directly in 3D and consists of three stages. In the first stage, the images are filtered to reduce noise and segmented using a region‐growing algorithm in order to obtain a properly defined boundary of the arterial lumen walls. In the second stage, a surface triangulation representing the vessel walls is generated using a direct tessellation of the boundary voxels. This surface is then smoothed and the quality of the resulting triangulation is improved. Finally, in the third stage, the triangulation is subdivided into so‐called discrete surface patches for surface gridding, the desired element size distribution is defined and the finite element grid generated. Copyright © 2001 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.     

Automatic solver for non‐linear partial differential equations with implicit local laws: Application to unilateral contact

In general, non‐linear continuum mechanics combine global balance equations and local constitutive laws. In this work, frictionless contact between a rigid tool and a thin elastic shell is considered. This class of boundary value problems involves two non‐linear algebraic laws: the first one gives explicitly the stress field as a function of the strain throughout the continuum part, whereas the second one is a non‐linear equation relating the contact forces and the displacement at the boundary.Given the fact that classical computational approaches sometimes require significant effort in implementation of complex non‐linear problems, a computation technique based on automatic differentiation of constitutive laws is presented in this paper. The procedure enables to compute automatically the higher‐order derivatives of these constitutive laws and thereafter to define the Taylor series that are the basis of the continuation technique called asymptotic numerical method. The algorithm is about the same with an explicit or implicit constitutive relation. In the modelling of forming processes, many tool shapes can be encountered. The presented computational technique permits an easy implementation of these complex surfaces, for instance in a finite element code: the user is only required to define the tool geometry and the computer is able to obtain the higher‐order derivatives. Copyright © 2013 John Wiley & Sons, Ltd.     

Use of higher‐order shape functions in the scaled boundary finite element method

The scaled boundary finite element method is a novel semi‐analytical technique, whose versatility, accuracy and efficiency are not only equal to, but potentially better than the finite element method and the boundary element method for certain problems. This paper investigates the possibility of using higher‐order polynomial functions for the shape functions. Two techniques for generating the higher‐order shape functions are investigated. In the first, the spectral element approach is used with Lagrange interpolation functions. In the second, hierarchical polynomial shape functions are employed to add new degrees of freedom into the domain without changing the existing ones, as in the p‐version of the finite element method. To check the accuracy of the proposed procedures, a plane strain problem for which an exact solution is available is employed. A more complex example involving three scaled boundary subdomains is also addressed. The rates of convergence of these examples under p‐refinement are compared with the corresponding rates of convergence achieved when uniform h‐refinement is used, allowing direct comparison of the computational cost of the two approaches. The results show that it is advantageous to use higher‐order elements, and that higher rates of convergence can be obtained using p‐refinement instead of h‐refinement. Copyright © 2005 John Wiley & Sons, Ltd.     

A modified scaled boundary finite element method for three‐dimensional dynamic soil‐structure interaction in layered soil

This paper is devoted to the analysis of elastodynamic problems in 3D‐layered systems which are unbounded in the horizontal direction. For this purpose, a finite element model of the near field is coupled to a scaled boundary finite element model (SBFEM) of the far field. The SBFEM is originally based on describing the geometry of a half‐space or full‐space domain by scaling the geometry of the near field / far field interface using a radial coordinate. A modified form of the SBFEM for waves in a 2D layer is also available. None of these existing formulations can be used to describe a 3D‐layered medium. In this paper, a modified SBFEM for the analysis of 3D‐layered continua is derived. Based on the use of a scaling line instead of a scaling centre, a suitable scaled boundary transformation is proposed. The derivation of the corresponding scaled boundary finite element (SBFE) equations in displacement and stiffness is presented in detail. The latter is a nonlinear differential equation with respect to the radial coordinate, which has to be solved numerically for each excitation frequency considered in the analysis. Various numerical examples demonstrate the accuracy of the new method and its correct implementation. These include rigid circular and square foundations embedded in or resting on the surface of layered homogeneous or inhomogeneous 3D soil deposits over rigid bedrock. Hysteretic damping is assumed in some cases. The dynamic stiffness coefficients calculated using the proposed method are compared with analytical solutions or existing highly accurate numerical results. Copyright © 2011 John Wiley & Sons, Ltd.     

A boundary condition in Padé series for frequency‐domain solution of wave propagation in unbounded domains

A boundary condition satisfying the radiation condition at infinity is frequently required in the numerical simulation of wave propagation in an unbounded domain. In a frequency domain analysis using finite elements, this boundary condition can be represented by the dynamic stiffness matrix of the unbounded domain defined on its boundary. A method for determining a Padé series of the dynamic stiffness matrix is proposed in this paper. This method starts from the scaled boundary finite‐element equation, which is a system of ordinary differential equations obtained by discretizing the boundary only. The coefficients of the Padé series are obtained directly from the ordinary differential equations, which are not actually solved for the dynamic stiffness matrix. The high rate of convergence of the Padé series with increasing order is demonstrated numerically. This technique is applicable to scalar waves and elastic vector waves propagating in anisotropic unbounded domains of irregular geometry. It can be combined seamlessly with standard finite elements. Copyright © 2006 John Wiley & Sons, Ltd.     

Unconstrained plastering—Hexahedral mesh generation via advancing‐front geometry decomposition

The generation of all‐hexahedral finite element meshes has been an area of ongoing research for the past two decades and remains an open problem. Unconstrained plastering is a new method for generating all‐hexahedral finite element meshes on arbitrary volumetric geometries. Starting from an unmeshed volume boundary, unconstrained plastering generates the interior mesh topology without the constraints of a pre‐defined boundary mesh. Using advancing fronts, unconstrained plastering forms partially defined hexahedral dual sheets by decomposing the geometry into simple shapes, each of which can be meshed with simple meshing primitives. By breaking from the tradition of previous advancing‐front algorithms, which start from pre‐meshed boundary surfaces, unconstrained plastering demonstrates that for the tested geometries, high quality, boundary aligned, orientation insensitive, all‐hexahedral meshes can be generated automatically without pre‐meshing the boundary. Examples are given for meshes from both solid mechanics and geotechnical applications. Copyright © 2009 John Wiley & Sons, Ltd.     

A p‐hierarchical adaptive procedure for the scaled boundary finite element method

This paper describes a p‐hierarchical adaptive procedure based on minimizing the classical energy norm for the scaled boundary finite element method. The reference solution, which is the solution of the fine mesh formed by uniformly refining the current mesh element‐wise one order higher, is used to represent the unknown exact solution. The optimum mesh is assumed to be obtained when each element contributes equally to the global error. The refinement criteria and the energy norm‐based error estimator are described and formulated for the scaled boundary finite element method. The effectivity index is derived and used to examine quality of the proposed error estimator. An algorithm for implementing the proposed p‐hierarchical adaptive procedure is developed. Numerical studies are performed on various bounded domain and unbounded domain problems. The results reflect a number of key points. Higher‐order elements are shown to be highly efficient. The effectivity index indicates that the proposed error estimator based on the classical energy norm works effectively and that the reference solution employed is a high‐quality approximation of the exact solution. The proposed p‐hierarchical adaptive strategy works efficiently. Copyright © 2007 John Wiley & Sons, Ltd.     

hp‐Generalized FEM and crack surface representation for non‐planar 3‐D cracks

A high‐order generalized finite element method (GFEM) for non‐planar three‐dimensional crack surfaces is presented. Discontinuous p‐hierarchical enrichment functions are applied to strongly graded tetrahedral meshes automatically created around crack fronts. The GFEM is able to model a crack arbitrarily located within a finite element (FE) mesh and thus the proposed method allows fully automated fracture analysis using an existing FE discretization without cracks. We also propose a crack surface representation that is independent of the underlying GFEM discretization and controlled only by the physics of the problem. The representation preserves continuity of the crack surface while being able to represent non‐planar, non‐smooth, crack surfaces inside of elements of any size. The proposed representation also provides support for the implementation of accurate, robust, and computationally efficient numerical integration of the weak form over elements cut by the crack surface. Numerical simulations using the proposed GFEM show high convergence rates of extracted stress intensity factors along non‐planar curved crack fronts and the robustness of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

Stress analysis of solids with open-ended tubular holes by BFM

Two kinds of special surface elements, curvilinear tube element and triangular element with negative parts, are proposed for modeling solids containing many slender open-ended tubular shaped holes in the framework of boundary face method. The surfaces of each tubular hole in the solid are modeled by the tube elements, and the outer surfaces that intersected and trimmed by the holes are modeled by triangular elements with negative parts. Substantial savings in both modeling effort and computational cost have been achieved. In addition, all the special surface elements are defined in the parametric space of the surface, and the exact geometry data are obtained directly from a CAD model of the solid. Therefore, automatic analysis is possible. Several numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method.     

Element‐local level set method for three‐dimensional dynamic crack growth

An approximate level set method for three‐dimensional crack propagation is presented. In this method, the discontinuity surface in each cracked element is defined by element‐local level sets (ELLSs). The local level sets are generated by a fitting procedure that meets the fracture directionality and its continuity with the adjacent element crack surfaces in a least‐square sense. A simple iterative procedure is introduced to improve the consistency of the generated element crack surface with those of the adjacent cracked elements. The discrete discontinuity is treated by the phantom node method which is a simplified version of the extended finite element method (XFEM). The ELLS method and the phantom node technology are combined for the solution of dynamic fracture problems. Numerical examples for three‐dimensional dynamic crack propagation are provided to demonstrate the effectiveness and robustness of the proposed method. Copyright © 2009 John Wiley & Sons, Ltd.     

Finite element formulation of exact non‐reflecting boundary conditions for the time‐dependent wave equation

A modified version of an exact Non‐reflecting Boundary Condition (NRBC) first derived by Grote and Keller is implemented in a finite element formulation for the scalar wave equation. The NRBC annihilate the first N wave harmonics on a spherical truncation boundary, and may be viewed as an extension of the second‐order local boundary condition derived by Bayliss and Turkel. Two alternative finite element formulations are given. In the first, the boundary operator is implemented directly as a ‘natural’ boundary condition in the weak form of the initial–boundary value problem. In the second, the operator is implemented indirectly by introducing auxiliary variables on the truncation boundary. Several versions of implicit and explicit time‐integration schemes are presented for solution of the finite element semidiscrete equations concurrently with the first‐order differential equations associated with the NRBC and an auxiliary variable. Numerical studies are performed to assess the accuracy and convergence properties of the NRBC when implemented in the finite element method. The results demonstrate that the finite element formulation of the (modified) NRBC is remarkably robust, and highly accurate. Copyright © 1999 John Wiley & Sons, Ltd.