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The design of shell-infill structures has been a focal point in the topology optimization community due to their advantages in energy absorption characteristics, strength-to weight ratio and bucking resistance. This paper introduces a phase field-based topology optimization method for designing shell-infill structures. Interface-related issues can be easily addressed through the phase field function. A coupled topology optimization process is proposed to establish the connection between the shell and infill, facilitating the generation of optimized structures. The shell thickness, infill pattern and infill volume percentage, can be naturally controlled by different model parameters. Additionally, multiscale phase field topology optimization integrates the numerical homogenization method to evaluate the effective elasticity matrix of the microstructural infill. The approach is introduced for a uniform, periodical microstructure layout in the infill region, thereby achieving superior mechanical properties. Numerical results indicate the effectiveness of the proposed method in the design of both 2D and 3D shell-infill structures. [ABSTRACT FROM AUTHOR]

This paper introduces three dimensional (3D) topology optimization specifically tailored for designing spatially-varying primitive-cubic (CP) type lattice structures. The developed design process consists of three steps: pre-processing, main processing, and post-processing. In the pre-processing step, a surrogate model between lattice geometry variables and effective elasticity tensor is constructed using an artificial neural network. This surrogate model is essential because it enables the quick calculation of the effective elasticity tensor of lattice microstructure. In the subsequent main processing step, multiscale topology optimization is performed. During this step, lattice microstructure size and rotation design variables are optimized together with the macrostructure density design variables. Macrostructures are designed using a well-established three-field density scheme with a SIMP formulation. Formulations for spatially-varying 3D lattice microstructure are newly developed based on a unit quaternion rotation representation scheme. In the final post-processing step, a spatially-varying microstructure is restored using a de-homogenization scheme, newly developed particularly for the primitive-cubic (CP) type 3D lattice structures. The effectiveness and robustness of the proposed formulations are validated through three design examples for the compliance minimization problem. Moreover, a designed lattice structure is fabricated using an additive manufacturing machine. [ABSTRACT FROM AUTHOR]

Abstract This paper presents a novel computational framework for the numerical simulation of the electromechanical response of the myocardium during the cardiac cycle. The paper presents the following main novelties. (1) Two new mixed formulations, tailor-made for active stress and active strain coupling approaches, have been developed and used in conjunction with two different ionic models, namely Bueno-Orovio et al. (2008) and Ten Tusscher et al. (2004). Taking as a reference the mixed formulations introduced by Bonet et al. (2015) in the context of nonlinear elasticity, the proposed formulations include as unknown fields the geometry and the transmembrane potential (and possibly a Lagrange multiplier enforcing weakly the incompressibility constraint) as well as the deformation gradient tensor, its cofactor, its determinant, the gradient of the transmembrane potential and their respective work conjugates. The Finite Element implementation of these formulations is shown in this paper, where a static condensation procedure is presented in order to yield an extremely competitive computational approach. (2) A comprehensive and rigorous study of different ionic models (i.e Bueno-Orovio and Ten Tusscher) and electromechanical activation couplings (i.e active strain and active stress) has been carried out. (3) An analytical and numerical analysis of the possible loss of ellipticity and polyconvexity of one of the most widely used constitutive models in the context of cardiac mechanics is carried out in this paper, putting forward possible polyconvexifications of the existing model. (4) In addition, an invariant representation of Guccione's constitutive model is proposed. Finally, a series of numerical examples are included in order to demonstrate the applicability and robustness of the proposed formulations. [ABSTRACT FROM AUTHOR]

This paper presents a high order finite element implementation of the convex multi-variable electro-elasticity for large deformations large electric fields analyses and its particularisation to the case of small strains through a staggered scheme. With an emphasis on accurate geometrical representation, a high performance curvilinear finite element framework based on an a posteriori mesh deformation technique is developed to accurately discretise the underlying displacement-potential variational formulation. The performance of the method under near incompressibility and bending actuation scenarios is analysed with extremely thin and highly stretched components and compared to the performance of mixed variational principles recently reported by Gil and Ortigosa (2016) and Ortigosa and Gil (2016). Although convex multi-variable constitutive models are elliptic hence, materially stable for the entire range of deformations and electric fields, other forms of physical instabilities are not precluded in these models. In particular, physical instabilities present in dielectric elastomers such as pull-in instability, snap-through and the formation, propagation and nucleation of wrinkles and folds are numerically studied with a detailed precision in this paper, verifying experimental findings. For the case of small strains, the essence of the approach taken lies in guaranteeing the objectivity of the resulting work conjugates, by starting from the underlying convex multi-variable internal energy, whence avoiding the need for further symmetrisation of the resulting Maxwell and Minkowski-type stresses at small strain regime. In this context, the nonlinearity with respect to electrostatic counterparts such as electric displacements is still retained, hence resulting in a formulation similar but more competitive with the existing linearised electro-elasticity approaches. Virtual prototyping of many application-oriented dielectric elastomers are carried out with an eye on pattern forming in soft robotics and other potential medical applications. [ABSTRACT FROM AUTHOR]

This paper addresses some problems of modelling incompressible media using meshfree methods with a stabilised mixed formulation. The aims of this paper are twofold: to avoid the volumetric locking that occurs at the incompressible limit; and to circumvent the inf–sup condition while approximating both displacement and pressure with the same set of nodes and shape functions. We employ a modified polynomial pressure projection stabilisation method to stabilise and filter spurious pressure modes, where the gradient is corrected as proposed by Duan et al. (2014). The gradient correction is used to counteract the lack of variational consistency when numerical integration of the rational shape functions is performed, a threat that could undermine the stabilisation effort. The resulting novel method passes the mixed patch tests, and is demonstrated to be effective and optimally accurate in linear elasticity and Stokes flow test cases. [ABSTRACT FROM AUTHOR]

We propose an FFT-based iterative algorithm for solving the Lippmann–Schwinger equation in the context of periodic homogenization of infinitely contrasted linear elastic composites. Our work initially reformulates the Moulinec–Suquet, Eyre–Milton and Monchiet–Bonnet schemes using a residual formulation. Subsequently, we introduce an enhanced scheme, termed Adaptive Eyre–Milton (AEM), as a natural extension of the EM scheme where we optimize a relaxation parameter to minimize the residual. We demonstrate the unconditional linear convergence of the AEM scheme, regardless of initialization and the chosen reference material. The paper further extends the AEM scheme to handle composites with both pores and infinitely rigid inclusions. Practical implementation aspects and illustrative applications in two- and three-dimensional settings are discussed, highlighting the efficiency of the proposed AEM scheme. We particularly emphasize the scheme's robustness for materials with infinitely large contrasts in elastic properties. [ABSTRACT FROM AUTHOR]

In this paper, we present a first-order Stress-Hybrid Virtual Element Method (SH-VEM) on six-noded triangular meshes for linear plane elasticity. We adopt the Hellinger–Reissner variational principle to construct a weak equilibrium condition and a stress based projection operator. In each element, the stress projection operator is expressed in terms of the nodal displacements, which leads to a displacement based formulation. This stress-hybrid approach assumes a globally continuous displacement field while the stress field is discontinuous across each element. The stress field is initially represented by divergence-free tensor polynomials based on Airy stress functions, but we also present a formulation that uses a penalty term to enforce the element equilibrium conditions, referred to as the Penalty Stress-Hybrid Virtual Element Method (PSH-VEM). Numerical results are presented for PSH-VEM and SH-VEM, and we compare their convergence to the composite triangle FEM and B-bar VEM on benchmark problems in linear elasticity. The SH-VEM converges optimally in the L 2 norm of the displacement, energy seminorm, and the L 2 norm of hydrostatic stress. Furthermore, the results reveal that PSH-VEM converges in most cases at a faster rate than the expected optimal rate, but it requires the selection of a suitably chosen penalty parameter. [ABSTRACT FROM AUTHOR]

In this paper, a hybrid variational framework for the Virtual Element Method (VEM) is proposed and a family of polygonal elements for plane elasticity is developed. Under specific assumptions, it is proved that the minimization of Total Potential Energy and the projection operation typical of enhanced VEM can be deduced from the stationary condition of the Hellinger–Reissner mixed functional. Since the designed elements can be regarded as either enhanced VEM or hybrid finite elements, they are named as Hybrid Virtual Element Method (HVEM). The primary variables are the displacements along the element boundary and the stress field within the element domain. The assumed stress field is expressed on a polynomial basis that satisfies the divergence-free condition. In the HVEM formulation, stabilization-free elements can be obtained using two concepts, namely hyper-stability and iso-stability. In particular, the iso-stable cases show the best solution in recovering both displacement and stress fields. Several numerical applications are developed, assessing the stability for a single distorted element. The proposed family of HVEM proves to be accurate, also if coarse meshes are used. Additionally, the effectiveness of the proposed HVEM is demonstrated for typical structural elements, testing the convergence rate and comparing the results with analytic or other numerical solutions. [ABSTRACT FROM AUTHOR]

This paper presents an enhanced theoretical formulation and associated computational framework for brittle fracture in elastic solids within the context of configurational mechanics, building on the authors’ previous paper, Kaczmarczyk et al. (2014). The local form of the first law of thermodynamics provides an equilibrium condition for the crack front, expressed in terms of the configurational forces. Applying the principle of maximal energy dissipation, it is shown that the direction of the crack propagation is given by the direction of the configurational forces. In combination with a fracture criterion, these are utilised to determine the position of the continuously evolving crack front. This exploitation of the crack front equilibrium condition leads to a completely new, implicit, crack propagation formulation. A monolithic solution strategy is adopted, solving simultaneously for both the material displacements (i.e. crack extension) and the spatial displacements. The resulting crack path is resolved as a discrete displacement discontinuity, where the material displacements of the nodes on the crack front change continuously, without the need for mesh splitting or the use of enrichment techniques. In order to trace the dissipative loading path, an arc-length procedure is adopted that controls the incremental crack area growth. In order to maintain mesh quality, smoothing of the mesh is undertaken as a continuous process, together with face flipping, node merging and edge splitting where necessary. Hierarchical basis functions of arbitrary polynomial order are adopted to increase the order of approximation without the need to change the finite element mesh. Performance of the formulation is demonstrated by means of three representative numerical simulations, demonstrating both accuracy and robustness. [ABSTRACT FROM AUTHOR]

The paper presents a theoretical framework for the shape sensitivity analysis of systems governed by partial differential equations. The proposed approach, based on geometrical concepts borrowed from differential geometry, shows that sensitivity of a performance function (i.e. any function of the solution of the problem) with respect to a given design variable can be represented mathematically as a Lie derivative, i.e. the derivative of that performance function along a flow representing the continuous shape modification of the geometrical model induced by the variation of the considered design variable. Theoretical formulae to express sensitivity analytically are demonstrated in detail in the paper, and applied to a nonlinear magnetostatic and a linear elastic problem, following both the direct and the adjoint approaches. Following the analytical approach, one linear system of which only the right-hand side needs be evaluated (the system matrix being known already) has to be solved for each of the design variables in the direct approach, or for each performance functions in the adjoint approach. A substantial gain in computation time is obtained this way compared to a finite difference evaluation of sensitivity, which requires solving a second nonlinear system for each design variable. This is the main motivation of the analytical approach. There is some freedom in the definition of the auxiliary flow that represents the shape modification. We present a method that makes benefit of this freedom to express sensitivity locally as a volume integral over a single layer of finite elements connected to both sides of the surfaces undergoing shape modification. All sensitivity calculations are checked with a finite difference in order to validate the analytic approach. Convergence is analyzed in 2D and 3D, with first and second order finite elements. [ABSTRACT FROM AUTHOR]

Arc-length methods based on Newton–Raphson (NR) iterative numerical solvers are indispensable in non-linear mechanics to trace the equilibrium curves of systems with snap-back or snap-through behaviours. An example is a crack propagating in a rubber-like solid, which can generate extremely sharp snap-backs when the elastic energy is released. A standard arc-length solver satisfies the arc-length constraint which only limits the increment in displacements and in load level to a fixed amount. Thus, it cannot guarantee the satisfaction of the energy balance in the rate form. This work proposes an arc-length based on the J -integral, and assumes the Griffith balance as a constraint equation. Firstly, a fracture criterion (critical load and direction of crack increment) is formulated, consistent with the thermodynamic principle of maximum dissipation: a crack propagates in the direction of the maximum strain energy release rate. For this purpose, this paper provides an explicit and simple expression of the J -integral for varying angles. This fracture criterion does not require the computation of mixed mode Stress Intensity Factors, or asymptotic solutions or derivatives of the tangent stiffness matrix. By doing so, it is shown that the proposed method easily reconciles with the theory of Configurational Forces. Secondly, because of the explicit expression of J provided in this paper, it is easy to linearize the discretized equations of motions consistently. The method proves able to handle very sudden snap-backs occurring under large strains, and both brittle and ductile crack propagation. A particular example of such behaviour is kirigami (paper-cuts) structures made of graphene. The proposed arc-length can capture, for these structures, the transitions from brittle to ductile crack propagation for the same crack patterns, but different pre-crack lengths. [ABSTRACT FROM AUTHOR]

This paper presents a structural optimisation method in three-dimensional acoustic–elastic coupled problems. The proposed optimisation method finds an optimal allocation of elastic materials which reduces the sound level on some fixed observation points. In the process of the optimisation, configuration of the elastic materials is expressed with a level set function, and the distribution of the level set function is iteratively updated with the help of the topological derivative. The topological derivative is associated with state and adjoint variables which are the solutions of the acoustic–elastic coupled problems. In this paper, the acoustic–elastic coupled problems are solved by a BEM–FEM coupled solver, in which the fast multipole method (FMM) and a multi-frontal solver for sparse matrices are efficiently combined. Along with the detailed formulations for the topological derivative and the BEM–FEM coupled solver, we present some numerical examples of optimal designs of elastic sound scatterer to manipulate sound waves, from which we confirm the effectiveness of the present method. [ABSTRACT FROM AUTHOR]

In this paper, we study the performance of the Generalized Finite Element Method (GFEM) applied to the Poisson problem with crack singularities. Recently, a GFEM with modified Heaviside enrichments was proposed in Gupta et al. (2013) to approximate the solution of a 2D elasticity problem with a crack. It was shown that the GFEM is indeed a Stable GFEM (SGFEM), i.e., it yields the optimal order of convergence and its conditioning is not worse than that of the standard finite element method (FEM). However, the robustness of the conditioning of the GFEM with respect to the position of the mesh relative to the crack was not addressed in Gupta et al. (2013). In this paper, we observed that the conditioning of the GFEM with the enrichments used in Gupta et al. (2013) is not robust when applied to approximate the solution of the Poisson problem. Moreover, the order of convergence may not be optimal when the position of the mesh is changed with respect to the crack interfaces. We proposed using additional singular enrichments at the nodes close to the crack near the crack-tip. We proved that the GFEM, with enrichments proposed in this paper, yields optimal order of convergence irrespective of the position of the mesh. Moreover, with a local orthogonalization procedure, we have shown through numerical experiments that the conditioning of this GFEM is not worse than that of the standard FEM and the conditioning is robust with respect to the position of the mesh. Thus the GFEM, with the enrichments suggested in this paper, is indeed an SGFEM when applied to a Poisson problem with the crack singularities. [ABSTRACT FROM AUTHOR]

The peridynamic theory of solid mechanics is a nonlocal reformulation of the classical continuum mechanics theory. At the continuum level, it has been demonstrated that classical (local) elasticity is a special case of peridynamics. Such a connection between these theories has not been extensively explored at the discrete level. This paper investigates the consistency between nearest-neighbor discretizations of linear elastic peridynamic models and finite difference discretizations of the Navier–Cauchy equation of classical elasticity. Although nearest-neighbor discretizations in peridynamics have been numerically observed to present grid-dependent crack paths or spurious microcracks, this paper focuses on a different, analytical aspect of such discretizations. We demonstrate that, even in the absence of cracks, such discretizations may be problematic unless a proper selection of weights is used. Specifically, we demonstrate that using the standard meshfree approach in peridynamics, nearest-neighbor discretizations do not reduce, in general, to discretizations of corresponding classical models. We study nodal-based quadratures for the discretization of peridynamic models, and we derive quadrature weights that result in consistency between nearest-neighbor discretizations of peridynamic models and discretized classical models. The quadrature weights that lead to such consistency are, however, model-/discretization-dependent. We motivate the choice of those quadrature weights through a quadratic approximation of displacement fields. The stability of nearest-neighbor peridynamic schemes is demonstrated through a Fourier mode analysis. Finally, an approach based on a normalization of peridynamic constitutive constants at the discrete level is explored. This approach results in the desired consistency for one-dimensional models, but does not work in higher dimensions. The results of the work presented in this paper suggest that even though nearest-neighbor discretizations should be avoided in peridynamic simulations involving cracks, such discretizations are viable, for example for verification or validation purposes, in problems characterized by smooth deformations. Moreover, we demonstrate that better quadrature rules in peridynamics can be obtained based on the functional form of solutions. [ABSTRACT FROM AUTHOR]

This paper proposes an identification strategy based on the principles of the Modified Constitutive Relation Error (M-CRE). The basic idea is to construct mechanical fields and material parameters that are a trade-off between all the available information but with no further hypothesis. The paper focuses on the identification of elastic properties from DIC (Digital Image Correlation) data in statics and the dealing of various kinds of boundary conditions (unknown or known) is especially discussed. We study our methodology on numerical examples in both homogeneous and heterogeneous isotropic elasticity, but a target application is the orthotopic behavior of aeronautic composites. [ABSTRACT FROM AUTHOR]

This work is concerned with the construction of statistical surrogates for concurrent multiscale modeling in structures comprising nonlinear random materials. The development of surrogates approximating a homogenization operator is a fairly classical topic that has been addressed through various methods, including polynomial- and deep-learning-based models. Such approaches, and their extensions to probabilistic settings, remain expensive and hard to deploy when the nonlinear upscaled quantities of interest exhibit large statistical variations (in the case of non-separated scales, for instance) and potential non-locality. The aim of this paper is to present a methodology that addresses this particular setting from the point of view of probabilistic learning. More specifically, we formulate the approximation problem using conditional statistics, and use probabilistic learning on manifolds to draw samples of the nonlinear constitutive model at mesoscale. Two applications, relevant to inverse problem solving and forward propagation, are presented in the context of nonlinear elasticity. We show that the framework enables accurate predictions (in probability law), despite the small amount of training data and the very high levels of nonlinearity and stochasticity in the considered system. [ABSTRACT FROM AUTHOR]

Because of the best approximation property, traditional Bubnov–Galerkin numerical methods have proven immensely successful in modeling self-adjoint problems, such as heat conduction, elasticity, and so on. However, a numerical instability arises in these (and central finite difference) methods for problems with strong convection. In this class of problems, the convective transport term can lead to large spurious oscillations but can be handled by the class of Petrov–Galerkin methods. In particular, the upwind-type schemes and their variational and subgrid descendants have been substantially developed over the years for an effective weak-form Galerkin solution that precludes these instabilities. Nevertheless, the scale of development of upwind methods for strong-form collocation is substantially smaller, where numerical oscillations are also observed when they are straightforwardly applied to convection-dominated problems without special treatment. To this end, this paper presents a new upwind collocation method. First, the connection between the upwind finite difference scheme and the gradient smoothing technique in meshfree methods is established. It is then shown that selecting the collocation points as meshfree nodal points is not optimal; selecting the collocation points according to the flow direction and Péclet number is then studied. The upwind effect is achieved without introducing artificial parameters and is trivial to generalize for multi-dimensional cases. Cross-wind diffusion is also not observed in the solution. An error analysis is presented, and the effectiveness of the proposed methodology is well demonstrated by the steady and unsteady numerical examples. • Severe oscillations in node-based collocation methods have been found in convection-dominated problems. • Relationship between upwind stabilization and gradient smoothing technique is built. • Upwind reproducing kernel collocation method is designed. • No artificial parameter for stabilization is required in the entire formulation. [ABSTRACT FROM AUTHOR]

This paper presents an approach to the numerical estimation of effective properties of highly porous materials based on the scaled boundary finite element method (SBFEM). The latter can be formulated on quadtree meshes with hanging nodes and thus facilitates the efficient mesh generation and analysis of a large number of randomly created samples. To generate the corresponding Representative Volume Elements (RVEs), an improved Random Sequential Addition (RSA) method with overlap is used. Moreover, an alternative image-based digital matrix method for generating random assemblies of circular particles of the same size is proposed. In addition, a modified smoothing algorithm to treat the staircase boundaries present in a quadtree mesh is developed in order to accurately and efficiently capture the sinter-neck features of overlapping circular particles. The proposed image-based mesh generation and analysis procedure is used to evaluate the relationship between microstructural parameters and effective Young's modulus considering a large number of samples. In this context, a new dimensionless morphological parameter N f c is proposed. The latter represents the morphology and connectivity of voids and is shown to be a useful indicator with respect to the effective elastic properties of highly porous ceramic matrix material. • Scaled boundary finite element method for homogenization analysis of highly porous materials. • Model generation method for large number of Representative Volume Elements (RVE) of porous ceramic matrix material. • New automatic quadtree meshing and smoothing algorithm with high accuracy and efficiency for complex structures. • New dimensionless parameter that exhibits a stronger correlation with the material's geometric properties and effective elastic properties than porosity. [ABSTRACT FROM AUTHOR]

The use of Electro-Active Polymers (EAPs) for the fabrication of evermore sophisticated miniaturised soft robotic actuators has seen an impressive development in recent years. The incorporation of crystallographic anisotropic micro-architectures, within an otherwise nearly uniform isotropic soft polymer matrix, has shown great potential in terms of advanced three-dimensional actuation (i.e. stretching, bending, twisting), especially at large strains, that is, beyond the onset of geometrical pull-in instabilities. From the computational point of view, the design of accurate and robust albeit efficient constitutive models is a very active area of research. This paper introduces a novel polyconvex phenomenological invariant-based transversely isotropic formulation (and relevant computational frameworks) for the simulation of transversely isotropic EAPs at large strains, where the ab initio satisfaction of polyconvexity is exploited to ensure the robustness of numerical results for any range of deformations and applied electric fields. The paper also presents key important results both in terms of the existence of minimisers and material stability of coupled electro-mechanics, enhancing previous works in the area of large strain elasticity. In addition, a comprehensive series of selected numerical examples is included in order to demonstrate the effect that the anisotropic orientation and the contrast of material properties, as well as the level of deformation and electric field, have upon the response of the EAP when subjected to large three-dimensional stretching, bending and torsion, including the possible development of wrinkling and the potential loss of ellipticity in ill-posed constitutive models. [ABSTRACT FROM AUTHOR]

In this paper, a single stress integral equation is presented for solving multi-medium elasticity problems, and by using a newly proposed method for treating arbitrarily high order of singular boundary integrals, a new method is developed for computing the stresses on the interfaces of multi-media. Comparing to conventional multi-domain boundary element methods, the presented Interface Integral BEM (IIBEM) is more efficient in computational time, data preparing, and program coding. However, a big issue is encountered in IIBEM when computing the stresses on interfaces since the commonly used traction-recovery method in computing outer boundary stresses cannot be applied on the interfaces. Therefore, a direct method for handling a hyper-singular stress interface integral equation has to be used to obtain the interface stresses. In the direct method used in the paper, singularities are analytically removed by expressing the non-singular part of the integration kernel as a power series in a local distance defined on a projection line/plane, and the stresses on the interfaces can be evaluated precisely. Numerical examples are given to verify the correctness of the derived boundary-interface integral equations. [ABSTRACT FROM AUTHOR]

This paper presents a novel computational formulation for large strain polyconvex elasticity. The formulation, based on the original ideas introduced by Schröder et al. (2011), introduces the deformation gradient (the fibre map), its adjoint (the area map) and its determinant (the volume map) as independent kinematic variables of a convex strain energy function. Compatibility relationships between these variables and the deformed geometry are enforced by means of a multi-field variational principle with additional constraints. This process allows the use of different approximation spaces for each variable. The paper extends the ideas presented in Schröder et al. (2011) by introducing conjugate stresses to these kinematic variables which can be used to define a generalised convex complementary energy function and a corresponding complementary energy principle of the Hellinger–Reissner type, where the new conjugate stresses are primary variables together with the deformed geometry. Both compressible and incompressible or nearly incompressible elastic models are considered. A key element to the developments presented in the paper is the new use of a tensor cross product, presented for the first time by de Boer (1982), page 76, which facilitates the algebra associated with the adjoint of the deformation gradient. For the numerical examples, quadratic interpolation of the displacements, piecewise linear interpolation of strain and stress fields and piecewise constant interpolation of the Jacobian and its stress conjugate are considered for compressible cases. In the case of incompressible materials two formulations are presented. First, continuous quadratic interpolation for the displacement together with piecewise constant interpolation for the pressure and second, linear continuous interpolation for both displacement and pressure stabilised via a Petrov–Galerkin technique. [ABSTRACT FROM AUTHOR]

In this paper we propose a numerical scheme for the calculation of stresses and corresponding consistent tangent moduli for hyperelastic material models, which are derived in terms of the first and second derivatives of a strain energy function. This numerical scheme provides a compact model-independent framework, which means that once the framework is implemented, any other hyperelastic material model can be incorporated by solely modifying the energy function. The method is based on the numerical calculation of strain energy derivatives using hyper-dual numbers and thus referred to as hyper-dual step derivative (HDSD). The HDSD does neither suffer from roundoff errors nor from truncation errors and is thereby a highly accurate method with high stability being insensitive to perturbation values. Furthermore, it enables the calculation of derivatives of arbitrary order. This is a great advantage compared to other numerical approaches as, e.g., the finite difference approximation which is highly sensitive with respect to the perturbation value and which thus only yields accurate approximations for a small regime of perturbation values. Another alternative, the complex-step derivative approximation enables highly accurate derivatives for a wide range of small perturbation values, but it only provides first derivatives and is thus not able to calculate stresses and moduli at once. In this paper, representative numerical examples using an anisotropic model are provided showing the performance of the proposed method. In detail, an introductory example shows the insensitivity with respect to the perturbation values and the higher accuracy compared to the finite difference scheme. Furthermore, examples demonstrate the robustness and simple implementation of the HDSD scheme in finite element software. It turns out that the higher accuracy compared with other approaches can still be achieved in reasonable computing time. [ABSTRACT FROM AUTHOR]

This paper introduces a new computational framework for the analysis of large strain fast solid dynamics. The paper builds upon previous work published by the authors (Gil et al., 2014) [48] , where a first order system of hyperbolic equations is introduced for the simulation of isothermal elastic materials in terms of the linear momentum, the deformation gradient and its Jacobian as unknown variables. In this work, the formulation is further enhanced with four key novelties. First, the use of a new geometric conservation law for the co-factor of the deformation leads to an enhanced mixed formulation, advantageous in those scenarios where the co-factor plays a dominant role. Second, the use of polyconvex strain energy functionals enables the definition of generalised convex entropy functions and associated entropy fluxes for solid dynamics problems. Moreover, the introduction of suitable conjugate entropy variables enables the derivation of a symmetric system of hyperbolic equations, dual of that expressed in terms of conservation variables. Third, the new use of a tensor cross product [61] greatly facilitates the algebraic manipulations of expressions involving the co-factor of the deformation. Fourth, the development of a stabilised Petrov–Galerkin framework is presented for both systems of hyperbolic equations, that is, when expressed in terms of either conservation or entropy variables. As an example, a polyconvex Mooney–Rivlin material is used and, for completeness, the eigen-structure of the resulting system of equations is studied to guarantee the existence of real wave speeds. Finally, a series of numerical examples is presented in order to assess the robustness and accuracy of the new mixed methodology, benchmarking it against an ample spectrum of alternative numerical strategies, including implicit multi-field Fraeijs de Veubeke–Hu–Washizu variational type approaches and explicit cell and vertex centred Finite Volume schemes. [ABSTRACT FROM AUTHOR]

In this paper, we develop a computational approach based on variational principles combined with FFT technique to determine the effective elastic properties of composite materials. The unit cell problem of the composite is recast in a weak and integral form by making use of classic variational approaches, based on the strain and the stress potential, and the Hashin–Shtrikman variational principles. The problem is discretized with Fourier series and the stationary point is computed numerically by means of an iterative scheme. These algorithms use a representation of the local elastic tensor on the double grid (twice the size of the discretization) which is introduced by the integral formulation; it advantageously accounts for the exact geometry of the microstructure and allows the derivation of rigorous bounds of the effective elasticity coefficients. In the second part of the paper, we establish a hierarchy between the different FFT solutions: the strain, the stress based solutions coming from the classic variational formulations and the polarization based solution derived from the Hashin–Shtrikman principle, which depends on the choice of the elastic moduli of the reference material. It is proved that the strain and the stress based solutions deliver optimal bounds since they are always better than those obtained from the Hashin–Shtrikman variational principle when the same discretization is used. Alternatively, when the elastic moduli of the reference material is negative, the polarization based solution provides an estimate of the effective properties which is comprised between the strain based upper bound and the stress based lower bound. [ABSTRACT FROM AUTHOR]

• Multi-material topology optimization framework with anisotropic strengths is developed. • Systematical guideline for preselecting candidate materials before optimization is proposed. • Efficiency of candidate materials on strength performance can be quantified as E σ ¯ / ρ 3. • Existence condition of material i in multi-material designs is stated as: E i σ ¯ i < E i + 1 σ ¯ i + 1 and. (E i σ ¯ i − E i − 1 σ ¯ i − 1 ) / (ρ i 3 − ρ i − 1 3) ≥ (E i + 1 σ ¯ i + 1 − E i − 1 σ ¯ i − 1 ) / (ρ i + 1 3 − ρ i − 1 3) • Both 2D and 3D numerical examples are investigated in this paper. Multi-material structures offer the superior performance and broader designable freedom, and have gained increasing interest benefitting from the advances in additive manufacturing (AM) technique. However, the effect of anisotropic yield strength of these AM-fabricated parts on topology optimization of multi-material structures has so far received little attention. In the current work, an anisotropic strength-based topology optimization method for multi-material structures is developed for maximizing strength performance under total mass constraint, which allows for the consideration of directional dependence and tension-compression asymmetry of yield strengths based on the Tsai-Wu yield criterion. Another challenge for multi-material structures design is that the possible permutations of numerous candidate materials are unlimited, it is impossible that all combinations of multiple candidate materials are tested for the optimal strength performance of multi-material structures. To solve this problem, we propose a preselection method for the inclusion or exclusion of candidate materials based on their elastic properties, yield strengths and densities before optimization. Afterwards, our topology optimization method can automatically find the optimal combination of volume fractions of different candidate materials. Several 2D and 3D numerical examples are investigated for minimizing the maximum of failure indexes of strength for fixed total mass. The advantages of multi-material structures are verified by comparing to single-material designs. The effectiveness of the preselection method of candidate materials is also verified. [ABSTRACT FROM AUTHOR]

We consider the linear elasticity equation, discretized with multi-patch Isogeometric Analysis. A standard discretization error analysis is based on Korn's inequality, which degrades for certain geometries, such as long and thin cantilevers. This phenomenon is known as geometry locking. We observe that high-order methods, like Isogeometric Analysis is beneficial in such a setting. The main focus of this paper is the construction and analysis of a domain decomposition solver, namely an Isogeometric Tearing and Interconnecting (IETI) solver, where we prove that the convergence behavior does not depend on the constant of Korn's inequality for the overall domain, but only on the corresponding constants for the individual patches. Moreover, our analysis is explicit in the choice of the spline degree. Numerical experiments are provided which demonstrates the efficiency of the proposed solver. • High-order methods help mitigating the locking effects related to Korn's inequality. • Isogeometric Analysis is a high-order method with a dof count as a low-order method. • We propose and analyze a domain decomposition solver for the compressible case. • Its performance only mildly depends on problem and discretization parameters. [ABSTRACT FROM AUTHOR]

Abstract: The recent interest in dielectric elastomers has given rise to a pressing need for predictive non-linear electromechanical coupling models. Since elastomers behave elastically and can sustain large deformations, the constitutive laws are naturally based on the formulation of adequate free energy functions. Due to the coupling, such functions include terms which combine the strain tensor and the electric field. In contrast to existing frameworks, this paper proposes to establish the electromechanical coupling by the multiplicative split of the deformation gradient into a part related to the elastic behavior of the material and further one which describes the deformation induced by the electric field. Already available and well tested functions of elastic free energy functions can be immediately deployed without any modifications provided the argument of the function is the strain tensor alone which in turn is defined by the elastic part of the deformation gradient only. An appropriate constitutive relation is formulated for the electrically induced part of the deformation gradient. The paper discusses in depth such a formulation. The approach is elegant, straightforward and above all, provides clear physical insight. The paper presents also a numerical formulation of the theoretical framework based on a meshfree method. Various numerical examples of highly non-linear coupled deformations demonstrate the potential and strength of the theory. [Copyright &y& Elsevier]

Inspired by the introduction of Volumetric Modeling via volumetric representations (V-reps) by Massarwi and Elber in 2016, in this paper we present a novel approach for the construction of isogeometric numerical methods for elliptic PDEs on trimmed geometries, seen as a special class of more general V-reps. We develop tools for approximation and local re-parameterization of trimmed elements for three dimensional problems, and we provide a theoretical framework that fully justifies our algorithmic choices. We validate our approach both on two and three dimensional problems, for diffusion and linear elasticity. • We present a novel approach for constructing IGA methods for elliptic PDEs on V-reps. • Approximation and local re-parameterization tools for 3D cut elements are presented. • We provide a theoretical framework that fully justifies our algorithmic choices. • The developed methods guarantee optimal convergence rates. • Our approach is validated in 2D and 3D for diffusion and linear elasticity problems. [ABSTRACT FROM AUTHOR]

The ability to make decisions based on quantities of interest that depend on variables inferred from measurement finds application in different fields of mechanics and physics. The evaluation of the inferred variables, and hence the quantities of interest, from the measurement typically requires the solution of an inverse problem. For example, in medical imaging the elastic heterogeneity of a tumor and its nonlinear elastic response can be used to distinguish benign tumors from their malignant counterparts. These images of linear and nonlinear elastic parameters of tissue are typically obtained by using a measured displacement field and solving a complex inverse elasticity problem. In this paper we consider circumventing the solution of the inverse problem by using measured displacements as input to a deep convolutional neural network (CNN) and training it to classify tumors on the basis of their elastic heterogeneity and nonlinearity. For a simple, 5-layer CNN trained with 8,000 samples for heterogeneity, and a 4-layer CNN trained with 4,000 samples for nonlinear elasticity we report classification accuracies in the range of 99. 7 % − 99. 9 %. The training and testing data are both obtained from the forward solution of finite element models of samples. We also analyze the weights of the trained model to understand the process through which the network extracts features of elastic moduli from the input displacement images. Finally, we apply the nonlinear elasticity classifier, which is trained entirely using simulated data, to displacement images obtained from ten patients with breast lesions and note that it correctly classifies eight out of ten cases. This application illustrates how data from physics-based models can be used in improving the performance of a data-driven algorithm in data-sparse scenarios. • Learning-based framework to bypass the solution of inverse problems in mechanics. • Analysis of learning process of CNNs in physical and Fourier space. • Demonstration of transfer learning in medical imaging with physics-based simulation. [ABSTRACT FROM AUTHOR]

Abstract This paper proposes an efficient numerical procedure for computing T-stresses of cracks in three-dimensional, linearly elastic, infinite media. The technique is established in a broad framework allowing a medium made of generally anisotropic materials and cracks of arbitrary shape and under general loading conditions to be treated. A pair of weakly singular, weak-form, displacement and traction boundary integral equations is utilized to formulate the key equations governing the unknown crack-face fields. Besides the basic benefits such as the reduction of spatial dimensions of the solution space and the efficient treatment of unbounded domains and remote boundary data, use of such integral equations in the formulation offers additional positive features including the computational simplicity resulting from the weakly singular nature of all involved integrals and the involvement of a complete set of crack-face displacement fields. A weakly singular symmetric Galerkin boundary element method together with the special near-front approximation is utilized to solve for the unknown relative crack-face displacement whereas the sum of the crack-face displacement is obtained from the displacement boundary integral equation for cracks via the Galerkin technique. The latter step is one of the essential aspects of the present study that provides the direct means for determining the T-stress data in terms of the sum of the crack-face displacement in the neighborhood of the crack-front. An extensive numerical study is conducted for various scenarios and a selected set of results is reported to demonstrate both the accuracy and capability of the proposed technique. Highlights • A numerical method is established to determine T-stress for cracks in 3D media. • The formulation is based on a pair of weakly singular weak-form integral equations. • A special element is used for the near-front relative crack-face displacement. • Explicit formulae are used to extract the T-stress from the crack-face data. • Highly accurate T-stress results are obtained using relatively coarse meshes. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate the construction and identification of a new random field model for representing the constitutive behavior of laminated composites. Here, the material is modeled as a random hyperelastic medium characterized by a spatially dependent, stochastic and anisotropic strain energy function. The latter is parametrized by a set of material parameters, modeled as non-Gaussian random fields. From a probabilistic standpoint, the construction is first achieved by invoking information theory and the principle of maximum entropy. Constraints related to existence theorems in finite elasticity are, in particular, accounted for in the formulation. The identification of the parameters defining the random fields is subsequently addressed. This issue is attacked as a two-step problem where the mean model is calibrated in a first step, by imposing a match between the linearized model and nominal values proposed in the literature. The remaining parameters controlling the fluctuations are next estimated by solving an inverse problem in which principal component analysis and the maximum likelihood method are combined. The whole framework is illustrated considering an experimental database where multi-axial measurements are performed on a carbon-epoxy laminate. This work constitutes a first step towards the development of an integrated framework that will support decision making under uncertainty for the design, certification and qualification of composite materials and structures. [ABSTRACT FROM AUTHOR]

Abstract The motivation of this work is to propose a general methodology to deal with complex nonlinear mechanical behaviors in the context of identification and model updating problems. We follow here the principle of the modified Constitutive Relation Error that is an energy-based functional suited to the solution of inverse problems, and originally used in the context of elasticity and elasto-dynamics problems with potentially highly corrupted experimental data. In the paper, we develop and analyze an extended formulation of the modified Constitutive Relation Error functional in order to deal with a wide class of nonlinear mechanical behaviors. The general idea in the construction of the functional is to ensure a strong mechanical content by referring to the thermodynamical framework. In addition, a dedicated numerical process sharing similarities with the LaTIn method is proposed in order to allow effective inversion with reasonable computational cost. The performance of the developed procedure is analyzed on 2D elastic-damage and 3D elasto-visco-plastic cases. Highlights • The first natural extension of the mCRE concept to nonlinear material models is developed. • The introduced functional is based on strong and sound thermodynamics principles. • A dedicated effective numerical strategy is proposed to drive the inversion process. • The methodology is applied and analyzed on representative 2D/3D numerical experiments. [ABSTRACT FROM AUTHOR]

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

Abstract: This paper concerns the adaptation of reduced-order models during simulations of series of elastoviscoplastic problems. In continuation with previous works, this paper aimed at extending the A Priori Hyper-Reduction method (APHR method) for nonlinear thermal problems to nonlinear mechanical problems involving internal variables. This method is an a priori approach because full incremental responses of detailed models are not forecasted in order to build reduced-order models. The recent extension of the Hyper-Reduction method to reduction of mechanical models involving internal variables makes possible the reduction of degrees of freedom and the reduction of integration points. A multi-level formulation is introduced to focus on the capability of the method to perform efficient parallel computations to adapt reduced-order models. [Copyright &y& Elsevier]

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

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

Abstract: This paper presents a new topology optimization scheme for nonlinear electrostatic systems actuated by Coulomb’s forces. For successful optimization, computational issues such as (i) alternating governing equations with respect to spatially defined design variables, (ii) imposing interaction boundary conditions between insulator (air) and conductor (solid) and (iii) control of minimum geometrical feature sizes must be addressed. To address the first two issues, the paper presents a monolithic formulation based on continuum mechanics theory which simultaneously calculates the electric potential and structural displacements. To interpolate between insulator and conductor with continuous design variables, the monolithic approach distinguishes between the permittivity value of the electric Poisson’s equation and that of Maxwell’s stress tensor. For the optimization, standard material interpolations are used for Young’s modulus and permittivity values. Moreover, a recently developed morphology filter is applied to control electrode gaps and other geometrical features. [Copyright &y& Elsevier]

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

Abstract: In the case of linear elasticity, a direct connection between the concept of reduced integration with hourglass stabilization and a mixed method can usually be established. In the non-linear case, this is in general not possible. To overcome this difficulty we suggest in this paper a new concept based on a Taylor expansion of the constitutively dependent quantities with respect to the centre of the element. The push-forward of the second (linear) term of the Taylor series for the first Piola–Kirchhoff stress tensor to the current configuration determines the so-called hourglass stabilization part of the residual force vector. Due to the fact that the element uses only one Gauss point and the hourglass stabilization part is computed by means of a simple functional evaluation, the present element technology is very efficient from the computational point of view. In contrast to the 2D case the computation of the Jacobi determinant only in the centre of the 3D element does not yield the correct volume, if the element shape deviates from being a parallelipiped. It is shown in the paper that the error becomes negligibly small for a relatively coarse discretization. The formulation is free of volumetric locking and can compete with shell formulations up to an aspect ratio of about hundred. For bending-dominated problems, at least two elements over the thickness are needed in order to compute the onset of plastification correctly. The element behaves very robustly in finite elasticity and inelasticity, also when large element distortions occur. [Copyright &y& Elsevier]

In this paper, we study the deep Ritz method for solving the linear elasticity equation from a numerical analysis perspective. A modified Ritz formulation using the H 1 / 2 (Γ D) norm is introduced and analyzed for linear elasticity equation in order to deal with the (essential) Dirichlet boundary condition. We show that the resulting deep Ritz method provides the best approximation among the set of deep neural network (DNN) functions with respect to the "energy" norm. Furthermore, we demonstrate that the total error of the deep Ritz simulation is bounded by the sum of the network approximation error and the numerical integration error, disregarding the algebraic error. To effectively control the numerical integration error, we propose an adaptive quadrature-based numerical integration technique with a residual-based local error indicator. This approach enables efficient approximation of the modified energy functional. Through numerical experiments involving smooth and singular problems, as well as problems with stress concentration, we validate the effectiveness and efficiency of the proposed deep Ritz method with adaptive quadrature. [ABSTRACT FROM AUTHOR]

Adjoint-based error estimation provides the ability to approximate the discretization error for a functional quantity of interest, such as point-wise displacements or stresses. Mesh adaptation provides the ability to control the discretization error to obtain more accurate solutions while still remaining computationally feasible. In this paper, we develop an approach for adjoint-based error estimation and mesh adaptation for nonlinear finite deformation elasticity using a mixed stabilized finite element method. We apply our developed technique to a well known test case, the Cook’s membrane problem, to validate and demonstrate its effectiveness. We then demonstrate the utility of adjoint-based error estimation and mesh adaptation for a three-dimensional example motivated by the study of a cell embedded in a matrix. [ABSTRACT FROM AUTHOR]

Pentamode metamaterials (PMMs) are a new class of three-dimensional (3D) mechanical metamaterials, engineered to have unusual elastic property of vanishing shear modulus. Here 'penta' denotes five, referring to only one non-zero but five vanishing eigenvalues in the elasticity tensor of isotropic materials. PMMs gain their properties from their rationally designed structural architecture rather than their composition, mimicking the behavior of fluids but are solid, hard to compress yet easy to deform. Compared to most up-to-date design methods based on rigid-body double-cone concept, this paper is to propose, for the first time, a new generative design method using topology optimization to find novel micro-lattice architectures, to enable pentamode properties through the overall elastic deformation of the micro-lattice. The design problem is then formulated to make the micro-lattice have a large but realistically attainable ratio of effective bulk modulus compared to the shear modulus, corresponding to the isotropic microstructure with the effective Poisson's ratio close to 0.5. The larger of the ratio, the better of the PMM solids to simulate liquids. Several numerical cases with the additive manufacture technique (SLM: selective laser melting) are used to demonstrate the effectiveness of the proposed topological design method in this paper. [ABSTRACT FROM AUTHOR]

The paper is concerned with the development of efficient and accurate solution procedures for the isogeometric boundary element method (BEM) when applied to problems that contain inclusions that have elastic properties different to the computed domain. This topic has been addressed in previous papers but the approach presented here is a considerable improvement in terms of efficiency and accuracy. One innovation is that initial stresses instead of body forces are used, resulting in discretised equations that can be written using matrix–vector multiplications. This allowed a one step solution to be implemented for elastic inclusions. In addition, a novel approach is used for the computation of strains, that avoids the use of highly singular fundamental solutions. Finally, a new type of inclusion is presented that can be used to model reinforcement bars or rock bolts and where analytical integration can be used. Test examples, where results are compared with Finite Element simulations, show that the proposed approach is sound. • A novel approach to deal with inclusions. • Computation of strains inside the inclusion using strain recovery to avoid highly singular integration. • Introduction of a new inclusion type that is able to represent reinforcement bars or rock bolts, where the integrals can be solved analytically. • Introduction of a one step solution for the case where the inclusions are elastic. [ABSTRACT FROM AUTHOR]

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

This paper addresses an efficient and robust automatic adaptive local multilevel mesh refinement strategy for unilateral frictional contact problems in elastostatics. The proposed strategy couples the Local Defect Correction multigrid method LDC (Hackbusch, 1984) with the ZZ (Zienkiewicz and Zhu, 1987) a posteriori error estimator. An extension of LDC method to frictional contact problems is introduced. An interesting feature of this extended LDC algorithm is that it still only lies on interpolations of displacement fields. Neither forces conservation nor exchange of contact status is required between the refinement levels. The ZZ a posteriori error estimator is exploited to automatically build the sub-grids of the LDC method. A criterion linked to the relative error in stress is used. The efficiency of the proposed strategy is analyzed on examples derived from nuclear engineering. Practical numerical choices are proposed and justified. The refinement process automatically varies and stops with respect to a given tolerance. Post-treatments show that the sub-grids focus around the contact areas and that the converged LDC solution always respects the prescribed tolerance. [ABSTRACT FROM AUTHOR]

This paper provides a finite element treatment for a generalized hyperelastic theory in which the deformation gradient is subject to the multiplicative two-factor decomposition that is commonly used in large strain plasticity. The multiplicative decomposition in the present context is motivated by a theory of balanced internal variables, one of which is the first factor in the decomposition. Previous studies have shown how the associated continuum mechanical formulation then leads not only to the stress equations of equilibrium but also to an additional tensor balance equation that serves to determine the individual factors in the decomposition. Such a theory has been shown to predict loading thresholds associated with the emergence of slip surfaces at grip locations in well-posed boundary value problems. However, previous numerical treatments focused on shooting method procedures that failed to resolve loading plateaus and other key features associated with the emergence of such singular surfaces. In the present work we describe a finite element treatment that resolves such previously obscure details. Additional finite element simulations illustrate new aspects of singular surface emergence in this type of theory, including the emergence of singular surfaces in the interior of the computational domain. [ABSTRACT FROM AUTHOR]

The phase field crystal (PFC) model was extended to the modified phase field crystal (MPFC) model, which is a sixth-order nonlinear damped wave equation, to include not only diffusive dynamics but also elastic interactions. In this paper, we present temporally first- and second-order accurate methods for the MPFC equation, which are based on an appropriate splitting of the energy for the PFC equation. And we use the Fourier spectral method for the spatial discretization. The first- and second-order methods are shown analytically to be unconditionally stable with respect to the energy and pseudoenergy of the MPFC equation, respectively. Numerical experiments are presented demonstrating the accuracy and energy stability of the proposed methods. [ABSTRACT FROM AUTHOR]

Based on the concept of equivalent eigenstrain, a low-order accurate eXtended Finite Element Method (XFEM) is presented. The goal is the determination of the stress intensity factors of cracked homogeneous specimens. The proposed approach differs from the conspicuous amount of existing contributions on this topic. The findings of the present paper highlight aspects so far neglected in the literature, such as the mechanical meaning of additional fields and equations specific to the XFEM approximation of the displacement field. Moreover, based on the plane strain examples simulated in the present study, the proposed XFEM is generally computationally more robust and accurate than existing comparable XFEMs, while keeping a minimal implementation effort. [ABSTRACT FROM AUTHOR]

In this paper, we propose an isogeometric method for solving the linear nearly-incompressible elasticity problem. The method is similar to the B ̄ formulation where the volumetric strain is projected on a lower degree spline space in order to prevent volumetric locking. In our method, we adopt a local projection on a coarser mesh, chosen in order to guarantee optimal convergence. Moreover the locality of the projector allows to maintain the sparsity of the stiffness matrix, that is, the efficiency of the method. The analysis of the method is based on the inf–sup stability of the associated mixed formulation via a macro-element technique for spline functions. The numerical tests confirm the theory of the method. [ABSTRACT FROM AUTHOR]

In this paper, we study efficient numerical schemes of the classical phase field elastic bending energy model that has been widely used to describe the shape deformation of biological lipid vesicles, in which the free energy of the system consists of an elastic bending energy, a surface area constraint and a volume constraint. One major challenge in solving such model numerically is how to design appropriate temporal discretizations in order to preserve energy stability with large time step sizes at the semi-discrete level. We develop a first order and a second order time stepping scheme for this highly nonlinear and stiff parabolic PDE system based on the “Invariant Energy Quadratization” approach. In particular, the resulted semi-discretizations lead to linear systems in space with symmetric positive definite operators at each time step, thus can be efficiently solved. In addition, the proposed schemes are rigorously proved to be unconditionally energy stable. Various numerical experiments in 2D and 3D are presented to demonstrate the stability and accuracy of the proposed schemes. [ABSTRACT FROM AUTHOR]