Loss Of Positive Definiteness Research Articles

A stabilisation approach for topology optimisation of hyperelastic structures with the SIMP method

This paper presents a novel computational approach for SIMP-based Topology Optimisation (TO) of hyperelastic materials at large strains. During the TO process for structures subjected to very large deformations, and especially in the presence of intermediate density regions, the standard Newton-solver (or its arc length variant) have been reported not to converge (refer to References Wang et al. (2014), Lahuerta et al. (2013) and Liu et al. (2017)). In this paper, the new TO stabilisation technique proposed in Ortigosa et al. (2019) in the context of level-set TO, initially devised to alleviate numerical instabilities inherent to level-set TO, is extended for the TO by means of the SIMP method. The success of the methodology rests on the combination of two distinct key ingredients. First, the nonlinear equilibrium equations of motion for intermediate TO design stages are solved in a non-exact albeit consistent incrementally linearised fashion by splitting the design load into a number of load increments. Second, the resulting linearised tangent elasticity tensor is locally stabilised (regularised) in order to prevent its loss of positive definiteness and, thus, avoid the loss of convexity of the discrete tangent operator. This solution strategy is shown to be extremely robust in the context of density-based TO, where the constitutive law of the underlying evolving solid structure is a mixture of solid and void constituents, the latter classically defined by means of a fictitious strain energy. The robustness and applicability of this TO methodological approach are thoroughly demonstrated through an ample spectrum of challenging numerical examples, ranging from benchmark two-dimensional (plane stress) examples to larger scale three-dimensional applications. Crucially, the performance of all the final designs has been tested at a post-processing stage without adding any source of artificial stiffness. Specifically, an arc-length Newton–Raphson method has been employed in conjunction with a ratio of the material parameters for void and solid regions of 10−12.

A new stabilisation approach for level-set based topology optimisation of hyperelastic materials

This paper introduces a novel computational approach for level-set based topology optimisation of hyperelastic materials at large strains. This, to date, is considered an unresolved open problem in topology optimisation due to its extremely challenging nature. Two computational strategies have been proposed to address this problem. The first strategy resorts to an arc-length in the pre-buckling region of intermediate topology optimisation (TO) iterations where numerical difficulties arise (associated with nucleation, disconnected elements, etc.), and is then continued by a novel regularisation technique in the post-buckling region. In the second strategy, the regularisation technique is used for the entire loading process at each TO iteration. The success of both rests on the combination of three distinct key ingredients. First, the nonlinear equilibrium equations of motion are solved in a consistent incrementally linearised fashion by splitting the design load into a number of load increments. Second, the resulting linearised tangent elasticity tensor is stabilised (regularised) in order to prevent its loss of positive definiteness and, thus, avoid the loss of convexity of the discrete tangent operator. Third, and with the purpose of avoiding excessive numerical stabilisation, a scalar degradation function is applied on the regularised linearised elasticity tensor, based on a novel regularisation indicator field. The robustness and applicability of this new methodological approach are thoroughly demonstrated through an ample spectrum of challenging numerical examples, ranging from benchmark two-dimensional (plane stress) examples to larger scale three-dimensional applications. Crucially, the performance of all the designs has been tested at a post-processing stage without adding any source of artificial stiffness. Specifically, an arc-length Newton-Raphson method has been employed in conjunction with a ratio of the material parameters for void and solid regions of 10− 12.

Square-Root Consider Filters with Hyperbolic Householder Reflections

Maintaining positive-definiteness of a filter’s error covariance estimate is of paramount importance to any stable and reliable estimation scheme. Two widely used methods for protecting a filter from loss of positive-definiteness are that of consider filtering and square-root filtering. Despite their widespread use in estimation applications, a filtering methodology that combines the two techniques has yet to be presented in a unified, streamlined architecture. This paper develops a new approach to square-root filtering using hyperbolic Householder reflections. A specialization of the method is considered to permit computationally efficient applications, and a numerical example pertaining to a ballistic reentry is presented to validate the proposed approaches.

An anisotropic phase‐field model for transversely isotropic barium titanate with bounded moduli

AbstractThis contribution presents a phase‐field model for transversely isotropic barium titanate, which allows for the adjustment of the full set of anisotropic material parameters. It is a direct extension of the work by Schrade et al. [1] who proposed a phase‐field model for ferroelectrics in the framework of invariant theory. In the present contribution, the loss of positive definiteness is avoided by formulating energetic terms that provide upper and lower bounds for all material moduli involved. We show the characteristics of the formulation by a set of numerical examples in two and three dimensions. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Electromechanical instabilities in fiber-constrained, dielectric-elastomer composites subjected to all-around dead-loading

In this work we investigate the possible development of instabilities in a certain class of dielectric-elastomer composites (DECs) subjected to all-around dead electromechanical loading. The DECs consist of a dielectric elastomer matrix phase constrained to plane strain deformations by means of aligned, long, rigid dielectric fibers of elliptical cross-section that are also aligned but randomly distributed in the transverse plane. Two types of instabilities are considered: loss of positive definiteness (LPD), and loss of strong ellipticity (LE). LPD simply corresponds to the loss of local convexity of the homogenized electroelastic stored-energy function for the DECs and can be of two types depending on the resulting instability modes. When the modes are aligned with the ‘principal’ solution, the instability corresponds to a maximum in the nominal electric field, possibly followed by snapping behavior. Alternatively, when the modes are orthogonal to the principal solution, the instability corresponds to a bifurcation from the principal solution. The LE, on the other hand, corresponds to LPD of the electromechanical acoustic tensor and manifests itself by the onset of highly localized shear band instabilities. Our results show that the stability, as well as the type of instability (when stability is lost), of the DECs depends sensitively on the loading conditions and is also affected by the microstructure of the DECs (i.e. the volume fraction of the fibers and their aspect ratio). In particular, it is found that the non-aligned LPD instabilities typically precede aligned LPD and LE instabilities, especially for in-plane isotropic microstructures, but aligned (limit load) LPD and LE instabilities are also possible for composites with anisotropic microstructures. In this context, it should be noted that the possible development of non-aligned LPD bifurcation instabilities appears to have been ignored by prior work. Fortunately, however, such instabilities can usually be avoided by orienting the long (in-plane) axis of the fibers parallel to the tensile stress direction and orthogonal to the applied electric field.

Fiber-constrained, dielectric-elastomer composites: Finite-strain response and stability analysis

This paper presents homogenization estimates for the finite-strain effective response of a certain class of dielectric elastomer composites (DECs) subjected to electromechanical loading conditions. The DECs consist of a dielectric elastomer matrix phase constrained to undergo plane strain deformations by means of aligned, long, rigid-dielectric fibers of elliptical cross section that are also aligned but randomly distributed in the transverse plane. The estimates for the effective electro-active response are obtained by means of available estimates for the purely mechanical response of such composites, together with a partial decoupling strategy/approximation. Such homogenization estimates can then be used to assess the effect of various microstructural parameters, such as the concentration and cross-sectional shape of the fibers, on the overall electromechanical response of the DECs, when subjected to an electric potential difference across suitably positioned soft electrodes. In addition, three different instability and failure mechanisms are investigated: loss of positive definiteness, loss of strong ellipticity and dielectric breakdown, with the objective of finding an optimal design of the microstructure and constituent properties for maximal electrostriction before failure.

Asymptotic Stability Analysis for Sheet Metal Forming—Part I: Theory

In this paper is presented a general methodology for predicting puckering instabilities in sheet metal forming applications. A novel approach is introduced which does not use shell theory approximations. The starting point is Hill’s stability functional for a three-dimensional rate-independent stressed solid which is modified for contact. By using a multiple scale asymptotic technique with respect to the small dimensionless thickness parameter ε, one can derive the two-dimensional version of the stability functional which is accurate up to Oε4, thus taking into account bending effects. Loss of positive definiteness of this functional indicates possibility of a puckering instability in a sheet metal forming problem with a known stress and deformation state. An advantage of the proposed method is that the puckering investigation is independent of the algorithm used for calculating the deformed state of the sheet. [S0021-8936(00)00804-7]

Simulation of time-dependent viscoelastic channel Poiseuille flow at high Reynolds numbers

The development of a fully spectral, three-dimensional (in space) and time-dependent simulation for viscoelastic flows is described. For smooth flow geometries, such as the plane Poiseuille flow which is studied here, the method combines the advantages of a high accuracy, exponentially fast convergent with mesh refinement and spectral approximation with those of an efficient numerical implementation well suited for a vector/parallel computer architecture. This approach is shown here to lead to a stable numerical approximation for We ≈ 0(1) provided the size of the time step is kept small enough. Three different viscoelastic constitutive equations of the differential type are considered here: the upper convected Maxwell, the Oldroyd-B and the Chilcott-Rallison models. At large enough times, and before a stationary state is reached, numerical instabilities set in depending on the time-step size and initial conditions. Before the onset of the numerical instabilities, the conformation tensor is always observed to lose its positive-definiteness. For the constitutive equations utilized here this is due to the accumulation of numerical error since it can be shown that, if the constitutive equations are integrated in time exactly, the conformation tensor remains always positive definite. For the UCM model, this loss of positive definiteness is associated with the loss of evolutionarity of the governing equations. This is presumed to be responsible for the breakdown of the numerical simulations. Examination of the results of the three-dimensional simulations has shown similar results for the rms values of the velocity field and the Reynolds stress as compared to the Newtonian case. However, sharp boundary layers have been observed for all the components of the conformation tensor, for both their mean and rms values. These results were obtained at a Reynolds number of 5000, for initial perturbations to the steady Poiseuille flow solution in the form of the most unstable eigenvectors with a relative amplitude of the order of 1 2% in the energy norm.