A coupling extended multiscale finite element method (CEMsFEM) is developed for the dynamic analysis of heterogeneous saturated porous media. The coupling numerical base functions are constructed by a unified method with an equivalent stiffness matrix. To improve the computational accuracy, an additional coupling term that could reflect the interaction of the deformations among different directions is introduced into the numerical base functions. In addition, a kind of multi-node coarse element is adopted to describe the complex high-order deformation on the boundary of the coarse element for the two-dimensional dynamic problem. The coarse element tests show that the coupling numerical base functions could not only take account of the interaction of the solid skeleton and the pore fluid, but also consider the effect of the inertial force in the dynamic problems. On the other hand, based on the static balance condition of the coarse element, an improved downscaling technique is proposed to directly obtain the satisfying microscopic solutions in the CEMsFEM. Both one and two-dimensional numerical examples of the heterogeneous saturated porous media are carried out and the results verify the validity and the efficiency of the CEMsFEM by comparing with the conventional finite element method. This article is protected by copyright. All rights reserved.

A novel approach is presented based upon the Linear Matching Method framework in order to directly calculate the ratchet limit of structures subjected to *arbitrary* thermo-mechanical load histories. Traditionally, ratchet analysis methods have been based upon the fundamental premise of decomposing the cyclic load history into cyclic and constant components respectively, in order to assess the magnitude of additional constant loading a structure may accommodate before ratcheting occurs. The method proposed in this paper, for the first time, accurately and efficiently calculates the ratchet limit with respect to a proportional variation between the cyclic primary *and* secondary loads, as opposed to an additional primary load only. The method is a strain based approach and utilises a novel convergence scheme in order to calculate an approximate ratchet boundary based upon a predefined target magnitude of ratchet strain per cycle. The ratcheting failure mechanism evaluated by the method leads to less conservative ratchet boundaries compared to the traditional Bree solution. The method yields the total and plastic strain ranges as well as the ratchet strains for various levels of loading between the ratchet and limit load boundaries. Two example problems have been utilised in order to verify the proposed methodology. This article is protected by copyright. All rights reserved.

The extended finite element method (XFEM) is extended to allow computation of the limit load of cracked structures. In the paper it is demonstrated that the linear elastic tip enrichment basis with and without radial term may be used in the framework of limit analysis, but the six-function enrichment basis based on the well-known Hutchinson-Rice-Rosengren (HRR) asymptotic fields appears to be the best. The discrete kinematic formulation is cast in the form of a second-order cone problem, which can be solved using highly efficient interior-point solvers. Finally, the proposed numerical procedure is applied to various benchmark problems, showing that the present results are in good agreement with those in the literature. This article is protected by copyright. All rights reserved.

The low-rank damping term appears commonly in quadratic eigenvalue problems arising from physical simulations. To exploit the low-rank damping property, we propose a Padé Approximate Linearization (PAL) algorithm. The advantage of the PAL algorithm is that the dimension of the resulting linear eigenvalue problem is only *n* + *ℓ**m*, which is generally substantially smaller than the dimension 2*n* of the linear eigenvalue problem produced by a direct linearization approach, where *n* is the dimension of the quadratic eigenvalue problem, *ℓ* and *m* are the rank of the damping matrix and the order of a Padé approximant, respectively. Numerical examples show that by exploiting the low-rank damping property, the PAL algorithm runs 33 – 47% faster than the direct linearization approach for solving modest size quadratic eigenvalue problems. This article is protected by copyright. All rights reserved.

The computational complexity behind the bi-level optimization problem has led the researchers to adopt Karush – Kuhn – Tucker (KKT) optimality conditions. However, the problem function has more number of complex constraints to be satisfied. Classical optimization algorithms are impotent to handle the function. This paper presents a simplified minimization function, in which both the profit maximization problem and the ISO market clearance problem are considered, but with no KKT optimality conditions. Subsequently, this paper solves the minimization function using a hybrid optimization algorithm. The hybrid optimization algorithm is developed by combining the operations of Group Search Optimizer (GSO) and Genetic Algorithm (GA). The hybridization enables the dispersion process of GSO to be a new mutated dispersion process for improving the convergence rate. We evaluate the methodology by experimenting on IEEE 14 and IEEE 30 bus systems. The obtained results are compared with the outcomes of bidding strategies that are based on GSO, PSO and GA. The results demonstrate that the hybrid optimization algorithm solves the minimization function better than PSO, GA and GSO. Hence, the profit maximization in the proposed methodology is relatively better than that of the conventional algorithms. This article is protected by copyright. All rights reserved.

Since the inception of Discrete Element Method (DEM) over 30 years ago, significant algorithmic developments have been made to enhance the performance of DEM while emphasizing simulation fidelity. Nevertheless, DEM is still a computationally expensive numerical method for simulation of granular materials. In this study, a new impulse-based DEM (iDEM) approach is introduced that uses collision impulse instead of contact force, and directly handles velocity while bypassing integration of acceleration. Contact force required for engineering applications is retrieved with reasonable fidelity via an original proposed formulation. The method is robust, numerically stable and results in significant speed-up of almost two orders of magnitude over conventional DEM. The proposed iDEM allows for the simulation of large number of particles within reasonable run times on readily accessible computer hardware. This article is protected by copyright. All rights reserved.

In this paper, we propose a new component mode synthesis method by enhancing the Craig–Bampton (CB) method. To develop the enhanced CB method, the transformation matrix of the CB method is enhanced considering the effect of residual substructural modes and the unknown eigenvalue in the enhanced transformation matrix is approximated by using O'Callahan's approach in Guyan reduction. Using the newly defined transformation matrix, original finite element models can be more accurately approximated by reduced models. For this reason, the accuracy of the reduced models is significantly improved with a low additional computational cost. We here present the formulation details of the enhanced CB method and demonstrate its performance through several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.

This paper discusses a method that provides the direct identification of constitutive model parameters by intimately integrating the finite element method (FEM) with digital image correlation (DIC), namely, directly connecting the experimentally obtained images for all time increments to the unknown material parameters. The problem is formulated as a single minimization problem that incorporates all the experimental data. It allows for precise specification of the unknowns, which can be, but are not limited to, the unknown material properties. The tight integration between FEM and DIC enables for identification while providing necessary regularization of the DIC procedure, making the method robust and noise insensitive. Through this approach, the versatility of the FE method is extended to the experimental realm, enhancing the analyses of existing experiments and opening new experimental opportunities. Copyright © 2015 John Wiley & Sons, Ltd.

This work deals with the question of the resolution of nonlinear problems for many different configurations in order to build a ‘virtual chart’ of solutions. The targeted problems are three-dimensional structures driven by Chaboche-type elastic-viscoplastic constitutive laws. In this context, parametric analysis can lead to highly expensive computations when using a direct treatment. As an alternative, we present a technique based on the use of the time-space proper generalized decomposition in the framework of the LATIN method. To speed up the calculations in the parametrized context, we use the fact that at each iteration of the LATIN method, an approximation over the entire time-space domain is available. Then, a global reduced-order basis is generated, reused and eventually enriched, by treating, one-by-one, all the various parameter sets. The novelty of the current paper is to develop a strategy that uses the reduced-order basis for any new set of parameters as an initialization for the iterative procedure. The reduced-order basis, which has been built for a set of parameters, is reused to build a first approximation of the solution for another set of parameters. An error indicator allows adding new functions to the basis only if necessary. The gain of this strategy for studying the influence of material or loading variability reaches the order of 25 in the industrial examples that are presented. Copyright © 2015 John Wiley & Sons, Ltd.

This report presents a numerical study of reduced-order representations for simulating incompressible Navier–Stokes flows over a range of physical parameters. The reduced-order representations combine ideas of approximation for nonlinear terms, of local bases, and of least-squares residual minimization. To construct the local bases, temporal snapshots for different physical configurations are collected automatically until an error indicator is reduced below a user-specified tolerance. An adaptive time-integration scheme is also employed to accelerate the generation of snapshots as well as the simulations with the reduced-order representations. The accuracy and efficiency of the different representations is compared with examples with parameter sweeps. Copyright © 2015 John Wiley & Sons, Ltd.

This paper studies the static fracture problems of an interface crack in linear piezoelectric bimaterial by means of the extended finite element method (X-FEM) with new crack-tip enrichment functions. In the X-FEM, crack modeling is facilitated by adding a discontinuous function and crack-tip asymptotic functions to the classical finite element approximation within the framework of the partition of unity. In this work, the coupled effects of an elastic field and an electric field in piezoelectricity are considered. Corresponding to the two classes of singularities of the aforementioned interface crack problem, namely, *ϵ* class and *κ* class, two classes of crack-tip enrichment functions are newly derived, and the former that exhibits oscillating feature at the crack tip is numerically investigated. Computation of the fracture parameter, i.e., the *J*-integral, using the domain form of the contour integral, is presented. Excellent accuracy of the proposed formulation is demonstrated on benchmark interface crack problems through comparisons with analytical solutions and numerical results obtained by the classical FEM. Moreover, it is shown that the geometrical enrichment combining the mesh with local refinement is substantially better in terms of accuracy and efficiency. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, a non-intrusive stochastic model reduction scheme is developed for polynomial chaos representation using proper orthogonal decomposition. The main idea is to extract the optimal orthogonal basis via inexpensive calculations on a coarse mesh and then use them for the fine-scale analysis. To validate the developed reduced-order model, the method is implemented to: (1) the stochastic steady-state heat diffusion in a square slab; (2) the incompressible, two-dimensional laminar boundary-layer over a flat plate with uncertainties in free-stream velocity and physical properties; and (3) the highly nonlinear Ackley function with uncertain coefficients. For the heat diffusion problem, the thermal conductivity of the slab is assumed to be a stochastic field with known exponential covariance function and approximated via the Karhunen–Loève expansion. In all three test cases, the input random parameters are assumed to be uniformly distributed, and a polynomial chaos expansion is found using the regression method. The Sobol's quasi-random sequence is used to generate the sample points. The numerical results of the three test cases show that the non-intrusive model reduction scheme is able to produce satisfactory results for the statistical quantities of interest. It is found that the developed non-intrusive model reduction scheme is computationally more efficient than the classical polynomial chaos expansion for uncertainty quantification of stochastic problems. The performance of the developed scheme becomes more apparent for the problems with larger stochastic dimensions and those requiring higher polynomial order for the stochastic discretization. Copyright © 2015 John Wiley & Sons, Ltd.

Among numerous finite element techniques, few models can perfectly (without any numerical problems) break through MacNeal's theorem: any 4-node, 8-DOF membrane element will either lock in in-plane bending or fail to pass a *C*_{0} patch test when the element's shape is an isosceles trapezoid. In this paper, a 4-node plane quadrilateral membrane element is developed following the unsymmetric formulation concept, which means two different sets of interpolation functions for displacement fields are simultaneously used. The first set employs the shape functions of the traditional 4-node bilinear isoparametric element, while the second set adopts a novel composite coordinate interpolation scheme with analytical trail function method, in which the Cartesian coordinates (*x*,*y*) and the second form of quadrilateral area coordinates (QACM-II) (*S*,*T*) are applied together. The resulting element US-ATFQ4 exhibits amazing performance in rigorous numerical tests. It is insensitive to various serious mesh distortions, free of trapezoidal locking, and can satisfy both the classical first-order patch test and the second-order patch test for pure bending. Furthermore, because of usage of the second form of quadrilateral area coordinates (QACM-II), the new element provides the invariance for the coordinate rotation. It seems that the behaviors of the present model are beyond the well-known contradiction defined by MacNeal's theorem. Copyright © 2015 John Wiley & Sons, Ltd.

An algorithm is derived for the computation of eigenpair derivatives of asymmetric quadratic eigenvalue problem with distinct and repeated eigenvalues. In the proposed method, the eigenvector derivatives of the damped systems are divided into a particular solution and a homogeneous solution. By introducing an additional normalization condition, we construct two extended systems of linear equations with nonsingular coefficient matrices to calculate the particular solution. The method is numerically stable, and the homogeneous solutions are computed by the second-order derivatives of the eigenequations. Two numerical examples are used to illustrate the validity of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.

The material-point method models continua by following a set of unconnected material points throughout the deformation of a body. This set of points provides a Lagrangian description of the material and geometry. Information from the material points is projected onto a background grid where equations of motion are solved. The grid solution is then used to update the material points. This paper describes how to use this method to solve quasi-static problems. The resulting discrete equations are a coupled set of nonlinear equations that are then solved with a Jacobian-free, Newton–Krylov algorithm. The technique is illustrated by examining two problems. The first problem simulates a compact tension test and includes a model of material failure. The second problem computes effective, macroscopic properties of a polycrystalline thin film. Copyright © 2015 John Wiley & Sons, Ltd.

In discrete element method simulations, multi-sphere particle is extensively employed for modeling the geometry shape of non-spherical particle. A contact detection algorithm for multi-sphere particles has been developed through two-level-grid-searching. In the first-level-grid-searching, each multi-sphere particle is represented by a bounding sphere, and global space is partitioned into identical square or cubic cells of size *D*, the diameter of the greatest bounding sphere. The bounding spheres are mapped into the cells in global space. The candidate particles can be picked out by searching the bounding spheres in the neighbor cells of the bounding sphere for the target particle. In the second-level-grid-searching, a square or cubic local space of size (*D* + *d*) is partitioned into identical cells of size *d*, the diameter of the greatest element sphere. If two bounding spheres of two multi-sphere particles are overlapped, the contacts occurring between the element spheres in the target multi-sphere particle and in the candidate multi-sphere particle are checked. Theoretical analysis and numerical tests on the memory requirement and contact detection time of this algorithm have been performed to verify the efficiency of this algorithm. The results showed that this algorithm can effectively deal with the contact problem for multi-sphere particles. Copyright © 2015 John Wiley & Sons, Ltd.

This paper describes an approach to numerically approximate the time evolution of multibody systems with flexible (compliant) components. Its salient attribute is that at each time step, both the formulation of the system equations of motion and their numerical solution are carried out using parallel computing on graphics processing unit cards. The equations of motion are obtained using the absolute nodal coordinate formulation, yet any other multibody dynamics formalism would fit equally well the overall solution strategy outlined herein. The implicit numerical integration method adopted relies on a Newton–Krylov methodology and a parallel direct sparse solver to precondition the underlying linear system. The proposed approach, implemented in a software infrastructure available under an open-source BSD-3 license, leads to improvements in overall simulation times of up to one order of magnitude when compared with matrix-free parallel solution approaches that do not use preconditioning. Copyright © 2015 John Wiley & Sons, Ltd.

A robust computational framework for the solution of fluid–structure interaction problems characterized by compressible flows and highly nonlinear structures undergoing pressure-induced dynamic fracture is presented. This framework is based on the finite volume method with exact Riemann solvers for the solution of multi-material problems. It couples a Eulerian, finite volume-based computational approach for solving flow problems with a Lagrangian, finite element-based computational approach for solving structural dynamics and solid mechanics problems. Most importantly, it enforces the governing fluid–structure transmission conditions by solving local, one-dimensional, fluid–structure Riemann problems at evolving structural interfaces, which are embedded in the fluid mesh. A generic, comprehensive, and yet effective approach for representing a fractured fluid–structure interface is also presented. This approach, which is applicable to several finite element-based fracture methods including inter-element fracture and remeshing techniques, is applied here to incorporate in the proposed framework two different and popular approaches for computational fracture in a seamless manner: the extended FEM and the element deletion method. Finally, the proposed embedded boundary computational framework for the solution of highly nonlinear fluid–structure interaction problems with dynamic fracture is demonstrated for one academic and three realistic applications characterized by detonations, shocks, large pressure, and density jumps across material interfaces, dynamic fracture, flow seepage through narrow cracks, and structural fragmentation. Correlations with experimental results, when available, are also reported and discussed. For all four considered applications, the relative merits of the extended FEM and element deletion method for computational fracture are also contrasted and discussed. Copyright © 2015 John Wiley & Sons, Ltd.

Predicting the frequency response of a complex vibro-acoustic system becomes extremely difficult in the mid-frequency regime. In this work, a novel hybrid face-based smoothed finite element method/statistical energy analysis (FS-FEM/SEA) method is proposed, aiming to further improve the accuracy of ‘mid-frequency’ predictions. According to this approach, the whole vibro-acoustic system is divided into a combination of a plate subsystem with statistical behaviour and an acoustic cavity subsystem with deterministic behaviour. The plate subsystem is treated using the recently developed FS-FEM, and the cavity subsystem is dealt with using the SEA. These two different types of subsystems can be coupled and interacted through the so-called diffuse field reciprocity relation. The ensemble average response of the system is calculated, and the uncertainty is confined and treated in the SEA subsystems. The use of FS-FEM ‘softens’ the well-known ‘overly stiff’ behaviour in the standard FEM and reduces the inherent numerical dispersion error. The proposed FS-FEM/SEA approach is verified and its features are examined by various numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.

We propose a family of methods for simulating two-dimensional incompressible, low Reynolds number flow around a moving obstacle whose motion is prescribed. The methods make use of a universal mesh: a fixed background mesh that adapts to the geometry of the immersed obstacle at all times by adjusting a few elements in the neighborhood of the obstacle's boundary. The resulting mesh provides a conforming triangulation of the fluid domain over which discretizations of any desired order of accuracy in space and time can be constructed using standard finite element spaces together with off-the-shelf time integrators. We demonstrate the approach by using Taylor-Hood elements to approximate the fluid velocity and pressure. To integrate in time, we consider implicit Runge-Kutta schemes as well as a fractional step scheme. We illustrate the methods and study their convergence numerically via examples that involve flow around obstacles that undergo prescribed deformations. Copyright © 2015 John Wiley & Sons, Ltd.

Generating matching meshes for problems with complex boundaries is often an intricate process, and the use of non-matching meshes appears as an appealing solution. Yet, enforcing boundary conditions on non-matching meshes is not a straightforward process, especially when prescribing those of Dirichlet type. By combining a type of Generalized Finite Element Method (GFEM) with the Lagrange multiplier method, a new technique for the treatment of essential boundary conditions on non-matching meshes is introduced in this manuscript. The new formulation yields a symmetric stiffness matrix and is straightforward to implement. As a result, the methodology makes possible the analysis of problems with the use of simple structured meshes, irrespective of the problem domain boundary. Through the solution of linear elastic problems, we show that the optimal rate of convergence is preserved for piecewise linear finite elements. Yet, the formulation is general and thus it can be extended to other elliptic boundary value problems. Copyright © 2015 John Wiley & Sons, Ltd.

The computational burden associated to finite element based digital image correlation methods is mostly due to the inversion of finite element systems and to image interpolations. A non-overlapping dual domain decomposition method is here proposed to rationalise the computational cost of high resolution finite element digital image correlation measurements when dealing with large images. It consists in splitting the global mesh into submeshes and the reference and deformed states images into subset images. Classic finite element digital image correlation formulations are first written in each subdomain independently. The displacement continuity at the interfaces is enforced by introducing a set of Lagrange multipliers. The problem is then condensed on the interface and solved by a conjugate gradient algorithm. Three different preconditioners are proposed to accelerate its convergence. The proposed domain decomposition method is here exemplified with real high resolution images. It is shown to combine the metrological performances of finite element based digital image correlation and the parallelisation ability of subset based methods. Copyright © 2015 John Wiley & Sons, Ltd.

A hybrid finite element formulation is used to model the hygro-thermo-chemical process of cement hydration in high performance concrete. The temperature and the relative humidity fields are directly approximated in the domain of the element using naturally hierarchical bases independent of the mapping used to define its geometry. This added flexibility in modeling implies the independent approximation of the heat and moisture flux fields on the boundary of the element, the typical feature of hybrid finite element formulations. The formulation can be implemented using coarse and, eventually, unstructured meshes, which may contain elements with high aspect ratios, an option that can be advantageously used in the simulation of the casting of concrete structural elements. The resulting solving system is sparse and well suited to adaptive refinement and parallelization. It is solved coupling a trapezoidal time integration rule with an adaptation of the Newton–Raphson method designed to preserve symmetry. The relative performance of the formulation is assessed using a set of testing problems supported by experimental data and results obtained with conventional (conform) finite elements. Copyright © 2015 John Wiley & Sons, Ltd.

Based on the one-dimensional differential matrix derived from the Lagrange series expansion, the finite block method was recently developed to solve both the elasticity and transient heat conduction problems of anisotropic and functionally graded materials. In this paper, the formulation of the Lagrange finite block method with boundary type in the strong form is presented and applied to non-conforming contact problems for the functionally graded materials subjected to either static or dynamic loads. The first order partial differential matrices are only needed both in the governing equations and in the Neumann boundary condition. By introducing the mapping technique, a block of quadratic type is transformed from the Cartesian coordinate of global system to the normalized coordinate with eight seeds. Time dependent partial differential equations are analyzed in the Laplace transformed domain and the Durbin's inversion method is applied to determine all the physical values in the time domain. Conforming and non-conforming contacts are investigated by using the iterative algorithm with full load technique. Illustrative numerical examples are given and comparisons have been made with analytical solutions. Copyright © 2015 John Wiley & Sons, Ltd.

We develop an acceleration method for material-dominated calculations based on phase-space simplicial interpolation of the relevant material-response functions. This process of interpolation constitutes an approximation scheme by which an exact material-response function is replaced by a sequence of approximating response functions. The terms in the sequence are increasingly accurate, thus ensuring the convergence of the overall solution. The acceleration ratio depends on the dimensionality, the complexity of the deformation, the time-step size, and the fineness of the phase-space interpolation. We ascertain these trade-offs analytically and by recourse to selected numerical tests. The numerical examples with piecewise-quadratic interpolation in phase space confirm the analytical estimates. Copyright © 2015 John Wiley & Sons, Ltd.

The properties and numerical performance of reduced-order models based on trajectory piecewise linearization (TPWL) and proper orthogonal decomposition (POD) are assessed. The target application is subsurface flow modeling, although our findings should be applicable to a range of problems. The errors arising at each step in the POD–TPWL procedure are described. The impact of constraint reduction on accuracy and stability is considered in detail. Constraint reduction entails projection of the overdetermined system into a low-dimensional subspace, in which the system is solvable. Optimality conditions for constraint reduction, in terms of error minimization, are derived. Galerkin and Petrov–Galerkin projections are shown to correspond to optimality in norms that involve weighting with the Jacobian matrix. Two new treatments, inverse projection and weighted inverse projection, are suggested. These methods minimize error in appropriate norms, although they require substantial preprocessing computations. Numerical results are presented for oil reservoir simulation problems. Galerkin projection provides reasonable accuracy for simpler oil–water systems, although it becomes unstable in more challenging cases. Petrov–Galerkin projection is observed to behave stably in all cases considered. Weighted inverse projection also behaves stably, and it provides the highest accuracy. Runtime speedups of 150–400 are achieved using these POD–TPWL models. Copyright © 2015 John Wiley & Sons, Ltd.

This paper addresses the problem of finding a stationary point of a nonlinear dynamical system whose state variables are under inequality constraints. Systems of this type often arise from the discretization of PDEs that model physical phenomena (e.g., fluid dynamics) in which the state variables are under realizability constraints (e.g., positive pressure and density). We start from the popular pseudo-transient continuation method and augment it with nonlinear inequality constraints. The constraint handling technique does not help in situations where no steady-state solution exists, for example, because of an under-resolved discretization of PDEs. However, an often overlooked situation is one in which the steady-state solution exists but cannot be reached by the solver, which typically fails because of the violation of constraints, that is, a non-physical state error during state iterations. This is the shortcoming that we address by incorporating physical realizability constraints into the solution path from the initial condition to steady state. Although we focus on the DG method applied to fluid dynamics, our technique relies only on implicit time marching and hence can be extended to other spatial discretizations and other physics problems. We analyze the sensitivity of the method to a range of input parameters and present results for compressible turbulent flows that show that the constrained method is significantly more robust than a standard unconstrained method while on par in terms of cost. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, a novel finite element approach is presented to solve three-dimensional problems using trimmed hexahedral elements generated by cutting a simple block consisting of regular hexahedral elements with a computer-aided design (CAD) surface. Trimmed hexahedral elements, which are polyhedral elements with curved faces, are placed at the boundaries of finite element models, and regular hexahedral elements remain in the interior regions. Shape functions for trimmed hexahedral elements are developed by using moving least square approximation with harmonic weight functions based on an extension of Wachspress coordinates to curved faces. A subdivision of polyhedral domains into tetrahedral sub-domains is performed to construct shape functions for trimmed hexahedral elements, and numerical integration of the weak form can be carried out consistently over the tetrahedral sub-domains. Trimmed hexahedral elements have similar properties to conventional finite elements regarding the continuity, the completeness, the node–element connectivity, and the inter-element compatibility. Numerical examples for three-dimensional linear elastic problems with complex geometries show the efficiency and effectiveness of the present method. Copyright © 2015 John Wiley & Sons, Ltd.

This paper presents a novel formulation of a hereditary cohesive zone model able to effectively capture rate-dependent crack propagation along a defined interface, over a wide range of applied loading rates and with a single set of seven input parameters only, as testified by the remarkable agreement with experimental results in the case of a double cantilever beam made of steel adherends bonded along a rubber interface. The formulation relies on the assumption that the measured fracture energy is the sum of a rate-independent ‘rupture’ energy, related to the rupture of primary bonds at the atomic or molecular level, and of additional dissipation caused by other rate-dependent dissipative mechanisms present in the material and occurring simultaneously to rupture. The first contribution is accounted for by introducing a damage-type internal variable, whose evolution follows a rate-independent law for consistency with the assumption of rate independence of the rupture energy. To account for the additional dissipation, a fractional-calculus-based linear viscoelastic model is used, because for many polymers, it is known to capture the material response within an extremely wide range of strain rates much more effectively than classic models based on an exponential kernel. To the authors' knowledge, this is the first application of fractional viscoelasticity to the simulation of fracture. © 2015 The Authors. *International Journal for Numerical Methods in Engineering* published by John Wiley & Sons Ltd.

Nonlinear fracture analysis of rubber-like materials is computationally challenging due to a number of complicated numerical problems. The aim of this paper is to study finite strain fracture problems based on appropriate enrichment functions within the extended finite element method. Two-dimensional static and quasi-static crack propagation problems are solved to demonstrate the efficiency of the proposed method. Complex mixed-mode problems under extreme large deformation regimes are solved to evaluate the performance of the proposed extended finite element analysis based on different tip enrichment functions. Finally, it is demonstrated that the logarithmic set of enrichment functions provides the most accurate and efficient solution for finite strain fracture analysis. Copyright © 2015 John Wiley & Sons, Ltd.

The computational efficiency of a typical, projection-based, nonlinear model reduction method hinges on the efficient approximation, for explicit computations, of the scalar projections onto a subspace of a residual vector. For implicit computations, it also hinges on the additional efficient approximation of similar projections of the Jacobian of this residual with respect to the solution. The computation of both approximations is often referred to in the literature as hyper reduction. To this effect, this paper focuses on the analysis and comparative performance study for nonlinear finite element reduced-order models of solids and structures of the recently developed energy-conserving mesh sampling and weighting (ECSW) hyper reduction method. Unlike most alternative approaches, this method approximates the scalar projections of residuals and/or Jacobians directly, instead of approximating first these vectors and matrices then projecting the resulting approximations onto the subspaces of interest. In this paper, it is shown that ECSW distinguishes itself furthermore from other hyper reduction methods through its preservation of the Lagrangian structure associated with Hamilton's principle. For second-order dynamical systems, this enables it to also preserve the numerical stability properties of the discrete system to which it is applied. It is also shown that for a fixed set of parameter values, the approximation error committed *online* by ECSW is bounded by its counterpart error committed *off-line* during the training of this method. Therefore, this error can be estimated in this case *a priori* and controlled. The performance of ECSW is demonstrated first for two academic but nevertheless interesting nonlinear dynamic response problems. For both of them, ECSW is shown to preserve numerical stability and deliver the desired level of accuracy, whereas the discrete empirical interpolation method and its recently introduced unassembled variant are shown to be susceptible to failure because of numerical instability. The potential of ECSW for enabling the effective reduction of nonlinear finite element dynamic models of solids and structures is also highlighted with the realistic simulation of the nonlinear transient dynamic response of a complete car engine to thermal and combustion pressure loads using an implicit scheme. For this simulation, ECSW is reported to enable the reduction of the CPU time required by the high-dimensional nonlinear finite element dynamic analysis by more than four orders of magnitude, while achieving a very good level of accuracy. Copyright © 2015 John Wiley & Sons, Ltd.

The post-treatment of (3D) displacement fields for the identification of spatially varying elastic material parameters is a large inverse problem that remains out of reach for massive 3D structures. We explore here the potential of the constitutive compatibility method for tackling such an inverse problem, provided an appropriate domain decomposition technique is introduced. In the method described here, the statically admissible stress field that can be related through the known constitutive symmetry to the kinematic observations is sought through minimization of an objective function, which measures the violation of constitutive compatibility. After this stress reconstruction, the local material parameters are identified with the given kinematic observations using the constitutive equation. Here, we first adapt this method to solve 3D identification problems and then implement it within a domain decomposition framework which allows for reduced computational load when handling larger problems. Copyright © 2015 John Wiley & Sons, Ltd.

We present a robust method for generating high-order nodal tetrahedral curved meshes. The approach consists of modifying an initial linear mesh by first, introducing high-order nodes, second, displacing the boundary nodes to ensure that they are on the computer-aided design surface, and third, smoothing and untangling the mesh obtained after the displacement of the boundary nodes to produce a valid curved high-order mesh. The smoothing algorithm is based on the optimization of a regularized measure of the mesh distortion relative to the original linear mesh. This means that whenever possible, the resulting mesh preserves the geometrical features of the initial linear mesh such as shape, stretching, and size. We present several examples to illustrate the performance of the proposed algorithm. Furthermore, the examples show that the implementation of the optimization problem is robust and capable of handling situations in which the mesh before optimization contains a large number of invalid elements. We consider cases with polynomial approximations up to degree ten, large deformations of the curved boundaries, concave boundaries, and highly stretched boundary layer elements. The meshes obtained are suitable for high-order finite element analyses. Copyright © 2015 John Wiley & Sons, Ltd.

This paper provides a comparison between one particular *phase-field* damage model and a *thick level set* (TLS) damage model for the simulation of brittle and quasi-brittle fractures. The TLS model is recasted in a variational framework, which allows comparison with the phase-field model. Using this framework, both the equilibrium equations and the damage evolution laws are guided by the initial choice of the potential energy. The potentials of the phase-field model and of the TLS model are quite different. TLS potential enforces *a priori* a bound on damage gradient whereas the phase-field potential does not. The TLS damage model is defined such that the damage profile fits to the one of the phase-field model for a beam of infinite length. The model parameters are calibrated to obtain the same surface fracture energy. Numerical results are provided for unidimensional and bidimensional tests for both models. Qualitatively, similar results are observed, although TLS model is observed to be less sensible to boundary conditions. Copyright © 2015 John Wiley & Sons, Ltd.

Numerical techniques are suggested in this paper, in order to improve the computational efficiency of the spectral boundary integral method, initiated by Clamond & Grue [D. Clamond and J. Grue. A fast method for fully nonlinear water-wave computations. *J*. *Fluid Mech*. 2001; **447**: 337–355] for simulating nonlinear water waves. This method involves dealing with the high order convolutions by using Fourier transform or inverse Fourier transform and evaluating the integrals with weakly singular integrands. A de-singularity technique is proposed here to help in efficiently evaluating the integrals with weak singularity. An anti-aliasing technique is developed in this paper to overcome the aliasing problem associated with Fourier transform or inverse Fourier transform with a limited resolution. This paper also presents a technique for determining a critical value of the free surface, under which the integrals can be neglected. Numerical tests are carried out on the numerical techniques and on the improved method equipped with the techniques. The tests will demonstrate that the improved method can significantly accelerate the computation, in particular when waves are strongly nonlinear. Copyright © 2015 John Wiley & Sons, Ltd.

A new numerical approach for solving incompressible two-phase flows is presented in the framework of the recently developed Consistent Particle Method (CPM). In the context of the Lagrangian particle formulation, the CPM computes spatial derivatives based on the generalized finite difference scheme and produces good results for single-phase flow problems. Nevertheless, for two-phase flows, the method cannot be directly applied near the fluid interface because of the abrupt discontinuity of fluid density resulting in large change in pressure gradient. This problem is resolved by dealing with the pressure gradient normalized by density, leading to a two-phase CPM of which the original singlephase CPM is a special case. In addition, a new adaptive particle selection scheme is proposed to overcome the problem of ill-conditioned coefficient matrix of pressure Poisson equation when particles are sparse and non-uniformly spaced. Numerical examples of Rayleigh–Taylor instability, gravity current flow, water-air sloshing and dam break are presented to demonstrate the accuracy of the proposed method in wave profile and pressure solution. Copyright © 2015 John Wiley & Sons, Ltd.

Previous studies have shown that the commonly used quadrature schemes for polygonal and polyhedral finite elements lead to consistency errors that persist under mesh refinement and subsequently render the approximations non-convergent. In this work, we consider minimal perturbations to the gradient field at the element level in order to restore polynomial consistency and recover optimal convergence rates when the weak form integrals are evaluated using quadrature. For finite elements of arbitrary order, we state the accuracy requirements on the underlying volumetric and boundary quadrature rules and discuss the properties of the resulting corrected gradient operator. We compare the proposed approach with the pseudo-derivative method developed by Belytschko and co-workers and, for linear elliptic problems, with our previous remedy that involves splitting of polynomial and non-polynomial of elemental energy bilinear form. We present several numerical results for linear and nonlinear elliptic problems in two and three dimensions that not only confirm the recovery of optimal convergence rates but also suggest that the global error levels are close to those of approximations obtained from exact evaluation of the weak form integrals. Copyright © 2015 John Wiley & Sons, Ltd.

The aim of this paper is to propose a continuous–discontinuous computational homogenization–localization framework to upscale microscale localization toward the onset and propagation of a cohesive discontinuity at the macroscale. The major novelty of this contribution is the development of a fully coupled micro–macro solution strategy, where the solution procedure for the macroscopic domain is based on the extended finite element method. The proposed approach departs from classical computational homogenization, which allows to derive the effective stress–strain response before the onset of localization. Upon strain localization, the microscale is characterized by a strain localization band where damage grows and by two adjacent unloading bulk regions at each side of the localization zone. The microscale localization band is lumped into a macroscopic cohesive crack, accommodated through discontinuity enriched macroscale kinematics. The governing response of the continuum with a discontinuity is obtained numerically based on proper scale transition relations in terms of the traction–separation law and the stress–strain description of the continuous surrounding material at both sides of the discontinuity. The potential of the method is demonstrated with a numerical example, which illustrates the onset and propagation of a macroscale cohesive crack emerging from microstructural damage within the underlying microstructure. Copyright © 2015 John Wiley & Sons, Ltd.

A numerical model to deal with nonlinear elastodynamics involving large rotations within the framework of the finite element based on NURBS (Non-Uniform Rational B-Spline) basis is presented. A comprehensive kinematical description using a corotational approach and an orthogonal tensor given by the exact polar decomposition is adopted. The state equation is written in terms of corotational variables according to the hypoelastic theory, relating the Jaumann derivative of the Cauchy stress to the Eulerian strain rate.

The generalized-*α* method (G*α*) method and Generalized Energy-Momentum Method with an additional parameter (GEMM+*ξ*) are employed in order to obtain a stable and controllable dissipative time-stepping scheme with algorithmic conservative properties for nonlinear dynamic analyses.

The main contribution is to show that the energy–momentum conservation properties and numerical stability may be improved once a NURBS-based FEM in the spatial discretization is used. Also it is shown that high continuity can postpone the numerical instability when GEMM+*ξ* with consistent mass is employed; likewise, increasing the continuity class yields a decrease in the numerical dissipation. A parametric study is carried out in order to show the stability and energy budget in terms of several properties such as continuity class, spectral radius and lumped as well as consistent mass matrices. Copyright © 2015 John Wiley & Sons, Ltd.

The extended finite element method (X-FEM) has proven to be an accurate, robust method for solving embedded interface problems. With a few exceptions, the X-FEM has mostly been used in conjunction with piecewise-linear shape functions and an associated piecewise-linear geometrical representation of interfaces. In the current work, the use of spline-based finite elements is examined along with a Nitsche technique for enforcing constraints on an embedded interface. To obtain optimal rates of convergence, we employ a hierarchical local refinement approach to improve the geometrical representation of curved interfaces. We further propose a novel weighting for the interfacial consistency terms arising in the Nitsche variational form with B-splines. A qualitative dependence between the weights and the stabilization parameters is established with additional element level eigenvalue calculations. An important consequence of this weighting is that the bulk as well as the interfacial fields remain well behaved in the presence of large heterogeneities as well as elements with arbitrarily small volume fractions. We demonstrate the accuracy and robustness of the proposed method through several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.

A stabilized discontinuous Galerkin method is developed for general hyperelastic materials at finite strains. Starting from a mixed method incorporating Lagrange multipliers along the interface, the displacement formulation is systematically derived through a variational multiscale approach whereby the numerical fine scales are modeled via edge bubble functions. Analytical expressions that are free from user-defined parameters arise for the weighted numerical flux and stability tensor. In particular, the specific form taken by these derived quantities naturally accounts for evolving geometric nonlinearity as well as discontinuous material properties. The method is applicable both to problems containing nonconforming meshes or different element types at specific interfaces and to problems consisting of fully discontinuous numerical approximations. Representative numerical tests involving large strains and rotations are performed to confirm the robustness of the method. Copyright © 2015 John Wiley & Sons, Ltd.

In this work, the extended finite element method (XFEM) is for the first time coupled with face-based strain-smoothing technique to solve three-dimensional fracture problems. This proposed method, which is called face-based smoothed XFEM here, is expected to combine both the advantages of XFEM and strain-smoothing technique. In XFEM, arbitrary crack geometry can be modeled and crack advance can be simulated without remeshing. Strain-smoothing technique can eliminate the integration of singular term over the volume around the crack front, thanks to the transformation of volume integration into area integration. Special smoothing scheme is implemented in the crack front smoothing domain. Three examples are presented to test the accuracy, efficiency, and convergence rate of the face-based smoothed XFEM. From the results, it is clear that smoothing technique can improve the performance of XFEM for three-dimensional fracture problems. Copyright © 2015 John Wiley & Sons, Ltd.

Non-probabilistic *convex models* need to be provided only the changing boundary of parameters rather than their exact probability distributions; thus, such models can be applied to uncertainty analysis of complex structures when experimental information is lacking. The *interval* and the *ellipsoidal models* are the two most commonly used modeling methods in the field of non-probabilistic convex modeling. However, the former can only deal with independent variables, while the latter can only deal with dependent variables. This paper presents a more general non-probabilistic convex model, the *multidimensional parallelepiped model*. This model can include the independent and dependent uncertain variables in a unified framework and can effectively deal with complex ‘multi-source uncertainty’ problems in which dependent variables and independent variables coexist. For any two parameters, the concepts of the correlation angle and the correlation coefficient are defined. Through the marginal intervals of all the parameters and also their correlation coefficients, a multidimensional parallelepiped can easily be built as the uncertainty domain for parameters. Through the introduction of affine coordinates, the parallelepiped model in the original parameter space is converted to an interval model in the affine space, thus greatly facilitating subsequent structural uncertainty analysis. The parallelepiped model is applied to structural uncertainty propagation analysis, and the response interval of the structure is obtained in the case of uncertain initial parameters. Finally, the method described in this paper was applied to several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.

The vector potential formulation is a promising solution method for nonlinear electromechanically coupled boundary value problems. However, one of the drawbacks of this formulation is the non-uniqueness of the (electric) vector potential in three dimensions. The present paper focuses on the Coulomb gauging method to overcome this problem. In particular, the corresponding gauging boundary conditions and their consistency with the physical boundary conditions are examined in detail. Furthermore, certain topological features like cavities and multiply connectedness of the domain of analysis are taken into account. Different variational/weak formulations being appropriate for finite element implementation are described. Finally, the suitability of these formulations is demonstrated in several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.

A discrete hyperelastic model was developed in this paper for a single atomic layer of graphene structure that was originally planar. This model can be viewed as an extension to the well-known continuum hyperelastic model. Based on the discrete nature of the atomic structure, the notion of discrete mapping and the concept of spatial secant were introduced. The spatial secant served as a deformation measure that provided a geometric exact mapping in the discrete sense between the atomistic and continuum representations. By incorporating a physics-based interatomic potential, the corresponding discrete hyperelastic model was then established. After an introduction of the model, the computational implementation using the Galerkin finite element and/or meshfree method was outlined. The computational framework was then applied to study of the mechanics of graphene sheets. Extensive comparisons with full-scale molecular mechanics simulations and experimental measurement were made to illustrate the robustness of this approach. Copyright © 2014 John Wiley & Sons, Ltd. Copyright © 2015 John Wiley & Sons, Ltd.

A novel meshless method based on the Shepard and Taylor interpolation method (STIM) and the hybrid boundary node method (HBNM) is proposed. Based on the Shepard interpolation method and Taylor expansion, the STIM is developed to construct the shape function of the HBNM. In the STIM, the Shepard shape function is used as the basic function, which is the zero-level shape function, and the high-power basic functions are constructed through Taylor expansion. Four advantages of the STIM are the interpolation property, the arbitrarily high-order consistency, the absence of inversion for the whole process of shape function construction, and the low computational expense. These properties are desirable in the implementation of meshless methods. By combining the STIM and the HBNM, a much more effective meshless method is proposed to solve the elasticity problems. Compared with the traditional HBNM, the STIM can improve accuracy because of the use of high-power basic functions and can also improve the computational efficiency because there is no inversion for the shape function construction process. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.

Dielectric materials like electro-active polymers (EAPs) exhibit coupled electro-mechanical behavior at large strains. They respond by a deformation to an applied electrical field and are used in advanced industrial environments as sensors and actuators, for example, in robotics, biomimetics and smart structures. In field-activated or electronic EAPs, the electric activation is driven by Coulomb-type electrostatic forces, resulting in Maxwell stresses. These materials are able to provide finite actuation strains, which can even be improved by optimizing their composite microstructure. However, EAPs suffer from different types of instabilities. This concerns *global structural instabilities*, such as buckling and wrinkling of EAP devices, as well as *local material instabilities*, such as limit-points and bifurcation-points in the constitutive response, which induce snap-through and fine scale localization of local states. In this work, we outline variational-based definitions for structural and material stability, and design algorithms for accompanying stability checks in typical finite element computations. The formulation starts from stability criteria for a *canonical energy minimization principle* of electro-elasto-statics, and then shifts them over to representations related to an *enthalpy-based saddle point principle* that is considered as the most convenient setting for numerical implementation. Here, global structural stability is analyzed based on a *perturbation of the total electro-mechanical energy*, and related to statements of positive definiteness of incremental finite element tangent arrays. We base the local material stability on an *incremental quasi-convexity condition* of the electro-mechanical energy, inducing the positive definiteness of both the incremental electro-mechanical moduli as well as a generalized acoustic tensor. It is shown that the incremental arrays to be analyzed in the stability criteria appear within the enthalpy-based setting in a *distinct diagonal form*, with pure mechanical and pure electrical partitions. Applications of accompanying stability analyses in finite element computations are demonstrated by means of representative model problems. Copyright © 2015 John Wiley & Sons, Ltd.

Zero-thickness interface elements are commonly used in computational mechanics to model material interfaces or to introduce discontinuities. The latter class requires the existence of a non-compliant interface prior to the onset of fracture initiation. This is accomplished by assigning a high dummy stiffness to the interface prior to cracking. This dummy stiffness is known to introduce oscillations in the traction profile when using Gauss quadrature for the interface elements, but these oscillations are removed when resorting to a Newton-Cotes integration scheme 1. The traction oscillations are aggravated for interface elements that use B-splines or non-uniform rational B-splines as basis functions (isogeometric interface elements), and worse, do not disappear when using Newton-Cotes quadrature. An analysis is presented of this phenomenon, including eigenvalue analyses, and it appears that the use of lumped integration (at the control points) is the only way to avoid the oscillations in isogeometric interface elements. New findings have also been obtained for standard interface elements, for example that oscillations occur in the relative displacements at the interface irrespective of the value of the dummy stiffness. Copyright © 2015 John Wiley & Sons, Ltd.

A strategy for a two-dimensional contact analysis involving finite strain plasticity is developed with the aid of variable-node elements. The variable-node elements, in which nodes are added freely where they are needed, make it possible to transform the non-matching meshes into matching meshes directly. They thereby facilitate an efficient analysis, maintaining node-to-node contact during the contact deformation. The contact patch test, wherein the contact patch is constructed out of variable-node elements, is thus passed, and iterations for equilibrium solutions reach convergence faster in this scheme than in the conventional approach based on the node-to-surface contact. The effectiveness and accuracy of the proposed scheme are demonstrated through several numerical examples of elasto-plastic contact analyses. Copyright © 2015 John Wiley & Sons, Ltd.

In this work, we present a discrete beam lattice model with embedded discontinuities capable of simulating rock failure as a result of propagating cracks through rock mass. The developed model is a two-dimensional (plane strain) microscale representation of rocks as a two-phase heterogeneous material. Phase I is chosen for intact rock part, while phase II stands for pre-existing microcracks and other defects. The proposed model relies on Timoshenko beam elements enhanced with additional kinematics to describe localized failure mechanisms. The model can properly take into account the fracture process zone with pre-existing microcracks coalescence, along with localized failure modes, mode I of tensile opening and mode II of shear sliding. Furthermore, we give the very detailed presentation for two different approaches to capturing the evolution of modes I and II, and their interaction and combination. The first approach is to deal with modes I and II separately, where mode II can be activated but compression force may still be transferred through rock mass which is not yet completely damaged. The second approach is to represent both modes I and II being activated simultaneously at a point where complete failure is reached. A novel numerical procedure for dealing with two modes failure within framework of method of incompatible modes is presented in detail and validated by a set of numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.

This paper introduces a theoretical and algorithmic reduced model approach to efficiently evaluate time responses of complex dynamic systems. The proposed approach combines four main components: analytical expressions of the average of the system's transfer functions in the frequency domain, precise and convergent rational approximations of these exact expressions, exact evaluation of these approximations through model reduction in rational Krylov subspaces and semi-analytical interpolation at just a few frequency points. The resulting algorithmic principles to evaluate the time response of a particular system are relatively straightforward: one first evaluates the response of the system with slight additional damping at a few frequencies and one then projects or reduces the system in the subspace spanned by these responses. The time response of the reduced model implicitly provides a precise evaluation of that of the original system. The properties of the reduced models and the precision of the proposed approach are studied, and applications on complex matrix systems are presented and discussed. While the theory and numerical algorithms are presented in a matrix context, they are also transposable in a continuous functional context. Copyright © 2015 John Wiley & Sons, Ltd.

In this work, a new, unconditionally stable, time marching procedure for dynamic analyses is presented. The scheme is derived from the standard central difference approximation, with stabilization being provided by a consistent perturbation of the original problem. Because the method only involves constitutive variables that are already available from computations at previous time steps, iterative procedures are not required to establish equilibrium when nonlinear models are focused, allowing more efficient analyses to be obtained. The theoretical properties of the proposed scheme are discussed taking into account standard stability and accuracy analyses, indicating the excellent performance of the new technique. At the end of the contribution, representative nonlinear numerical examples are studied, further illustrating the effectiveness of the new technique. Numerical results obtained by the standard central difference procedure and the implicit constant average acceleration method are also presented along the text for comparison. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we establish the connections between the virtual element method (VEM) and the hourglass control techniques that have been developed since the early 1980s to stabilize underintegrated *C*^{0} Lagrange finite element methods. In the VEM, the bilinear form is decomposed into two parts: a consistent term that reproduces a given polynomial space and a correction term that provides stability. The essential ingredients of -continuous VEMs on polygonal and polyhedral meshes are described, which reveals that the variational approach adopted in the VEM affords a generalized and robust means to stabilize underintegrated finite elements. We focus on the heat conduction (Poisson) equation and present a virtual element approach for the isoparametric four-node quadrilateral and eight-node hexahedral elements. In addition, we show quantitative comparisons of the consistency and stabilization matrices in the VEM with those in the hourglass control method of Belytschko and coworkers. Numerical examples in two and three dimensions are presented for different stabilization parameters, which reveals that the method satisfies the patch test and delivers optimal rates of convergence in the *L*^{2} norm and the *H*^{1} seminorm for Poisson problems on quadrilateral, hexahedral, and arbitrary polygonal meshes. Copyright © 2015 John Wiley & Sons, Ltd.

The control-volume mixed finite element method is formulated for and applied to a computational domain consisting of a tetrahedral partitioning to solve the steady groundwater flow equations. Test functions consistent with piecewise constant and piecewise linear pressure distributions are used in the formulation. Comparisons are made with a standard mixed finite element formulation using lowest-order Raviart Thomas basis functions. Results suggest that the control-volume based formulation is a viable alternative to the standard formulation. Copyright © 2015 John Wiley & Sons, Ltd.

In bending problems of Mindlin–Reissner plate, the resultant forces often vary dramatically within a narrow range near free and soft simply-supported (SS1) boundaries. This is so-called the edge effect or the boundary layer effect, a challenging problem for conventional finite element method. In this paper, an effective finite element method for analysis of such edge effect is developed. The construction procedure is based on the *hybrid displacement function* (HDF) element method [1], a simple hybrid-Trefftz stress element method proposed recently. What is different is that an additional displacement function *f* related to the edge effect is considered, and its analytical solutions are employed as the additional trial functions for the first time. Furthermore, the free and the SS1 boundary conditions are also applied to modify the element assumed resultant fields. Then, two new special elements, HDF-P4-Free and HDF-P4-SS1, are successfully constructed. These new elements are allocated along the corresponding boundaries of the plate, while the other region is modeled by the usual HDF plate element HDF-P4-11 *β* [1]. Numerical tests demonstrate that the present method can effectively capture the edge effects and exactly satisfy the corresponding boundary conditions by only using relatively coarse meshes. Copyright © 2015 John Wiley & Sons, Ltd.

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

We provide a constructive and numerically implementable proof that synchronized groups of coupled, self-sustaining oscillators can be represented as a single effective Perturbation Projection Vector (PPV) (or Phase Response Curve) phase macromodel – in other words, that a group of synchronized oscillators behaves as a single effective oscillator with respect to external influences. This result constitutes a foundation for understanding and predicting synchronization/timing hierarchically in large, complex systems that arise in nature and engineering. We apply this result hierarchically to networks of synchronized oscillators, thereby enabling their efficient and scalable analysis. We illustrate our theory and numerical methods with examples from electronics (networks of three-stage ring oscillators), biology (Fitzhugh–Nagumo neurons) and mechanics (pendulum clocks). Our experiments demonstrate that effective PPVs extracted hierarchically can capture emergent phenomena, such as pattern formation, in coupled oscillator networks. Copyright © 2015 John Wiley & Sons, Ltd.

Materials have a hierarchical nature, deriving often their most useful properties from microscale or nanoscale constituents. Multiresolution analysis, a generalized continuum mechanics-based theory, uses extra degrees of freedom to account for an arbitrary number of these nested length scales. In the past, multiresolution analyses have focused mostly on metal alloys. While this article addresses recent advances in image-based multiresolution analysis of metal alloys, it also highlights extensions to multiresolution theory for modeling of bone mechanics and multiresolution analysis of polymers and polymer nanocomposites. A strong link between molecular dynamics simulations and macroscale multiresolution analyses is shown for both polymers and polymer nanocomposites. The forthcoming work is greatly indebted to the pioneering advances of Ted Belytschko in many areas of computational mechanics; his influence on our work and on the field of finite elements as a whole is substantial Copyright © 2015 John Wiley & Sons, Ltd.

Recently, graphics processing units (GPUs) have been increasingly leveraged in a variety of scientific computing applications. However, architectural differences between CPUs and GPUs necessitate the development of algorithms that take advantage of GPU hardware. As sparse matrix vector (SPMV) multiplication operations are commonly used in finite element analysis, a new SPMV algorithm and several variations are developed for unstructured finite element meshes on GPUs. The effective bandwidth of current GPU algorithms and the newly proposed algorithms are measured and analyzed for 15 sparse matrices of varying sizes and varying sparsity structures. The effects of optimization and differences between the new GPU algorithm and its variants are then subsequently studied. Lastly, both new and current SPMV GPU algorithms are utilized in the GPU CG solver in GPU finite element simulations of the heart. These results are then compared against parallel PETSc finite element implementation results. The effective bandwidth tests indicate that the new algorithms compare very favorably with current algorithms for a wide variety of sparse matrices and can yield very notable benefits. GPU finite element simulation results demonstrate the benefit of using GPUs for finite element analysis and also show that the proposed algorithms can yield speedup factors up to 12-fold for real finite element applications. Copyright © 2015 John Wiley & Sons, Ltd.

A time-domain meshless algorithm based on vector potentials is introduced for the analysis of transient electromagnetic fields. The proposed numerical algorithm is a modification of the radial point interpolation method, where radial basis functions are used for local interpolation of the vector potentials and their derivatives. In the proposed implementation, solving the second-order vector potential wave equation intrinsically enforces the divergence-free property of the electric and magnetic fields. Furthermore, the computational effort associated with the generation of a dual node distribution (as required for solving the first-order Maxwell's equations) is avoided. The proposed method is validated with several examples of 2D waveguides and filters, and the convergence is empirically demonstrated in terms of node density or size of local support domains. It is further shown that inhomogeneous node distributions can provide increased convergence rates, that is, the same accuracy with smaller number of nodes compared with a solution for homogeneous node distribution. A comparison of the magnetic vector potential technique with conventional radial point interpolation method is performed, highlighting the superiority of the divergence-free formulation. Copyright © 2015 John Wiley & Sons, Ltd.

An adaptive refinement scheme is presented to reduce the geometry discretization error and provide higher-order enrichment functions for the interface-enriched generalized FEM. The proposed method relies on the *h*-adaptive and *p*-adaptive refinement techniques to reduce the discrepancy between the exact and discretized geometries of curved material interfaces. A thorough discussion is provided on identifying the appropriate level of the refinement for curved interfaces based on the size of the elements of the background mesh. Varied techniques are then studied for selecting the quasi-optimal location of interface nodes to obtain a more accurate approximation of the interface geometry. We also discuss different approaches for creating the integration sub-elements and evaluating the corresponding enrichment functions together with their impact on the performance and computational cost of higher-order enrichments. Several examples are presented to demonstrate the application of the adaptive interface-enriched generalized FEM for modeling thermo-mechanical problems with intricate geometries. The accuracy and convergence rate of the method are also studied in these example problems. Copyright © 2015 John Wiley & Sons, Ltd.

An *anchored* analysis of variance (ANOVA) method is proposed in this paper to decompose the statistical moments. Compared to the *standard* ANOVA with mutually orthogonal component functions, the anchored ANOVA, with an arbitrary choice of the *anchor point*, loses the orthogonality if employing the same measure. However, an advantage of the anchored ANOVA consists in the considerably reduced number of deterministic solver's computations, which renders the uncertainty quantification of real engineering problems much easier. Different from existing methods, the *covariance decomposition* of the output variance is used in this work to take account of the interactions between non-orthogonal components, yielding an exact variance expansion and thus, with a suitable numerical integration method, provides a strategy that converges. This convergence is verified by studying academic tests. In particular, the sensitivity problem of existing methods to the choice of anchor point is analyzed via the Ishigami case, and we point out that covariance decomposition survives from this issue. Also, with a truncated anchored ANOVA expansion, numerical results prove that the proposed approach is less sensitive to the anchor point. The *covariance-based sensitivity indices* (SI) are also used, compared to the *variance-based SI*. Furthermore, we emphasize that the covariance decomposition can be generalized in a straightforward way to decompose higher-order moments. For academic problems, results show the method converges to exact solution regarding both the skewness and kurtosis. Finally, the proposed method is applied on a realistic case, that is, estimating the chemical reactions uncertainties in a hypersonic flow around a space vehicle during an atmospheric reentry. Copyright © 2015 John Wiley & Sons, Ltd.

We discuss recent advances on robust unfitted finite element methods on cut meshes. These methods are designed to facilitate computations on complex geometries obtained, for example, from computer-aided design or image data from applied sciences. Both the treatment of boundaries and interfaces and the discretization of PDEs on surfaces are discussed and illustrated numerically. © 2014 The Authors. *International Journal for Numerical Methods in Engineering* published by John Wiley & Sons Ltd.

A numerical technique that is based on the integration of the asymptotic solution in the numerical framework for computing the local singular behavior of Stokes flow near a sharp corner is presented. Moffat's asymptotic solution is used, and special enriched shape functions are developed and integrated in the extended finite element method (X-FEM) framework to solve the Navier–Stokes equations. The no-slip boundary condition on the walls of the corner is enforced via the use of Lagrange multipliers. Flows around corners with different angles are simulated, and the results are compared with both those of the known analytic solution and the X-FEM with no special enrichment near the corner. The results of the present technique are shown to greatly reduce the error made in computing the pressure and velocity fields near a corner tip without the need for mesh refinement near the corner. The method is then applied to the estimation of the permeability of a network of fibers, where it is shown that the local small-scale pressure singularities have a large impact on the large-scale network permeability. Copyright © 2014 John Wiley & Sons, Ltd.

Nonlinear geometrically exact rod dynamics is of great interest in many areas of engineering. In recent years, the research was focused towards Timoshenko-type rod theories where shearing is of importance. However, in many general model of mechanisms and spatial deformations, it is desirable to have a displacement-only formulation, which brings us back to the classical Bernoulli beam. While it is well known for linear analysis, the Bernoulli beam is not as common in geometrically exact models of dynamics, especially when we want to incorporate the rotational inertia into the model. This paper is about the development of an energy-momentum integration scheme for the geometrically exact Bernoulli-type rod. We will show that the task is achievable and devise a general framework to do so. Copyright © 2014 John Wiley & Sons, Ltd.

This paper primarily deals with the computational aspects of chemical dissolution-front instability problems in two-dimensional fluid-saturated porous media under non-isothermal conditions. After the dimensionless governing partial differential equations of the non-isothermal chemical dissolution-front instability problem are briefly described, the formulation of a computational procedure, which contains a combination of using the finite difference and finite element method, is derived for simulating the morphological evolution of chemical dissolution fronts in the non-isothermal chemical dissolution system within two-dimensional fluid-saturated porous media. To ensure the correctness and accuracy of the numerical solutions, the proposed computational procedure is verified through comparing the numerical solutions with the analytical solutions for a benchmark problem. As an application example, the verified computational procedure is then used to simulate the morphological evolution of chemical dissolution fronts in the supercritical non-isothermal chemical dissolution system. The related numerical results have demonstrated the following: (1) the proposed computational procedure can produce accurate numerical solutions for the planar chemical dissolution-front propagation problem in the non-isothermal chemical dissolution system consisting of a fluid-saturated porous medium; (2) the Zhao number has a significant effect not only on the dimensionless propagation speed of the chemical dissolution front but also on the distribution patterns of the dimensionless temperature, dimensionless pore-fluid pressure, and dimensionless chemical-species concentration in a non-isothermal chemical dissolution system; (3) once the finger penetrates the whole computational domain, the dimensionless pore-fluid pressure decreases drastically in the non-isothermal chemical dissolution system.