Stochastic modeling in mechanics: course material

Learning material (2013)

Measure transformation and efficient quadrature in reduced-dimensional stochastic modeling of coupled problems

in International Journal for Numerical Methods in Engineering (2012), 92

Coupled problems with various combinations of multiple physics, scales, and domains are found in numerous areas of science and engineering. A key challenge in the formulation and implementation of ... [more ▼]

Coupled problems with various combinations of multiple physics, scales, and domains are found in numerous areas of science and engineering. A key challenge in the formulation and implementation of corresponding coupled numerical models is to facilitate the communication of information across physics, scale, and domain interfaces, as well as between the iterations of solvers used for response computations. In a probabilistic context, any information that is to be communicated between subproblems or iterations should be characterized by an appropriate probabilistic representation. Although the number of sources of uncertainty can be expected to be large in most coupled problems, our contention is that exchanged probabilistic information often resides in a considerably lower-dimensional space than the sources themselves. In this work, we thus propose to use a dimension reduction technique for obtaining the representation of the exchanged information, and we propose a measure transformation technique that allows subproblem implementations to exploit this dimension reduction to achieve computational gains. The effectiveness of the proposed dimension reduction and measure transformation methodology is demonstrated through a multiphysics problem relevant to nuclear engineering. [less ▲]

Stochastic Dimension Reduction Techniques for Uncertainty Quantification of Multiphysics Systems

Conference (2012, April 02)

Uncertainty quantification of multiphysics systems represents numerous mathematical and computational challenges. Indeed, uncertainties that arise in each physics in a fully coupled system must be ... [more ▼]

Uncertainty quantification of multiphysics systems represents numerous mathematical and computational challenges. Indeed, uncertainties that arise in each physics in a fully coupled system must be captured throughout the whole system, the so-called curse of dimensionality. We present techniques for mitigating the curse of dimensionality in network-coupled multiphysics systems by using the structure of the network to transform uncertainty representations as they pass between components. Examples from the simulation of nuclear power plants will be discussed. [less ▲]

Dimension Reduction and Measure Transformation in Stochastic Analysis of Coupled Systems

Scientific conference (2011, September 29)

Uncertain Handshaking for Coupled Physics

Conference (2011, July 18)

Dimension reduction and measure transformation in stochastic multiphysics modeling

Scientific conference (2011, March 31)

A variational-inequality approach to stochastic boundary value problems with inequality constraints and its application to contact and elastoplasticity

in International Journal for Numerical Methods in Engineering (2011)

This paper is concerned with stochastic boundary value problems (SBVPs) whose formulation involves inequality constraints. A class of stochastic variational inequalities (SVIs) is defined, which is well ... [more ▼]

This paper is concerned with stochastic boundary value problems (SBVPs) whose formulation involves inequality constraints. A class of stochastic variational inequalities (SVIs) is defined, which is well adapted to characterize the solution of specified inequality-constrained SBVPs. A methodology for solving such SVIs is proposed, which involves their discretization by projection onto polynomial chaos and collocation of the inequality constraints, followed by the solution of a finite-dimensional constrained optimization problem. Simulation studies in contact and elastoplasticity are provided to demonstrate the proposed framework. [less ▲]

Identification of Bayesian posteriors for coefficients of chaos expansions

in Journal of Computational Physics (2010), 229(9), 3134-3154

This article is concerned with the identification of probabilistic characterizations of random variables and fields from experimental data. The data used for the identification consist of measurements of ... [more ▼]

This article is concerned with the identification of probabilistic characterizations of random variables and fields from experimental data. The data used for the identification consist of measurements of several realizations of the uncertain quantities that must be characterized. The random variables and fields are approximated by a polynomial chaos expansion, and the coefficients of this expansion are viewed as unknown parameters to be identified. It is shown how the Bayesian paradigm can be applied to formulate and solve the inverse problem. The estimated polynomial chaos coefficients are hereby themselves characterized as random variables whose probability density function is the Bayesian posterior. This allows to quantify the impact of missing experimental information on the accuracy of the identified coefficients, as well as on subsequent predictions. An illustration in stochastic aeroelastic stability analysis is provided to demonstrate the proposed methodology. [less ▲]

Probabilistic equivalence and stochastic model reduction in multiscale analysis

in Computer Methods in Applied Mechanics & Engineering (2008), 197(43-44), 3584-3592

This paper presents a probabilistic upscaling of mechanics models. A reduced-order probabilistic model is constructed as a coarse-scale representation of a specified fine-scale model whose probabilistic ... [more ▼]

This paper presents a probabilistic upscaling of mechanics models. A reduced-order probabilistic model is constructed as a coarse-scale representation of a specified fine-scale model whose probabilistic structure can be accurately determined. Equivalence of the fine- and coarse-scale representations is identified such that a reduction in the requisite degrees of freedom can be achieved while accuracy in certain quantities of interest is maintained. A significant stochastic model reduction can a priori be expected if a separation of spatial and temporal scales exists between the fine- and coarse-scale representations. The upscaling of probabilistic models is subsequently formulated as an optimization problem suitable for practical computations. An illustration in stochastic structural dynamics is provided to demonstrate the proposed framework. [less ▲]

3D periodic BE–FE model for various transportation structures interacting with soil

in Computers & Geotechnics (2008), 35(1), 22-32

A three-dimensional model for soil-transportation structures is presented. This model exploits the geometrical periodicity of the system and takes into account the dynamic soil–structure interaction with ... [more ▼]

A three-dimensional model for soil-transportation structures is presented. This model exploits the geometrical periodicity of the system and takes into account the dynamic soil–structure interaction with a methodology coupling a boundary element method for the soil and a finite element formulation for the structure. A general overview of this approach is given based on several real transportation structures. Moreover, comparative studies between the different structures have been carried out. Then the model is improved by introducing a general rule for the determination of the optimal number of cells. Finally, the periodic modes propagation is investigated offering a first seizing of the significant dynamical phenomena in the soil–structure system. [less ▲]