Showing papers in "Advanced Modeling and Simulation in Engineering Sciences in 2019"

An extension of assumed stress finite elements to a general hyperelastic framework

Abstract: Assumed stress finite elements are known for their extraordinary good performance in the framework of linear elasticity. In this contribution we propose a mixed variational formulation of the Hellinger–Reissner type for hyperelasticity. A family of hexahedral shaped elements is considered with a classical trilinear interpolation of the displacements and different piecewise discontinuous interpolation schemes for the stresses. The performance and stability of the new elements are investigated and demonstrated by the analysis of several benchmark problems. In addition the results are compared to well known enhanced assumed strain elements.

Reduced-order modelling of parametric systems via interpolation of heterogeneous surrogates

Abstract: This paper studies parametric reduced-order modeling via the interpolation of linear multiple-input multiple-output reduced-order, or, more general, surrogate models in the frequency domain. It shows that realization plays a central role and two methods based on different realizations are proposed. Interpolation of reduced-order models in the Loewner representation is equivalent to interpolating the corresponding frequency response functions. Interpolation of reduced-order models in the (real) pole-residue representation is equivalent to interpolating the positions and residues of the poles. The latter pole-matching approach proves to be more natural in approximating system dynamics. Numerical results demonstrate the high efficiency and wide applicability of the pole-matching method. It is shown to be efficient in interpolating surrogate models built by several different methods, including the balanced truncation method, the Krylov method, the Loewner framework, and a system identification method. It is even capable of interpolating a reduced-order model built by a projection-based method and a reduced-order model built by a data-driven method. Its other merits include low computational cost, small size of the parametric reduced-order model, relative insensitivity to the dimension of the parameter space, and capability of dealing with complicated parameter dependence.

Thermodynamic properties of muscle contraction models and associated discrete-time principles

Abstract: Considering a large class of muscle contraction models accounting for actin-myosin interaction, we present a mathematical setting in which solution properties can be established, including fundamental thermodynamic balances. Moreover, we propose a complete discretization strategy for which we are also able to obtain discrete versions of the thermodynamic balances and other properties. Our major objective is to show how the thermodynamics of such models can be tracked after discretization, including when they are coupled to a macroscopic muscle formulation in the realm of continuum mechanics. Our approach allows to carefully identify the sources of energy and entropy in the system, and to follow them up to the numerical applications.

Some robust integrators for large time dynamics

Abstract: This article reviews some integrators particularly suitable for the numerical resolution of differential equations on a large time interval. Symplectic integrators are presented. Their stability on exponentially large time is shown through numerical examples. Next, Dirac integrators for constrained systems are exposed. An application on chaotic dynamics is presented. Lastly, for systems having no exploitable geometric structure, the Borel–Laplace integrator is presented. Numerical experiments on Hamiltonian and non-Hamiltonian systems are carried out, as well as on a partial differential equation.

Benchmark cases for robust explicit time integrators in non-smooth transient dynamics

Abstract: This article introduces benchmark cases for time integrators devoted to non-smooth impact dynamics. It focuses on numerical properties of explicit integrators. Each case tests one necessary numerical property in computational impact dynamics: energy behaviour at impact, angular momentum conservation, non-linear behaviour. The cases are easy to implement and analyse, providing a benchmark well-suited to first numerical studies. We rewrite explicit schemes for non-smooth impact dynamics with unified notations, and analyse them with the benchmark cases.

Best theory diagrams for multilayered structures via shell finite elements

Abstract: Composite structures are convenient structural solutions for many engineering fields, but their design is challenging and may lead to oversizing due to the significant amount of uncertainties concerning the current modeling capabilities. From a structural analysis standpoint, the finite element method is the most used approach and shell elements are of primary importance in the case of thin structures. Current research efforts aim at improving the accuracy of such elements with limited computational overheads to improve the predictive capabilities and widen the applicability to complex structures and nonlinear cases. The present paper presents shell elements with the minimum number of nodal degrees of freedom and maximum accuracy. Such elements compose the best theory diagram stemming from the combined use of the Carrera Unified Formulation and the Axiomatic/Asymptotic Method. Moreover, this paper provides guidelines on the choice of the proper higher-order terms via the introduction of relevance factor diagrams. The numerical cases consider various sets of design parameters such as the thickness, curvature, stacking sequence, and boundary conditions. The results show that the most relevant set of higher-order terms are third-order and that the thickness plays the primary role in their choice. Moreover, certain terms have very high influence, and their neglect may affect the accuracy of the model significantly.

Non-intrusive proper generalized decomposition involving space and parameters: application to the mechanical modeling of 3D woven fabrics

Abstract: In our former works we proposed different Model Order Reduction strategies for alleviating the complexity of computational simulations. In fact we proved that separated representations are specially appealing for addressing many issues, in particular, the treatment of 3D models defined in degenerated domains (those involving very different characteristic dimensions, like beams, plate and shells) as well as the solution of parametrized models for calculating their parametric solutions. However it was proved that the efficiency of solvers based on the construction of such separated representations strongly depends on the affine decompositions (separability) of operators, parameters and geometry. Even if our works proved that different techniques exists for performing such beneficial separation prior of applying the separated representation constructor, the complexity of the solver increases in certain circumstances too much, as the one involving the space separation of complex microstructures concerned by 3D woven fabrics. In this paper we explore an alternative route that allows circumventing the just referred difficulties. Thus, instead of following the standard procedure that consists of introducing the separated representation of the unknown field prior to discretize the models, the strategy here proposed consists of proceeding inversely: first the model is discretized and then the separated representation of the discrete unknown field is enforced. Such a procedure enables the consideration of very complex and non separable features, like complex domains, boundary conditions and microstructures as the ones concerned by homogenized models of complex and rich 3D woven fabrics. It will be proved that such a procedure can be also easily coupled with a non-intrusive treatment of the parametric dimensions by using a sparse hierarchical collocation technique.

Evaluation of shear and membrane locking in refined hierarchical shell finite elements for laminated structures

Abstract: Shear and membrane locking phenomena are fundamental issues of shell finite element models. A family of refined shell elements for laminated structures has been developed in the framework of Carrera Unified Formulation, including hierarchical elements based on higher-order Legendre polynomial expansions. These hierarchical elements were reported to be relatively less prone to locking phenomena, yet an exhaustive evaluation of them regarding the mitigation of shear and membrane locking on laminated shells is still essential. In the present article, numerically efficient integration schemes for hierarchical elements, including also reduced and selective integration procedures, are discussed and evaluated through single-element p-version finite element models. Both shear and membrane locking are assessed quantitatively through the estimation of strain energy components. The numerical results show that the fully integrated hierarchical shell elements can overcome the shear and membrane locking effectively when a sufficiently high polynomial degree is reached. Reduced and selective integration schemes can help with the mitigation of locking on lower-order hierarchical shell elements.

On the coupling of local 3D solutions and global 2D shell theory in structural mechanics

Abstract: Most of mechanical systems and complex structures exhibit plate and shell components. Therefore, 2D simulation, based on plate and shell theory, appears as an appealing choice in structural analysis as it allows reducing the computational complexity. Nevertheless, this 2D framework fails for capturing rich physics compromising the usual hypotheses considered when deriving standard plate and shell theories. To circumvent, or at least alleviate this issue, authors proposed in their former works an in-plane-out-of-plane separated representation able to capture rich 3D behaviors while keeping the computational complexity of 2D simulations. However, that procedure it was revealed to be too intrusive for being introduced into existing commercial softwares. Moreover, experience indicated that such enriched descriptions are only compulsory locally, in some regions or structure components. In the present paper we propose an enrichment procedure able to address 3D local behaviors, preserving the direct minimally-invasive coupling with existing plate and shell discretizations. The proposed strategy will be extended to inelastic behaviors and structural dynamics.

Some comparisons and analyses of time or space discontinuous Galerkin methods applied to elastic wave propagation in anisotropic and heterogeneous media

Abstract: This research work presents some comparisons and analyses of the time discontinuous space–time Galerkin method and the space discontinuous Galerkin method applied to elastic wave propagation in anisotropic and heterogeneous media. Mechanism of both methods to ensure their stability using time or space discontinuities of unknown fields is analyzed and compared. The most general case of anisotropic and heterogeneous media with physical interfaces of discontinuous material properties is considered, especially for the space discontinuous Galerkin method. A new stability result is proved. Numerical applications to different elastic media, more particularly polycrystalline materials containing a large number of physical interfaces, are also presented to confirm theoretical analyses.

A minimally-intrusive fully 3D separated plate formulation in computational structural mechanics

Abstract: Most of mechanical systems and complex structures exhibit plate and shell components. Therefore, 2D simulation, based on plate and shell theory, appears as an appealing choice in structural analysis as it allows reducing the computational complexity. Nevertheless, this 2D framework fails for capturing rich physics compromising the usual hypotheses considered when deriving standard plate and shell theories. To circumvent, or at least alleviate this issue, authors proposed in their former works an in-plane–out-of-plane separated representation able to capture rich 3D behaviors while keeping the computational complexity the one of 2D simulations. In the present paper we propose an efficient integration of fully 3D descriptions into existing plate software.

Modeling of cylindrical composite shell structures based on the Reissner’s Mixed Variational Theorem with a variable separation method

Abstract: This work deals with the modeling of laminated composite and sandwich shells through a variable separation approach based on a Reissner’s Variational Mixed Theorem (RMVT). Both the displacement and transverse stress fields are approximated as a sum of products of separated functions of the in-plane coordinates and the transverse coordinate. This approach yields to a non-linear problem that is solved by an iterative process, in which 2D and 1D problems are separately considered at each iteration. In the thickness direction, a fourth-order expansion in each layer is used. For the in-plane description, classical Finite Element method is used. Numerical examples involving several representative shell configurations (deep/shallow, thick/thin) are addressed to show the accuracy of the present method. It is shown that it can provide quasi-3D results less costly than classical LW computations. In particular, the estimation of the transverse stresses, which are of major importance for damage analysis, is very good.