On large deformations of thin elasto-plastic shells: Implementation of a finite rotation model for quadrilateral shell element

A large-deformation model for thin shells composed of elasto-plastic material is presented in this work, Formulation of the shell model, equivalent to the two-dimensional Cosserat continuum, is developed from the three-dimensional continuum by employing standard assumptions on the distribution of the displacement held in the shell body, A model for thin shells is obtained by an approximation of terms describing the shell geometry. Finite rotations of the director field are described by a rotation vector formulation. An elasto-plastic constitutive model is developed based on the von Mises yield criterion and isotropic hardening. In this work, attention is restricted to problems where strains remain small allowing for all aspects of material identification and associated computational treatment, developed for small-strain elastoplastic models, to be transferred easily to the present elasto-plastic thin-shell model. A finite element formulation is based on the four-noded isoparametric element. A particular attention is devoted to the consistent linearization of the shell kinematics and elasto-plastic material model, in order to achieve quadratic rate of asymptotic convergence typical for the Newton-Raphson-based solution procedures. To illustrate the main objective of the present approach-namely the simulation of failures of thin elastoplastic shells typically associated with buckling-type instabilities and/or bending-dominated shell problems resulting in formation of plastic hinges-several numerical examples are presented, Numerical results are compared with the available experimental results and representative numerical simulations

Weighted Sobolev spaces and regularity for polyhedral domains

We prove a regularity result for the Poisson problem $-\Delta u = f$, u |\_{\pa \PP} = g on a polyhedral domain \PP \subset \RR^3 using the \BK\ spaces \Kond{m}{a}(\PP). These are weighted Sobolev spaces in which the weight is given by the distance to the set of edges \cite{Babu70, Kondratiev67}. In particular, we show that there is no loss of \Kond{m}{a}--regularity for solutions of strongly elliptic systems with smooth coefficients. We also establish a "trace theorem" for the restriction to the boundary of the functions in \Kond{m}{a}(\PP)

On non-linear dynamics of shells : implementation of energy-momentum conserving algorithm for a finite rotation shell model

Continuum and numerical formulations for non-linear dynamics of thin shells are presented in this work. An elastodynamic shell model is developed from the three-dimensional continuum by employing standard assumptions of the first-order shear-deformation theories. Motion of the shell-directior is described by a singularity-free formulation based on the rotation vector. Temporal discretization is performed by an implicit, one-step, second-order accurate, time-integration scheme. In this work, an energy and momentum conserving algorithm, which exactly preserves the fundamental constants of the shell motion and guaranties unconditional algorithmic stability, is used. It may be regarded as a modification of the standard mid-point rule. Spatial discretization is based on the four-noded isoparametric element. Particular attention is devoted to the consistent linearization of the weak form of the initial boundary value problem discretized in time and space, in order to achieve a quadratic rate of asymptotic convergence typical for the Newton-Raphson based solution procedures. An unconditionally stable time finite element formulation suitable for the long-term dynamic computations of flexible shell-like structures, which may be undergoing large displacements, large rotations and large motions is therefore obtained. A set of numerical examples is presented to illustrate the present approach and the performance of the isoparametric four-noded shell finite element in conjunction with the implicit energy and momentum conserving time-integration algorithm

Stable Generalized Finite Element Method (SGFEM)

The Generalized Finite Element Method (GFEM) is a Partition of Unity Method (PUM), where the trial space of standard Finite Element Method (FEM) is augmented with non-polynomial shape functions with compact support. These shape functions, which are also known as the enrichments, mimic the local behavior of the unknown solution of the underlying variational problem. GFEM has been successfully used to solve a variety of problems with complicated features and microstructure. However, the stiffness matrix of GFEM is badly conditioned (much worse compared to the standard FEM) and there could be a severe loss of accuracy in the computed solution of the associated linear system. In this paper, we address this issue and propose a modification of the GFEM, referred to as the Stable GFEM (SGFEM). We show that the conditioning of the stiffness matrix of SGFEM is not worse than that of the standard FEM. Moreover, SGFEM is very robust with respect to the parameters of the enrichments. We show these features of SGFEM on several examples.Comment: 51 pages, 4 figure

Constructing Parsimonious Analytic Models for Dynamic Systems via Symbolic Regression

Developing mathematical models of dynamic systems is central to many disciplines of engineering and science. Models facilitate simulations, analysis of the system's behavior, decision making and design of automatic control algorithms. Even inherently model-free control techniques such as reinforcement learning (RL) have been shown to benefit from the use of models, typically learned online. Any model construction method must address the tradeoff between the accuracy of the model and its complexity, which is difficult to strike. In this paper, we propose to employ symbolic regression (SR) to construct parsimonious process models described by analytic equations. We have equipped our method with two different state-of-the-art SR algorithms which automatically search for equations that fit the measured data: Single Node Genetic Programming (SNGP) and Multi-Gene Genetic Programming (MGGP). In addition to the standard problem formulation in the state-space domain, we show how the method can also be applied to input-output models of the NARX (nonlinear autoregressive with exogenous input) type. We present the approach on three simulated examples with up to 14-dimensional state space: an inverted pendulum, a mobile robot, and a bipedal walking robot. A comparison with deep neural networks and local linear regression shows that SR in most cases outperforms these commonly used alternative methods. We demonstrate on a real pendulum system that the analytic model found enables a RL controller to successfully perform the swing-up task, based on a model constructed from only 100 data samples

Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods

Use of the stochastic Galerkin finite element methods leads to large systems of linear equations obtained by the discretization of tensor product solution spaces along their spatial and stochastic dimensions. These systems are typically solved iteratively by a Krylov subspace method. We propose a preconditioner which takes an advantage of the recursive hierarchy in the structure of the global matrices. In particular, the matrices posses a recursive hierarchical two-by-two structure, with one of the submatrices block diagonal. Each one of the diagonal blocks in this submatrix is closely related to the deterministic mean-value problem, and the action of its inverse is in the implementation approximated by inner loops of Krylov iterations. Thus our hierarchical Schur complement preconditioner combines, on each level in the approximation of the hierarchical structure of the global matrix, the idea of Schur complement with loops for a number of mutually independent inner Krylov iterations, and several matrix-vector multiplications for the off-diagonal blocks. Neither the global matrix, nor the matrix of the preconditioner need to be formed explicitly. The ingredients include only the number of stiffness matrices from the truncated Karhunen-Lo\`{e}ve expansion and a good preconditioned for the mean-value deterministic problem. We provide a condition number bound for a model elliptic problem and the performance of the method is illustrated by numerical experiments.Comment: 15 pages, 2 figures, 9 tables, (updated numerical experiments

Raviart Thomas Petrov-Galerkin Finite Elements

The general theory of Babu\v{s}ka ensures necessary and sufficient conditions for a mixed problem in classical or Petrov-Galerkin form to be well posed in the sense of Hadamard. Moreover, the mixed method of Raviart-Thomas with low-level elements can be interpreted as a finite volume method with a non-local gradient. In this contribution, we propose a variant of type Petrov-Galerkin to ensure a local computation of the gradient at the interfaces of the elements. The in-depth study of stability leads to a specific choice of the test functions. With this choice, we show on the one hand that the mixed Petrov-Galerkin obtained is identical to the finite volumes scheme "volumes finis \`a 4 points" ("VF4") of Faille, Gallo\"uet and Herbin and to the condensation of mass approach developed by Baranger, Maitre and Oudin. On the other hand, we show the stability via an inf-sup condition and finally the convergence with the usual methods of mixed finite elements.Comment: arXiv admin note: text overlap with arXiv:1710.0439