An hp‐adaptive finite element method for scattering problems in computational electromagnetics

An hp‐adaptive finite element (FE) approach is presented for a reliable, efficient and accurate solution of 3D electromagnetic scattering problems. The radiation condition in the far field is satisfied automatically by approximation with infinite elements (IE). Near optimal discretizations that can effectively resolve local rapid variations in the scattered field are sought adaptively by mesh refinements blended with graded polynomial enrichments. The p‐enrichments need not be spatially isotropic. The discretization error can be controlled by a self‐adaptive process, which is driven by implicit or explicit a posteriori error estimates. The error may be estimated in the energy norm or in a quantity of interest. A radar cross section (RCS) related linear functional is used in the latter case. Adaptively constructed solutions are compared to pure uniform p approximations. Numerical, highly accurate, and fairly converged solutions for a number of generic problems are given and compared to previously published results. Copyright © 2004 John Wiley & Sons, Ltd.     

A goal‐oriented hp‐adaptive finite element method with electromagnetic applications. Part I: electrostatics

We describe the development and application of a finite element (FE) self‐adaptive hp goal‐oriented algorithm for elliptic problems. The algorithm delivers (without any user interaction) a sequence of optimal hp‐grids. This sequence of grids minimizes the error of a prescribed quantity of interest with respect to the problem size. The refinement strategy is an extension of a fully automatic, energy‐norm based, hp‐adaptive algorithm. We illustrate the efficiency of the method with 2D numerical results. Among other problems, we apply the goal‐oriented hp‐adaptive strategy to simulate direct current (DC) resistivity logging instruments (including through casing resistivity tools) in a borehole environment and for the assessment of rock formation properties. Copyright © 2005 John Wiley & Sons, Ltd.     

An a posteriori error estimator for the p‐ and hp‐versions of the finite element method

An a posteriori error estimator is proposed in this paper for the p‐ and hp‐versions of the finite element method in two‐dimensional linear elastostatic problems. The local error estimator consists in an enhancement of an error indicator proposed by Bertóti and Szabó (Int. J. Numer. Meth. Engng. 1998; 42 :561–587), which is based on the minimum complementary energy principle. In order to obtain the local error estimate, this error indicator is corrected by a factor which depends only on the polynomial degree of the element. The proposed error estimator shows a good effectivity index in meshes with uniform and non‐uniform polynomial distributions, especially when the global error is estimated. Furthermore, the local error estimator is reliable enough to guide p‐ and hp‐adaptive refinement strategies. Copyright © 2004 John Wiley & Sons, Ltd.     

A goal‐oriented hp‐adaptive discontinuous Galerkin approach for biharmonic problems

A goal‐oriented algorithm is developed and applied for hp‐adaptive approximations given by the discontinuous Galerkin finite element method for the biharmonic equation. The methodology is based on the dual problem associated with the target functional. We consider three error estimators and analyse their properties as basic tools for the design of the hp‐adaptive algorithm. To improve adaptation, the combination of two different error estimators is used, each one at its best efficiency, to guide the tasks of where and how to adapt the approximation spaces. The performance of the resulting hp‐adaptive schemes is illustrated by numerical experiments for two benchmark problems. Copyright © 2013 John Wiley & Sons, Ltd.     

hp‐adaptive extended finite element method

This paper discusses higher‐order extended finite element methods (XFEMs) obtained from the combination of the standard XFEM with higher‐order FEMs. Here, the focus is on the embedding of the latter into the partition of unity method, which is the basis of the XFEM. A priori error estimates are discussed, and numerical verification is given for three benchmark problems. Moreover, methodological aspects, which are necessary for hp‐adaptivity in XFEM and allow for exponential convergence rates, are summarized. In particular, the handling of hanging nodes via constrained approximation and an hp‐adaptive strategy are presented. Copyright © 2011 John Wiley & Sons, Ltd.     

Wave propagation in a simplified modelled poroelastic continuum: fundamental solutions and a time domain boundary element formulation

In finite element formulations for poroelastic continua a representation of Biot's theory using the unknowns solid displacement and pore pressure is preferred. Such a formulation is possible either for quasi‐static problems or for dynamic problems if the inertia effects of the fluid are neglected. Contrary to these formulations a boundary element method (BEM) for the general case of Biot's theory in time domain has been published (Wave Propagation in Viscoelastic and Poroelastic Continua: A Boundary Element Approach. Lecture Notes in Applied Mechanics. Springer: Berlin, Heidelberg, New York, 2001.). If the advantages of both methods are required it is common practice to couple both methods. However, for such a coupled FE/BE procedure a BEM for the simplified dynamic Biot theory as used in FEM must be developed. Therefore, here, the fundamental solutions as well as a BE time stepping procedure is presented for the simplified dynamic theory where the inertia effects of the fluid are neglected. Further, a semi‐analytical one‐dimensional solution is presented to check the proposed BE formulation. Finally, wave propagation problems are studied using either the complete Biot theory as well as the simplified theory. These examples show that no significant differences occur for the selected material. Copyright © 2005 John Wiley & Sons, Ltd.     

3D‐based hp‐adaptive first‐order shell finite element for modelling and analysis of complex structures—Part 2: Application to structural analysis

The paper presents a 3D‐based adaptive first‐order shell finite element to be applied to hierarchical modelling and adaptive analysis of complex structures. The main feature of the element is that it is equipped with 3D degrees of freedom, while its mechanical model corresponds to classical first‐order shell theory. Other useful features of the element are its modelling and adaptive capabilities. The element is assigned to hierarchical modelling and hpq‐adaptive analysis of shell parts of complex structures consisting of solid, thick‐ and thin‐shell parts, as well as of transition zones, where h, p and q denote the mesh density parameter and the longitudinal and transverse orders of approximation, respectively. The proposed hp‐adaptive first‐order shell element can be joined with 3D‐based hpq‐adaptive hierarchical shell elements or 3D hpp‐adaptive solid elements by means of the family of 3D‐based hpq/hp‐ or hpp/hp‐adaptive transition elements. The main objective of the first part of our research, presented in the first part of the paper, was to provide non‐standard information on the original parts of the element algorithm. Here we describe the second part of the research, devoted to the methodology and results of the application of the element to various plate and shell problems. The main objective of this part is to verify algorithms of the element and to show its usefulness in modelling and adaptive analysis of shell and plate parts of complex structures. In order to do that, there is a presentation of the results of a comparative analysis of model plate and shell problems using the classical and our elements, and equidistributed and integrated Legendre shape functions. For the plate problem a comparison of the results obtained from the adaptive and non‐adaptive analysis is also included. Additionally, some advantages of the application of our element are shown through a comparative analysis of p‐convergence of the thin plate problem and an adaptive analysis of the exemplary complex structure. Copyright © 2006 John Wiley & Sons, Ltd.     

C1 macroelements in adaptive finite element methods

Some interesting properties of composite basis functions for C¹ macroelements are investigated, including their use for constructing conforming C¹ function spaces on non‐conforming adaptively refined meshes. Of particular interest is the classical cubic Hsieh–Clough–Tocher 3‐split triangle because of its simplicity and convergence properties in fourth‐order problems, as compared with the Powell–Sabin–Heindl 6‐split and 12‐split triangles. A posteriori error indicators for adaptive refinement are developed. Numerical experiments demonstrate convergence rates, and adaptive refinement performance based on a simplified error indicator is tested. Extensibility to analogous three‐dimensional tetrahedral elements is briefly discussed. Copyright © 2006 John Wiley & Sons, Ltd.     

An improved nonintrusive global–local approach for sharp thermal gradients in a standard FEA platform

Several classes of important engineering problems—in this case, problems exhibiting sharp thermal gradients—have solution features spanning multiple spatial scales and, therefore, necessitate advanced hp finite element discretizations. Although hp‐FEM is unavailable off‐the‐shelf in many predominant commercial analysis software packages, the authors herein propose a novel method to introduce these capabilities via a generalized FEM nonintrusively in a standard finite element analysis (FEA) platform. The methodology is demonstrated on two verification problems as well as a representative, industrial‐scale problem. Numerical results show that the techniques utilized allow for accurate resolution of localized thermal features on structural‐scale meshes without hp‐adaptivity or the ability to account for complex and very localized loads in the FEA code itself. This methodology enables the user to take advantage of all the benefits of both hp‐FEM discretizations and the appealing features of many available computer‐aided engineering /FEA software packages to obtain optimal convergence for challenging multiscale problems. Copyright © 2012 John Wiley & Sons, Ltd.     

A time‐integration method for the viscoelastic–viscoplastic analyses of polymers and finite element implementation

The present study introduces a time‐integration algorithm for solving a non‐linear viscoelastic–viscoplastic (VE–VP) constitutive equation of isotropic polymers. The material parameters in the constitutive models are stress dependent. The algorithm is derived based on an implicit time‐integration method (Computational Inelasticity. Springer: New York, 1998) within a general displacement‐based finite element (FE) analysis and suitable for small deformation gradient problems. Schapery's integral model is used for the VE responses, while the VP component follows the Perzyna model having an overstress function. A recursive‐iterative method (Int. J. Numer. Meth. Engng 2004; 59 :25–45) is employed and modified to solve the VE–VP constitutive equation. An iterative procedure with predictor–corrector steps is added to the recursive integration method. A residual vector is defined for the incremental total strain and the magnitude of the incremental VP strain. A consistent tangent stiffness matrix, as previously discussed in Ju (J. Eng. Mech. 1990; 116 :1764–1779) and Simo and Hughes (Computational Inelasticity. Springer: New York, 1998), is also formulated to improve convergence and avoid divergence. Available experimental data on time‐dependent and inelastic responses of high‐density polyethylene are used to verify the current numerical algorithm. The time‐integration scheme is examined in terms of its computational efficiency and accuracy. Numerical FE analyses of microstructural responses of polyethylene reinforced with elastic particle are also presented. Copyright © 2009 John Wiley & Sons, Ltd.     

3D‐based hp‐adaptive first‐order shell finite element for modelling and analysis of complex structures—Part 1: The model and the approximation

The paper presents a 3D‐based adaptive first‐order shell finite element to be applied to hierarchical modelling and adaptive analysis of complex structures. The main feature of the element is that it is equipped with 3D degrees of freedom, while its mechanical model corresponds to classical first‐order shell theory. Other useful features of the element are its modelling and adaptive capabilities. The element is assigned to hierarchical modelling and hpq‐adaptive analysis of shell parts of complex structures consisting of solid, thick‐ and thin‐shell parts, as well as of transition zones, where h, p and q denote the mesh density parameter and the longitudinal and transverse orders of approximation, respectively. The proposed hp‐adaptive first‐order shell element can be joined with 3D‐based hpq‐adaptive hierarchical shell elements or 3D hpp‐adaptive solid elements by means of the family of 3D‐based hpq/hp‐ or hpp/hp‐adaptive transition elements. The main objective of the first part of our research, presented in this paper, is to provide non‐standard information on the original parts of the element algorithm. In order to do that, we present the definition of shape functions necessary for p‐adaptivity, as well as the procedure for imposing constraints corresponding to the lack of elongation of the straight lines perpendicular to the shell mid‐surface, which is the procedure necessary for q‐adaptivity. The 3D version of constrained approximation presented next is the basis for h‐adaptivity of the element. The second part of our research, devoted to methodology and results of the numerical research on application of the element to various plate and shell problems, are described in the second part of this paper. Copyright © 2006 John Wiley & Sons, Ltd.     

An adaptive MsFEM for nonperiodic viscoelastic composites

We present a modification of the multiscale finite element method (MsFEM) for modeling of heterogeneous viscoelastic materials and an enhancement of this method by the adaptive generation of both meshes, ie, a macroscale coarse one and a microscale fine one. The fine mesh refinements are performed independently within coarse elements adjusting the microscale discretization to the microstructure, whereas the coarse mesh adaptation optimizes the macroscale approximation. Besides the coupling of the hp‐adaptive finite element method with the MsFEM we propose a modification of the MsFEM to accommodate for the analysis of transient nonlinear problems. We illustrate the efficiency and accuracy of the new approach for a number of benchmark examples, including the modeling of functionally graded material, and demonstrate the potential of our improvement for upscaling nonperiodic and nonlinear composites.     

Adaptive finite element approximation of multiphysics problems: A fluid–structure interaction model problem

In this paper we consider finite element simulation of the mechanical response of an elastic solid immersed into a viscous incompressible fluid flow. For simplicity, we assume that the mechanics of the solid is governed by linear elasticity and the motion of the fluid by the Stokes equation. For this one‐way coupled multiphysics problem we derive an a posteriori error estimate using duality techniques. Based on the estimate we propose an adaptive algorithm that automatically constructs a suitable mesh for the fluid and solid computational domains given a specific goal quantity for the elastic problem. Copyright © 2010 John Wiley & Sons, Ltd.     

On computational methods for variational inequalities

In this note, we focus on optimised mesh design for the Finite Element (FE) method for variational inequalities using global norm estimates for local error control. The strategies are based on the so called dual-weighted-residual (DWR) approach to a posteriori error control for FE-schemes (see, e.g., Rannacher et al. [19, 6, 2]), where error control for the primal problem is established by solving an auxiliary (dual) problem. In this context we blamed (cf. e.g., Rannacher and Suttmeier [18, 19]) global norm estimates being not that useful in applications. But having a closer look at the DWR-concept, one observes that in fact global (energy) error bounds can be employed to establish local error control. Our ideas and techniques are illustrated at the socalled obstacle problem.It turns out, that reliable and efficient energy error control is one main ingredient to establish useful a posteriori error bounds for local quantities. Therefore, in addition, we derive an unified approach to a posteriori error control in the energy norm for elliptic variational inequalities of first kind. Eventually, this framework is applied to Signorinis problem.     

An adaptive hp-version of the finite element method applied to flame propagation problems

This paper describes an adaptive hp-version mesh refinement strategy and its application to the finite element solution of one-dimensional flame propagation problems. The aim is to control the spatial and time discretization errors below a prescribed error tolerance at all time levels. In the algorithm, the optimal time step is first determined in an adaptive manner by considering the variation of the computable error in the reaction zone. Later, the method uses a p-version refinement till the computable a posteriori error is brought down below the tolerance. During the p-version, if the maximum allowable degree of approximation is reached in some elements of the mesh without satisfying the global error tolerance criterion, then conversion from p- to h-version is performed. In the conversion procedure, a gradient based non-uniform h-version refinement has been introduced in the elements of higher degree approximation. In this way, p-version and h-version approaches are used alternately till the a posteriori error criteria are satisfied. The mesh refinement is based on the element error indicators, according to a statistical error equi-distribution procedure. Numerical simulations have been carried out for a linear parabolic problem and premixed flame propagation in one-space dimension. © 1997 John Wiley & Sons, Ltd.     

Adaptive methods for thermomechanical coupled contact problems

In this paper an adaptive method for the analysis of thermomechanical coupled multi‐body contact problems is presented. The method is applied to non‐linear elastic solids undergoing finite (thermal) deformations. The contact model considers non‐linear pressure‐dependent heat flux as well as frictional heating in the interface. A time–space‐finite element discretization of the governing equations is formulated including unilateral constraints due to contact. A staggered solution algorithm has been constructed that allows an independent spatial discretization of the coupled subproblems. A posteriori projection‐based error estimators, which enforce implicitly the special boundary conditions due to thermal contact, are used to control the spatial discretization as well as the adaptive time stepping. Numerical examples are presented to corroborate the applicability of the adaptive algorithm to the considered problem type. Copyright © 2004 John Wiley & Sons, Ltd.     

On the a priori and a posteriori error analysis of a two‐fold saddle‐point approach for nonlinear incompressible elasticity

In this paper, we reconsider the a priori and a posteriori error analysis of a new mixed finite element method for nonlinear incompressible elasticity with mixed boundary conditions. The approach, being based only on the fact that the resulting variational formulation becomes a two‐fold saddle‐point operator equation, simplifies the analysis and improves the results provided recently in a previous work. Thus, a well‐known generalization of the classical Babu?ka–Brezzi theory is applied to show the well‐posedness of the continuous and discrete formulations, and to derive the corresponding a priori error estimate. In particular, enriched PEERS subspaces are required for the solvability and stability of the associated Galerkin scheme. In addition, we use the Ritz projection operator to obtain a new reliable and quasi‐efficient a posteriori error estimate. Finally, several numerical results illustrating the good performance of the associated adaptive algorithm are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

A posteriori error estimates of mixed methods for miscible displacement problems

The miscible displacement of one incompressible fluid by another in a porous medium is governed by a system of two equations. One is an elliptic equation of the pressure and the other is a parabolic equation of the concentration of one of the fluids. Since the pressure appears in the concentration only through its velocity field, we choose a mixed finite element method to approximate the pressure equation and for the concentration we use the standard Galerkin method. We shall obtain an explicit a posteriori error estimator in L²(L²) for the semi‐discrete scheme applied to the non‐linear coupled system. Copyright © 2007 John Wiley & Sons, Ltd.     

An adaptive non‐conforming finite‐element method for Reissner–Mindlin plates

Adaptive algorithms are important tools for efficient finite‐element mesh design. In this paper, an error controlled adaptive mesh‐refining algorithm is proposed for a non‐conforming low‐order finite‐element method for the Reissner–Mindlin plate model. The algorithm is controlled by a reliable and efficient residual‐based a posteriori error estimate, which is robust with respect to the plate's thickness. Numerical evidence for this and the efficiency of the new algorithm is provided in the sense that non‐optimal convergence rates are optimally improved in our numerical experiments. Copyright © 2003 John Wiley & Sons, Ltd.     

Port reduction in parametrized component static condensation: approximation and a posteriori error estimation

We introduce a port (interface) approximation and a posteriori error bound framework for a general component‐based static condensation method in the context of parameter‐dependent linear elliptic partial differential equations. The key ingredients are as follows: (i) efficient empirical port approximation spaces—the dimensions of these spaces may be chosen small to reduce the computational cost associated with formation and solution of the static condensation system; and (ii) a computationally tractable a posteriori error bound realized through a non‐conforming approximation and associated conditioner—the error in the global system approximation, or in a scalar output quantity, may be bounded relatively sharply with respect to the underlying finite element discretization. Our approximation and a posteriori error bound framework is of particular computational relevance for the static condensation reduced basis element (SCRBE) method. We provide several numerical examples within the SCRBE context, which serve to demonstrate the convergence rate of our port approximation procedure as well as the efficacy of our port reduction error bounds. Copyright © 2013 John Wiley & Sons, Ltd.