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Abstract: This is a paper out of a series of three papers devoted to model- and hpq-adaptive finite element methods assigned for modeling and analysis of complex structures. In this paper we focus our attention on hierarchical models and hierarchical approximations, while the issues of error estimation and adaptivity control will be presented in the next papers of the series. In all the papers we consider the elastic structures of complex mechanical description. We apply 3D or 3D-based mechanical models, hierarchical modeling, and hierarchical approximations within our finite element formulation. In the proposed approach, the mechanical model and discretization parameters, such as the size h of the element, and the longitudinal and transverse approximation orders, p and q, can vary locally, i.e. they can be different in each finite element. The a posteriori error estimation is based on the residual equilibration methods applied to the assessment of the modeling and approximation errors. The error-controlled adaptive procedures are derived from a four-step strategy, with possible iterations on h- and p-steps. [ABSTRACT FROM AUTHOR]

Abstract: This paper introduces a recovery-type error estimator yielding upper bounds of the error in energy norm for linear elastic fracture mechanics problems solved using the extended finite element method (XFEM). The paper can be considered as an extension and enhancement of a previous work in which the upper bounds of the error were developed in a FEM framework. The upper bound property requires the recovered solution to be equilibrated and continuous. The proposed technique consists of using a recovery technique, especially adapted to the XFEM framework that yields equilibrium at a local level (patch by patch). Then a postprocess based on the partition of unity concept is used to obtain continuity. The result is a very accurate but only nearly-statically admissible recovered stress field, with small equilibrium defaults introduced by the postprocess. Sharp upper bounds are obtained using a new methodology accounting for the equilibrium defaults, as demonstrated by the numerical tests. [Copyright &y& Elsevier]

Abstract: Sensitivity Analysis is an essential issue in the structural optimization field. The calculation of the derivatives of the most relevant quantities (displacements, stresses, strains) in optimum design of structures allows to estimate the structural response when changes in the design variables are introduced. This essential information is used by the most frequent conventional optimization algorithms (SLP, MMA, Feasible directions) in order to reach the optimal solution. According to this idea, the Sensitivity Analysis of the stress constraints in Topology Optimization problems is a crucial aspect to obtain the optimal solution when stress constraints are considered. Maximum stiffness approaches usually involve one linear constraint and one non-linear objective function. Thus, the computation of the required sensitivity analysis does not mean a crucial limitation. However, in the topology optimization problem with stress constraints, efficient and accurate computation of the derivatives is needed in order to reach appropriate optimal solutions. In this paper, a complete analytic and efficient procedure to obtain the Sensitivity Analysis of the stress constraints in topology optimization of continuum structures is analyzed. First order derivatives and second order directional derivatives of the stress constraints are analyzed and included in the optimization procedure. In addition, topology optimization problems usually involve thousands of design variables and constraints. Thus, an efficient implementation of the algorithms used in the computation of the Sensitivity Analysis is developed in order to reduce the computational cost required. Finally, the sensitivity analysis techniques presented in this paper are tested by solving some application examples. [Copyright &y& Elsevier]

Abstract: A temporal finite element discretization of a boundary value problem has several advantages compared to a time-integrating evolution form for optimized target movement simulations. The paper gives some basic aspects on how such a finite element form can be stated, with both displacements and controls discretized and seen as unknowns. Aspects on the resulting formulations are discussed. Important issues are the order, continuity and fineness of the discretizations. When the formulation is seen in an optimization context, minimizing the effort for a prescribed movement, the discretization affects the results obtained in several manners, where some aspects of results are artifacts. The paper discusses these effects from basic principles, but also verifies them in numerical simulations. [Copyright &y& Elsevier]

Abstract: This paper considers the analysis and numerical simulation of strong discontinuities in partially saturated solids. The goal is to study observed localized failures in such media like shear bands and similar. The developments consider a fully coupled partially saturated elastoplastic model for the (continuum) bulk response of the solid formulated in effective stresses, identifying the necessary mathematical conditions for the appearance of strong discontinuities (that is, discontinuities in the displacement field leading to singular strains) as well as the proper treatment for the fields characterizing the flow of the different fluid phases, namely, the fluid contents of these phases and their individual pore pressures. The geometrically linear range of infinitesimal strains is considered. These developments allow the formulation of multiphase cohesive laws along the strong discontinuity, capturing in this way the coupled localized dissipation observed in the aforementioned failures. Furthermore, the paper also presents the formulation of enhanced finite elements capturing all these discontinuous solutions in general unstructured meshes. In particular, the finite elements capture the strong discontinuity through the proper enhancements of the discrete element strains, allowing for a complete local resolution of these effects. This results in a particularly efficient computational approach, easily accommodated in an existing finite element code. Different representative numerical simulations are presented illustrating the performance of the proposed formulation, as well as its use in practical applications like the modeling of the excavation of tunnels in variably saturated media. [Copyright &y& Elsevier]

Abstract: The Arlequin method is a generic numerical method that allows, by local superposition and coupling of models, to address multimodel and multiscale mechanical problems. In particular, this method has already been used to super-impose cracked patches on sound structures, reducing this way the global simulation resources a classical finite element approach would have required. In this paper, one of the key features of the extended-finite element method, namely the heaviside enrichment function, is used within the Arlequin framework to further reduce the costs of crack propagation simulations. The main goal of the paper is to describe the proposed methodology and to assess its performance through numerical experiments. [Copyright &y& Elsevier]

Abstract: In the western hemisphere almost the half of all events of death are caused by cardiovascular diseases, e.g. strokes and heart attacks. The latter are consequences of arteriosclerosis leading to abnormal deposits (plaque) in blood vessels. In order to avoid the serious symptoms discussed in the above or to hold affected blood vessels open, tubular structures made of metallic or polymeric materials (stents) are implanted. In the paper we discuss the modelling of a new kind of stents, so-called shape memory polymer (SMP) stents. The first part of the paper is devoted to the thermo-mechanical modelling of these materials. Aspects as the transition from entropy to energy elasticity are included. The constitutive equations are derived in the framework of large strains. We follow both, a purely macroscopic as well as a micromechanically motivated approach. In the second part of the work representative examples based on realistic stent structures are used to validate the model. [Copyright &y& Elsevier]

Abstract: This paper presents a numerical multiscale modelling strategy for simulating fracturing in materials where the fine-scale heterogeneities are fully resolved, with a particular focus on concrete. The fine-scale is modelled using a hybrid-Trefftz stress formulation for modelling propagating cohesive cracks. The very large system of algebraic equations that emerges from detailed resolution of the fine-scale structure requires an efficient iterative solver with a preconditioner that is appropriate for fracturing heterogeneous materials. This paper proposes a two-grid strategy for construction of the preconditioner that utilizes scale transition techniques derived for computational homogenization and represents an adaptation of the work of Miehe and Bayreuther and its extension to fracturing heterogeneous materials. For the coarse scale, this paper investigates both classical -continuous displacement-based finite elements as well as -continuous elements. The preconditioned GMRES Krylov iterative solver with dynamic convergence tolerance is integrated with a constrained Newton method with local arc-length control and line searches. The convergence properties and performance of the parallel implementation of the proposed solution strategy is illustrated on two numerical examples. [Copyright &y& Elsevier]

Abstract: This paper describes a novel simplification and fairing method for NURBS (Non-Uniform Rational B-Spline) curves and surfaces using orthogonal non-uniform spline wavelets. The compression nature of wavelets is utilized for NURBS simplification. Sufficient conditions for - and -continuity at the shared boundary of multiple NURBS patches are proved and utilized to preserve the boundary continuity. Our fairing method filters out unwanted imperfections and noise on the NURBS curves and surfaces while preserving geometric details via wavelet decomposition and selective reconstruction. The continuity between neighboring patches is preserved automatically during fairing. Our algorithm is accurate, automatic, robust, efficient, and can be extensively applied to any NURBS models in various research fields such as Computer Aided Design (CAD), Reverse Engineering (RE), finite element analysis and isogeometric analysis. Several examples are presented in this paper, including a propeller CAD model, a Navy boat model and a -continuous NURBS model of the thoracic aorta constructed from medical CT (Computed Tomography) data. [Copyright &y& Elsevier]

Abstract: This paper focuses on the description and analysis of a method and its application for simulating the mold filling process in Resin Transfer Molding (RTM) within an updated Lagrangian framework. For this purpose, we have employed a well established meshless technique known as Natural Element Method (NEM). This technique presents some advantages over finite element simulations, such as no remeshing requirements and the fact that the interpolation accuracy is not significantly affected by the nodal distribution. The use of a meshless technique in RTM allows avoiding the numerical difficulties associated to the fluid properties transport through the whole domain in fixed mesh simulations. The position of the flow front or the geometry of the fluid domain is handled by invoking the geometrical concept of the -shape of the cloud of nodes, using new strategies to negotiate the position, forms and evolution of the flow front advance. The paper includes some examples that illustrate the potential of the method. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the multiscale computation of the eigenvalue problems for the elastic equations in composite materials. The multiscale finite element method for solving these problems is presented. Numerous numerical results validate the proposed method of this paper. [Copyright &y& Elsevier]

Abstract: This paper presents a general explicit time integration scheme for dynamics simulations using the eXtended Finite Element Method with standard critical time step. We use the generalized mass lumping technique proposed in Part I of this paper. This technique allows us to consider arbitrary enrichment functions in the X-FEM for explicit dynamics simulations. In this second part, the proposed approach allows the use of standard finite elements critical time step estimates. For that purpose, we develop a classical element-by-element strategy that couples the standard central difference scheme with the unconditionally stable-explicit scheme proposed by Chang [S.Y. Chang, An explicit method with improved stability property, Int. J. Numer. Methods Engrg. 77 (8) (2008) 1100–1120]. This scheme coupled with X-FEM allows us to recover FE critical time steps independently of the enrichment functions considered. Furthermore, a study of the stability property of this new explicit scheme is proposed in the X-FEM framework, both with fixed enrichments and evolving enrichments with time. Some examples illustrate the good properties of the new explicit numerical time scheme and some applications to dynamics crack growth are given. [Copyright &y& Elsevier]

Abstract: Finite element (FE) response sensitivity analysis is an essential tool for gradient-based optimization methods used in various sub-fields of civil engineering such as structural optimization, reliability analysis, system identification, and finite element model updating. Furthermore, stand-alone sensitivity analysis is invaluable for gaining insight into the effects and relative importance of various system and loading parameters on system response. The direct differentiation method (DDM) is a general, accurate and efficient method to compute FE response sensitivities to FE model parameters. In this paper, the DDM-based response sensitivity analysis methodology is applied to a pressure independent multi-yield-surface J 2 plasticity material model, which has been used extensively to simulate the nonlinear undrained shear behavior of cohesive soils subjected to static and dynamic loading conditions. The complete derivation of the DDM-based response sensitivity algorithm is presented. This algorithm is implemented in a general-purpose nonlinear finite element analysis program. The work presented in this paper extends significantly the framework of DDM-based response sensitivity analysis, since it enables numerous applications involving the use of the multi-yield-surface J 2 plasticity material model. The new algorithm and its software implementation are validated through two application examples, in which DDM-based response sensitivities are compared with their counterparts obtained using forward finite difference (FFD) analysis. The normalized response sensitivity analysis results are then used to measure the relative importance of the soil constitutive parameters on the system response. [Copyright &y& Elsevier]

Abstract: The paper investigates the possibility of solving beam and thin-walled structures with an automatic hp-adaptive Finite Element Method for three-dimensional linear elasticity. This technique generates sequences of meshes with nonuniform distribution of element sizes h and their spectral orders p to minimize the interpolation error for the hypothetical exact solution with constant number of degrees-of-freedom. The intent of using hp method for beam and thin-walled structures is that it should automatically create FE meshes with adequate distribution of the approximation orders to prevent the numerical locking phenomenon and to provide accurate solutions. The method also reduces the human work spent on building the numerical model to generation of the description of the geometry. We discuss energy based and goal-oriented versions of hp-adaptivity. The paper presents a broad spectrum of examples of engineering applications. [Copyright &y& Elsevier]

Abstract: In this paper, heat transfer problems exhibiting sharp thermal gradients are analyzed using the classical and generalized finite element methods. The effect of solution roughness on the ability of the methods to obtain accurate approximations is investigated. Convergence studies show that low order (linear and quadratic) elements require strongly refined meshes for acceptable accuracy. We propose a generalized FEM with global–local enrichments for the class of problems investigated in the paper. In this procedure, a global solution space defined on a coarse mesh is enriched through the partition of unity framework of the generalized FEM with solutions of local boundary value problems. The local problems are defined using the same procedure as in the global–local FEM, where boundary conditions are provided by a coarse scale global solution. Coarse, uniform, global meshes are acceptable even at regions with thermal spikes that are orders of magnitude smaller than the element size. Convergence on these discretizations was achieved even when no or limited convergence was observed in the local problems. Two approaches are proposed to improve the boundary conditions prescribed on local problems and their convergence. The use of the corresponding improved local solutions as enrichments for the global problem extends the range of target error level for the enriched global problem. The two-way information transfer provided by the proposed generalized FEM is appealing to several classes of problems, especially those involving multiple spatial scales. The proposed methodology brings the benefits of generalized FEM to problems where limited or no information about the solution is known a priori. [Copyright &y& Elsevier]

Abstract: We begin the mathematical study of the k-method utilizing the theory of Kolmogorov n-widths. The k-method is a finite element technique where spline basis functions of higher-order continuity are employed. It is a fundamental feature of the new field of isogeometric analysis. In previous works, it has been shown that using the k-method has many advantages over the classical finite element method in application areas such as structural dynamics, wave propagation, and turbulence. The Kolmogorov n-width and sup–inf were introduced as tools to assess the effectiveness of approximating functions. In this paper, we investigate the approximation properties of the k-method with these tools. Following a review of theoretical results, we conduct a numerical study in which we compute the n-width and sup–inf for a number of one-dimensional cases. This study sheds further light on the approximation properties of the k-method. We finish this paper with a comparison study of the k-method and the classical finite element method and an analysis of the robustness of polynomial approximation. [Copyright &y& Elsevier]

Abstract: This paper is a sequel on the topic of localized Lagrange multipliers (LLM) for applications of fluid–structure interaction (FSI) between finite-element models of an acoustic fluid and an elastic structure. The prequel paper formulated the spatial-discretization methods, the LLM interface treatment, the time-marching partitioned analysis procedures, and the application to 1D benchmark problems. Here, we expand on formulation aspects required for successful application to more realistic 2D and 3D problems. Additional topics include duality relations at the fluid–structure interface, partitioned vibration analysis, reduced-order modeling, handling of curved interface surfaces, and comparison of LLM with other coupling methods. Emphasis is given to non-matching fluid–structure meshes. We present benchmark examples that illustrate the benefits and drawbacks of competing interface treatments. Realistic application problems involving the seismic response of two existing dams are considered. These include 2D modal analyses of the Koyna gravity dam, transient-response analyses of that dam with and without reduced-order modeling, incorporation of nonlinear cavitation effects, and the 3D transient-response analysis of the Morrow Point arch dam. [Copyright &y& Elsevier]

From its early days the SPH method has been criticised for its shortcomings namely tensile instability and consistency. Without thorough understanding of the method attempts were made to make the classical SPH method consistent and stable which resulted in the local and Total Lagrangian forms of SPH similar to the finite element method. In this paper we derived and analysed a consistent nonlocal SPH which has similarity with Bazant's imbricate continuum. In addition, the paper provides comparison and discussion of different SPH forms including: Classical SPH, Nonlocal, Local and Mixed SPH. The partition of unity approach was used to define the following two mixed forms: Local–Nonlocal and Local–Classical SPH. These mixed forms were intended for modelling of physical processes characterised with local and nonlocal effects (local and nonlocal constitutive equations), e.g. progressive damage and failure. The stabilising effect of the Local form on the Classical SPH, which is inherently unstable (tensile instability), are also illustrated. The stability analysis, presented in appendices A and B, demonstrate stability of the continuous and discrete form of the nonlocal SPH based on Eulerian kernels for elastic continuum. • Derivation of a Nonlocal SPH formulation with no tensile instability. • Derivation of a Mixed Local–Nonlocal SPH formulation based on partition of unity. • Proposal of a Mixed Local–Classical SPH formulation based on partition of unity. • Von Neumann stability analysis the Nonlocal SPH. • Stability requirements for the SPH kernel function. [ABSTRACT FROM AUTHOR]

Abstract: The subject of the paper is an optimal choice of material parameters characterizing the core layer of sandwich plates within the framework of the conventional plate theory in which the core layer is treated as soft in the in-plane direction. The mathematical description is similar to the Hencky–Reissner model of plates with transverse shear deformation. Here, however, the bending stiffnesses and the transverse shear stiffnesses can be designed independently. The present paper deals only with optimal design of the core layer to make the plate compliance minimal. Two core materials are at our disposal, which leads to the ill-posed problem. To consider it one should relax this problem by admitting composite domains and characterize their overall properties by the homogenization formulae. The numerical approach is based on this relaxed formulation thus making it mesh-independent. The equilibrium problem is solved by the DSG3 finite element method. The optimization results are found with using the convergent updating schemes of the COC method. [Copyright &y& Elsevier]

Abstract: This paper describes an original time-domain formulation to analyse saturated porous media. Standard finite element procedures are employed to numerically discretize the spatial domain of the model and a time-marching scheme based on the mechanical Green’s function of the problem is considered. The Green’s function matrices are implicitly and numerically evaluated, taking into account the Newmark method. The present methodology allows the system of equations of the solid and fluid phase to be treated separately, providing a more efficient and accurate solution procedure (smaller, simpler and better conditioned systems of equations are analysed). At the end of the paper, numerical examples are presented, illustrating the potentialities of the new approach. [Copyright &y& Elsevier]

Abstract: A spatial stabilization via bubble functions of linear finite element methods for nonlinear evolutionary convection–diffusion equations is discussed. The method of lines with SUPG discretization in space leads to numerical schemes that are not only difficult to implement, when considering nonlinear evolutionary equations, but also do not produce satisfactory results. The method we propose can be seen as an alternative to this kind of methods. Once the numerical approximation belonging to a linear finite element space enriched with bubble functions is obtained, the bubble part is discarded. The linear part is shown to give a stabilized approximation to the solution being approached. The bubble functions are deduced using a linear steady convection–diffusion model in such a way that the linear part of the approximation to the linear steady model (after static condensation of bubbles) reproduces the SUPG method in the convection-dominated regime. However, for the nonlinear evolutionary equations we consider in the paper the method we propose is not equivalent to the SUPG method. Some numerical experiments are provided in the paper to show the efficiency of the procedure. [Copyright &y& Elsevier]

Abstract: This paper recovers the original spirit of the continuous crack approaches, where displacements jumps across the crack are smeared over the affected elements and the behaviour is established through a softening stress–(total) strain law, using standard finite element displacement interpolations and orthotropic local constitutive models. The paper focuses on the problem of shear locking observed in the discrete problem when orthotropic models are used. The solution for this drawback is found in the form of a mesh corrected crack model where the structure of the inelastic strain tensor is linked to the geometry of the cracked element. The discrete model is formulated as a non-symmetric orthotropic local damage constitutive model, in which the softening modulus is regularized according to the material fracture energy and the element size. The resulting formulation is easily implemented in standard non-linear FE codes and suitable for engineering applications. Numerical examples show that the results obtained using this crack model do not suffer from dependence on the mesh directional alignment, comparing very favourably with those obtained using related standard isotropic or orthotropic damage models. [Copyright &y& Elsevier]

Abstract: In the present paper, adaptive finite element methods are studied for time-harmonic eddy current problems in the case of three-dimensional isotropic and linear materials. We adopt the formulation based on the magnetic field and a magnetic scalar potential in this paper, since it needs the least number of unknowns. Reliable and efficient a posteriori error estimates are obtained and the efficiency of the adaptive algorithm is demonstrated by numerical and engineering experiments. [Copyright &y& Elsevier]

Abstract: The intrinsic inability to withstand compression is a challenge in numerical simulation of a membrane with the finite element method. Basically there are two possibilities to deal with this problem. On the one hand wrinkles would be detected in detail by a refined mesh of shell elements once a bending formulation is used. On the other hand this problem is computed on a coarser mesh of membrane elements augmented with a wrinkling model, which considers effects of unresolved deformation patterns. This contribution focuses on the material modification approach, which is considered as a member of the latter. Within this category, works in [P. Contri, B.A. Schrefler, A geometrically nonlinear finite element analysis of wrinkled membrane surfaces by a no-compression material model, Commun. Appl. Numer. Methods 4 (1988) 5–15; X. Liu, C.H. Jenkins, W.W. Schur, Large deflection analysis of pneumatic envelopes using a penalty parameter modified material model, Finite Elements Anal. Des. 37 (2001) 233–251; X. Liu, Fine scale analysis of wrinkled membrane, Int. J. Numer. Comput. Engrg. Sci. 1 (2) (2000) 1017–1038; R. Rossi, M. Lazzari, R. Vitaliani, E. Onate, Simulation of light-weight membrane structures by wrinkling model, Int. J. Numer, Methods Engrg. 62 (15) (2005) 2127–2153; R. Rossi, E. Onate, Convergence of the modified material model for wrinkling simulation of light-weight membrane structures, in: E. Onate, B. Kröpin (Eds.), Textile Composites and Inflatable Structures (Structural Membranes 2003), CIMNE, 2003; R. Rossi, Light Weight Structures: Structural Analysis and Coupling Issues, Ph.D. Diss., The university of Bologna, 2005] are attractive because their wrinkling models on a coarse grid are simple and intuitive. The basic idea of this model is as follows: whenever compression occurs in a membrane, the corresponding components of the constitutive tensor in the direction of the compressive stress are penalized to weaken the compressive stiffness of the membrane. However, such abrupt change causes inevitably an unphysical oscillation of stress redistribution. Therefore, an algorithm to stabilize this oscillation is required. The first objective of this paper is to give a systematic verification of a wrinkling model based on the material modification approach by means of an analogy between wrinkling and plasticity. The second one is to include this model in a static analysis for both isotropic and orthotropic membranes. This model holds following advantages: it is computationally inexpensive and virtually able to reproduce the exact stress field, caused by wrinkling, on a macroscopic scale. To the author’s knowledge applications of a material modification approach with orthotropic materials are scarcely available. This paper demonstrates the potential of the model by means of numerical examples, compared to ones in literature. [Copyright &y& Elsevier]

Abstract: This paper presents a procedure to build enrichment functions for partition of unity methods like the generalized finite element method and the hp cloud method. The procedure combines classical global–local finite element method concepts with the partition of unity approach. It involves the solution of local boundary value problems using boundary conditions from a global problem defined on a coarse discretization. The local solutions are in turn used to enrich the global space using the partition of unity framework. The computations at local problems can be parallelized without difficulty allowing the solution of large problems very efficiently. The effectiveness of the approach in terms of convergence rates and computational cost is investigated in this paper. We also analyze the effect of inexact boundary conditions applied to local problems and the size of the local domains on the accuracy of the enriched global solution. Key aspects of the computational implementation, in particular, the numerical integration of generalized FEM approximations built with global–local enrichment functions, are presented. The method is applied to fracture mechanics problems with multiple cracks in the domain. Our numerical experiments show that even on a serial computer the method is very effective and allows the solution of complex problems. Our analysis also demonstrates that the accuracy of a global problem defined on a coarse mesh can be controlled using a fixed number of global degrees of freedom and the proposed global–local enrichment functions. [Copyright &y& Elsevier]

This paper deals with the use of the extended Finite Element Method (X-FEM) for rapid dynamic problems. To solve the equations of motion, a common technique is the explicit direct integration with a Newmark scheme. Since this temporal scheme is only conditionally stable, the critical time step must be determined. It is generally induced by mesh constraints. The idea of the paper is to weaken constraints on mesh generation algorithms so that the critical time step is as large as possible. Using the X-FEM one allows a non-conformity between mesh and discontinuities such as cracks, holes or interfaces. In a first part, we present a summary about direct integration schemes and about the eXtended Finite Element Method. Then, we focus on the theoretical description of a 1D X-FEM finite element and its generalization to 2D and 3D finite elements. Then, dynamic numerical simulations are shown. They concern structures under impact with holes or external boundaries not exactly matched by the mesh. Comparisons are made with numerical results coming from the ABAQUS software. It shows that developments are satisfactory. We conclude with some outlooks concerning this work. [Copyright &y& Elsevier]

Abstract: The accurate analysis and characterization of waveguide discontinuities is an important issue in microwave engineering. In some cases, the discontinuities are an unavoidable result of mechanical or electric transitions which effects have to be minimized; in other cases, the discontinuities are deliberately introduced into the waveguide to perform a certain electric function. This is the first of two papers analyzing different types of rectangular waveguide discontinuities by using a fully automatic hp-adaptive finite element method. In this paper, a fully automatic energy-norm based hp-adaptive Finite Element (FE) strategy applied to a number of relevant waveguide structures, is presented. The methodology produces exponential convergence rates in terms of the energy-norm error of the solution against the problem size (number of degrees of freedom). Extensive numerical results demonstrate the suitability of the hp-method for solving different rectangular waveguide discontinuities. These results illustrate the flexibility, reliability, and high-accuracy of the method. The self-adaptive hp-FEM provides similar (sometimes more) accurate results than those obtained with semi-analytical techniques such as the mode matching method, for problems where semi-analytical methods can be applied. At the same time, the hp-FEM provides the flexibility of modeling more complex waveguide structures and including the effects of dielectrics, metallic screws, round corners, etc., which cannot be easily considered when using semi-analytical techniques. In the second part, the work is extended by presenting a fully automatic goal-oriented hp-adaptive FE in terms of the scattering or S-parameters which are widely used in microwave engineering for the characterization of microwave devices. [Copyright &y& Elsevier]

Abstract: This work investigates the possibility of using equivalent continua within a multilevel strategy for the analysis of brick masonry walls, as prototypes for the in-plane behavior of masonry buildings. This strategy, proposed and discussed in detail in Part I of the same paper, lies in an iterative solution scheme which uses two levels, local and global, of description: the former accounts for the interaction between the single bricks; the latter, modeled as a coarse Finite Element discretization, accounts for global interactions. Within the same Finite Element format and strategy, the paper compares the performances of schemes implementing the identification procedures based on equivalent continua with those implementing the algebraic re-parametrization of the local model proposed in Part I. The former, while suitable for a synthetic representation of masonry behavior, does not prove to be convenient for a multilevel solution strategy. [Copyright &y& Elsevier]

In our earlier work [E. Stein, S. Ohnimus, Coupled model-and solution-adaptivity in the finite-element method, Comput. Methods Appl. Mech. Engrg. 150 (1997) 327–350; E. Stein, S. Ohnimus, Anisotropic discretization-and model-error estimation in solid mechanics by local Neumann problems, Comput. Methods Appl. Mech. Engrg. 176 (1999) 363–385; E. Stein, M. Rüter, S. Ohnimus, Adaptive finite element analysis and modelling of solids and structures. Findings, problems and trends, Int. J. Numer. Methods Engrg. 60 (2004) 103–138.], residual-based combined model and finite element discretization error estimators as a basis for adaptivity were derived that rely on the solutions of Neumann problems on the element level. In particular, these estimators control the errors obtained while expanding the model hierarchically from 1D-to 2D-and from 2D-to 3D-elastic theories in the (global) energy norm. In this paper, we extend both types of estimates to goal-oriented a posteriori error estimation within the framework of nonlinear structural mechanics. Structural model adaptivity is characterized by the fact that the dimensions, in general the constitutive equations and especially the related subdomains—and thus the test and solution spaces of the process-dependent models considered—are different. This has the important consequence that consistent prolongations of kinematic and static nodal quantities of finite elements of the coarser model “m” to those of the refined model “m +1” are necessary. Therefore, such prolongation operators are discussed in detail. Furthermore, duality techniques are used for the model and discretization error estimates, using system and load-dependent quantities of interest. In this paper, we also consider different finite element meshes for the primal and the dual problem. Finally, some numerical examples demonstrate the expected effects, realizing verification and even validation in structural mechanics, i.e. approaching a new paradigm in Computational Mechanics for achieving reliability and efficiency. [Copyright &y& Elsevier]

One-step finite element method (also called inverse approach) is more and more widely used in the automobile industry because of its unique advantages. Initial solution is an essential issue to ensure the success of the non-linear resolution in the implicit static one-step finite element solver. In order to speed up the convergence of the Newton–Raphson iterations, different kinds of initial solution methods are known. These are studied and compared in this paper. Several examples are followed to show their performance and efficiency. However most of these methods are based on geometric considerations like the geometric mapping method or the radial length development method. This kind of geometry based mapping methods could not reflect all aspects of the actual forming process. Therefore, an energy based mapping algorithm was implemented and coupled with the reverse deformation method which is based on the assumption of linear elastic deformation. This novel algorithm is proposed to provide the initial solution of one-step finite element method. For a complicated sheet forming modeling initial solutions obtained by different energy based algorithms coupled with the reverse deformation method are then compared in this paper. The results show that the Desbrun quadratic energy method and the accordant parameterization method combined with the inverse deformation method respectively are universal, efficient and robust initial solution schemes for the one-step finite element method. [Copyright &y& Elsevier]

Abstract: This paper describes an efficient scheme for coupling the FE and the BE methods. The domain of the original problem is subdivided into two sub-domains, which are independently modelled by the FEM and the BEM. Both sub-domains are analyzed separately and the coupling is enforced at their interface. The FEM time-marching process is carried out through an algorithm based on implicit Green’s functions of the related mechanical system and the convolution process of the BEM is compressed by proper interpolation of the time generated matrices. The efficiency of the proposed scheme is improved by a special procedure to deal with different time-step choices in each sub-domain. At the end of the paper numerical examples are presented in order to illustrate the accuracy and efficiency of the present approach. [Copyright &y& Elsevier]

Abstract: Large-scale numerical simulations of the flow and associated transport phenomena governed by the Navier–Stokes and Energy equations are routinely calculated in engineering practice. Nevertheless, the uncertainty due to spatial discretization limits the confidence of practitioners in numerical solutions. An approach to provide information about the accuracy of the quantity of interest is proposed here-in. The novel a posteriori error estimation technique – the bound method – is based on relaxing Lagrange multipliers that enforces continuity between sub-domains. The method provides fast, efficient, asymptotic but reliable lower and upper bounds to the output of underlying partial differential equations (PDEs). Herein, we highlight the method when applied to outputs of the steady incompressible Navier–Stokes and Energy equations. The bound method in this paper follows the directly equilibrated hybrid-flux approach for the flux calculation between sub-domains and uses the Crouzeix–Raviart () approximation spaces. To improve the effectiveness of the bound method, an adaptive sub-domain refinement strategy leading to sharper bounds is adopted. A convective heat transfer problem in a series of electronic chip devices is investigated. The novelty of this paper is to present bounds using adaptive domain decomposition for outputs associated to a complex three-dimensional field solution of the Navier–Stokes and Energy equations. [Copyright &y& Elsevier]

Abstract: This paper is devoted to the solution of the instationary Maxwell equations with charges. The geometry of the domain can be singular, in the sense that its boundary can include reentrant corners or edges. The difficulties arise from the fact that those geometrical singularities generate, in their neighborhood, strong electromagnetic fields. The time-dependency of the divergence of the electric field, is addressed. To tackle this problem, some new theoretical and practical results are presented, on curl-free singular fields, and on singular fields with L 2 (non-vanishing) divergence. The method, which allows to compute the instationary electromagnetic field, is based on a splitting of the spaces of solutions into a two-term direct sum. First, the subspace of regular fields: it coincides with the whole space of solutions, provided that the domain is either convex, or with a smooth boundary. Second, a singular subspace, defined and characterized via the singularities of the Laplace operator. Several numerical examples are presented, to illustrate the mathematical framework. This paper is the generalization of the singular complement method. [Copyright &y& Elsevier]

Abstract: The paper deals with application of the inverse analysis to the identification of friction and rheological models in metal forming. The direct model based on the finite element approach and the inverse algorithm are described. The method is validated by performing the inverse computations for different plastometric tests: uniaxial compression, ring compression, cube compression, plane strain compression and channel test. The analysis concerns both hot tests and room temperature tests. It is shown in the paper that, in general, when inverse analysis is applied the results of identification are insensitive to the type of the test, sample dimensions etc. This observation is not valid for the channel test, in which constrain of spread fosters initiation of micro-shear bands and leads to a decrease of the flow stress. Parameter sensitivity analysis is the objective of the second part of the paper. Two aspects of sensitivity are considered. The first is evaluation of the sensitivity of the output (measured) parameters of the tests with respect to rheological and friction parameters. This analysis shows that the loads are most sensitive to the rheological parameters in the uniaxial compression and the shape of the sample is the most sensitive to friction in the ring compression. The second aspect of the analysis focused on evaluation of sensitivity of the results of the inverse analysis with respect to the coefficients, which are assumed in the calculations. It is observed that the plane strain compression test is the least sensitive to the errors in evaluation of the friction coefficient. On the other hand, when friction coefficient is identified, the ring compression is almost not sensitive to the evaluation of the flow stress of the material. Various optimization techniques were used in the analysis and efficiency of the methods is compared. It is emphasized that drawing the general conclusion regarding the best optimization strategy for the inverse analysis is difficult. Nevertheless, some recommendations are given in the paper. Application of the multicriterion optimization is one of the possible efficient approaches. [Copyright &y& Elsevier]

Abstract: The paper describes the application of the finite element method to predict the thermal, material and mechanical effects of welding; Computational Welding Mechanics (CWM). The most common numerical approach used in CWM is only described shortly. The paper focuses on different modelling aspects. The most important modelling issues are the models for heat input and material behaviour. Finally, some recent applications are reviewed and future developments are discussed. [Copyright &y& Elsevier]

Abstract: In this paper we present a new approach for computing elastic spring-back based on a strictly three-dimensional, high order, solid, finite element formulation for curved, thin-walled structures allowing for an anisotropic discretization. In combination with appropriate mesh design, the p-version yields an exponential rate of convergence in the error in energy norm in contrast to low-order elements, which yield only an algebraic rate of convergence. Anisotropic Ansatz spaces based on high order elements lead to very efficient discretizations. The structural behavior of three-dimensional thin-walled continua can be predicted with a similar number of degrees of freedom as in the two-dimensional case, yet significantly more accurately because of the three-dimensional model. We also introduce an approach for the efficient computation of the relevant geometrically nonlinear problem. Furthermore the paper describes the necessary model conversion from a low-order deep drawing simulation to a spring-back computation based on the p-version of the FEM. [Copyright &y& Elsevier]

Abstract: Computer simulations of metal forming processes using the finite element method (FEM) are, today, well established. This form of simulation uses an increasing number of sophisticated geometrical and material models, relying on a certain number of input data, which are not always readily available. The aim of inverse problems, which will be considered here, is to determine one or more of the input data relating to these forming process simulations, thereby leading to a desired result. In this paper, we will focus on two categories of such inverse problems. The first category consists of parameter identification inverse problems. These involve evaluating the material parameters for material constitutive models that would lead to the most accurate results with respect to physical experiments, i.e. minimizing the difference between experimental results and FEM simulations. The second category consists of shape/process optimization inverse problems. These involve determining the initial geometry of the specimen and/or the shape of the forming tools, as well as some parameters of the process itself, in order to provide the desired final geometry after the forming process. These two categories of inverse problems can be formulated as optimization problems in a similar way, i.e. by using identical optimization algorithms. In this paper, we intend firstly to solve these two types of optimization problems by using different non-linear gradient based optimization methods and secondly to compare their efficiency and robustness in a variety of numerical applications. [Copyright &y& Elsevier]

Abstract: A general finite element-based formulation is presented for the analysis of the mean and mean square response of stochastic structural systems whose material properties are described by random fields. A flexibility-based formulation is followed that does not involve any approximations. Closed-formed integral expressions for the mean and mean square value of the response displacement of statically indeterminate stochastic structures are introduced in this paper. These integral expressions involve the spectral density function of a stochastic field modeling the uncertain material properties and two new quantities named the mean response function (MRF) and the variability response function for the mean square response (VRF1). The MRF and the VRF1 have many similarities with the classical variability response function (VRF) used to estimate the variance of the response displacement. The existence of the MRF and the VRF1 depends on a conjecture that is validated numerically using a brute force Monte Carlo simulation approach. A so-called finite element method-based fast Monte Carlo simulation procedure (FEM-FMCS) is introduced for the accurate and efficient numerical evaluation of the MRF and the VRF1. This methodology is established for the analysis of stochastic beam/frame structures, as well as for more general stochastic finite element systems (i.e., plane stress and shell problems). Numerical examples are provided including a statically indeterminate beam, a plane stress problem and a shell structure. The MRF and VRF1 are used to perform sensitivity/parametric analyses with respect to various probabilistic characteristics involved in the problem (i.e., correlation distance, standard deviation) and to establish realizable upper bounds on the mean value of the response displacement. Throughout this paper, the VRF is computed in parallel to the MRF and the VRF1. It should be noted that although the concept of the VRF has been introduced in an earlier work, this is the first time that a flexibility-based approach has been used to estimate the VRF for two-dimensional problems. [Copyright &y& Elsevier]

Abstract: This paper describes Petrov–Galerkin finite element methods for solving the steady-state Boltzmann transport equation. The methods in their most general form are non-linear and therefore capable of accurately resolving sharp gradients in the solution field. In contrast to previously developed non-linear Petrov–Galerkin (shock-capturing) schemes the methods described in this paper are streamline based. The methods apply a finite element treatment for the internal domain and a Riemann approach on the boundary of the domain. This significantly simplifies the numerical application of the scheme by circumventing the evaluation of complex half-range angular integrals at the boundaries of the domain. The underlying linear Petrov–Galerkin scheme is compared with other Petrov–Galerkin methods found in the radiation transport literature such as those based on the self-adjoint angular flux equation (SAAF) [G.C. Pomraning, Approximate methods of solution of the monoenergetic Boltzmann equation, Ph.D. Thesis, MIT, 1962] and the even-parity (EP) equation [V.S. Vladmirov, Mathematical methods in the one velocity theory of particle transport, Atomic Energy of Canada Limited, Ontario, 1963]. The relationship of the linear method to Riemann methods is also explored. The Petrov–Galerkin methods developed in this paper are applied to a variety of 2-D steady-state and fixed source radiation transport problems. The examples are chosen to cover the range of different radiation regimes from optically thick (diffusive and/or highly absorbing) to transparent and semi-transparent media. These numerical examples show that the non-linear Petrov–Galerkin method is capable of producing accurate, oscillation free solutions across the full spectrum of radiation regimes. [Copyright &y& Elsevier]

Abstract: In this paper, we provide an analysis on the collocation methods (CM), which uses a large scale of admissible functions such as orthogonal polynomials, trigonometric functions, radial basis functions and particular solutions, etc. The admissible functions can be chosen to be piecewise, i.e., different functions are used in different subdomains. The key idea is that the collocation method can be regarded as the least squares method involving integration approximation, and optimal convergence rates can be easily achieved based on the traditional analysis of the finite element method. The key analysis is to prove the uniformly V h -elliptic inequality and some inverse inequalities used. This paper explores the interesting fact that for the collocation methods given in this paper, the integration rules only affect on the uniformly V h -elliptic inequality, but not on the solution accuracy. The advantage of the CM is to formulate easily the associated algebraic equations, which can be solved from the collocation equations directly by the least squares method, thus to greatly reduce the condition number of the associated matrix. Moreover, the new effective condition number is proposed to provide a better upper bound of condition number, and to show a good stability for real problems solved by the collocation methods. Note that the boundary approximation method in Li [Z.C. Li, Combined Methods for Elliptic Equations with Singularities, Interfaces and Infinities, Kluwer Academic Publishers, Boston, London, 1998] is a special case of the CM, where the admissible functions satisfy the equations exactly. Numerical experiments are also carried for Poisson’s problem to support the analysis made. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to present the residual-free bubbles (RFB) method and to show its relationship with the streamline-upwind Petrov/Galerkin method (SUPG). The paper is organized as follows. After the Introduction, we show how the failure of classical finite element methods for convection-dominated convection–diffusion problem can be explained by energy considerations. Then we describe the residual-free bubbles ideas, and in the last section we show the equivalence of RFB and SUPG in a simple case. [Copyright &y& Elsevier]

Abstract: Goal-oriented a posteriori error estimators are presented in this paper for the error obtained while approximately evaluating the J-integral in linear elastic fracture mechanics using the finite element method. The J-integral provides a criterion for crack propagation of a pre-existing crack. Therefore, its reliable and accurate evaluation is of considerable importance in engineering applications to prevent failure of structures. This can be achieved by goal-oriented adaptive finite element methods based on the well-established strategy of solving an auxiliary dual problem in order to control the error of a (non)linear functional, here the J-integral. In this paper, two principally different goal-oriented adaptive strategies are presented. The first strategy rests upon equilibrated residual error estimators based on local Neumann problems, whereas the second one makes use of averaging techniques. The paper is concluded by a numerical example that illustrates our theoretical results. [Copyright &y& Elsevier]

Abstract: Most structures in the real life are subject to multiple load cases. This paper aims at extending the evolutionary structural optimization (ESO) algorithm to optimal contact shape design for elastic bodies under the multiple load cases. To evaluate the reference stresses of each contact node in a finite element framework, an extreme stress criterion (the worst case design) and a weighted average criterion (Pareto design) are presented. In the extreme stress method, the highest nodal contact stress under all load cases is adopted as the reference level. In the weighted average method, the weighted sum of nodal contact stresses over all the load cases is regarded as the reference. It is found that these two criteria can produce different results. In this paper, the examples are presented to demonstrate some new features of contact shape optimization in the presence of the multiple load cases. [Copyright &y& Elsevier]

Abstract: This paper presents an application of multi-scale finite element methods to the solution of the multi-dimensional Fokker–Planck equation. The Fokker–Planck, or forward Kolmogorov, equation is a degenerate convective–diffusion equation arising in Markov-Process theory. It governs the evolution of the transition probability density function of the response of a broad class of dynamical systems driven by Gaussian noise, and completely describes the response process. Analytical solutions for the Fokker–Planck equation have been developed for only a limited number of low-dimensional systems, leading to a large body of approximation theory. One such approach successfully applied to the solution of these problems in the past is the finite element method, though for systems of dimension three or less. In this paper, a multi-scale finite element method is applied to the Fokker–Planck equation in an effort to develop a formulation that can yield higher accuracy on cruder spatial discretizations, thus reducing the computational overhead associated with large scale problems that arise in higher dimensions. [Copyright &y& Elsevier]

Abstract: The objective of this paper is to critically review the emerging non-probabilistic approaches for uncertainty treatment in finite element analysis. The paper discusses general theoretical and practical aspects of both the interval and fuzzy finite element analysis. First, the applicability of the non-probabilistic concepts for numerical uncertainty analysis is discussed from a theoretical viewpoint. The necessary conditions for a useful application of the non-probabilistic concepts are determined, and are proven to be complementary rather than competitive to the classical probabilistic approach. The second part of the paper focuses on numerical aspects of the interval finite element method. It describes two principal strategies for the implementation, i.e., the anti-optimisation and the interval arithmetic approach, and gives a state-of-the-art of the interval finite element algorithms available from literature. It is shown how the application of the interval arithmetic approach to the classical finite element procedure can result in a severe overestimation of the uncertainty on the output, and the main sources of this conservatism are identified. A numerical example in the final part of the paper illustrates the capabilities of the different strategies on an eigenfrequency analysis of a built-up benchmark structure. [Copyright &y& Elsevier]

This paper presents and evaluates several variationally consistent state determination algorithms for geometric and material nonlinear beam finite elements based on the Hellinger–Reissner (HR) assumed stress variational principle. The specifically targeted element type is derived from the two-field (displacement and generalized stress) form of this principle. Momentum balance and strain compatibility are expressed in a weak-form, and the constitutive equations are satisfied directly at each integration point (i.e., section) along the element length. A specific formulation of one element of this type is presented in a companion paper. Interelement compatibility is not enforced in the dual (stress resultant) variables for this element, and thus various state determination algorithms can be formulated at the local (i.e., element and section) levels. Four state determination algorithms, referred to as the L–L, L–N, N–L and N–N procedures, are considered. The first symbol in these names indicates the element level and the second denotes the section level. Furthermore, the symbol N indicates that local nonlinear iteration is performed prior to returning to the higher level, whereas the symbol L indicates that only the linearized equations are satisfied at the corresponding local level. The L–L algorithm corresponds to conventional Newton iteration on both the element and section levels, whereas the N–N algorithm entails the iterative satisfaction of the nonlinear element compatibility and section constitutive equations for each global iteration. The performance of these algorithms is evaluated and compared. It is found that the algorithms in which the local nonlinear strain–displacement compatibility conditions are solved iteratively at the element level for every global iteration lead to fewer global iterations at the expense of iterations at the element level. In particular, the N–N algorithm, is overall the most efficient of the four algorithms for the problems tested. Also, the N–N algorithm has substantially smaller element storage requirements. The N–N algorithm is expected to be particularly advantageous for cases in which a large number of elements remain elastic and inelasticity is concentrated in a few elements. [Copyright &y& Elsevier]

The paper presents combination of discrete element method (DEM) and finite element method (FEM) for dynamic analysis of geomechanics problems. Combined models can employ spherical (or cylindrical in 2D) rigid elements and finite elements in the discretization of different parts of the system. The FEM is a suitable tool to model soil/rock materials while the FEM in many cases can be a better choice to model other parts of the system considered. A typical example can be an idealization of rock cutting with a tool discretized with finite elements and rock or soil samples modelled with discrete elements. The FEM presented in the paper allows large elasto-plastic deformations in the solid regions. Such problems require the use of stabilized FEM to deal with the incompressibility constraint in order to eliminate the volumetric locking defect, especially when using triangular and tetrahedra elements with equal order interpolation for the displacement and the pressure variables. In the paper a stabilization based on the finite calculus (FIC) approach is used. Both theoretical algorithms of DEM and stabilized FEM are implemented in an explicit dynamic code. The paper presents some details of both formulations. A combined numerical algorithm is described finally. Selected numerical results illustrate the possibilities and performance of discrete/finite element analysis in geomechanics problems. [Copyright &y& Elsevier]

In this paper, we investigate a new approach for the two-dimensional fully non-linear depth-averaged shallow water equations by coupling continuous and discontinuous Galerkin methods. We apply a discontinuous Galerkin method to the primitive continuity and momentum equations and couple this to a continuous method discretizing the so-called “wave continuity” and momentum equation. Herein we present the formulation and derive an a priori error estimate, and demonstrate some preliminary numerical results. This work represents the culmination of a series of papers exploring coupled methods for the shallow water equations. [Copyright &y& Elsevier]

The paper introduces a general procedure for computational analysis of a wide class of multiscale problems in mechanics using a finite calculus (FIC) formulation. The FIC approach is based in expressing the governing equations in mechanics accepting that the domain where the standard balance laws are established has a finite size. This introduces naturally additional terms into the classical equations of infinitesimal theory in mechanics which are useful for the numerical solution of problems involving different scales in the physical parameters. The discrete nodal values obtained with the FIC formulation and the finite element method (FEM) can be effectively used as the starting point for obtaining a more refined solution in zones where high gradients of the relevant variables occur using hierarchical or enriched FEM. Typical multiscale problems in mechanics which can be solved with the FIC method include convection–diffusion-reaction problems with high localized gradients, incompressible problems in solid and fluid mechanics, localization problems such as prediction of shear bands in solids and shock waves in compressible fluids, turbulence, etc. The paper presents an introduction of the treatment of multiscale problems using the FIC approach in conjunction with the FEM. Examples of application of the FIC/FEM formulation to the solution of simple multiscale convection–diffusion problems are given. [Copyright &y& Elsevier]

The main goal of the paper is to present theoretical aspects and the finite element method implementation of sensitivity analysis in homogenization of composite materials with linear elastic components using the effective modules approach. The sensitivity analysis of effective material properties is presented in a general form for n-component periodic composite and is illustrated using the examples of periodic 1D as well as 2D heterogeneous structures. The sensitivity coefficients are determined for the effective Young’s modulus and the effective elasticity tensor components. The structural response functional for the fiber-reinforced elastic composite is proposed in the form of total strain energy resulting from some uniform strain state of the composite representative volume element (RVE). The results of sensitivity analysis presented in the paper confirm the usefulness of the homogenization method in computational analysis of composite materials and its application in composite optimization, identification, shape sensitivity studies and, after some probabilistic extensions, in stochastic analysis of random composites. [Copyright &y& Elsevier]