pinecone/core-2020-05-10-deduplication huggingface dataset:
Time-dependent fractional advection–diffusion equations by an implicit MLS meshless method

Recently, many new applications in engineering and science are governed by a series of fractional partial differential equations (FPDEs). Unlike the normal partial differential equations (PDEs), the differential order in a FPDE is with a fractional order, which will lead to new challenges for numerical simulation, because most existing numerical simulation techniques are developed for the PDE with an integer differential order. The current dominant numerical method for FPDEs is Finite Difference Method (FDM), which is usually difficult to handle a complex problem domain, and also hard to use irregular nodal distribution. This paper aims to develop an implicit meshless approach based on the moving least squares (MLS) approximation for numerical simulation of fractional advection-diffusion equations (FADE), which is a typical FPDE. The discrete system of equations is obtained by using the MLS meshless shape functions and the meshless strong-forms. The stability and convergence related to the time discretization of this approach are then discussed and theoretically proven. Several numerical examples with different problem domains and different nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of the FADE
Output error estimation strategies for discontinuous Galerkin discretizations of unsteady convection‐dominated flows

We study practical strategies for estimating numerical errors in scalar outputs calculated from unsteady simulations of convection‐dominated flows, including those governed by the compressible Navier–Stokes equations. The discretization is a discontinuous Galerkin finite element method in space and time on static spatial meshes. Time‐integral quantities are considered for scalar outputs and these are shown to superconverge with temporal refinement. Output error estimates are calculated using the adjoint‐weighted residual method, where the unsteady adjoint solution is obtained using a discrete approach with an iterative solver. We investigate the accuracy versus computational cost trade‐off for various approximations of the fine‐space adjoint and find that exact adjoint solutions are accurate but expensive. To reduce the cost, we propose a local temporal reconstruction that takes advantage of superconvergence properties at Radau points, and a spatial reconstruction based on nearest‐neighbor elements. This inexact adjoint yields output error estimates at a computational cost of less than 2.5 times that of the forward problem for the cases tested. The calculated error estimates account for numerical error arising from both the spatial and temporal discretizations, and we present a method for identifying the percentage contributions of each discretization to the output error. Copyright © 2011 John Wiley & Sons, Ltd
The thin-layer method in a cross-anisotropic 3-D space

This article presents a generalization of the thin-layer method to three dimensions, a tool that allows assessing layered media subjected to loads eliciting nonsymmetrical wave fields. It is based on a formulation that fully couples the three components of motion, and allows finding effective solutions to either stationary or moving loads of arbitrary shape that act on (or within) horizontally layered media. In particular, it is an ideal tool for finding analytical solutions to the so-called 2.5D problem, which entails loads with arbitrary distribution in one horizontal coordinate direction together with a harmonic (sinusoidal) variation in the other. Inasmuch as the Green's functions for the latter case are found explicitly in the spatial domain without recourse to numerical integration, it allows using such functions — most likely in the context of the boundary element method — as fundamental solutions for problems of soil–structure interaction where the structure is invariant in one horizontal direction, such as a railroad track resting on an embankment. The method entails solving at each frequency of interest two uncoupled eigenvalue problems for generalized SH and SVP waves (i.e. horizontally and vertically polarized shear and pressure waves), after which the fundamental solutions are obtained in closed-form at any desired point in space. Inasmuch as the proposed technique dispenses with at least one of the two inverse Fourier transforms into the spatial domain, in due time the methodology presented is likely to become the preferred tool for a wide class of problems. The technique is first benchmarked against the known solution for a point load and then applied to a rectangular and triangular load distribution. Copyright © 2011 John Wiley & Sons, Ltd
Model Order Reduction based on Proper Generalized Decomposition for the Propagation of Uncertainties in Structural Dynamics

International audienceA priori model reduction methods based on separated representations are introduced for the prediction of the low frequency response of uncertain structures within a parametric stochastic framework. The Proper Generalized Decomposition method is used to construct a quasi-optimal separated representation of the random solution at some frequency samples. At each frequency, an accurate representation of the solution is obtained on reduced bases of spatial functions and stochastic functions. An extraction of the deterministic bases allows for the generation of a global reduced basis yielding a reduced order model of the uncertain structure which appears to be accurate on the whole frequency band under study and for all values of input random parameters. This strategy can be seen as an alternative to traditional constructions of reduced order models in structural dynamics in the presence of parametric uncertainties. This reduced order model can then be used for further analyses such as the computation of the response at unresolved frequencies or the computation of more accurate stochastic approximations at some frequencies of interest. The dynamic response being highly nonlinear with respect to the input random parameters, a second level of separation of variables is introduced for the representation of functions of multiple random parameters, thus allowing the introduction of very ﬁne approximations in each parametric dimension even when dealing with high parametric dimension
PMLs: A direct approach

This brief article outlines a new and rather simple method for obtaining the finite element matrices for a perfectly matched layer used for elastic wave propagation in the context of a frequency-domain formulation. For this purpose, we introduce a fairly mild simplification, which allows applying the stretching functions directly to the mass and stiffness matrices obtained via conventional methods (i.e., elastic elements), a novel strategy that allows circumventing the use of integration via Gaussian quadrature. In essence, the technique proposed herein is equivalent to a direct application of the method of weighted residuals in stretched space, followed by a conversion of the linear dimensions into position-dependent complex-values. Most importantly, numerical tests demonstrate that the technique does work as intended, and in fact, splendidly so
Development of shear locking-free shell elements using an enhanced assumed strain formulation

The degenerated approach for shell elements of Ahmad and co-workers is revisited in this paper. To avoid transverse shear locking effects in four-node bilinear elements, an alternative formulation based on the enhanced assumed strain (EAS) method of Simo and Rifai is proposed directed towards the transverse shear terms of the strain field. In the first part of the work the analysis of the null transverse shear strain subspace for the degenerated element and also for the selective reduced integration (SRI) and assumed natural strain (ANS) formulations is carried out. Locking effects are then justified by the inability of the null transverse shear strain subspace, implicitly defined by a given finite element, to properly reproduce the required displacement patterns. Illustrating the proposed approach, a remarkably simple single-element test is described where ANS formulation fails to converge to the correct results, being characterized by the same performance as the degenerated shell element. The adequate enhancement of the null transverse shear strain subspace is provided by the EAS method, enforcing Kirchhoff hypothesis for low thickness values and leading to a framework for the development of shear-locking-free shell elements. Numerical linear elastic tests show improved results obtained with the proposed formulation. Copyright (C) 2001 John Wiley Sons, Ltd
Arbitrary Lagrangian-Eulerian (ALE) formulation for hyperelastoplasticity

The arbitrary Lagrangian-Eulerian (ALE) description in non-linear solid mechanics is nowadays standard for hypoelastic-plastic models. An extension to hyperelastic-plastic models is presented here. A fractional-step method - a common choice in ALE analysis - is employed for time-marching: every time-step is split into a Lagrangian phase, which accounts for material effects, and a convection phase, where the relative motion between the material and the finite element mesh is considered. In contrast to previous ALE formulations of hyperelasticity or hyperelastoplasticity, the deformed configuration at the beginning of the time-step, not the initial undeformed configuration, is chosen as the reference configuration. As a consequence, convecting variables are required in the description of the elastic response. This is not the case in previous formulations, where only the plastic response contains convection terms. In exchange for the extra convective terms, however, the proposed ALE approach has a major advantage: only the quality of the mesh in the spatial domain must be ensured by the ALE remeshing strategy; in previous formulations, it is also necessary to keep the distortion of the mesh in the material domain under control. Thus, the full potential of the ALE description as an adaptive technique can be exploited here. These aspects are illustrated in detail by means of three numerical examples: a necking test, a coining test and a powder compaction test.Peer ReviewedPostprint (author’s final draft
A novel Bayesian strategy for the identification of spatially-varying
  material properties and model validation: an application to static
  elastography

The present paper proposes a novel Bayesian, computational strategy in the
context of model-based inverse problems in elastostatics. On one hand we
attempt to provide probabilistic estimates of the material properties and their
spatial variability that account for the various sources of uncertainty. On the
other hand we attempt to address the question of model fidelity in relation to
the experimental reality and particularly in the context of the material
constitutive law adopted. This is especially important in biomedical settings
when the inferred material properties will be used to make decisions/diagnoses.
We propose an expanded parametrization that enables the quantification of model
discrepancies in addition to the constitutive parameters. We propose scalable
computational strategies for carrying out inference and learning tasks and
demonstrate their effectiveness in numerical examples with noiseless and noisy
synthetic data.Comment: in International Journal for Numerical Methods in Engineering 201
On a multiscale strategy and its optimization for the simulation of
  combined delamination and buckling

This paper investigates a computational strategy for studying the
interactions between multiple through-the-width delaminations and global or
local buckling in composite laminates taking into account possible contact
between the delaminated surfaces. In order to achieve an accurate prediction of
the quasi-static response, a very refined discretization of the structure is
required, leading to the resolution of very large and highly nonlinear
numerical problems. In this paper, a nonlinear finite element formulation along
with a parallel iterative scheme based on a multiscale domain decomposition are
used for the computation of 3D mesoscale models. Previous works by the authors
already dealt with the simulation of multiscale delamination assuming small
perturbations. This paper presents the formulation used to include geometric
nonlinearities into this existing multiscale framework and discusses the
adaptations that need to be made to the iterative process in order to ensure
the rapid convergence and the scalability of the method in the presence of
buckling and delamination. These various adaptations are illustrated by
simulations involving large numbers of DOFs
Variable kinematic plate elements coupled via Arlequin method

In this work, plate elements based on different kinematic assumptions and variational principles are combined through the Arlequin method. Computational costs are reduced assuming refined models only in those zones with a quasi-three-dimensional stress field, whereas computationally cheap, low-order elements are used in the remaining parts of the plate. Plate elements are formulated on the basis of a unified formulation (UF). Via UF, higher-order, layer-wise and mixed theories can be easily formulated. Classical theories, such as Kirchhoff's and Reissner's models, can be obtained as particular cases. UF is extended to the Arlequin method to derive the matrices that account for the coupling between different theories. Multi-layered composite plates are investigated. Variable kinematic multiple models solutions are assessed towards mono-model results and three-dimensional exact results. Numerical investigation has shown that Arlequin method in the context of UF effectively couples sub-domains having finite elements based upon different theories, reducing the computational costs without loss of accurac
A full-discontinuous Galerkin formulation of non-linear Kirchhoff-Love shells: elasto-plastic finite deformations, parallel computation & fracture applications

Due to its ability to take into account discontinuities, the discontinuous Galerkin (DG) method presents some advantages for modeling crack initiations and propagations. This concept has been recently applied to 3D simulations and to elastic thin bodies. In this last case, the assumption of small elastic deformations before crack initiations or propagations reduces drastically the applicability of the framework to a reduced number of materials.
To remove this limitation, a full-DG formulation of non-linear Kirchhoff-Love shells is presented and is used in combination with an elasto-plastic finite deformations model. The results obtained by this new formulation are in agreement with other continuum elasto-plastic shell formulations.
Then this full-DG formulation of Kirchhoff-Love shells is coupled with the cohesive zone model to
perform thin body fracture simulations. As this method allows considering elasto-plastic constitutive laws in combination with the cohesive model, accurate results compared to the experiments are found. In particular, the crack path and propagation rate of a blasted cylinder are shown to match experimental results. One of the main advantages of this framework is its ability to run in parallel with a high speed-up factor, allowing the simulation of ultra fine meshes.Peer reviewe
A partition of unity FEM for time-dependent diffusion problems using multiple enrichment functions.

An enriched partition of unity FEM is developed to solve time-dependent diffusion problems. In the present formulation, multiple exponential functions describing the spatial and temporal diffusion decay are embedded in the finite element approximation space. The resulting enrichment is in the form of a local asymptotic expansion. Unlike previous works in this area where the enrichment must be updated at each time step, here, the temporal decay in the solution is embedded in the asymptotic expansion. Thus, the system matrix that is evaluated at the first time step may be decomposed and retained for every next time step by just updating the right-hand side of the linear system of equations. The advantage is a significant saving in the computational effort where, previously, the linear system must be reevaluated and resolved at every time step. In comparison with the traditional finite element analysis with p-version refinements, the present approach is much simpler, more efficient, and yields more accurate solutions for a prescribed number of DoFs. Numerical results are presented for a transient diffusion equation with known analytical solution. The performance of the method is analyzed on two applications: the transient heat equation with a single source and the transient heat equation with multiple sources. The aim of such a method compared with the classical FEM is to solve time-dependent diffusion applications efficiently and with an appropriate level of accuracy
A new hole insertion method for level set based structural topology optimization

Structural shape and topology optimization using level set functions is becoming increasingly popular. However, traditional methods do not naturally allow for new hole creation and solutions can be dependent on the initial design. Various methods have been proposed that enable new hole insertion; however, the link between hole insertion and boundary optimization can be unclear. The new method presented in this paper utilizes a secondary level set function that represents a pseudo third dimension in two-dimensional problems to facilitate new hole insertion. The update of the secondary function is connected to the primary level set function forming a meaningful link between boundary optimization and hole creation. The performance of the method is investigated to identify suitable parameters that produce good solutions for a range of problems
The bipenalty method for arbitrary multipoint constraints

In finite element (FE) analysis, traditional penalty methods impose constraints by adding virtual stiffness to the FE system. In dynamics, this can decrease the critical time step of the system when conditionally stable time integration schemes are used by introducing spurious modes with high eigenfrequencies. Recent studies have shown that using mass penalties alongside traditional stiffness penalties can mitigate this effect for systems with a one single-point constraint. In the present work, we extend this finding to include systems with an arbitrary set of multipoint constraints. By analysing the generalised eigenvalue problem, we show that the values of spurious eigenfrequencies may be controlled by the choice of stiffness and mass penalty parameters. The method is demonstrated using numerical examples, including a one-dimensional contact–impact formulation and a two-dimensional crack propagation analysis. The results show that constraint imposition using the bipenalty method can be employed such that the critical time step of an analysis is unaffected, whereas also displaying superiority over the mass penalty method in terms of accuracy and versatility.Peer ReviewedPostprint (author’s final draft
Nonlinear manifold learning for meshfree finite deformation thin shell analysis

Calculations on general point-set surfaces are attractive because of their flexibility and simplicity in the preprocessing but present important challenges. The absence of a mesh makes it nontrivial to decide if two neighboring points in the three-dimensional embedding are nearby or rather far apart on the manifold. Furthermore, the topology of surfaces is generally not that of an open two-dimensional set, ruling out global parametrizations. We propose a general and simple numerical method analogous to the mathematical theory of manifolds, in which the point-set surface is described by a set of overlapping charts forming a complete atlas. We proceed in four steps: (1) partitioning of the node set into subregions of trivial topology; (2) automatic detection of the geometric structure of the surface patches by nonlinear dimensionality reduction methods; (3) parametrization of the surface using smooth meshfree (here maximum-entropy ) approximants; and (4) gluing together the patch representations by means of a partition of unity. Each patch may be viewed as a meshfree macro-element. We exemplify the generality, flexibility, and accuracy of the proposed approach by numerically approximating the geometrically nonlinear Kirchhoff–Love theory of thin-shells. We analyze standard benchmark tests as well as point-set surfaces of complex geometry and topology.Peer ReviewedPostprint (author's final draft
Novel basis functions for the partition of unity boundary element method for Helmholtz problems.

The BEM is a popular technique for wave scattering problems given its inherent ability to deal with infinite domains. In the last decade, the partition of unity BEM, in which the approximation space is enriched with a linear combination of plane waves, has been developed; this significantly reduces the number of DOFs required per wavelength. It has been shown that the element ends are more susceptible to errors in the approximation than the mid-element regions. In this paper, the authors propose that this is due to the use of a collocation approach in combination with a reduced order of continuity in the Lagrangian shape function component of the basis functions. It is demonstrated, using numerical examples, that choosing trigonometric shape functions, rather than classical polynomial shape functions (quadratic in this case), provides accuracy benefits. Collocation schemes are investigated; it is found that the somewhat arbitrary choice of collocating at equally spaced points about the surface of a scatterer is better than schemes based on the roots of polynomials or consideration of the Fock domain
Nonincremental proper generalized decomposition solution of parametric uncoupled models defined in evolving domains

This work addresses the recurrent issue related to the existence of reduced bases related to the solution of parametric models defined in evolving domains. In this first part of the work, we address the case of decoupled kinematics, that is, models whose solution does not affect the domain in which they are defined. The chosen framework considers an updated Lagrangian description of the kinematics, solved by using natural neighbor Galerkin methods within a nonincremental space–time framework that can be generalized for addressing parametric models. Examples showing the performance and potentialities of the proposed methodology are included.International audienceThis work addresses the recurrent issue related to the existence of reduced bases related to the solution of parametric models defined in evolving domains. In this first part of the work, we address the case of decoupled kinematics, that is, models whose solution does not affect the domain in which they are defined. The chosen framework considers an updated Lagrangian description of the kinematics, solved by using natural neighbor Galerkin methods within a nonincremental space–time framework that can be generalized for addressing parametric models. Examples showing the performance and potentialities of the proposed methodology are included