Symmetric coupling of multi‐zone curved Galerkin boundary elements with finite elements in elasticity

The coupling of Finite Element Method (FEM) with a Boundary Element Method (BEM) is a desirable result that exploits the advantages of each. This paper examines the efficient symmetric coupling of a Symmetric Galerkin Multi‐zone Curved Boundary Element Analysis method with a Finite Element Method for 2‐D elastic problems. Existing collocation based multi‐zone boundary element methods are not symmetric. Thus, when they are coupled with FEM, it is very difficult to achieve symmetry, increasing the computational work to solve the problem. This paper uses a fully Symmetric curved Multi‐zone Galerkin Boundary Element Approach that is coupled to an FEM in a completely symmetric fashion. The symmetry is achieved by symmetrically converting the boundary zones into equivalent ‘macro finite elements’, that are symmetric, so that symmetry in the coupling is retained. This computationally efficient and fast approach can be used to solve a wide range of problems, although only 2‐D elastic problems are shown. Three elasticity problems, including one from the FEM‐BEM literature that explore the efficacy of the approach are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

三维实体结构NURBS等几何分析

等几何分析是近年来在有限元法基础上发展起来的一种新的数值方法,它消除了有限元的几何误差,具有高阶连续性。该文研究了三维结构等几何分析中NURBS几何体的表示方式及载荷、约束的施加方法,分别从计算精度和仿真效率两个方面对比了等几何分析的计算结果与有限元法一阶单元和二阶单元的计算结果,展示了等几何分析相对于标准有限元法的优势,并以厚壁圆筒模型算例验证了等几何分析的实用性。将NURBS单元应用于几何形状精度要求高的齿轮和变截面圆筒,数值结果表明三维NURBS等几何分析方法在复杂三维结构的仿真计算中具有较好的灵活性和适用性,可得到连续的应力场,有望在工程中得到广泛应用。     

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.     

Viscoplastic plane‐strain analysis of infinite solids using elastic supports

A highly efficient novel Finite Element Boundary Element Method (FEBEM) is proposed for the elasto‐viscoplastic plane‐strain analysis of displacements and stresses in infinite solids. The proposed method takes advantage of both the Finite Element Method (FEM) and the Boundary Element Method (BEM) to achieve higher efficiency and accuracy by using the concept of elastic supports to simulate the effects of unbounded solid mass surrounding the region of interest. The BEM is used to compute the stiffnesses of elastic supports and to estimate the location of the truncation boundary for the finite element model. As compared to the conventional coupled FEBEM, the proposed method has three main computational advantages. Firstly, the symmetrical and highly banded form of the standard finite element stiffness matrix is not disturbed. Secondly, the proposed technique may be implemented simply by using standard codes for elasto‐viscoplastic finite element analysis and elastic boundary element analysis. Thirdly, the yielded zone is approximately located in advance by using the BEM and hence, an unnecessarily large extent of the domain does not have to be discretized for the finite element modelling. The efficiency and accuracy of the proposed method are demonstrated by computing elastic and elasto‐plastic displacements and stresses around ‘deep’ underground openings in rock mass subject to hydrostatic and non‐hydrostatic in situ stresses. Results obtained by the proposed method are compared with ‘exact’ solutions and with those obtained by using a BEM and a coupled FEBEM. Copyright © 1999 John Wiley & Sons, Ltd.     

Generalized boundary element method for galerkin boundary integrals

A meshless approach to the Boundary Element Method in which only a scattered set of points is used to approximate the solution is presented. Moving Least Square approximations are used to build a Partition of Unity on the boundary and then used to construct, at low cost, trial and test functions for Galerkin approximations. A particular case in which the Partition of Unity is described by linear boundary element meshes, as in the Generalized Finite Element Method, is then presented. This approximation technique is then applied to Galerkin boundary element formulations. Finally, some numerical accuracy and convergence solutions for potential problems are presented for the singular, hypersingular and symmetric approaches.     

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.     

A new particular solution strategy for hyperbolic boundary value problems using hybrid‐Trefftz displacement elements

A new indirect approach to the problem of approximating the particular solution of non‐homogeneous hyperbolic boundary value problems is presented. Unlike the dual reciprocity method, which constructs approximate particular solutions using radial basis functions, polynomials or trigonometric functions, the method reported here uses the homogeneous solutions of the problem obtained by discarding all time‐derivative terms from the governing equation. Nevertheless, what typifies the present approach from a conceptual standpoint is the option of not using these trial functions exclusively for the approximation of the particular solution but to fully integrate them with the (Trefftz‐compliant) homogeneous solution basis. The particular solution trial basis is capable of significantly improving the Trefftz solution even when the original equation is genuinely homogeneous, an advantage that is lost if the basis is used exclusively for the recovery of the source terms. Similarly, a sufficiently refined Trefftz‐compliant basis is able to compensate for possible weaknesses of the particular solution approximation. The method is implemented using the displacement model of the hybrid‐Trefftz finite element method. The functions used in the particular solution basis reduce most terms of the matrix of coefficients to boundary integral expressions and preserve the Hermitian, sparse and localized structure of the solving system that typifies hybrid‐Trefftz formulations. Even when domain integrals are present, they are generally easy to handle, because the integrand presents no singularity. Copyright © 2015 John Wiley & Sons, Ltd.     

Multi-region Trefftz boundary element method for fracture analysis in plane piezoelectricity

This paper presents a multi-region Trefftz boundary element method for fracture analysis in plane piezoelectricity. To model the sub-region that contains the crack, a special set of Trefftz functions that satisfy the traction-free and charge-free conditions along the crack faces are constructed. To model the remaining sub-regions, the basic set of Trefftz functions co-derived previously by the authors are employed. With the two sets of Trefftz functions, the multi-region Trefftz boundary element method is formulated by point collocation. The special set of Trefftz functions exempts all the boundary treatment of the crack faces and enables the direct determination of the electromechanical intensity factors. Numerical examples are presented to illustrate the efficacy of the formulation.     

Dynamic behavior of frame structures by boundary integral procedures

The Boundary Element Method is applied to synthesize a set of Boundary Integral Equations representing the uncoupled axial and flexural dynamic behavior of rectilinear Bernoulli–Euler beam elements in the frequency domain. In the sequence, these structural elements are coupled by the sub-region technique to model two-dimensional frame structures, in which the axial and flexural behaviors are coupled. This methodology is used to accurately recover modal data, eigenfrequencies and eigenmodes, of two frame structures. The usual Boundary Element procedure is recast to deliver simultaneously the values of variables at the element boundaries and at an arbitrary number of internal nodes. The inclusion of internal nodes allow to recover the structure eigenmodes and makes feasible the coupling of the assembled systems with a surrounding environment, for instance, an acoustic field. The results obtained are compared with a standard Finite Element eigenvalue analysis. It is shown that for increasing response frequencies, the Boundary Element scheme delivers modal data within a degree of accuracy, which is only obtained by the conventional Finite Element Method with considerable finer meshes.     

An efficient Wave Based Method for solving Helmholtz problems in three-dimensional bounded domains

This paper discusses the use of the Wave Based Method for the analysis of time-harmonic three-dimensional (3D) interior acoustic problems. Conventional element-based prediction methods, such as the Finite Element Method, are most commonly used for these types of problems, but they are restricted to low-frequency applications. The Wave Based Method is an alternative deterministic technique which is based on the indirect Trefftz approach. Up to now, this method's very high computational efficiency has been illustrated mainly for two-dimensional (2D) problem settings, allowing the analysis of problems at higher frequencies. The numerical validation examples presented in this work shows that the enhanced computational efficiency of the Wave Based Method in comparison with conventional element-based methods is kept when the method is extended to 3D case with and without the presence of material damping.     

Embedded interfaces by polytope FEM

In this paper, the Polytope Finite Element Method is employed to model an embedded interface through the body, independent of the background FEM mesh. The elements that are crossed by the embedded interface are decomposed into new polytope elements which have some nodes on the interface line. The interface introduces discontinuity into the primary variable (strong) or into its derivatives (weak). Both strong and weak discontinuities are studied by the proposed method through different numerical examples including fracture problems with traction‐free and cohesive cracks, and heat conduction problems with Dirichlet and Dirichlet–Neumann types of boundary conditions on the embedded interface. For traction‐free cracks which have tip singularity, the nodes near the crack tip are enriched with the singular functions through the eXtended Finite Element Method. The concept of Natural Element Coordinates (NECs) is invoked to drive shape functions for the produced polytopes. A simple treatment is proposed for concave polytopes produced by a kinked interface and also for locating crack tip inside an element prior to using the singularity enrichment. The proposed method pursues some implementational details of eXtended/Generalized Finite Element Methods for interfaces. But here the additional DOFs are constructed on the interface lines in contrast to X/G‐FEM, which attach enriched DOFs to the previously existed nodes. Copyright © 2011 John Wiley & Sons, Ltd.     

Elastic wave propagation in unsaturated porous media using hybrid‐Trefftz stress elements

The stress model of the hybrid‐Trefftz finite element is formulated for the analysis of elastodynamic problems defined on unsaturated porous media. The supporting mathematical model is the theory of mixtures with interfaces and considers the full coupling between the solid, fluid and gas phases, including the effect of seepage acceleration. Hybrid‐Trefftz stress elements use the free‐field regular solutions of the homogeneous Navier (or Beltrami) equation to construct the approximation of the generalized stresses in the domain of the element. The influence of non‐homogeneous terms in the Navier equation is modelled using solutions of the corresponding static problem. The resulting elements are highly convergent under p‐refinement and robust to both low and high excitation frequencies, as the trial functions embody relevant physical information on the modelled phenomenon. Copyright © 2013 John Wiley & Sons, Ltd.     

Trefftz‐based dual reciprocity method for hyperbolic boundary value problems

A new dual reciprocity‐type approach to approximating the solution of non‐homogeneous hyperbolic boundary value problems is presented in this paper. Typical variants of the dual reciprocity method obtain approximate particular solutions of boundary value problems in two steps. In the first step, the source function is approximated, typically using radial basis, trigonometric or polynomial functions. In the second step, the particular solution is obtained by analytically solving the non‐homogeneous equation having the approximation of the source function as the non‐homogeneous term. However, the particular solution trial functions obtained in this way typically have complicated expressions and, in the case of hyperbolic problems, points of singularity. Conversely, the method presented here uses the same trial functions for both source function and particular solution approximations. These functions have simple expressions and need not be singular, unless a singular particular solution is physically justified. The approximation is shown to be highly convergent and robust to mesh distortion. Any boundary method can be used to approximate the complementary solution of the boundary value problem, once its particular solution is known. The option here is to use hybrid‐Trefftz finite elements for this purpose. This option secures a domain integral‐free formulation and endorses the use of super‐sized finite elements as the (hierarchical) Trefftz bases contain relevant physical information on the modeled problem. Copyright © 2015 John Wiley & Sons, Ltd.     

The Discontinuity‐Enriched Finite Element Method

We introduce a new methodology for modeling problems with both weak and strong discontinuities independently of the finite element discretization. At variance with the eXtended/Generalized Finite Element Method (X/GFEM), the new method, named the Discontinuity‐Enriched Finite Element Method (DE‐FEM), adds enriched degrees of freedom only to nodes created at the intersection between a discontinuity and edges of elements in the mesh. Although general, the method is demonstrated in the context of fracture mechanics, and its versatility is illustrated with a set of traction‐free and cohesive crack examples. We show that DE‐FEM recovers the same rate of convergence as the standard FEM with matching meshes, and we also compare the new approach to X/GFEM.     

Hybrid‐Trefftz stress element for bounded and unbounded poroelastic media

The equations that govern the dynamic response of saturated porous media are first discretized in time to define the boundary value problem that supports the formulation of the hybrid‐Trefftz stress element. The (total) stress and pore pressure fields are directly approximated under the condition of locally satisfying the domain conditions of the problem. The solid displacement and the outward normal component of the seepage displacement are approximated independently on the boundary of the element. Unbounded domains are modelled using either unbounded elements that locally satisfy the Sommerfeld condition or absorbing boundary elements that enforce that condition in weak form. As the finite element equations are derived from first‐principles, the associated energy statements are recovered and the sufficient conditions for the existence and uniqueness of the solutions are stated. The performance of the element is illustrated with the time domain response of a biphasic unbounded domain to show the quality of the modelling that can be attained for the stress, pressure, displacement and seepage fields using a high‐order, wavelet‐based time integration procedure. Copyright © 2010 John Wiley & Sons, Ltd.     

Testing complete and compact radial basis functions for solution of eigenvalue problems using the boundary element method with direct integration

This paper analyses the performance of the main radial basis functions in the formulation of the Boundary Element Method (DIBEM). This is an alternative for solving problems modeled by non-adjoint differential operators, since it transforms domain integrals in boundary integrals using radial basis functions. The solution of eigenvalue problem was chosen to performance evaluation. Natural frequencies are calculated numerically using several radial functions and their accuracy is evaluated by comparison with the available analytical solutions and with the Finite Element Method as well. The standard radial basis functions have presented similar performance to compact radial functions, being even slightly superior.     

An effectively modified direct Trefftz method for 2D potential problems considering the domain''s characteristic length

The present paper proposes a new modification of the direct Trefftz method by taking the characteristic length of the problem domain into account, whose inclusion into the T-complete bases ensures that the modified direct Trefftz method is stable, because the condition number of the resulting linear equations system can be greatly reduced over 12 orders. Then, the boundary element method and the Fourier series method are used to derive the linear equations system to determine the unknown coefficients, which can be employed to solve the mixed-boundary value 2D potential problems. We use numerical examples to explore why the conventional direct Trefftz method is unstable and the modified one is stable and workable. The direct Trefftz method is applicable to the case where the problem size is smaller or with its maximum length near to 1 and using suitable elements number or bases number. Under this condition, the modified method still has a great advantage to improve the accuracy up to two or three orders.