37 research outputs found
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)
Parametric design evolution for production setups; a case study for welding fixtures
Automated design evolution and optimization are concepts of utmost interest to the industry for reducing design time and cost. This paper addresses the development of an automated design solution based on an evolutionary algorithm for a parametric design problem. The design problem addresses welding fixture designs as an integrated geometric design and configuration problem. The suggested automation approach is based on the parameterization of the fixture geometry and the welding process to minimize collisions across moving and static bodies during the welding operation. The aim is to produce a welding fixture design simulation that can be adjusted for specific requirements based on expert knowledge without re-iterating the entire composition of the problem. The modularity is demonstrated by modifications of the design problem formulation aimed at altering the exploration spectrum and design proposals of the welding fixture. An early system integration shows a reduction of designer efforts to 30% compared to a manual process, with an increase in overall process time due to software computational times.</p