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The neural operator method is reported to be able to rapidly solve partial differential equations (PDEs) in applied mechanics and engineering. However, the great efficiency is not fully realized in practice as the pre-trained models can hardly deal with the varied computational domain. In this work, we propose local neural operator (LNO) to approximate a local-related and shift-invariant time-marching operator when solving transient PDEs, in which the neural operators learn equations separated from case-specific conditions and thus make the domain flexible. It comes with a handy implementation including boundary treatments, enabling one pre-trained LNO to solve on varied domains. For demonstration, we let LNO learn typical transient PDEs, including viscous Burgers' equation, wave equation, and Navier–Stokes (N–S) equation from randomly generated field samples. Then, the pre-trained LNO models solve demo problems on unseen domains different from training ones. Significantly, the pre-trained LNO solves practical flows (governed by N–S equation) three orders of magnitude faster than the conventional finite element method at most. This advantage of LNO is attributed to its pure explicit approximation for the large-interval time-marching operator. [ABSTRACT FROM AUTHOR]

Comments are provided on the recent paper by Ebadi et al. (2024) which demonstrates that the formulated model that was solved contains misconceptions or errors that render the work unsuitable for describing the evolution of interfacial areas in two-fluid porous medium systems. The need for kinematic equations is described and components of a theoretically consistent approach are summarized. [ABSTRACT FROM AUTHOR]

Various uncertainties exist in engineering practice, which brings adverse effects to the reliability of complicated engineering systems. Considering the case that interval and fuzzy uncertainties exist simultaneously, a new reliability estimation model is proposed based on the level cut strategy and volume ratio theory. The new reliability model is better than the traditional one which is very conservative. Moreover, a sequential optimization and reliability assessment (SORA) approach for multidisciplinary systems under hybrid interval and fuzzy uncertainties is developed to decouple the reliability analysis from the deterministic multidisciplinary design optimization (MDO). In the framework of SORA, the deterministic MDO and reliability analysis are executed sequentially, thus the efficiency can be improved. For the multidisciplinary uncertainty analysis, the first order Taylor expansion method and the interval vertex method are formulated. The calculation of the safety possibility under the volume ratio theory and the calculation of the shifting distance are deduced. Both numerical and engineering examples are employed to demonstrate the validity of the proposed method. [ABSTRACT FROM AUTHOR]

Electroactive polymers such as dielectric elastomers (DEs) have attracted significant attention in recent years. Computational techniques to solve the coupled electromechanical system of equations for this class of materials have universally centered around fully coupled monolithic formulations, which while generating good accuracy requires significant computational expense. However, this has significantly hindered the ability to solve large scale, fully three-dimensional problems involving complex deformations and electromechanical instabilities of DEs. In this work, we provide theoretical basis for the effectiveness and accuracy of staggered explicit–implicit finite element formulations for this class of electromechanically coupled materials, and elicit the simplicity of the resulting staggered formulation. We demonstrate the stability and accuracy of the staggered approach by solving complex electromechanically coupled problems involving electroactive polymers, where we focus on problems involving electromechanical instabilities such as creasing, wrinkling, and bursting drops. In all examples, essentially identical results to the fully monolithic solution are obtained, showing the accuracy of the staggered approach at a significantly reduced computational cost. [ABSTRACT FROM AUTHOR]

This study presents the weak form of peridynamic (PD) governing equations which permit the direct imposition of nonlocal essential and natural boundary conditions. It also presents a variational approach to derive the PD form of first- and second-order derivatives of a field variable at a point which is not symmetrically located in its domain of interaction. This capability enables the nonlocal PD representation of the internal force vector and the stress components without any calibration procedure. Furthermore, it removes the concern of truncated domain of interaction for a point near the surface. Thus, the solution is free of nonlocal boundary forces and surface effects. The numerical solution of the resulting equations can be achieved by considering an unstructured nonuniform discretization. The implicit solution to the discrete form of the equations is achieved by employing BiConjugate Gradient Stabilized (BICGSTAB) method which is an iterative technique for solving sparse non-symmetric linear systems. The explicit analysis is performed by constructing a global diagonal mass matrix, and using a hybrid implicit/explicit time integration scheme. The accuracy of this approach is demonstrated by considering an elastic isotropic plate with or without a cutout subjected to a combination of different types of boundary conditions under plane stress conditions. [ABSTRACT FROM AUTHOR]

We develop a unified continuum modeling framework using the Gibbs free energy as the thermodynamic potential. This framework naturally leads to a pressure primitive variable formulation for the continuum body, which is well-behaved in both compressible and incompressible regimes. Our derivation also provides a rational justification of the isochoric–volumetric additive split of free energies in nonlinear elasticity. The variational multiscale analysis is performed for the continuum model to construct a foundation for numerical discretization. We first consider the continuum body instantiated as a hyperelastic material and develop a variational multiscale formulation for the hyper-elastodynamic problem. The generalized- α method is applied for temporal discretization. A segregated algorithm for the nonlinear solver, based on the original idea introduced in Scovazzi et al. (2016), is carefully analyzed. Second, we apply the new formulation to construct a novel unified formulation for fluid–solid coupled problems. The variational multiscale formulation is utilized for spatial discretization in both fluid and solid subdomains. The generalized- α method is applied for the whole continuum body, and optimal high-frequency dissipation is achieved in both fluid and solid subproblems. A new predictor multi-corrector algorithm is developed based on the segregated algorithm. The efficacy of the new formulations is examined in several benchmark problems. The results indicate that the proposed modeling and numerical methodologies constitute a promising technology for biomedical and engineering applications, particularly those necessitating incompressible models. [ABSTRACT FROM AUTHOR]

In this paper, an h ∕ p spectral element method with least-square formulation for parabolic interface problem will be presented. The regularity result of the parabolic interface problem is proven for non-homogeneous interface data. The differentiability estimates and the main stability estimate theorem, using non-conforming spectral element functions, are proven. Error estimates are derived for h and p versions of the proposed method. Specific numerical examples are given to validate the theory. [ABSTRACT FROM AUTHOR]

This article investigates an explicit a-posteriori error estimator for the finite element approximation of the compressible Navier–Stokes equations. The proposed methodology employs the Variational Multi-Scale framework, and specifically, the idea is to use the variational subscales to estimate the error. These subscales are defined to be orthogonal to the finite element space, dynamic and non-linear, and both the subscales in the interior of the element and on the element boundaries are considered. Another particularity of the model is that we define some norms that lead to a dimensionally consistent measure of the compressible flow solution error inside each element; a scaled L 2 -norm, and the calculation of a physical entropy measure, are both studied in this work. The estimation of the error is used to drive the adaptive mesh refinement of several compressible flow simulations. Numerical results demonstrate good accuracy of the local error estimate and the ability to drive the adaptative mesh refinement to minimize the error through the computational domain. [ABSTRACT FROM AUTHOR]

In this work an isogeometric solid-shell model for geometrically nonlinear analyses is proposed. It is based on a linear interpolation through the thickness and a NURBS interpolation on the middle surface of the shell for both the geometry and the displacement field. The Green–Lagrange strains are linearized along the thickness direction and a modified generalized constitutive matrix is adopted to easily eliminate thickness locking without introducing any additional unknowns and to model multi-layered composite shells. Reduced integration schemes, which take into account the high continuity of the shape functions, are investigated to avoid interpolation locking and to increase the computational efficiency. The relaxation of the constitutive equations at each integration point is adopted in the iterative scheme in order to reconstruct the equilibrium path using large steps and a low number of iterations, even for very slender structures. This strategy makes it possible to minimize the number of stiffness matrix evaluations and decompositions and it turns out to be particularly convenient in isogeometric analyses. [ABSTRACT FROM AUTHOR]

The aim of this work is to present a fast and viable approach for taking into account turbulence in topology optimization of complex fluid flow systems, without resorting to any simplifying assumptions in the derivation of discrete adjoints. Topology optimization is an iterative gradient-based design process which minimizes an objective and satisfies a set of selected design constraints by distributing material in a design domain. The gradients are obtained using adjoint sensitivity analysis which requires solutions of a forward state problem and an additional adjoint problem. In the presented article the forward solver is based on finite volume discretized Reynolds-averaged Navier–Stokes equations coupled with either one- or two-equation turbulence closure models, and the adjoint solver is obtained via automatic differentiation. The presented approach is demonstrated on the optimization of several 2D and 3D examples including a detailed comparison to designs and sensitivities obtained with different turbulence models and under a frozen turbulence assumption. The results demonstrate the importance of exact sensitivity analysis and open new possibilities for the design of large scale multiphysics problems involving turbulent flows. [ABSTRACT FROM AUTHOR]

The Young's Double-Slit Experiment (YDSE) optimizer is a newly developed swarm intelligence algorithm inspired by Young's double-slit experiment that reveals the wave nature of light. Due to its simple structure and efficiency, the researchers employed the algorithm to address real-world optimization problems in multiple domains. However, in large-scale mechanical and engineering applications, it still needs to balance exploration and exploitation effectively and easily fall into local optimum. Therefore, in order to ameliorate these shortcomings, a multi-strategy improved YDSE optimizer named IYDSE is developed in this study. In IYDSE, introducing the Bernoulli map strategy ensures a higher quality initial population in the initialization, thus improving the convergence speed. Then, Levy flight is utilized as a local development engine to avoid local search stagnation of the optimal solution during optimization. In addition, mirror reflection learning is designed to enhance group diversity. Finally, the information entanglement strategy is introduced to enhance the information exchange between light and dark fringes, thus balancing the exploration-development capability. The superiority of the proposed IYDSE is comprehensively demonstrated by comparing it with state-of-the-art algorithms on the CEC2019 and CEC2020 test suites, respectively. Meanwhile, the practicability of IYDSE is also verified by solving five specific engineering design cases and 50 different types of engineering optimization suites. Finally, the IYDSE algorithm is applied to three truss topology optimization problems. The experimental results demonstrate the applicability and development potential of the proposed algorithm in practical engineering applications. Therefore, IYDSE is potentially an effective and competitive algorithm for solving applied mechanics and engineering. [ABSTRACT FROM AUTHOR]

Our goal in this work is to develop and compare modified Newton methods for fully monolithic quasi-static brittle phase-field fracture propagation. In variational phase-field fracture, a smoothed phase-field indicator variable denoting the crack path is coupled to elasticity. Moreover, a crack irreversibility condition is incorporated. To develop a fully monolithic scheme is an extremely challenging task since the underlying problem is non-convex and the Jacobian of Newton’s method is indefinite. To split the problem using alternating minimization, thus a partitioned approach, is a possible resort. However, there are good reasons to consider monolithic approaches such as for example robustness and efficiency. Although an error-oriented Newton method can cope with a larger variety of configurations, it appears that this method is not always robust and also not always efficient. Inspired by nonlinear flow problems, as alternative, we develop a modified Newton scheme in which globalization is based on a dynamic modification of the Jacobian matrix rather than utilizing line-search or trust-region strategies. This variation switches smoothly between full Newton and Newton-like steps. In several 2D and 3D numerical examples, all of them with different characteristic features, our modified Newton solver is compared to a backtracking line-search Newton method, another line-search method monitoring the global energy and allowing for negative curvatures, and to already published results of an error-oriented version. These computations also include further modifications of Newton’s method and detailed discussions why certain schemes either work or fail. Revisiting all findings, the main outcome of this paper is that the modified Newton scheme with Jacobian modification is currently the only method that works in a robust and efficient way for all provided examples, whereas line-search schemes or the error-oriented scheme show deficiencies for certain configurations. [ABSTRACT FROM AUTHOR]

This paper presents a novel ‘h-p adaptive’ generalized analysis of variance decomposition (G-ANOVA) for high dimensional stochastic computations. Firstly, an iterative scheme based on variance based global sensitivity analysis is proposed to determine the optimal polynomial order and important component functions ( p adaptivity ) of G-ANOVA. Then, a homotopy algorithm is employed to determine the unknown expansion coefficients. Moreover, a novel distribution adaptive sequential experimental design ( h-adaptivity ) is utilized at each step and the accuracy of the G-ANOVA based surrogate model is computed using leave one out statistical test. It is observed that significantly less number of component functions are retained. As a consequence, the proposed approach is highly efficient and is applicable for solving problems involving large number of stochastic dimensions. Implementation of the proposed approach has been illustrated with four high dimensional applied mechanics problems. The proposed approach is found to yield accurate and efficient results. Finally, the proposed approach has been applied for structural reliability analysis of a large scale real life problem. [ABSTRACT FROM AUTHOR]

In this paper, Special Relativity Search (SRS) is developed for solving engineering problems. The interaction of particles in a magnetic field is considered as an optimizer model of the SRS in which the magnetic field is the feasible search space and the particles in this field are the possible design vectors. Particles interaction is investigated using the Lorentz force, velocity and distance between particles. Charged particles that move in opposite directions produce an inverse force and repel each other while charged particles that move in the same direction attract each other. The special relativity theory is implemented in an applicable way to develop SRS. In SRS, using the angular frequency, centripetal force, and inverse Lorentz transformations, main equations of the algorithm are developed. The design variables are randomly selected as a candidate for optimal answers from the initial population and updated their position in each iteration. This process continues until the global optimum is obtained. In this paper, to validate the SRS performance in solving engineering problems, total twenty-nine CEC-2017 Boundary Constraint (BC) problems and 12 engineering problems are optimized. To evaluate the superiority of the SRS algorithm, the results are compared with other well-known metaheuristic methods. Friedman's test is also used to rank the best algorithm. The results show a good performance of the SRS in most problems with the lowest computational cost. [Display omitted] • Special Relativity Search is developed. • Lorentz inverse transformations, the phenomenon of length contraction, and time dilation are main steps. • The new algorithm is utilized to solve real-world mechanical and structural engineering problems. [ABSTRACT FROM AUTHOR]

The present paper aims to present a new approach for controlling both maximum and minimum length scales of structural members in the Solid Isotropic Material with Penalization (SIMP)-based topology optimization framework. In the proposed approach, length scale control is achieved with help of structural skeleton, which is a key concept in mathematical morphology and a powerful tool for describing structural topologies. Numerical examples show that the proposed approach does have the capability to give a complete control of the length scales of an optimal structure in an explicit and local way. [ABSTRACT FROM AUTHOR]

Highlights: [•] Progressive PGD definitions are applied to the unsteady Navier–Stokes equations. [•] Minimax and progressive PGD definitions are applied to Stokes equations. [•] PGD results are compared to classical POD based ROMs. [•] Ability of PGD algorithm to enrich incomplete space bases is highlighted. [Copyright &y& Elsevier]

Abstract: The work presents two new numerical techniques devised for modeling propagating material failure, i.e. cracks in fracture mechanics or slip-lines in soil mechanics. The first one is termed crack-path-field technique and is conceived for the identification of the path of those cracks, or slip-lines, represented by strain-localization based solutions of the material failure problem. The second one is termed strain-injection, and consists of a procedure to insert, during specific stages of the simulation and in selected areas of the domain of analysis, goal oriented specific strain fields via mixed finite element formulations. In the approach, a first injection, of elemental constant strain modes (CSM) in quadrilaterals, is used, in combination of the crack-path-field technique, for obtaining reliable information that anticipates the position of the crack-path. Based on this information, in a subsequent stage, a discontinuous displacement mode (DDM) is efficiently injected, ensuring the required continuity of the crack-path across sides of contiguous elements. Combination of both techniques results in an efficient and robust procedure based on the staggered resolution of the crack-path-field and the mechanical failure problems. It provides the classical advantages of the “intra-elemental” methods for capturing complex propagating displacement discontinuities in coarse meshes, as E-FEM or X-FEM methods, with the non-code-invasive character of the crack-path-field technique. Numerical representative simulations of a wide range of benchmarks, in terms of the type of material and the failure problem, show the broad applicability, accuracy and robustness of the proposed methodology. The finite element code used for the simulations is open-source and available at http://www.cimne.com/compdesmat/. [Copyright &y& Elsevier]

Highlights: [•] We present a Mortar based approach for thermomechanical contact problems. [•] An isogeometric approach for the discretisation is considered. [•] A consistent thermodynamical model is used. [•] Constitutive laws for the thermomechanical contact interface are applied. [Copyright &y& Elsevier]

Highlights: [•] We study Hyper Reduced Order Models for non-linear problems with moving sources. [•] We examine different strategies for hyper-reduction of the nonlinear terms. [•] Separate hyper-reduction of residual terms improves compressibility and conditioning. [•] Moving frame improves compressibility and conditioning in moving source cases. [•] Nonlinear phase change and welding problems are particularly considered. [Copyright &y& Elsevier]

Highlights: [•] Elastic waves in structures are considered. [•] A scheme based on the Nitsche method is proposed for the coupling of 2D–1D regions. [•] The results are compared with those obtained by the penalty method. [•] The performance of the two methods is investigated via numerical examples. [Copyright &y& Elsevier]

Highlights: [•] Mixed element-finite element method for miscible displacement problem. [•] The velocity–pressure relation is described by the Darcy–Forchheimer equation. [•] Convergence analysis and numerical examples. [Copyright &y& Elsevier]

Abstract: This paper examines the computation of all the (infinitely many) solutions of the frictional wedging problem in the non-coercive context by an algorithm applied to its complementarity formulation. Our analysis offers insights regarding the mechanical and the geometrical meaning of the solution set. We find that the solution structure depends much on the value of the coefficient of friction. The evidence indicates that non-coercivity implies much longer computation times. Coefficients of friction at the onset of wedging are computed and related. [Copyright &y& Elsevier]

Abstract: This work focuses on the assessment of the relative performance of the so-called contact domain method, using either the Lagrange multiplier or the penalty strategies. The mathematical formulation of the contact domain method and the imposition of the contact constraints using a stabilized Lagrange multiplier method are taken from the seminal work (as cited later), whereas the penalty based implementation is firstly described here. Although both methods result into equivalent formulations, except for the difference in the constraint imposition strategy, in the Lagrange multiplier method the constraints are enforced using a stabilized formulation based on an interior penalty method, which results into a different estimation of the contact forces compared to the penalty method. Several numerical examples are solved to assess certain numerical intricacies of the two implementations. The results show that both methods perform similarly as one increases the value of the penalty parameter or decreases the value of the stabilization factor (in case of the Lagrange multiplier method). However there seems to exist a clear advantage in using the Lagrange multiplier based strategy in a few critical situations, where the penalty method fails to produce convincing results due to excessive penetration. [Copyright &y& Elsevier]

Abstract: In this paper, we consider residual and equilibrated error indicators for contact problems with Coulomb friction. The contact problem is handled within the abstract framework of saddle point problems. More precisely, the non-penetration constraint and the friction law is realized as a variationally consistent weak formulation in terms of a localized dual Lagrange multiplier space. Thus from the displacement, we can easily compute in a local post-process the Lagrange multiplier which acts as a Neumann condition on the possible contact zone. Having computed the discrete Lagrange multiplier, we can apply standard error estimators by replacing the unknown Neumann data by its approximation. As it is shown in , this results in an error estimator for a one-sided contact problem without friction. Here, we consider more general situations and discuss two additional contact terms which measure the non-conformity of the discrete Lagrange multiplier. Numerical results in two and three dimensions illustrate the flexibility of the approach and show the influence of the material parameters on the adaptive refinement process. [Copyright &y& Elsevier]

Abstract: Some mechanical problems involve inequality kinematic constraints. This study deals with an original approach to handle those difficult problems. The inequality constraint implies variational inequality since the area where the constraint is active is a priori unknown. The method, introduced here, is to find the exact constrained area iteratively starting from an initial trial one. Thanks to numerical tools such as level-set and X-FEM we turn the constrained minimization problem into a shape equilibrium problem. [Copyright &y& Elsevier]

Abstract: Dislocation nucleation is central to our understanding of the onset of plasticity during nanoindentation. The shear stress in small volumes beneath the nanoindenter can achieve the theoretical limit of a perfect crystal. The ensuing nonlinear elastic instability can trigger homogenous dislocation nucleation inside the crystal. Here we employ the interatomic potential finite element method to simulate nanoindentation and predict dislocation nucleation. Simulations are performed for indentation on the (111), (110) and (100) surfaces of Al, Cu, Ni single crystals. We quantify the critical conditions of dislocation nucleation, including the indentation load of nucleation, location of nucleation site, nucleation stress and activated slip system. We find these conditions sensitively depend on indentation orientation, but are consistent for different crystals. The results highlight the critical role of hyperelasticity (the nonlinear elasticity caused by elastic softening at large strain) and crystallography in dislocation nucleation in small material volumes. Our study also reveals the deficiency of commonly used nucleation criterion such as the critical resolved shear stress. [Copyright &y& Elsevier]

Abstract: Wavelet methods have been presented recently to solve the initial value problems of time-dependent ODEs. Before applying the current wavelet methods, a nonlinear vibration equation usually needs to be transformed to state equations, which may raise the storage and computation cost. In this paper, a modified form of wavelet approximation for the vibration solutions of linear/nonlinear MDOF systems is presented, based on which a multi-resolution AWCM (adaptive wavelet collocation method) is established for MDOF systems. The modified wavelet approximation’s wavelet coefficients explicitly include the boundary first-order time derivatives of the solutions, besides values of solutions on inner time locations of the time domain on which the solutions are defined. This can avoid to transform the dynamical equations of a linear/nonlinear vibrational MDOF system to state equations. Therefore, by using the modified wavelet approximation, the computation and storage cost required for wavelet collocation method may be reduced to at most of that needed by those based on state equations; and the computation efficiency may be at least twice faster than wavelet collocation methods based on state equations, because the number of solution components that are needed to be computed is only of the corresponding state equations. On the other hand, the implementation of initial conditions is straightforward by using the modified wavelet approximation proposed in this paper. The effectiveness of the modified wavelet approximation is tested by applying it to a linear and a nonlinear 10-DOF vibrational system. [Copyright &y& Elsevier]

Abstract: We consider an implicit a posteriori error estimation technique for the adaptive solution of the Maxwell equations with Nédélec edge finite element methods on three-dimensional domains. On each element of the tessellation an equation for the error is formulated and solved with a properly chosen local finite element basis. The discrete bilinear form of the local problems is shown to satisfy an inf–sup condition which ensures the well posedness of the error equations. An adaptive solution algorithm is developed based on the obtained error estimates. The performance of the method is tested on various problems including non-convex domains with non-smooth boundaries. The numerical results show that the estimated error, computed by the implicit a posteriori error estimation technique, correlates well with the actual error. On the meshes generated adaptively with the help of the error estimator, the achieved accuracy is higher than on globally refined meshes with comparable number degrees of freedom. Moreover, the rate of the error convergence on the locally adapted meshes is faster than that on the globally refined meshes. [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: A simplified two-level method for solving the steady two-dimensional incompressible Navier–Stokes equations is investigated in this article. The simplified two-level method involves solving one small nonlinear Navier–Stokes problem on the coarse mesh and one linear Stokes problem (hence, linear with positive definite symmetric part) on the fine mesh. The uniform optimal error estimates with respect to large for the standard Galerkin method and the two-level method are proven. Finally, some numerically tests to confirm the theoretical results of the simplified two-level method and the standard Galerkin method are provided. [Copyright &y& Elsevier]

Abstract: The main obstacle for obtaining fast domain decomposition solvers for spectral element discretizations of second order elliptic equations was the lack of fast solvers for local internal problems on subdomains of decomposition and their faces. As recently shown by Korneev and Rytov, such solvers can be derived on the basis of a specific interrelation between stiffness matrices of the spectral and hierarchical p reference elements (coordinate polynomials of the latter are tensor products of the integrated Legendre’s polynomials). This interrelation allows us to develop fast solvers for discretizations by spectral elements, which are quite similar in basic features to those developed for discretizations by hierarchical elements. Using these facts and preceding findings on the wire basket preconditioners, we present a domain decomposition preconditioner-solver for spectral element discretizations of second order elliptic equations in 3-d domains which is almost optimal in the total arithmetical cost. [Copyright &y& Elsevier]

Abstract: A fully geometric and constitutive nonlinear model for the description of the dynamic behavior of beam structures is developed. The proposed formulation is based on the geometrically exact formulation for beams due to Simo but, in this article an intermediate curved reference configuration is considered. The resulting deformation map belongs to a nonlinear differential manifold and, therefore, an appropriated version of Newmark’s scheme is used in updating the kinematics variables. Each material point of the cross-section is assumed to be composed of several simple materials with their own constitutive laws. The mixing rule is used to describe the resulting composite. An explicit expression for the objective measure of the strain rate acting on each material point is deduced in this article. Details about its numerical implementation in the time-stepping scheme are also addressed. Viscosity is included at the constitutive level by means of a thermodynamically consistent visco damage model developed in terms of the material description of the First Piola Kirchhoff stress vector. The constitutive part of the tangent tensor is deduced including the effect of rate dependent inelasticity. Finally, several numerical examples, validating the proposed formulation, are given. [Copyright &y& Elsevier]

Abstract: Based on two-point diagonal quadratic approximation (referred to as TDQA), developed by Kim et al. [Min-Soo Kim, Jong-Rip Kim, Jae-Young Jeon, Dong-Hoon Choi, Efficient mechanical system optimization using two-point diagonal approximation in the nonlinear intervening variable space, J. KSME 15 (2001) 1257–126], enhanced two-point approximation methods are proposed in this paper. The suggested methods reinforce TDQA with new quadratic correction terms using the concept of TANA-3. These methods overcome the disadvantage of TDQA when the derivatives at two design points have the same sign. In addition, the values and derivatives of the proposed approximation functions are completely equal to those of an original function at the two design points whether the derivatives at those points have the same sign or not. Several examples show the numerical performance and accuracy of the proposed methods compared to previous work, and optimization examples show that these methods can successfully reach the optimum via sequential approximation optimization. [Copyright &y& Elsevier]

Abstract: In this paper the algorithm needed for performing a crack growth analysis of a three-dimensional component by post-processing results from a standard finite element stress analysis is given. Weight functions are used for calculating the stress intensity factor for an embedded crack and a surface crack. Defects are generated in several nominally equal components, and crack growth calculations are carried out by using a short crack model to determine the probability of component fatigue failure. The algorithm has been implemented in a finite element post-processor. [Copyright &y& Elsevier]

Abstract: In this paper, we design a posteriori error estimators for the mixed finite element approximation of the convection–diffusion problems with dominant convection which exhibit solutions with internal layers and boundary layers. A discontinuous upstream weighting scheme in association with the mixed method is used for the convection term. The a posteriori error estimator yields global upper and local lower bounds on the error measured in the L 2-norm. We present several numerical results to show the performance of the error estimator. We find that our estimator is not only reliable and efficient but also robust for various numerical examples. [Copyright &y& Elsevier]

Abstract: This paper presents a new numerical method, in which self contact phenomena associated with a body undergoing large deformations and sliding can be described. In particular, the approach relies on a particular extension of the mortar approach appropriate for this class of problems. A bounding volume hierarchy (organized as a binary tree) is built for the self contact surface, based on the geometry and the mesh connectivity of the surface. A curvature criterion, using a new algorithm to detect subsurface adjacency, is used to accelerate the self contact searching procedure. To ensure that the mortar traction fields are properly defined on contiguous surface patches, a novel facet sorting algorithm is also proposed, based on the mesh connectivity of the contact element pairs found by the self contact searching algorithm. Several two- and three-dimensional numerical examples show the new self contact mortar formulation to be very efficient, and also demonstrate that it can be combined with multi-body contact algorithms to simulate a very general class of contact problems. [Copyright &y& Elsevier]

Abstract: We present a mathematical framework for general over- and under-determined hybrid (switched) systems of differential-algebraic equations (HDAEs). We give a systematic formulation of HDAEs and discuss existence and uniqueness of solutions, the treatment of the switch points and how to perform consistent initialization at switch points. We show how numerical solution methods for DAEs can be adapted for HDAEs and present a numerical results for these methods for the real world example of simulating an automatic gearbox. [Copyright &y& Elsevier]

Abstract: The generalized-α time integrator is considered for the simulation of mechatronic systems. In this context, the fundamental concept of numerical damping is analysed for coupled sets of first and second-order differential–algebraic equations. First, it appears that the algebraic variables do not influence the spectral properties of the dynamic variables. Second, we demonstrate that the coupling between the dynamic variables does not influence the high-frequency spectral response, so that the numerical damping can be determined as usual from elementary characteristic polynomials. Those results are exploited to assess the stability properties of the scheme and to select an algorithm with optimal damping properties. [Copyright &y& Elsevier]

Abstract: This paper addresses the inversion of probabilistic models for the dynamical behaviour of structures using experimental data sets of measured frequency-domain transfer functions. The inversion is formulated as the minimization, with respect to the unknown parameters to be identified, of an objective function that measures a distance between the data and the model. Two such distances are proposed, based on either the loglikelihood function, or the relative entropy. As a comprehensive example, a probabilistic model for the dynamical behaviour of a slender beam is inverted using simulated data. The methodology is then applied to a civil and environmental engineering case history involving the identification of a probabilistic model for ground-borne vibrations from real experimental data. [Copyright &y& Elsevier]

Abstract: A simple methodology to model shear bands as strong displacement discontinuities in an adaptive meshfree method is presented. The shear band is represented by a displacement jump at discrete particle positions. The displacement jump in normal direction is suppressed with penalty method. Loss of material stability is used as transition criterion from continuum to discontinuum. The method is two- and three-dimensional. Examples of complicated shear banding including transition from brittle-to-ductile failure are studied and compared to experimental data and other examples from the literature. [Copyright &y& Elsevier]

Abstract: In this paper, the guided wave propagation in multi-layered elastic slender systems is numerically analyzed. In this context, the applicability of the wave finite element (WFE) formulation is discussed for the description of the wave modes and their frequency evolutions. The WFE formulation suffers from a number of numerical issues, especially when multi-layered structures are concerned. To address this problem, a dynamic substructuring technique is proposed, which allows the dynamics of a typical layer cross-section to be projected on a reduced local wave mode basis with appropriate dimension. The formulation, termed modified wave finite element (MWFE), allows the standard wave motions of multi-layered systems to be correctly captured. Numerical simulations and comparisons with classic theories show the pertinence of the modelings. [Copyright &y& Elsevier]

Abstract: In this paper, we generalize the rectangular nonconforming Wilson element method proposed in [Z. Zhang, S. Zhang, Wilson element for the Reissner–Mindlin plate,Comput. Methods Appl. Mech. Engrg. 113 (1994) 55–65] for the Reissner–Mindlin plate problem to the general quadrilateral mesh and analyze the error. It is proved that this method converges at uniformly optimal rates with respect to both the energy and norms. These estimates improve those of [Z. Zhang, S. Zhang, Wilson element for the Reissner–Mindlin plate, Comput. Methods Appl. Mech. Engrg. 113 (1994) 55–65] in the sense that the requirement of regularity on the solution is dropped, and that the error estimate of this scheme is analyzed. The numerical examples at the end of this paper demonstrate the superiority of this method over the MITC4 [K.J. Bathe, E. Dvorkin, A four-node plate bending element based on Mindlin–Reissner plate theory and a mixed interpolation, Int. J. Numer. Methods Engrg. 21 (1985) 367–383]. [Copyright &y& Elsevier]

In this work, a boundary element based analysis methodology for the steady state uncoupled thermoelastic analysis of fiber reinforced three-dimensional solids is presented. For efficient analysis and modeling, the cylindrical shaped fibers are idealized by a system of curvilinear line elements with a prescribed diameter. The variations in the displacement, traction, temperature and flux fields in the circumferential direction are represented in terms of a trigonometric shape function together with a linear or quadratic variation in the longitudinal direction. This facilitates semi-analytical integration around the fiber which reduces the computational task significantly. The numerical accuracy of the proposed method of analysis is verified and some examples of thermoelastic analysis of fiber-reinforced solids are presented. [Copyright &y& Elsevier]

In this work a general framework for the simulation of sedimentary basins in presence of salt structures is addressed. Sediments and evaporites are modeled as non-Newtonian fluids and the thermal effects induced by the presence of salt are taken into account. The computational strategy is based on a Lagrangian methodology with intensive grid adaptivity, together with a kinematic modeling of faults and different kinds of boundary conditions representing sedimentation, erosion, basement evolution, lithospheric compression and extension. The proposed methodology is applied to simple test cases as well as to a geological reconstruction of industrial interest. [Copyright &y& Elsevier]

Multiscale finite element models are useful in some applications such as biological tissues or foams to determine the effective properties of the solids. In this paper, a model reduction method is proposed, that is applied in the post-buckling regime of cellular microstructures. The proposed approach combines a perturbation technique with the proper orthogonal decomposition (POD). The computation of finite series leads to an efficient continuation method, which enables handling limit points and instabilities naturally. Each term of the series is itself developed using POD. The size of the problems to be solved is thus drastically reduced. Computations involving two and three-dimensional microstructures in compression enable to evaluate the potential of the technique in view of applications to multiscale analyses. [Copyright &y& Elsevier]

A class of Lagrangian mixed finite elements is presented for applications to 2D structural problems based on a damage constitutive model. Attention is focused on localization and regularization issues as compared with the correspondent behavior of Lagrangian displacement-based elements. A non-local regularization procedure of integral type is adopted. A predictor–corrector technique is used to solve the evolution problem of the damage variable. The proposed elements show superior performances for typical structural applications. [Copyright &y& Elsevier]

New sets of hierarchical higher order basis functions in FEM for triangle elements are constructed using a systematic orthogonalization approach that yield better conditioning and investigated with different preconditioners: Jacobi, ICC, ILU, and SAINV. Presented theoretical and numerical results indicate that certain preconditioners are insensitive to the condition number of the basis functions. Advantageous properties of basis functions in relation to preconditioners are viewed from the perspective of similarity and congruence transforms of the basis functions. This provides an alternative view for a systematic construction of the basis functions that couples with a specific preconditioner to optimize the solution performance. [Copyright &y& Elsevier]

In this paper, we propose a new discontinuous finite element method to solve initial value problems for ordinary differential equations and prove that the finite element solution exhibits an optimal O(Δt p+1) convergence rate in the norm. We further show that the p-degree discontinuous solution of differential equation of order m and its first m −1 derivatives are O(Δt 2p+2−m ) superconvergent at the end of each step. We also establish that the p-degree discontinuous solution is O(Δt p+2) superconvergent at the roots of (p +1− m)-degree Jacobi polynomial on each step. Finally, we present several computational examples to validate our theory and construct asymptotically correct a posteriori error estimates. [Copyright &y& Elsevier]