Sparse bayesian polynomial chaos approximations of elasto-plastic material models

In this paper we studied the uncertainty quantification in a functional approximation form of elastoplastic models parameterised by material uncertainties. The problem of estimating the polynomial chaos coefficients is recast in a linear regression form by taking into consideration the possible sparsity of the solution. Departing from the classical optimisation point of view, we take a slightly different path by solving the problem in a Bayesian manner with the help of new spectral based sparse Kalman filter algorithms

Stochastic state estimation via incremental iterative sparse polynomial chaos based Bayesian-Gauss-Newton-Markov-Kalman filter

In this paper is proposed a novel incremental iterative Gauss-Newton-Markov-Kalman filter method for state estimation of dynamic models given noisy measurements. The mathematical formulation of the proposed filter is based on the construction of an optimal nonlinear map between the observable and parameter (state) spaces via a convergent sequence of linear maps obtained by successive linearisation of the observation operator in a Gauss-Newton-like form. To allow automatic linearisation of the dynamical system in a sparse form, the smoother is designed in a hierarchical setting such that the forward map and its linearised counterpart are estimated in a Bayesian manner given a forecasted data set. To improve the algorithm convergence, the smoother is further reformulated in its incremental form in which the current and intermediate states are assimilated before the initial one, and the corresponding posterior estimates are taken as pseudo-measurements. As the latter ones are random variables, and not deterministic any more, the novel stochastic iterative filter is designed to take this into account. To correct the bias in the posterior outcome, the procedure is built in a predictor-corrector form in which the predictor phase is used to assimilate noisy measurement data, whereas the corrector phase is constructed to correct the mean bias. The resulting filter is further discretised via time-adapting sparse polynomial chaos expansions obtained either via modified Gram-Schmidt orthogonalisation or by a carefully chosen nonlinear mapping, both of which are estimated in a Bayesian manner by promoting the sparsity of the outcomes. The time adaptive basis with non-Gaussian arguments is further mapped to the polynomial chaos one by a suitably chosen isoprobabilistic transformation. Finally, the proposed method is tested on a chaotic nonlinear Lorenz 1984 system

Parameter Estimation via Conditional Expectation --- A Bayesian Inversion

When a mathematical or computational model is used to analyse some system, it is usual that some parameters resp.\ functions or fields in the model are not known, and hence uncertain. These parametric quantities are then identified by actual observations of the response of the real system. In a probabilistic setting, Bayes's theory is the proper mathematical background for this identification process. The possibility of being able to compute a conditional expectation turns out to be crucial for this purpose. We show how this theoretical background can be used in an actual numerical procedure, and shortly discuss various numerical approximations

Bayesian parameter identification in plasticity

To evaluate the cyclic behaviour under different loading conditions using the kinematic and isotropic hardening theory of steel a Chaboche visco-plastic material model is employed. The parameters of a constitutive model are usually identified by minimization of the distance between model response and experimental data. However, measurement errors and differences in the specimens lead to deviations in the determined parameters. In this article the Choboche model is used and a stochastic simulation technique is applied to generate artificial data which exhibit the same stochastic behaviour as experimental data. Then the model parameters are identified by applying a variaty of Bayes’s theorem. Identified parameters are compared with the true parameters in the simulation and the efficiency of the identification method is discussed

A Review of Computational Stochastic Elastoplasticity

Heterogeneous materials at the micro-structural level are usually subjected to several uncertainties. These materials behave according to an elastoplastic model, but with uncertain parameters. The present review discusses recent developments in numerical approaches to these kinds of uncertainties, which are modelled as random elds like Young's modulus, yield stress etc. To give full description of random phenomena of elastoplastic materials one needs adequate mathematical framework. The probability theory and theory of random elds fully cover that need. Therefore, they are together with the theory of stochastic nite element approach a subject of this review. The whole group of di erent numerical stochastic methods for the elastoplastic problem has roots in the classical theory of these materials. Therefore, we give here the classical formulation of plasticity in very concise form as well as some of often used methods for solving this kind of problems. The main issues of stochastic elastoplasticity as well as stochastic problems in general are stochastic partial di erential equations. In order to solve them we must discretise them. Methods of solving and discretisation are called stochastic methods. These methods like Monte Carlo, Perturbation method, Neumann series method, stochastic Galerkin method as well as some other very known methods are reviewed and discussed here

Plasticity described by uncertain parameters - a variational inequality approach -

In this paper we consider the mixed variational formulation of the quasi-static stochastic plasticity with combined isotropic and kinematic hardening. By applying standard results in convex analysis we show that criteria for the existence, uniqueness, and convergence can be easily derived. In addition, we demonstrate the mathematical similarity with the corresponding deterministic formulation which further may be extended to a stochastic variational inequality of the first kind. The aim of this work is to consider the numerical approximation of variational inequalities by a “white noise analysis”. By introducing the random fields/processes used to model the displacements, stress and plastic strain and by approximating them by a combination of Karhunen-Lo`eve and polynomial chaos expansion, we are able to establish stochastic Galerkin and collocation methods. In the first approach, this is followed by a stochastic closest point projection algorithm in order to numerically solve the problem, giving an intrusive method relying on the introduction of the polynomial chaos algebra. As it does not rely on sampling, the method is shown to be very robust and accurate. However, the same procedure may be applied in another way, i.e. by calculating the residuum via high-dimensional integration methods (the second approach) giving a non-intrusive Galerkin techniques based on random sampling—Monte Carlo and related techniques—or deterministic sampling such as collocation methods. The third approach we present is in pure stochastic collocation manner. By highlighting the dependence of the random solution on the uncertain parameters, we try to investigate the influence of individual uncertain characteristics on the structure response by testing several numerical problems in plain strain or plane stress conditions

A deterministic filter for estimation of parameters describing inelastic heterogeneous media

We present a new, fully deterministic method to compute the updates for parameter estimates of quasi-static plasticity with combined kinematic and isotropic hardening from noisy measurements. The materials describing the elastic (reversible) and/or inelastic (irreversible) behaviour have an uncertain structure which further influences the uncertainty in the parameters such as bulk and shear modulus, hardening characteristics, etc. Due to this we formulate the problem as one of stochastic plasticity and try to identify parameters with the help of measurement data. However, in this setup the inverse problem is regarded as ill-posed and one has to apply some of regularisation techniques in order to ensure the existence, uniqueness and stability of the solution. Providing the apriori information next to the measurement data, we regularize the problem in a Bayesian setting which further allow us to identify the unknown parameters in a pure deterministic, algebraic manner via minimum variance estimator. The new approach has shown to be effective and reliable in comparison to most methods which take the form of integrals over the posterior and compute them by sampling, e.g. Markov chain Monte Carlo (MCMC)