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An energy‐based isotropic elastoplastic damage model is developed for investigating the elastoplastic damage responses and stress–strain relationships of nano‐silica incorporated concrete. The formulation employs a multiscale micromechanical framework to determine the effective elastic properties of composites at different scales. The stress–strain constitutive relation is derived by splitting the strain tensor into "elastic‐damage" and "plastic‐damage" parts while introducing the homogenized free potential energy function and the undamaged potential energy function. The elastoplastic damage response of the material is further characterized by elastic–plastic‐damage coupling. To construct realistic 3D three‐phase concrete mesostructures in numerical simulations, this paper introduces an encapsulation placement method that avoids particle overlap checking when placing aggregates. This methodology allows adjustments for the aggregate compactness as needed and enhances computational efficiency in concrete mesostructure construction. The numerical results of the modeling show good agreement with the experimental values in the open literature. Further, the influence of nano‐silica addition contents and ITZ (interfacial transition zone) thicknesses on the elastoplastic damage response of nano‐silica incorporated concrete are quantitatively and qualitatively investigated for the optimization of nano‐silica incorporated cementitious composites. The proposed model facilitates simulating and optimizing the mechanical characteristics of nano‐silica incorporated concrete and enhances the computational efficiency of 3D concrete modeling with the introduced encapsulation placement method. [ABSTRACT FROM AUTHOR]

Uniform grid solvers of the periodic Lippmann-Schwinger equation have been introduced by Moulinec and Suquet for the numerical homogenization of heterogeneous materials. Based on the fast Fourier transform, these methods use the strain as main unknown and usually do not produce displacement fields. While this is generally not perceived as a restriction for homogenization purposes, some tasks might require kinematically admissible displacement fields. In this paper, we show how the numerical solution to the periodic Lippmann-Schwinger equation can be post-processed to reconstruct a displacement field. Our procedure applies to any variant of the Moulinec-Suquet solver. The reconstruction is formulated as an auxiliary elastic equilibrium problem of a homogeneous material, which is solved with displacement-based finite elements. Taking advantage of periodicity, uniformity of the grid and homogeneity of the material, the resulting linear system is formulated and solved efficiently in Fourier space. The cost of our procedure is lower than that of one iteration of the Lippmann-Schwinger solver. Two applications are proposed, in two and three dimensions. In the first application, the reconstructed displacement field is used to compute a rigorous upper bound on the effective shear modulus. In the second application, the quality of the reconstruction is assessed quantitatively. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Computational aspects of a recently developed gradient elasticity model are discussed in this paper. This model includes the (Aifantis) strain gradient term along with two higher-order acceleration terms (micro-inertia contributions). It has been demonstrated that the presence of these three gradient terms enables one to capture the dispersive wave propagation with great accuracy. In this paper, the discretisation details of this model are thoroughly investigated, including both discretisation in time and in space. Firstly, the critical time step is derived that is relevant for conditionally stable time integrators. Secondly, recommendations on how to choose the numerical parameters, primarily the element size and time step, are given by comparing the dispersion behaviour of the original higher-order continuum with that of the discretised medium. In so doing, the accuracy of the discretised model can be assessed a priori depending on the selected discretisation parameters for given length-scales. A set of guidelines can therefore be established to select optimal discretisation parameters that balance computational efficiency and numerical accuracy. These guidelines are then verified numerically by examining the wave propagation in a one-dimensional bar as well as in a two-dimensional example. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Summary: The realistic representation of material degradation at a fully evolved crack is still one of the main challenges of the phase‐field method for fracture. An approach with realistic degradation behavior is only available for isotropic elasticity in the small deformation framework. In this paper, a variational framework is presented for the standard phase‐field formulation, which allows to derive the kinematically consistent material degradation. For this purpose, the concept of representative crack elements (RCE) is introduced to analyze the fully degraded material state. The realistic material degradation is further tested using the self‐consistency condition, where the behavior of the phase‐field model is compared to a discrete crack model. The framework is applied to isotropic elasticity, anisotropic elasticity and thermo‐elasticity, but not restricted to these constitutive formulations. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we propose an energy and momentum consistent time‐stepping scheme for nonlinear elastodynamics. The algorithm is based on a mixed Hu‐Washizu–type variational principle that is inspired by the concept of polyconvexity and relies on a tensor cross product of second‐order tensors. In addition, we introduce a new algorithmic stress formula in its eigenvalue representation to model the transient behavior of hyperelastic bodies of Ogden‐type materials. Finally, several numerical examples show the superior performance of the proposed formulation in terms of numerical robustness and stability. [ABSTRACT FROM AUTHOR]

Summary: In this paper, a purely displacement‐based formulation is presented within the framework of the scaled boundary finite element method to model compressible and nearly incompressible materials. A selective reduced integration technique combined with an analytical treatment in the nearly incompressible limit is employed to alleviate volumetric locking. The stiffness matrix is computed by solving the scaled boundary finite element equation. The salient feature of the proposed technique is that it neither requires a stabilization parameter nor adds additional degrees of freedom to handle volumetric locking. The efficiency and the robustness of the proposed approach is demonstrated by solving various numerical examples in two and three dimensions. [ABSTRACT FROM AUTHOR]

Summary: Computational analysis of gradient elasticity often requires the trial solution to be C1, yet constructing simple C1 finite elements is not trivial. In this paper, three four‐node 24‐DOF quadrilateral elements for gradient elasticity analysis are devised by generalizing some of the advanced element formulations for thin‐plate analysis. These include the discrete Kirchhoff method, a relaxed hybrid‐stress method, and the hybrid‐stress method with equilibrating internal force modes. The first two methods start with the derivation of a C0 displacement, which is quadratic complete in the Cartesian coordinates. In the first method, at the midside points are derived and interpolated together with those at the nodes. Strain is derived from the displacement interpolation yet the second‐order displacement derivatives are derived from the displacement‐gradient interpolation. In the second method, only the assumed constant double‐stress modes are employed to enforce the continuity of the normal derivative of the displacement. In the third method, the equilibrating internal force modes require the C1 displacement to be defined only along the element boundary and the domain interpolation can be avoided. Patch test involving linear stress and constant double stress as well as other tests are presented to validate the proposed elements. [ABSTRACT FROM AUTHOR]

Summary: This paper is concerned with the analysis and implementation of robust finite element approximation methods for mixed formulations of linear elasticity problems where the elastic body is almost incompressible. Several alternative a posteriori error estimators for the energy norm of the finite element error are introduced and analysed. We establish upper and lower bounds for the energy error in terms of these error estimators and show that the constants in the bounds are independent of the Lamé coefficients: The proposed estimators are robust in the incompressible limit. We also consider the requirement for pressure stabilisation when using lowest‐order conforming approximation. Computational results are presented that validate the theoretical estimates. The software used to generate these results is available online. [ABSTRACT FROM AUTHOR]

Summary: Quasicontinuum (QC) is a partitioned‐domain multiscale method where the full‐atomistic description is retained in highly deformable subdomains while the energy in the remaining region is approximated using a continuum constitutive relation called the Cauchy‐Born (CB) rule. While the CB rule can capture the full nonlinearity of the strain energy, its computational cost is one or two orders of magnitude higher than the linear elasticity (LE) constitutive relation. In this paper, we propose an extension to the QC method, referred to as "hybrid QC," where both the CB and LE relations are combined to increase the efficiency of the original QC method. This extension is summed up to a two‐step process. First, the continuum region is further divided into subregions with CB being restricted to subregions adjoining the atomistic region while LE is adopted in the remaining subregions. Second, a corrective scheme is applied to the LE subregion to recover the higher‐order accuracy. The hybrid QC method is validated through two examples: (i) a crystal containing a dipole of Lomer dislocations and (ii) a nanoindentation system. The simulation results show that the proposed hybrid QC method is numerically more efficient than the original QC method while maintaining virtually the same accuracy. [ABSTRACT FROM AUTHOR]

Summary: This paper presents a seminumerical homogenization framework for porous hyperelastic materials that is open for any hyperelastic microresponse. The conventional analytical homogenization schemes do apply to a limited number of elementary hyperelastic constitutive models. Within this context, we propose a general numerical scheme based on the homogenization of a spherical cavity in an incompressible unit hyperelastic solid sphere, which is denoted as the mesoscopic representative volume element (mRVE). The approach is applicable to any hyperelastic micromechanical response. The deformation field in the sphere is approximated via nonaffine kinematics proposed by Hou and Abeyaratne (JMPS 40:571‐592,1992). Symmetric displacement boundary conditions driven by the principal stretches of the deformation gradient are applied on the outer boundary of the mRVE. The macroscopic quantities, eg, stress and moduli expressions, are obtained by analytically derived pointwise geometric transformations. The macroscopic expressions are then computed numerically through quadrature rules applied in the radial and surface directions of the sphere. A three‐scale compressible microsphere model is derived from the developed seminumerical homogenization framework where the micro‐meso transition is based on the nonaffine microsphere model at every point of the mRVE. The numerical scheme developed for the derivation of macroscopic homogenized stresses and moduli terms as well as the modeling capability of the three‐scale microsphere model is investigated through representative boundary value problems. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we propose a stress recovery procedure for low‐order finite elements in 3D. For each finite element, the recovered stress field is obtained by satisfying equilibrium in an average sense and by projecting the directly calculated stress field onto a conveniently chosen space. Compared with existing recovery techniques, the current procedure gives more accurate stress fields, is simpler to implement, and can be applied to different types of elements without further modification. We demonstrate, through a set of examples in linear elasticity, that the recovered stresses converge at a higher rate than that of directly calculated stresses and that, in some cases, the rate of convergence is the same as that of the displacement field. [ABSTRACT FROM AUTHOR]

In this paper, a coupling technique is developed for the combination of the wavelet-Galerkin method (WGM) with the finite element method (FEM). In this coupled method, the WGM and FEM are respectively used in different sub-domains. The WGM sub-domain and the FEM sub-domain are connected by a transition region that is described by B-spline basis functions. The basis functions of WGM and FEM are modified in the transition region to ensure the basic polynomial reconstruction condition and the compatibility of displacements and their gradients. In addition, the elements of FEM and WGM are not necessary to conform to the transition region. This newly developed coupled method is applied to the stress analysis of 2D and 3D elasticity problems in order to investigate its performance and study parameters. Numerical results show that the present coupled method is accurate and stable. The new method has a promising potential for practical applications and can be easily extended for coupling of other numerical methods. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The scaled boundary radial point interpolation method (SBRPIM), a new semi-analytical technique, is introduced and applied to the analysis of the stress-strain problems. The proposed method combines the advantages of the scaled boundary finite element method and the boundary radial point interpolation method. In this method, no mesh is required, nodes are scattered only on the boundary of the domain, no fundamental solution is required, and as the shape functions constructed based on the radial point interpolation method possess the Kronecker delta function property, the boundary conditions of problems can be imposed accurately without additional efforts. The main ideas of the SBRPIM are introducing a new method based on boundary scattered nodes without the need to element connectivity information, satisfying Kronecker delta function property, and being capable to handle singular problems. The equations of the SBRPIM in stress-strain fields are outlined in this paper. Several benchmark examples of 2-D elastostatic are analyzed to validate the accuracy and efficiency of the proposed method. It is found that the SBRPIM is very easy to implement and the obtained results of the proposed method show a very good agreement with the analytical solution. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper considers stochastic hybrid stress quadrilateral finite element analysis of plane elasticity equations with stochastic Young's modulus and stochastic loads. Firstly, we apply Karhunen-Loève expansion to stochastic Young's modulus and stochastic loads so as to turn the original problem into a system containing a finite number of deterministic parameters. Then we deal with the stochastic field and the space field by k−version/ p−version finite element methods and a hybrid stress quadrilateral finite element method, respectively. We derive a priori error estimates, which are uniform with respect to the Lamè constant λ∈(0,+ ∞). Finally, we provide some numerical results. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

A large amount of research in computational mechanics has biased toward atomistic simulations. This trend, on one hand, is due to the increased demand to perform computations in nanoscale and, on the other hand, is due to the rather simple applications of pairwise potentials in modeling the interactions between atoms of a given crystal. The Cauchy-Born (CB) hypothesis has been used effectively to model the behavior of crystals under different loading conditions, in which the comparison with molecular dynamics simulations presents desirable coincidence between the results. A number of research works have been devoted to the validity of CB hypothesis and its application in post-elastic limit. However, the range of application of CB hypothesis is limited, and it remains questionable whether it is still applicable beyond the validity limit. In this paper, a multi-scale technique is developed for modeling of plastic deformations in nanoscale materials. The deformation gradient is decomposed into the plastic and elastic parts, i.e., F = F p F e. This decomposition is in contrast to the conventional decomposition, F = F e F p, generally encountered in continuum and crystal plasticity. It is shown that the former decomposition is more appropriate for the problem dealt within this work. Inspired by crystal plasticity, the plastic part is determined from the slip on potential slip systems. Based on the assumption that the CB hypothesis remains valid in the homogeneous deformation, the elastic deformation gradient resulting from the aforementioned decomposition is employed in conjunction with the CB hypothesis to update the state variables for face-centered cubic crystals. The assumption of homogeneity of elastic deformation gradient is justified by the fact that elastic deformations are considerably smaller than the plastic deformations. The computational algorithms are derived in details, and numerical simulations are presented through several examples to demonstrate the capability of the proposed computational algorithm in the modeling of golden crystals under different loading conditions. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The constantly rising demands on finite element simulations yield numerical models with increasing number of degrees-of-freedom. Due to nonlinearity, be it in the material model or of geometrical nature, the computational effort increases even further. For these reasons, it is today still not possible to run such complex simulations in real time parallel to, for example, an experiment or an application. Model reduction techniques such as the proper orthogonal decomposition method have been developed to reduce the computational effort while maintaining high accuracy. Nonetheless, this approach shows a limited reduction in computational time for nonlinear problems. Therefore, the aim of this paper is to overcome this limitation by using an additional empirical interpolation. The concept of the so-called discrete empirical interpolation method is translated to problems of solid mechanics with soft nonlinear elasticity and large deformations. The key point of the presented method is a further reduction of the nonlinear term by an empirical interpolation based on a small number of interpolation indices. The method is implemented into the finite element method in two different ways, and it is extended by using different solution strategies including a numerical as well as a quasi-Newton tangent. The new method is successfully applied to two numerical examples concerning hyperelastic as well as viscoelastic material behavior. Using the extended discrete empirical interpolation method combined with a quasi-Newton tangent enables reductions in computational time of factor 10 with respect to the proper orthogonal decomposition method without empirical interpolation. Negligibly, orders of error can be reached. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

SUMMARY The explicit residual-based error estimator originally proposed by Babuška and Miller (1987) for adaptive finite element analysis with application to problems of linear elasticity and fracture is known to be one of the most simple and inexpensive error estimators. Indeed, it provides a theoretically guaranteed upper bound on a discretization error, measured in the energy norm, and requires small (nearly negligible) post-processing computational effort. The main issue with this classical estimator, however, is that an upper error bound is explicitly computable up to an unknown multiplicative constant. In this paper, we track the source of this generic constant, revise the original derivation procedure and derive analytically four pre-computable constants that constitute an upper error bound, resulting in, what we then call, the constant-free error estimator of the Babuška-Miller type. The performance of this constant-free estimator, as well as an adaptive FEM based on it, are illustrated on regular and singular benchmark problems and on numerical examples featuring crack propagation. The special property of the estimator is a sharp upper bound of the error: effectivity indices in the range of 1.2-2.0 are obtained, what is treated as practically (very) acceptable. In terms of efficiency judged against simplicity and inexpensiveness, the proposed constant-free explicit estimator is superior to, for example, corresponding implicit residual estimators and may be seen as a very competitive one in comparison with other known, yet more intricate and complex error estimation techniques. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY Accurate simulations of large deformation hyperelastic materials by the FEM is still a challenging problem. In a total Lagrangian formulation, even when using a very fine initial mesh, the simulation can break down due to severe mesh distortion. Error estimation and adaptive remeshing on the initial geometry are helpful and can provide more accurate solutions but are not sufficient to attain very large deformations. The updated Lagrangian formulation where the geometry is periodically updated is then preferred. However, it requires data transfer from the old mesh to the new one and this is a very delicate issue. In this paper, we present an updated Lagrangian formulation where the error is estimated and adaptive remeshing is performed in order to reach high level of deformations while controlling both the accuracy of the solution and mesh distortion. Special attention is given to data transfer methods and a very accurate cubic Lagrange projection method is introduced. A continuation method is used to automatically pilot the complete algorithm including load increase, error estimation, adaptive remeshing, and data transfer. A number of examples will be presented and analyzed. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY We develop an efficient semi-local method for speeding up the solution of linear systems arising in spectral/hp element discretization of the linear elasticity equations. The main idea is to approximate the element-wise residual distribution with a localization operator we introduce in this paper, and subsequently solve the local linear system. Additionally, we decouple the three directions of displacement in the localization operator, hence enabling the use of an efficient low energy preconditioner for the conjugate gradient solver. This approach is effective for both nodal and modal bases in the spectral/hp element method, but here, we focus on the modal hierarchical basis. In numerical tests, we verify that there is no loss of accuracy in the semi-local method, and we obtain good parallel scalability and substantial speed-up compared to the original formulation. In particular, our tests include both structure-only and fluid-structure interaction problems, with the latter modeling a 3D patient-specific brain aneurysm. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY This paper presents a bubble-enhanced smoothed finite element formulation for the analysis of volume-constrained problems in two-dimensional linear elasticity. The new formulation is derived based on the variational multi-scale approach in which unequal order displacement-pressure pairs are used for the mixed finite element approximation and hierarchical bubble function is selected for the fine-scale displacement approximation. An area-weighted averaging scheme is employed for the two-scale smoothed strain calculation under the framework of edge-based smoothed FEM. The smoothed fine-scale solution is shown to naturally contain the stress field jump of the smoothed coarse-scale solution across the boundary of edge-based smoothing domain and thus provides the possibility to stabilize the global solution for volume-constrained problems. A global monolithic solution strategy is employed, and the fine-scale solution is solved without the consideration of approximating the strong form of the fine-scale equation. Several numerical examples are analyzed to demonstrate the accuracy of the present formulation. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper presents equilibrium elements for dual analysis. A traction-based equilibrium element is proposed in which tractions of an element instead of stresses are chosen as DOFs, and therefore, the interelement continuity and the Neumann boundary balance are directly satisfied. To be solvable, equilibrated tractions with respect to the space of rigid body motion are required for each element. As a result, spurious kinematic modes that may inflict troubles on stress-based equilibrium elements do not appear in the element because only equilibrium constraints on tractions are required. An admissible stress field is eventually constructed in terms of the equilibrated tractions for the element, and hence, equilibrium finite element procedures can proceed. The element is also generalized to accommodate non-zero body forces, nonlinear boundary tractions and curved Neumann boundaries. Numerical tests including a single equilibrium element, error estimation of a cantilever beam and an infinite plate with a circular hole are conducted, displaying excellent convergence and effectiveness of the element for error estimation. [ABSTRACT FROM AUTHOR]

SUMMARY A new finite heterogeneous element consisting of sliced microstructures (FHES) is applied in a multi-scale technique. The FHES represents a heterogeneous material with microscopic constituents without homogenization or microscopic finite element analysis. A representative volume element extracted from a heterogeneous structure is thinly sliced. Each slice is modeled as a combined spring to calculate properties of the FHES. Each FHES has the same number of nodes as an ordinary finite element, and the macroscopic analysis cost is the same as that for ordinary finite element analysis. However, the FHES retains information about the microscopic material layout (i.e., the distribution of a material's property) in itself that is lost during homogenization. In the proposed approach, materials are not homogenized. The FHES does not have a constant (homogenized) material property and can 'change stiffness' depending on its deformation behavior. This reduces error due to coarse-graining and allows us to calculate the macroscopic deformation behavior with sufficient accuracy even if a large gradient of strain is generated in the macroscopic field. The novelty of the research is the development of rational heterogeneous finite elements. The paper presents the theory behind the FHES and its practical application to a linear elastic problem. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY Numerical analysis of linear visco-elastic materials requires robust and stable methods to integrate partial differential equations in both space and time. In this paper, symmetric space-time finite element operators are derived for the first time for elementary linear elastic spring and linear viscous dashpot. These can thereafter be assembled in parallel and in series to simulate an arbitrarily complex linear visco-elastic behaviour. The flexibility of the proposed method allows the formulation of the behaviour, which closely reflects physical processes. An efficient algorithm is proposed to use the generated elementary matrices in a way that is comparable with finite difference schemes, in terms of both processor and memory costs. This unconditionally stable and convergent procedure is equally valid for space domains in which geometry or material properties evolve with time. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY The mapping method was introduced in Jeong et al. (2013) for highly accurate isogeometric analysis (IGA) of elliptic boundary value problems containing singularities. The mapping method is concerned with constructions of novel geometrical mappings by which push-forwards of B-splines from the parameter space into the physical space generate singular functions that resemble the singularities. In other words, the pullback of the singularity into the parameter space by the novel geometrical mapping (a non-uniform rational basis spline (NURBS) surface mapping) becomes highly smooth. One of the merits of IGA is that it uses NURBS functions employed in designs for the finite element analysis. However, push-forwards of rational NURBS may not be able to generate singular functions. Moreover, the mapping method is effective for neither the k-refinement nor the h-refinement. In this paper, highly accurate stress analysis of elastic domains with cracks and ∕ or corners are achieved by enriched IGA, in which push-forwards of NURBS via the design mapping are combined with push-forwards of B-splines via the novel geometrical mapping (the mapping technique). In a similar spirit of X-FEM (or GFEM), we propose three enrichment approaches: enriched IGA for corners, enriched IGA for cracks, and partition of unity IGA for cracks. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY This paper deals with the extension of proper generalized decomposition methods to non-linear problems, in particular, to hyperelasticity. Among the different approaches that can be considered for the linearization of the doubly weak form of the problem, we have implemented a new one that uses asymptotic numerical methods in conjunction with proper generalized decomposition to avoid complex consistent linearization schemes necessary in Newton strategies. This approach results in an approximation of the problem solution in the form of a series expansion. Each term of the series is expressed as a finite sum of separated functions. The advantage of this approach is the presence of only one tangent operator, identical for every term in the series. The resulting approach has proved to render very accurate results that can be stored in the form of a meta-model in a very compact format. This opens the possibility to use these results in real-time, reaching kHz feedback rates, or to be used in deployed, handheld devices such as smartphones and tablets. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY The consistency condition for the nodal derivatives in traditional meshfree Galerkin methods is only the differentiation of the approximation consistency (DAC). One missing part is the consistency between a nodal shape function and its derivatives in terms of the divergence theorem in numerical forms. In this paper, a consistency framework for the meshfree nodal derivatives including the DAC and the discrete divergence consistency (DDC) is proposed. The summation of the linear DDC over the whole computational domain leads to the so-called integration constraint in the literature. A three-point integration scheme using background triangle elements is developed, in which the corrected derivatives are computed by the satisfaction of the quadratic DDC. We prove that such smoothed derivatives also meet the quadratic DAC, and therefore, the proposed scheme possesses the quadratic consistency that leads to its name QC3. Numerical results show that QC3 is the only method that can pass both the linear and the quadratic patch tests and achieves the best performances for all the four examples in terms of stability, convergence, accuracy, and efficiency among all the tested methods. Particularly, it shows a huge improvement for the existing linearly consistent one-point integration method in some examples. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

A new computational framework for large strain elasticity in principal stretches is presented. Distinct from existing literature, the proposed formulation makes direct use of principal stretches rather than their squares that is, eigenvalues of Cauchy‐Green strain tensor. The proposed framework has three key features. First, the eigen‐decomposition of the tangent elasticity and initial (geometric) stiffness operators is obtained in closed‐form from principal information alone. Crucially, these newly found eigenvalues describe the general convexity conditions of isotropic hyperelastic energies. In other words, convexity is postulated concisely through tangent eigenvalues supplementing the original work of Ball (Arch Ration Mech Anal. 1976; 63(4): 337–403). Consequently, this novel finding opens the door for designing efficient automated Newton‐style algorithms with stabilised tangents via closed‐form semipositive definite projection or spectral shifting that converge irrespective of mesh resolution, quality, loading scenario and without relying on path‐following techniques. A critical study of closed‐form tangent stabilisation in the context of isotropic hyperelasticity is therefore undertaken in this work. Second, in addition to high order displacement‐based formulation, mixed Hu‐Washizu variational principles are formulated in terms of principal stretches by introducing stretch work conjugate Lagrange multipliers that enforce principal stretch‐stress compatibility. This is similar to enhanced strain methods. However, the resulting mixed finite element scheme is cost‐efficient, specially compared to approximating the entire strain tensors since the formulation is in the scalar space of singular values. Third, the proposed framework facilitates simulating rigid and stiff systems and those that are nearly‐inextensible in principal directions, a constituent of elasticity that cannot be easily studied using standard formulations. [ABSTRACT FROM AUTHOR]

Summary: Recently, the method of difference potentials has been extended to linear elastic fracture mechanics. The solution was calculated on a grid boundary belonging to the domain of an auxiliary problem, which must be solved multiple times. Singular enrichment functions, such as those used within the extended finite element method, were introduced to improve the approximation near the crack tip leading to near‐optimal convergence rates. Now, the method is further developed by significantly reducing the computation time. This is achieved via the implementation of a system of basis functions introduced along the physical boundary of the problem. The basis functions form an approximation of the trace of the solution at the physical boundary. This method has been proven efficient for the solution of problems on regular (Lipschitz) domains. By introducing the singularity into the finite element space, the approximation of the crack can be realised by regular functions. Near‐optimal convergence rates are then achieved for the enriched formulation. A solution algorithm using the fast Fourier transform is provided with the aim of further increasing the efficiency of the method. [ABSTRACT FROM AUTHOR]

The article presents a finite element mapping procedure for the coordinate change via higher‐order derivatives to compute the fundamental matrices of laminated thin plates with arbitrary domains in gradient elasticity. In this context, the approximate solution requires Hermite interpolating functions. Therefore, conforming and nonconforming formulations are needed for both membrane and bending degrees of freedom, which require respectively C1$$ {C}^1 $$ and C2$$ {C}^2 $$ continuity. The aim of the current procedure is the possibility to remove the limitations related to the regular rectangular shape which typically characterizes this kind of elements and to introduce arbitrary distortions, discussing the influence of structured and unstructured meshes. The accuracy and convergence features of the methodology are presented through some numerical tests and compared to relevant literature. [ABSTRACT FROM AUTHOR]

Summary: In this contribution we introduce a novel space‐time formulation for non‐linear elasticity, able to calculate large deformations and displacements with high efficiency using structured and unstructured meshes in the space‐time cylinder without changing the required regularity and thus, enabling the use of Lagrangian shape functions including tetrahedron and hypertetrahedron or tesseract elements. The common and indiscriminate treatment of spatial and temporal directions allows us to remove one of the major bottlenecks in parallel computations: The design of time‐stepping schemes, which can neither been parallelized efficiently in time nor allow for local refinements as no information transfer backwards in time is possible. Moreover, stability of time‐stepping schemes depend on the accumulation of local approximation errors in each time‐step, in contrast to space‐time formulation, characterized by enhanced stability and robustness. Eventually, we will demonstrate superiority in the convergence against classical time‐stepping schemes using various examples including highly sensitive systems in the context of non‐linear elasticity. [ABSTRACT FROM AUTHOR]

Summary: We propose a combined nodal integration and virtual element method for compressible and nearly incompressible elasticity, wherein the strain is averaged at the nodes from the strain of surrounding virtual elements. For the strain averaging procedure, a nodal averaging operator is constructed using a generalization to virtual elements of the node‐based uniform strain approach for finite elements. We refer to the proposed technique as the node‐based uniform strain virtual element method (NVEM). No additional degrees of freedom are introduced in this approach, thus resulting in a displacement‐based formulation. A salient feature of the NVEM is that the stresses and strains become nodal variables just like displacements, which can be exploited in nonlinear simulations. Through several benchmark problems in compressible and nearly incompressible elasticity as well as in elastodynamics, we demonstrate that the NVEM is accurate, optimally convergent and devoid of volumetric locking. [ABSTRACT FROM AUTHOR]

Motivated by nano-scale experimental evidence on the dispersion characteristics of materials with a lattice structure, a new multi-scale gradient elasticity model is developed. In the framework of gradient elasticity, the simultaneous presence of acceleration and strain gradients has been denoted as dynamic consistency. This model represents an extension of an earlier dynamically consistent model with an additional micro-inertia contribution to improve the dispersion behaviour. The model can therefore be seen as an enhanced dynamic extension of the Aifantis' 1992 strain-gradient theory for statics obtained by including two acceleration gradients in addition to the strain gradient. Compared with the previous dynamically consistent model, the additional micro-inertia term is found to improve the prediction of wave dispersion significantly and, more importantly, requires no extra computational cost. The fourth-order equations are rewritten in two sets of symmetric second-order equations so that [ABSTRACT FROM AUTHOR]

We provide theoretical investigations and empirical evidence that the effective stresses in computational homogenization of inelastic materials converge with a higher rate than the local solution fields. Due to the complexity of industrial‐scale microstructures, computational homogenization methods often utilize a rather crude approximation of the microstructure, favoring regular grids over accurate boundary representations. As the accuracy of such an approach has been under continuous verification for decades, it appears astonishing that this strategy is successful in homogenization, but is seldom used on component scale. A part of the puzzle has been solved recently, as it was shown that the effective elastic properties converge with twice the rate of the local strain and stress fields. Thus, although the local mechanical fields may be inaccurate, the averaging process leads to a cancellation of errors and improves the accuracy of the effective properties significantly. Unfortunately, the original argument is based on energetic considerations. The straightforward extension to the inelastic setting provides superconvergence of (pseudoelastic) potentials, but does not cover the primary quantity of interest: the effective stress tensor. The purpose of the work at hand is twofold. On the one hand, we provide extensive numerical experiments on the convergence rate of local and effective quantities for computational homogenization methods based on the fast Fourier transform. These indicate the superconvergence effect to be valid for effective stresses, as well. Moreover, we provide theoretical justification for such a superconvergence based on an argument that avoids energetic reasoning. [ABSTRACT FROM AUTHOR]

A graph convolutional network (GCN) is employed in the deep energy method (DEM) model to solve the momentum balance equation in three‐dimensional space for the deformation of linear elastic and hyperelastic materials due to its ability to handle irregular domains over the traditional DEM method based on a multilayer perceptron (MLP) network. The method's accuracy and solution time are compared to the DEM model based on a MLP network. We demonstrate that the GCN‐based model delivers similar accuracy while having a shorter run time through numerical examples. Two different spatial gradient computation techniques, one based on automatic differentiation (AD) and the other based on shape function (SF) gradients, are also accessed. We provide a simple example to demonstrate the strain localization instability associated with the AD‐based gradient computation and show that the instability exists in more general cases by four numerical examples. The SF‐based gradient computation is shown to be more robust and delivers an accurate solution even at severe deformations. Therefore, the combination of the GCN‐based DEM model and SF‐based gradient computation is potentially a promising candidate for solving problems involving severe material and geometric nonlinearities. [ABSTRACT FROM AUTHOR]

The present article, provides a novel mixed finite element formulation for elastoplasticity which is suitable for the discretization of thin‐walled structures using highly anisotropic volume elements. We present a thermodynamically consistent framework for the modeling of elastoplastic stress response, where dissipative effects are considered through a dissipation function instead of explicit flow rules. Along these lines, a mixed incremental principle, in which stresses are included as independent unknowns, is derived. We propose to employ tangential‐displacement normal‐normal‐stress (TDNNS) elements for a Galerkin discretization of the underlying variational problem. These elements were originally developed for linear elasticity, where it could be shown that they do not suffer from shear locking if the element's aspect ratio deteriorates. Excellent computational performance and accuracy of the proposed method are demonstrated in several benchmark problems. [ABSTRACT FROM AUTHOR]

We use the augmented Lagrangian formalism to derive discontinuous Galerkin (DG) formulations for problems in nonlinear elasticity. In elasticity, stress is typically a symmetric function of strain, leading to symmetric tangent stiffness matrices in Newton's method when conforming finite elements are used for discretization. By use of the augmented Lagrangian framework, we can also obtain symmetric tangent stiffness matrices in DG methods. We suggest two different approaches and give examples from plasticity and from large deformation hyperelasticity. [ABSTRACT FROM AUTHOR]

In this article, we propose a three‐dimensional meshing algorithm for discrete models combining the lattice approach with the polyhedral particle approach. Our aim is to be able to leverage readily available, well‐supported meshers to handle various geometries. We use them to generate a tetrahedral mesh, which our mesher then converts to a polyhedral mesh. The input mesh serves as geometrical support to generate the nodes of the discrete mesh. The desired shape is obtained with an assembly of convex polyhedral particles—without having to clip them. We show that this approach enables meshing convex or concave geometries with sharp edges, curved features, and more. Several three‐dimensional geometries are presented to support this claim and illustrate the capabilities of the mesher. We provide a detailed analysis of its isotropy. The geometric isotropy is studied by analyzing the orientation of the generated beams. The mechanical isotropy is verified by assessing the properties of the elasticity tensor. Finally, we show that the new mesh retains its ability to be a good support for the generation of realistic cracking patterns. [ABSTRACT FROM AUTHOR]

We present a machine learning framework to train and validate neural networks to predict the anisotropic elastic response of a monoclinic organic molecular crystal known as β$$ \beta $$‐HMX in the geometrical nonlinear regime. A filtered molecular dynamic (MD) simulations database is used to train neural networks with a Sobolev norm that uses the stress measure and a reference configuration to deduce the elastic stored free energy functional. To improve the accuracy of the elasticity tangent predictions originating from the learned stored free energy, a transfer learning technique is used to introduce additional tangential constraints from the data while necessary conditions (e.g., strong ellipticity, crystallographic symmetry) for the correctness of the model are either introduced as additional physical constraints or incorporated in the validation tests. Assessment of the neural networks is based on (1) the accuracy with which they reproduce the bottom‐line constitutive responses predicted by MD, (2) the robustness of the models measured by detailed examination of their stability and uniqueness, and (3) the admissibility of the predicted responses with respect to mechanics principles in the finite‐deformation regime. We compare the training efficiency of the neural networks under different Sobolev constraints and assess the accuracy and robustness of the models against MD benchmarks for β$$ \beta $$‐HMX. [ABSTRACT FROM AUTHOR]

Recently we have developed the optimal local truncation error method (OLTEM) for scalar PDEs on irregular domains and unfitted Cartesian meshes. Here, OLTEM is extended to a much more general case of a system of PDEs for the 2‐D time‐independent elasticity equations on irregular domains. Compact 9‐point uniform and nonuniform stencils (with the computational costs of linear finite elements) are used with OLTEM. The stencil coefficients are assumed to be unknown and are calculated by the minimization of the local truncation error. It is shown that the second order of accuracy is the maximum possible accuracy for 9‐point stencils independent of the numerical technique used for their derivations. The special treatment of the Neumann boundary conditions has been developed that does not increase the size of the stencils. The numerical examples are in agreement with the theoretical findings. They also show that due to the minimization of the local truncation error, OLTEM is much more accurate than linear finite elements and than quadratic finite elements (up to engineering accuracy of 0.1%–1%) at the same numbers of degrees of freedom. Due to the computational efficiency and trivial unfitted Cartesian meshes that are independent of irregular domains, the proposed technique with no remeshing for the shape change of irregular domains will be effective for many engineering applications. [ABSTRACT FROM AUTHOR]

This work proposes a simple but robust 4‐node 24‐DOF facet shell element for static analysis of small‐scale thin shell structures. To accommodate the size effects, the modified couple stress theory is employed as the theoretical basis. The element is constructed via two main innovations. First, the trial functions that can a priori satisfy related governing differential equations are adopted as the basic functions for formulating the element interpolations. Second, the generalized conforming theory and the penalty function method are employed to meet the C1 continuity requirement in weak sense for ensuring the computation convergence property. Several benchmarks of shells with different geometries are tested to assess the new facet shell element's capability. The numerical results reveal that the element can effectively simulate the size‐dependent mechanical behaviors of small‐scale thin shells, exhibiting satisfactory numerical accuracy and low susceptibility to mesh distortion. Moreover, as the shell element uses only six degrees of freedom per node, it can be incorporated into the commonly available finite element programs very readily. [ABSTRACT FROM AUTHOR]

Smoothed finite element method with the node‐based strain smoothing domains (NS‐FEM) is remarkable for the upper‐bound feature and insensitivity to the volumetric locking. As a mesh‐based methodology, its application is limited by the burdensome meshing for which elements align with the physical domain. We extend the strain smoothing in NS‐FEM to the numerical manifold method (NMM) with unfitted meshes, and propose a novel methodology named physical patch‐based smoothed NMM. Benchmark problems demonstrate the optimal convergence, stability in term of the condition number, upper‐bound property, suppression of the volumetric locking, as well as the stress stability. [ABSTRACT FROM AUTHOR]

Summary: This article introduces a novel error estimator for the proper generalized decomposition (PGD) approximation of parametrized equations. The estimator is intrinsically random: it builds on concentration inequalities of Gaussian maps and an adjoint problem with random right‐hand side, which we approximate using the PGD. The effectivity of this randomized error estimator can be arbitrarily close to unity with high probability, allowing the estimation of the error with respect to any user‐defined norm as well as the error in some quantity of interest. The performance of the error estimator is demonstrated and compared with some existing error estimators for the PGD for a parametrized time‐harmonic elastodynamics problem and the parametrized equations of linear elasticity with a high‐dimensional parameter space. [ABSTRACT FROM AUTHOR]

Summary: Peridynamics is a nonlocal theory which has been successfully applied to solid mechanics and crack propagation problems over the last decade. This methodology, however, may lead to large computational calculations which can soon become intractable for many problems of practical interest. In this context, a technique to couple—in a global/local sense–three‐dimensional peridynamics with one‐dimensional high‐order finite elements based on classical elasticity is proposed. The refined finite elements employed in this work are based on the well‐established Carrera Unified Formulation, which the previous literature has demonstrated to provide structural formulations with unprecedented accuracy and optimized computational efficiency. The coupling is realized by using Lagrange multipliers that guarantee versatility and physical consistency as shown by the numerical results, including the linear static analyses of solid and thin‐walled beams as well as of a reinforced panel of aeronautic interest. [ABSTRACT FROM AUTHOR]

Summary: Barycentric rational interpolation collocation method (BRICM) for solving plane elasticity problems with high accuracy is presented. The plane elasticity problems on a circular or rectangular domain can be solved directly by BRICM. Embedded the irregular domain into a regular (circular or rectangular) domain, the governing equations of plane elasticity on regular domain are discretized by the differentiation matrices based on barycentric rational interpolation to form a system of algebraic equations. Discrete boundary conditions are obtained using barycentric rational interpolation. The irregular boundary conditions are imposed by the additional method to form an over‐constraint linear system of algebraic equations. Numerical experiments are presented to illustrate the efficiency and high computing precision of proposed method. [ABSTRACT FROM AUTHOR]

Summary: A new C0 8‐node 48‐DOF hexahedral element is developed for analysis of size‐dependent problems in the context of the modified couple stress theory by extending the methodology proposed in our recent work (Shang et al., Int J Numer Methods Eng 119(9): 807‐825, 2019) to the three‐dimensional (3D) cases. There are two major innovations in the present formulation. First, the independent nodal rotation degrees of freedom (DOFs) are employed to enhance the standard 3D isoparametric interpolation for obtaining the displacement and strain test functions, as well as to approximatively design the physical rotation field for deriving the curvature test function. Second, the equilibrium stress functions instead of the analytical functions are used to formulate the stress trial function whilst the couple stress trial function is directly obtained from the curvature test function by using the constitutive relationship. Besides, the penalty function is introduced into the virtual work principle for enforcing the C1 continuity condition in weak sense. Several benchmark examples are examined and the numerical results demonstrate that the element can simulate the size‐dependent mechanical behaviors well, exhibiting satisfactory accuracy and low susceptibility to mesh distortion. [ABSTRACT FROM AUTHOR]

Summary: Homogenization methods are drawing increasing attention for simulation of heterogeneous materials like composites. For balancing the accuracy and the numerical efficiency of such strategies, we deal with both model and discretization errors of the finite element method (FEM) on a macroscale. Within a framework of goal‐oriented adaptivity, we consider linear elastic heterogeneous materials, for which first‐order homogenization schemes apply. A novel model hierarchy is proposed based on mean‐field and full‐field homogenization methods. For the former, we consider several well‐established schemes like Mori‐Tanaka or self‐consistent as basic models, and for the latter, as superior models, unit cell problems are solved via the FEM under an a priori chosen boundary condition. For a further stage of the model hierarchy, we consider hierarchical unit cells within the frame of the FEM toward an adaptive selection of the unit cell size. By means of several numerical examples, we illustrate the effectiveness of the proposed adaptive approach. [ABSTRACT FROM AUTHOR]

Summary: We develop a mixed formulation for incompressible hyperelastodynamics based on a continuum modeling framework recently developed in the work of Liu and Marsden and smooth generalizations of the Taylor‐Hood element based on nonuniform rational B‐splines (NURBSs). This continuum formulation draws a link between computational fluid dynamics and computational solid dynamics. This link inspires an energy stability estimate for the spatial discretization, which favorably distinguishes the formulation from the conventional mixed formulations for finite elasticity. The inf‐sup condition is utilized to provide a bound for the pressure field. The generalized‐α method is applied for temporal discretization, and a nested block preconditioner is invoked for the solution procedure. The inf‐sup stability for different pairs of NURBS elements is elucidated through numerical assessment. The convergence rate of the proposed formulation with various combinations of mixed elements is examined by the manufactured solution method. The numerical scheme is also examined under compressive and tensile loads for isotropic and anisotropic hyperelastic materials. Finally, a suite of dynamic problems is numerically studied to corroborate the stability and conservation properties. [ABSTRACT FROM AUTHOR]

Summary: A collocation method has been recently developed as a powerful alternative to Galerkin's method in the context of isogeometric analysis, characterized by significantly reduced computational cost, but still guaranteeing higher‐order convergence rates. In this work, we propose a novel adaptive isogeometric analysis meshfree collocation (IGAM‐C) for the two‐dimensional (2D) elasticity and frictional contact problems. The concept of the IGAM‐C method is based upon the correspondence between the isogeometric collocation and reproducing kernel meshfree approach, which facilitates the robust mesh adaptivity in isogeometric collocation. The proposed method reconciles collocation at the Greville points via the discretization of the strong form of the equilibrium equations. The adaptive refinement in collocation is guided by the gradient‐based error estimator. Moreover, the resolution of the nonlinear equations governing the contact problem is derived from a strong form to avoid the disadvantages of numerical integration. Numerical examples are presented to demonstrate the effectiveness, robustness, and straightforward implementation of the present method for adaptive analysis. [ABSTRACT FROM AUTHOR]

Summary: A stochastic approach to model crack propagation in random heterogeneous media, using mesoscopic representations of elastic and fracture properties, is presented. In order to obtain reference results, Monte‐Carlo simulations are first conducted on microstructural samples in which a pre‐existing crack is propagated by means of a phase‐field approach. These computations are used to estimate the subscale‐induced randomness on the macroscopic response of the domain. Mesoscopic descriptors are then introduced to investigate scale transition. Elasticity tensor random fields are specifically defined, at that stage, through a moving‐window upscaling approach. The mesoscopic fracture toughness, which is assumed homogeneous and deterministic, is identified by solving an inverse problem involving the macroscopic peak force. A stochastic model is subsequently constructed in which the mesoscopic elasticity is described as a non‐Gaussian random field. This model allows the multiscale‐informed elastic counterpart in the phase‐field formulation to be sampled without resorting to computational homogenization. The results obtained with the sample‐based and model‐based mesoscopic descriptions are finally compared with those corresponding to the full‐scale microscopic model. It is shown, in particular, that the mesoscopic elasticity‐phase‐field formulation associated with statically uniform boundary conditions enables the accurate predictions of the mean elastic response and mean peak force. [ABSTRACT FROM AUTHOR]