A domain decomposition method for discontinuous Galerkin discretizations of Helmholtz problems with plane waves and Lagrange multipliers

A nonoverlapping domain decomposition (DD) method is proposed for the iterative solution of systems of equations arising from the discretization of Helmholtz problems by the discontinuous enrichment method. This discretization method is a discontinuous Galerkin finite element method with plane wave basis functions for approximating locally the solution and dual Lagrange multipliers for weakly enforcing its continuity over the element interfaces. The primal subdomain degrees of freedom are eliminated by local static condensations to obtain an algebraic system of equations formulated in terms of the interface Lagrange multipliers only. As in the FETI‐H and FETI‐DPH DD methods for continuous Galerkin discretizations, this system of Lagrange multipliers is iteratively solved by a Krylov method equipped with both a local preconditioner based on subdomain data, and a global one using a coarse space. Numerical experiments performed for two‐ and three‐dimensional acoustic scattering problems suggest that the proposed DD‐based iterative solver is scalable with respect to both the size of the global problem and the number of subdomains. Copyright © 2009 John Wiley & Sons, Ltd.     

Trefftz boundary elements—multi‐region formulations

This paper is concerned with an effective numerical implementation of the Trefftz boundary element method, for the analysis of two‐dimensional potential problems, defined in arbitrarily shaped domains. The domain is first discretized into multiple subdomains or regions. Each region is treated as a single domain, either finite or infinite, for which a complete set of solutions of the problem is known in the form of an expansion with unknown coefficients. Through the use of weighted residuals, this solution expansion is then forced to satisfy the boundary conditions of the actual domain of the problem, leading thus to a system of equations, from which the unknowns can be readily determined. When this basic procedure is adopted, in the analysis of multiple‐region problems, proper boundary integral equations must be used, along common region interfaces, in order to couple to each other the unknowns of the solution expansions relative to the neighbouring regions. These boundary integrals are obtained from weighted residuals of the coupling conditions which allow the implementation of any order of continuity of the potential field, across the interface boundary, between neighbouring regions. The technique used in the formulation of the region‐coupling conditions drives the performance of the Trefftz boundary element method. While both of the collocation and Galerkin techniques do not generate new unknowns in the problem, the technique of Galerkin presents an additional and unique feature: the size of the matrix of the final algebraic system of equations which is always square and symmetric, does not depend on the number of boundary elements used in the discretization of both the actual and region‐interface boundaries. This feature which is not shared by other numerical methods, allows the Galerkin technique of the Trefftz boundary element method to be effectively applied to problems with multiple regions, as a simple, economic and accurate solution technique. A very difficult example is analysed with this procedure. The accuracy and efficiency of the implementations described herein make the Trefftz boundary element method ideal for the study of potential problems in general arbitrarily‐shaped two‐dimensional domains. Copyright © 1999 John Wiley & Sons, Ltd.     

Simulation of two dimensional unilateral contact using a coupled FE/EFG method

In this paper, simulation of two dimensional unilateral contact problems using a coupled finite element/element free Galerkin method is proposed. For the analysis, the element free Galerkin method and Galerkin formulation for two dimensional elasticity problems are considered. Then, the penalty method for imposition of contact constraint is proposed. The finite element shape functions are used in the penalty term of contact constraint. Finally, the accuracy of the presented method is verified through some examples. The numerical results have demonstrated that the presented approach is simple and accurate for frictionless contact analysis of 2D solids.     

An embedded mesh method for treating overlapping finite element meshes

A new technique for treating the mechanical interactions of overlapping finite element meshes is presented. Such methods can be useful for numerous applications, for example, fluid–solid interaction with a superposed meshed solid on an Eulerian background fluid grid. In this work, we consider the interaction of two elastic domains: one mesh is the foreground and defines the surface of interaction, the other is a background mesh and is often a structured grid. Many of the previously proposed methods employ surface defined Lagrange multipliers or penalties to enforce the boundary constraints. It has become apparent that these methods will cause mesh locking under certain conditions. Appropriately applied, the Nitsche method can overcome this locking, but, in its canonical form, is generally not applicable to non‐linear materials such as hyperelastics. The relationship between interior point penalty, discontinuous Galerkin and Nitsche's method is well known. Based on this relationship, a nonlinear theory analogous to the Nitsche method is proposed to treat nonlinear materials in an embedded mesh. Here, a discontinuous Galerkin derivative based on a lifting of the interface surface integrals provides a consistent treatment for non‐linear materials and demonstrates good behavior in example problems. Published 2012. This article is a US Government work and is in the public domain in the USA.     

A three‐dimensional surface stress tensor formulation for simulation of adhesive contact in finite deformation

A three‐dimensional surface adhesive contact formulation is proposed to simulate macroscale adhesive contact interaction characterized by the van der Waals interaction between arbitrarily shaped deformable continua under finite deformation. The proposed adhesive contact formulation uses a double‐layer surface integral to replace the conventional double volume integration to compute the adhesive contact force vector. Considering nonlinear finite deformation, we have derived the surface stress tensor and the corresponding tangent stiffness matrix in a Galerkin weak formulation. With the surface stress formulation, the adhesive contact problems are solved in the framework of nonlinear continuum mechanics by using the standard Lagrange finite element method. Surface stress tensors are formulated for both interacting bodies. Numerical examples show that the proposed surface contact algorithm is accurate, efficient, and reliable for three‐dimensional adhesive contact problems of large deformations for both quasi‐static and dynamic simulations. Copyright © 2015 John Wiley & Sons, Ltd.     

SGBEM with Lagrange multipliers applied to elastic domain decomposition problems with curved interfaces using non‐matching meshes

An original approach to the solution of linear elastic domain decomposition problems by the symmetric Galerkin boundary element method is developed. The approach is based on searching for the saddle‐point of a new potential energy functional with Lagrange multipliers. The interfaces can be either straight or curved, open or closed. The two coupling conditions, equilibrium and compatibility, along an interface are fulfilled in a weak sense by means of Lagrange multipliers (interface displacements and tractions), which enables non‐matching meshes to be used at both sides of interfaces between subdomains. The accuracy and robustness of the method is tested by several numerical examples, where the numerical results are compared with the analytical solution of the solved problems, and the convergence rates of two error norms are evaluated for h‐refinements of matching and non‐matching boundary element meshes. Copyright © 2010 John Wiley & Sons, Ltd.     

Buckling of folded plate structures subjected to partial in‐plane edge loads by the FSDT meshfree Galerkin method

This paper presents a meshfree Galerkin method that is based on the first‐order shear deformation theory (FSDT) to study the elastic buckling behaviour of stiffened and un‐stiffened folded plates under partial in‐plane edge loads. The un‐stiffened folded plates are modelled as assemblies of flat plates. The stiffness and initial stress matrices of the flat plates are derived by the meshfree Galerkin method. A treatment is implemented to modify the stiffness and initial stress matrices, and the matrices are then superposed to obtain the stiffness and initial stress matrix of the entire folded plate. The analytical process for stiffened folded plates is similar, except that the effects of the stiffeners must be taken into account. Because no mesh is required, the proposed method is superior for studying problems that would involve remeshing in the finite element method. Several examples are employed to show the convergence and accuracy of the proposed method. The results obtained show good agreement with the results computed from the finite element analysis software ANSYS. Copyright © 2005 John Wiley & Sons, Ltd.     

A high order hybrid finite element method applied to the solution of electromagnetic wave scattering problems in the time domain

The development of a hybrid high order time domain finite element solution procedure for the simulation of two dimensional problems in computational electromagnetics is considered. The chosen application area is that of electromagnetic scattering. The spatial approximation adopted incorporates both a continuous Galerkin spectral element method and a high order discontinuous Galerkin method. Temporal discretisation is achieved by means of a fourth order Runge–Kutta procedure. An exact analytical solution is employed initially to validate the procedure and the numerical performance is then demonstrated for a number of more challenging examples.     

A variational approach with embedded roughness for adhesive contact problems

Abstract

A new variational formulation is proposed for the solution of contact problems for profiles of arbitrary shape indenting a deformable half-plane, with special focus on rough indenters. The method exploits the new idea of accounting for the shape of roughness as a correction to the normal gap function. The resulting interface finite element with eMbedded Profile for Joint Roughness (MPJR interface finite element) is derived and its implementation is comprehensively described. The method is applied to a range of contact problems very challenging for traditional methods, opening new perspectives for the solution of contact problems with roughness and adhesion.     

A new method for meshless integration in 2D and 3D Galerkin meshfree methods

A method for the evaluation of regular domain integrals without domain discretization is presented. In this method, a domain integral is transformed into a boundary integral and a 1D integral. The method is then utilized for the evaluation of domain integrals in meshless methods based on the weak form, such as the element-free Galerkin method and the meshless radial point interpolation method. The proposed technique results in truly meshless methods with better accuracy and efficiency in comparison with their original forms. Some examples, including linear and large-deformation problems, are also provided to demonstrate the usefulness of the proposed method.     

On enrichment functions in the extended finite element method

This paper presents mathematical derivation of enrichment functions in the extended finite element method for numerical modeling of strong and weak discontinuities. The proposed approach consists in combining the level set method with characteristic functions as well as domain decomposition and reproduction technique. We start with the simple case of a triangular linear element cut by one interface across which displacement field suffers a jump. The main steps towards the derivation of enrichment functions are as follows: (1) extension of the subfields separated by the interface to the whole element domain and definition of complementary nodal variables; (2) construction of characteristic functions for describing the geometry and physical field; (3) determination of the sets of basic nodal variables; (4) domain decompositions according to Step 3 and then reproduction of the physical field in terms of characteristic functions and nodal variables; and (5) comparison of the piecewise interpolations formulated at Steps 3 and 4 with the standard extended finite element method form, which yields enrichment functions. In this process, the physical meanings of both the basic and complementary nodal variables are clarified, which helps to impose Dirichlet boundary conditions. Enrichment functions for weak discontinuities are constructed from deeper insights into the structure of the functions for strong discontinuities. Relationships between the two classes of functions are naturally established. Improvements upon basic enrichment functions for weak discontinuities are performed so as to achieve satisfactory convergence and accuracy. From numerical viewpoints, a simple and efficient treatment on the issue of blending elements is also proposed with implementation details. For validation purposes, applications of the derived functions to heterogeneous problems with imperfect interfaces are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

Ductile failure modeling and simulations using coupled FE–EFG approach

In the present work, a ductile fracture model has been employed to predict the failure of tensile specimen using coupled finite element–element free Galerkin (FE–EFG) approach. The fracture strain as a function of stress triaxiality has been evaluated by analyzing the notched tensile specimens. In the coupled approach, a small portion of the domain, where severe plastic deformation is expected, is modeled by EFG method whereas the rest of the domain is modeled by FEM to exploit the advantages of both the methods. A ramp function has been used in the interface region to maintain the continuity between FE and EFG domains. The nonlinear material behavior is modeled by von-Mises yield criterion and Hollomon’s power law. An implicit return mapping algorithm is employed for stress equilibrium in the plasticity model. The effect of geometric nonlinearity as a result of large deformation is captured by updated Lagrangian approach. The coupled approach is used to study the fracture behavior of two different cracked specimens in order to highlight its capabilities.     

Fracture mechanics analysis using the wavelet Galerkin method and extended finite element method

This paper presents fracture mechanics analysis using the wavelet Galerkin method and extended finite element method. The wavelet Galerkin method is a new methodology to solve partial differential equations where scaling/wavelet functions are used as basis functions. In solid/structural analyses, the analysis domain is divided into equally spaced structured cells and scaling functions are periodically placed throughout the domain. To improve accuracy, wavelet functions are superposed on the scaling functions within a region having a high stress concentration, such as near a hole or notch. Thus, the method can be considered a refinement technique in fixed‐grid approaches. However, because the basis functions are assumed to be continuous in applications of the wavelet Galerkin method, there are difficulties in treating displacement discontinuities across the crack surface. In the present research, we introduce enrichment functions in the wavelet Galerkin formulation to take into account the discontinuous displacements and high stress concentration around the crack tip by applying the concept of the extended finite element method. This paper presents the mathematical formulation and numerical implementation of the proposed technique. As numerical examples, stress intensity factor evaluations and crack propagation analyses for two‐dimensional cracks are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

A general discontinuous Galerkin method for finite hyperelasticity. Formulation and numerical applications

A discontinuous Galerkin formulation of the boundary value problem of finite‐deformation elasticity is presented. The primary purpose is to establish a discontinuous Galerkin framework for large deformations of solids in the context of statics and simple material behaviour with a view toward further developments involving behaviour or models where the DG concept can show its superiority compared to the continuous formulation. The method is based on a general Hu–Washizu–de Veubeke functional allowing for displacement and stress discontinuities in the domain interior. It is shown that this approach naturally leads to the formulation of average stress fluxes at interelement boundaries in a finite element implementation. The consistency and linearized stability of the method in the non‐linear range as well as its convergence rate are proven. An implementation in three dimensions is developed, showing that the proposed method can be integrated into conventional finite element codes in a straightforward manner. In order to demonstrate the versatility, accuracy and robustness of the method examples of application and convergence studies in three dimensions are provided. Copyright © 2006 John Wiley & Sons, Ltd.     

A discontinuous Galerkin finite element method for dynamic and wave propagation problems in non‐linear solids and saturated porous media

A time‐discontinuous Galerkin finite element method (DGFEM) for dynamics and wave propagation in non‐linear solids and saturated porous media is presented. The main distinct characteristic of the proposed DGFEM is that the specific P3–P1 interpolation approximation, which uses piecewise cubic (Hermite's polynomial) and linear interpolations for both displacements and velocities, in the time domain is particularly proposed. Consequently, continuity of the displacement vector at each discrete time instant is exactly ensured, whereas discontinuity of the velocity vector at the discrete time levels still remains. The computational cost is then obviously saved, particularly in the materially non‐linear problems, as compared with that required for the existing DGFEM. Both the implicit and explicit algorithms are developed to solve the derived formulations for linear and materially non‐linear problems. Numerical results illustrate good performance of the present method in eliminating spurious numerical oscillations and in providing much more accurate solutions over the traditional Galerkin finite element method using the Newmark algorithm in the time domain. Copyright © 2003 John Wiley & Sons, Ltd.     

Coupling of element free Galerkin and hybrid boundary element methods using modified variational formulation

 A novel method is proposed by coupling the element free Galerkin (EFG) and the hybrid boundary element (HBE) methods to achieve solution efficiency and accuracy for stress analysis in solids. A modified variational formulation is derived for the present coupled EFG/HBE method so that the continuity and compatibility can be preserved on the interface between the domains of EFG and HBE. The coupled EFG/HBE method has been coded in FORTRAN. The validity and efficiency of the proposed method are demonstrated through a number of example problems. It is found that the present method can take advantages of both EFG and HBE methods. The present method is very easy to implement, and very flexible for obtaining displacements and stresses of desired accuracy in solids, as the efforts for meshing the problem domain have been significantly reduced due to the use of boundary element method (BEM).     

A mortar‐finite element formulation for frictional contact problems

In this paper a finite element formulation is developed for the solution of frictional contact problems. The novelty of the proposed formulation involves discretizing the contact interface with mortar elements, originally proposed for domain decomposition problems. The mortar element method provides a linear transformation of the displacement field for each boundary of the contacting continua to an intermediate mortar surface. On the mortar surface, contact kinematics are easily evaluated on a single discretized space. The procedure provides variationally consistent contact pressures and assures the contact surface integrals can be evaluated exactly. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical simulation of bi-material interfacial cracks using EFGM and XFEM

In this paper, bi-material interfacial cracks have been simulated using element free Galerkin method (EFGM) and extended finite element method (XFEM) under mode-I and mixed mode loading conditions. Few crack interaction problems of dissimilar layered materials are also simulated using extrinsic partition of unity enriched approach. Material discontinuity has been modeled by a signed distance function whereas strong discontinuity has been modeled by two functions i.e. Heaviside and asymptotic crack tip enrichment functions. The stress intensity factors for bi-material interface cracks are numerically evaluated using the modified domain form of interaction integral. The results obtained by EFGM and XFEM for bi-material edge and center cracks are compared with those available in literature. In order to check the validity of simulations, the results have been obtained for two different ratio of Young’s modulus.