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We consider a model convection-diffusion problem and present our recent analysis and numerical results regarding mixed finite element formulation and discretization in the singular perturbed case when the convection term dominates the problem. Using the concepts of optimal norm and saddle point reformulation, we found new error estimates for the case of uniform meshes. We compare the standard linear Galerkin discretization to a saddle point least square discretization that uses quadratic test functions, and explain the non-physical oscillations of the discrete solutions. We also relate a known upwinding Petrov–Galerkin method and the stream-line diffusion discretization method, by emphasizing the resulting linear systems and by comparing appropriate error norms. The results can be extended to the multidimensional case in order to find efficient approximations for more general singular perturbed problems including convection dominated models. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new vectorized MATLAB-based algorithm for efficient serial computation of global matrix/force arising from finite element method (FEM) for meshes of any type and approximation order in linear elasticity. Because for-loops in MATLAB are very slow, we propose a modified process that takes advantage of vectorization and sparse assembly to achieve good performance while using the same memory as the standard algorithm. For this purpose, by using good programming practices, the implementation of this scheme is succinctly described and can be integrated into any MATLAB package dealing with FEM. Specifically, attention is paid to the calculation of the triplet (row index, column index, matrix components) as well as the assembly of the global stiffness matrix, mass matrix and force vector. Additionally, an extension of the proposed approach for Mindlin plate theory and functionally graded materials is outlined. Finally, the accuracy of this strategy is verified on selected numerical tests after comparing the obtained results with those of ABAQUS. In terms of performance, the study conducted on a set of meshes considering the standard algorithm and two other well-known MATLAB vectorized algorithms revealed that: (i) for a 2D beam problem meshed with P 1 -triangle elements, a speedup of about 8 and 15 is achieved with sparse and fsparse, respectively. (ii) for a 3D plate problem meshed with P 1 -tetrahedral elements, a speedup of about 4 and 8 is achieved with sparse and fsparse, respectively. When compared to ABAQUS performance, the proposed scheme results in a computational time that is about five times smaller. [ABSTRACT FROM AUTHOR]

Second-generation high-temperature superconducting (HTS) conductors, specifically rare earth-barium-copper-oxide (REBCO) coated conductor (CC) tapes, are promising candidates for high-energy and high-field superconducting applications. With respect to epoxy-impregnated REBCO composite magnets that comprise multilayer components, the thermomechanical characteristics of each component differ considerably under extremely low temperatures and strong electromagnetic fields. Traditional numerical models include homogenized orthotropic models, which simplify overall field calculation but miss detailed multi-physics aspects, and full refinement (FR) ones that are thorough but computationally demanding. Herein, we propose an extended multi-scale approach for analyzing the multi-field characteristics of an epoxy-impregnated composite magnet assembled by HTS pancake coils. This approach combines a global homogenization (GH) scheme based on the homogenized electromagnetic T-A model, a method for solving Maxwell's equations for superconducting materials based on the current vector potential T and the magnetic field vector potential A, and a homogenized orthotropic thermoelastic model to assess the electromagnetic and thermoelastic properties at the macroscopic scale. We then identify "dangerous regions" at the macroscopic scale and obtain finer details using a local refinement (LR) scheme to capture the responses of each component material in the HTS composite tapes at the mesoscopic scale. The results of the present GH-LR multi-scale approach agree well with those of the FR scheme and the experimental data in the literature, indicating that the present approach is accurate and efficient. The proposed GH-LR multi-scale approach can serve as a valuable tool for evaluating the risk of failure in large-scale HTS composite magnets. [ABSTRACT FROM AUTHOR]

Surface cracks are commonly observed in coatings and films. When structures with coatings are subject to stretching, opening mode cracks are likely to form on the surface, which may further lead to other forms of damage, such as interfacial delamination and substrate damage. Possible crack forms include cracks extending towards the interface and channeling across the film. In this paper, a two-dimensional numerical model is proposed to obtain the structural strain energy at arbitrary crack lengths for bilayer structures under uniaxial tension. The energy release rate and structural stress intensity factors can be obtained accordingly, and the effects of geometry and material features on fracture characteristics are investigated, with most crack patterns being confirmed as unstable. The proposed model can also facilitate the analysis of the stress distribution in periodic crack patterns of films. The results from the numerical model are compared with those obtained by the finite element method (FEM), and the accuracy of the theoretical results is demonstrated. [ABSTRACT FROM AUTHOR]

This paper develops a family of new weak Galerkin (WG) finite element methods (FEMs) for solving linear elasticity in the primal formulation. For a convex quadrilateral mesh, degree k ≥ 0 vector-valued polynomials are used independently in element interiors and on edges for approximating the displacement. No penalty or stabilizer is needed for these new methods. The methods are free of Poisson-locking and have optimal order (k + 1) convergence rates in displacement, stress, and dilation (divergence of displacement). Numerical experiments on popular test cases are presented to illustrate the theoretical estimates and demonstrate efficiency of these new solvers. Extension to cuboidal hexahedral meshes is briefly discussed. [ABSTRACT FROM AUTHOR]

A transient multi-physics model incorporated with an electromagneto-thermomechanical coupling is developed to capture the multi-field behavior of a single-pancake (SP) insert no-insulation (NI) coil in a hybrid magnet during the charging and discharging processes. The coupled problem is resolved by means of the finite element method (FEM) for the magneto-thermo-elastic behaviors and the Runge-Kutta method for the transient responses of the electrical circuits of the hybrid superconducting magnet system. The results reveal that the transient multi-physics responses of the insert NI coil primarily depend on the charging/discharging procedure of the hybrid magnet. Moreover, a reverse azimuthal current and a compressive hoop stress are induced in the insert NI coil during the charging process, while a forward azimuthal current and a tensile hoop stress are observed during the discharging process. The induced voltages in the insert NI coil can drive the currents flowing across the radial turns where the contact resistance exists. Therefore, it brings forth significant Joule heat, causing a temperature rise and a uniform distribution of this heat in the coil turns. Accordingly, a thermally/mechanically unstable or quenching event may be encountered when a high operating current is flowing in the insert NI coil. It is numerically predicted that a quick charging will induce a compressive hoop stress which may bring a risk of buckling instability in the coil, while a discharging will not. The simulations provide an insight of hybrid superconducting magnets under transient start-up or shutdown phases which are inevitably encountered in practical applications. [ABSTRACT FROM AUTHOR]

We investigate indentation of a linear viscoelastic half-space with periodically distributed heterogeneities by a smooth, rigid, axisymmetric indenter of an arbitrary profile. For this, we first show how such materials may be homogenized directly in the time-domain to yield a homogeneous,anisotropic, linear viscoelastic medium. For the layered half-space that we then focus upon homogenization leads to a transversely isotropic, homogeneous, viscoelastic medium. We then replace the layered half-space in the original indentation problem by its homogenized counterpart. For axisymmetric indenters, for loadings in which the contact area is non-decreasing, we solve this indentation problem analytically by extending the correspondence principle of Graham (in Q Appl Math 26:167–174, 1968. https://doi.org/10.1090/qam/99860). Finally, we compare the results of homogenization with direct finite element computations of both displacement and force controlled indentation tests. We find that the contact pressure, the depth of penetration and the displacement of the top surface of the layered half-space are well approximated by those obtained through homogenization, and the approximation improves as the layer thickness is reduced, which is precisely when FE computations become harder. The latter aspect makes the development of the present approach useful. [ABSTRACT FROM AUTHOR]

Rare earth barium copper oxide (REBCO) is the most researched and commercialized second-generation high-temperature superconducting material. Due to the anisotropic structure, strong deformation sensitivity, and central field errors caused by screening current effects, it is still a challenge for commercialization applications. In this study, the transversely isotropic constitutive relationship is selected as the mechanical model based on the structural characteristics of REBCO tapes, and suitable microelements are selected to equate the elastic constants using their average stress-strain relationships. Then, a two-dimensional axisymmetric model for coils wound by single-layer tapes is constructed to analyze the dependence of the electric-magnetic-force distribution in the tape on the strain. Finally, the anisotropic approximation of the homogenized bulk method is used to equate large-turn high-field coils, and the electric-magnetic-force distribution characteristics of the coils with/without screening effects and mechanical strain conditions are investigated, respectively. The results reveal that the mechanical strain has a weakening effect on the electromagnetic field distribution of superconducting tapes, but causes a significant enhancement in the force field distribution. In the presence of 0.5% mechanical strain, the maximum weakening of the peak value of the current density and the critical current density inside the high-field coil can reach about 8% and 13%, respectively, with a nearly 5 times increase in the peak stress. The screening current makes the current field distribution inside the coil improve by about 10 times. The screening current induced magnetic field can reach up to 0.8 T, making the relative error of the high-field coil center up to 7.8%. [ABSTRACT FROM AUTHOR]

In this paper, we design two classes of stabilized mixed finite element methods for linear elasticity on simplicial grids. In the first class of elements, we use H (div , Ω S)-Pk and L2 (Ω ; ℝn)- Pk-1 to approximate the stress and displacement spaces, respectively, for 1 ≤ k ≤ n and employ a stabilization technique in terms of the jump of the discrete displacement over the edges/faces of the triangulation under consideration; in the second class of elements, we use H01 (Ω ; ℝn)- Pk to approximate the displacement space for 1 ≤ k ≤ n, and adopt the stabilization technique suggested by Brezzi, Fortin, and Marini [19]. We establish the discrete inf-sup conditions, and consequently present the a priori error analysis for them. The main ingredient for the analysis are two special interpolation operators, which can be constructed using a crucial H(div) bubble function space of polynomials on each element. The feature of these methods is the low number of global degrees of freedom in the lowest order case. We present some numerical results to demonstrate the theoretical estimates.

The performance of the flexibility and stretchability of flexible electronics depends on the mechanical structure design, for which a great progress has been made in past years. The use of prestrain in the substrate, causing the compression of the transferred interconnects, can provide high elastic stretchability. Recently, the nonbuckling interconnects have been designed, where thick bar replaces thin ribbon layout to yield scissor-like in-plane deformation instead of in- or out-of-plane buckling modes. The nonbuckling interconnect design achieves significantly enhanced stretchability. However, combined use of prestrain and nonbuckling interconnects has not been explored. This paper aims to study the mechanical behavior of nonbuckling interconnects bonded to the prestrained substrate analytically and numerically. It is found that larger prestrain, longer straight segment, and smaller arc radius yield smaller strain in the interconnects. On the other hand, larger prestrain can also cause larger strain in the interconnects after releasing the prestrain. Therefore, the optimization of the prestrain needs to be found to achieve favorable stretchability. [ABSTRACT FROM AUTHOR]

The soft robotics displays huge advantages over their rigid counterparts when interacting with living organisms and fragile objects. As one of the most efficient actuators toward soft robotics, the soft pneumatic actuator (SPA) can produce large, complex responses with utilizing pressure as the only input source. In this work, a new approach that combines digital light processing (DLP) and injection-assisted post-curing is proposed to create SPAs that can realize different functionalities. To enable this, we develop a new class of photo-cross linked elastomers with tunable mechanical properties, good stretchability, and rapid curing speed. By carefully designing the geometry of the cavities embedded in the actuators, the resulting actuators can realize contracting, expanding, flapping, and twisting motions. In addition, we successfully fabricate a soft self-sensing bending actuator by injecting conductive liquids into the three-dimensional (3D) printed actuator, demonstrating that the present method has the potential to be used to manufacture intelligent soft robotic systems. [ABSTRACT FROM AUTHOR]

Quench and mechanical behaviors are critical issues in high temperature superconducting coils. In this paper, the quench characteristics in the rare earth barium copper oxide (REBCO) pancake coil at 4.2 K are analyzed, and a two-dimensional (2D) axisymmetric electro-magneto-thermal model is presented. The effects of the constituent materials, background field, and coil size are analyzed. An elastoplastic mechanical model is used to study the corresponding mechanical responses during the quench propagation. The variations of the temperature and strain in superconducting layers are compared. The results indicate that the radial strain evolutions can reflect the transverse quench propagation and the tensile hoop and radial stresses in superconducting layers increase with the quench propagation. The possible damages are discussed with the consideration of the effects of the background field and coil size. It is concluded that the high background field significantly increases the maximum tensile hoop and radial stresses in quenching coils and local damage may be caused. [ABSTRACT FROM AUTHOR]

We consider a preconditioning technique for mixed methods with a conforming test space and a nonconforming trial space. Our method is based on the classical saddle point disccretization theory for mixed methods and the theory of preconditioning symmetric positive definite operators. Efficient iterative processes for solving the discrete mixed formulations are proposed and choices for discrete compatible spaces are provided. For discretization, a basis is needed only for the test spaces and assembly of a global saddle point system is avoided. We provide approximation properties for the discretization and iteration errors and also provide a sharp estimate for the convergence rate of the proposed algorithm in terms of the condition number of the elliptic preconditioner and the discrete inf − sup and sup − sup constants of the pair of discrete spaces. We focus on applications to elliptic PDEs with discontinuous coefficients. Numerical results for two- and three-dimensional domains are included to support the proposed method. [ABSTRACT FROM AUTHOR]

The development of additive manufacturing techniques for composite structures brought the emergence of a new class of composite materials: the variable angle-tow composites. Additive manufacturing of reinforced polymers allows the tow to be placed along a curvilinear path in each lamina. Accordingly, optimised solutions with enhanced properties can be manufactured. In this work, the multi-scale two-level optimisation strategy for composites is exploited to optimise the strength of variable angle-tow composites subject to mechanical and manufacturing constraints. At the first step of the strategy, the laminate strength is described through a laminate-level failure criterion based on tensor invariants and on the first-order shear deformation theory. The lay-up design phase makes use of quasi-trivial solutions and integrates a check on the first-ply failure in order to ensure the integrity of the whole laminate. The effectiveness of the strategy as well as of the proposed failure criterion is proven on some meaningful test cases. [ABSTRACT FROM AUTHOR]

We investigate indentation by a smooth, rigid indenter of a two-dimensional half-space comprised of periodically arranged linear-elastic layers with different constitutive responses. Identifying the half-space's material parameters as periodic functions in space, we utilize the theory of periodic homogenization to approximate the layered heterogeneous material by a linear-elastic, homogeneous, but anisotropic medium. This approximation becomes exact as the layer thickness becomes infinitesimal. In this way, we reduce the original problem to the indentation of an anisotropic, homogeneous, linear-elastic half-space by a smooth, rigid indenter. The latter is solved analytically by formulating and resolving the corresponding matrix Riemann–Hilbert boundary-value problem in complex analysis. Thus, we obtain an approximate, but analytical solution for the indentation of a layered heterogeneous medium. We then compare this solution with finite element computations of the indentation on the original layered, heterogenous half-space. We conclude that (a) the contact pressure on the layered, heterogenous half-space is well approximated by that obtained through homogenization, and the approximation improves as the layer thickness is decreased, or if the indentation force is increased; (b) the upper bound of the difference between the two contact pressures depends only upon the ratio of the Young's moduli of the two materials constituting the heterogenous medium and their Poisson's ratio; and (c) the average variation of the discontinuous von Mises stress in the layered half-space is well approximated by the one found in the homogenized half-space. The approach presented here can be utilized for a diverse array of indentation and contact problems of finely mixed heterogeneous media, and is also amenable to systematic improvements. [ABSTRACT FROM AUTHOR]

This paper introduces an adaptive finite element method (AFEM) using the newest vertex bisection and marking exclusively according to the error estimator without special treatment of oscillation. By the combination of the global lower bound and the localized upper bound of the posteriori error estimator, perturbation of oscillation, and cardinality of the marked element set, it is proved that the AFEM is quasi-optimal for linear elasticity problems in two dimensions, and this conclusion is verified by the numerical examples. [ABSTRACT FROM AUTHOR]

We establish sharp well posedness and approximation estimates for variational saddle point systems at the continuous level. The main results of this note have been known to be true only in the finite dimensional case. Known spectral results from the discrete case are reformulated and proved using a functional analysis view, making the proofs in both cases, discrete and continuous, less technical than the known discrete approaches. We focus on analyzing the special case when the formis bounded, symmetric, and coercive, and the mixed formis bounded and satisfies a standardor LBB condition. We characterize the spectrum of the symmetric operators that describe the problem at the continuous level. For a particular choice of the inner product on the product space of, we prove that the spectrum of the operator representing the system at continuous level is. As consequences of the spectral description, we find the minimal length interval that contains the ratio between the norm of the data and the norm of the solution, and prove explicit approximation estimates that depend only on the continuity constant and the continuous and the discretecondition constants [ABSTRACT FROM AUTHOR]

This article is concerned with the numerical modeling of unilateral contact problems in an electro-elastic material with Tresca friction law and electrical conductivity condition. First, we prove the existence and uniqueness of the weak solution of the model. Rather than deriving a solution method for the full coupled problem, we present and study a successive iterative (decomposition) method. The idea is to solve successively a displacement subproblem and an electric potential subproblem in block Gauss-Seidel fashion. The displacement subproblem leads to a constraint non-differentiable (convex) minimization problem for which we propose an augmented Lagrangian algorithm. The electric potential unknown is computed explicitly using the Riesz's representation theorem. The convergence of the iterative decomposition method is proved. Some numerical experiments are carried out to illustrate the performances of the proposed algorithm. [ABSTRACT FROM PUBLISHER]

In this paper, we introduce a goal-oriented procedure for the updating of mechanical models. It is based as usual on information coming from measurement data, but these data are post-processed in a convenient way in order to firstly update model parameters which are the most influent for the prediction of a given quantity of interest. The objective is thus to perform a partial model calibration that enables to obtain an approximate value of the quantity of interest with sufficient accuracy and minimal model identification effort. The updating method uses the constitutive relation error framework as well as duality and adjoint techniques. It leads to a convenient strategy, mainly based on sensitivity analysis, that selects the relevant parameter set to be updated and also provides for useful quantitative tools in order to define optimal experiments. Performances of the approach are analyzed on examples involving linear elasticity and transient thermal models. [ABSTRACT FROM AUTHOR]

We extend the multiscale finite element method (MsFEM) as formulated by Hou and Wu in [Hou T.Y., Wu X.-H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 1997, 134(1), 169-189] to the PDE system of linear elasticity. The application, motivated by the multiscale analysis of highly heterogeneous composite materials, is twofold. Resolving the heterogeneities on the finest scale, we utilize the linear MsFEM basis for the construction of robust coarse spaces in the context of two-level overlapping domain decomposition preconditioners. We motivate and explain the construction and show that the constructed multiscale coarse space contains all the rigid body modes. Under the assumption that the material jumps are isolated, that is they occur only in the interior of the coarse grid elements, our numerical experiments show uniform convergence rates independent of the contrast in Young's modulus within the heterogeneous material. Elsewise, if no restrictions on the position of the high coefficient inclusions are imposed, robustness cannot be guaranteed any more. These results justify expectations to obtain coefficient-explicit condition number bounds for the PDE system of linear elasticity similar to existing ones for scalar elliptic PDEs as given in the work of Graham, Lechner and Scheichl [Graham I.G., Lechner P.O., Scheichl R., Domain decomposition for multiscale PDEs, Numer. Math., 2007, 106(4), 589-626]. Furthermore, we numerically observe the properties of the MsFEM coarse space for linear elasticity in an upscaling framework. Therefore, we present experimental results showing the approximation errors of the multiscale coarse space w.r.t. the fine-scale solution. [ABSTRACT FROM AUTHOR]

We present new rectangular mixed finite elements for linear elasticity. The approach is based on a modification of the Hellinger-Reissner functional in which the symmetry of the stress field is enforced weakly through the introduction of a Lagrange multiplier. The elements are analogues of the lowest order elements described in Arnold et al. (Math Comput 76:1699-1723, ). Piecewise constants are used to approximate the displacement and the rotation. The first order BDM elements are used to approximate each row of the stress field. [ABSTRACT FROM AUTHOR]

The main purpose of this study is to investigate the efficacy and usefulness of a class of recently proposed models that could be reasonable candidates for describing the response of brittle elastic materials. The class of models that are considered allows for a non-linear relationship between the linearized elastic strain and the Cauchy stress, and this allows one to describe situations wherein the stress increases while the strain yet remains small. Thus one would be in a position to model the response of brittle elastic bodies in the neighborhood of the tips of cracks and notches. In this paper we study the behavior of such models in a plate with a V-notch subject to a state of anti-plane stress. This geometrical simplification enables us to characterize the governing equation for the problem by means of the Airy stress function, though the constitutive relation is a non-linear relation between the linearized strain and the stress. We study the problem numerically by appealing to the finite element method. We find that the numerical solutions are stable. We are able to provide some information regarding the nature of the solution near the tip of the V-notch. In particular we find stress concentration in the vicinity of the singularity. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce and analyze Uzawa algorithms for non-symmetric saddle point systems. Convergence for the algorithms is established based on new spectral results about Schur complements. A new Uzawa type algorithm with optimal relaxation parameters at each new iteration is introduced and analyzed in a general framework. Numerical results supporting the efficiency of the algorithms are presented for finite element discretization of steady state Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

Abstract: We develop a residual-based a posteriori error analysis for the augmented mixed methods introduced in for the problem of linear elasticity in the plane. We prove that the proposed a posteriori error estimators are both reliable and efficient. Numerical experiments confirm these theoretical properties and illustrate the ability of the corresponding adaptive algorithms to localize the singularities and large stress regions of the solutions. [ABSTRACT FROM AUTHOR]

In order to image the shear modulus in soft tissue, for medical diagnosis, given one component of measured displacements as a function of time on an imaging plane, two related direct finite element-based inversion algorithms are presented. One algorithm is based on the governing equations expressed in the frequency domain, and the other is in the time domain. The algorithms consider the complete equations of isotropic, small deformation, elasto-dynamics, where the hydrostatic stress is also treated as an unknown. The algorithms reconstruct both the shear modulus and hydrostatic stress fields, and regularization is used to stabilize the hydrostatic stress recovery. An algorithm is also developed for reconstructing the second displacement component, while simultaneous finding a smooth approximation to the measured displacement component to reduce noise. Shear modulus reconstruction results from both algorithms, using experimental ultrasound measurements on a tissue-mimicking phantom, are presented, and the merits and drawbacks of each algorithm are discussed. [ABSTRACT FROM AUTHOR]

Abstract: For any continuous bilinear form defined on a pair of Hilbert spaces satisfying the compatibility Ladyshenskaya–Babušca–Brezzi condition, symmetric Schur complement operators can be defined on each of the two Hilbert spaces. In this paper, we find bounds for the spectrum of the Schur operators only in terms of the compatibility and continuity constants. In light of the new spectral results for the Schur complements, we review the classical Babušca–Brezzi theory, find sharp stability estimates, and improve a convergence result for the inexact Uzawa algorithm. We prove that for any symmetric saddle point problem, the inexact Uzawa algorithm converges, provided that the inexact process for inverting the residual at each step has the relative error smaller than 1/3. As a consequence, we provide a new type of algorithm for discretizing saddle point problems, which combines the inexact Uzawa iterations with standard a posteriori error analysis and does not require the discrete stability conditions. [Copyright &y& Elsevier]

In linear elasticity problems, the pressure is usually introduced for computing the incompressible state. In this paper is presented a technique which is based on a power series expansion of the displacement with respect to the inverse of Lamé's coefficient $\lambda$ . It does not require to introduce the pressure as an auxiliary unknown. Moreover, low degree finite elements can be used. The same technique can be applied to Stokes or Navier-Stokes equations, and can be extended to more general parameterized partial differential equations. Discretization error and convergence are analyzed and illustrated by some numerical results. [ABSTRACT FROM AUTHOR]